Transient analysis of an open-cell foam volumetric receiver

Miguel Ferreira Mora

Thesis to obtain the Master of Science Degree in

Engineering Physics

Supervisors: Prof. Luís Filipe Moreira MendesEng. João Pereira Cardoso

Examination Committee

Chairperson: Prof. Ilídio Pereira LopesSupervisor: Prof. Luís Filipe Moreira Mendes

Member of the Committee: Dr. Filipe Alexandre Ereira Mendes Marques

November 2019

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Acknowledgments

First, I would like to thank my supervisors, Professor Luıs Filipe Mendes and Engineer Joao Pereira

Cardoso, for their dedication. Their expertise and advises throughout this thesis have constantly chal-

lenged me to improve my work. Additionally, I would like to thank Professors Pedro Coelho and Joao

Fareleira, who were available to discuss the physical and mathematical model in its early stages.

Moreover, I would like to express my gratitude to my girlfriend Ana Almeida, who has kept me focused

and motivated during this process. Without her love, I wouldn’t achieved these results. To my family who

supported me and made my life easier, thank you for everything.

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Resumo

A energia solar com concentracao tem vindo a destacar-se como uma alternativa para producao de

eletricidade. Neste setor, as torres solares demonstram maior potencial, devido a grande capacidade

de armazenamento e concentracao. Contudo, esta tecnologia implica investimentos elevados, o que

torna crucial a analise e melhoramento do seu desempenho.

Este trabalho tem como principal objectivo a optimizacao do funcionamento de um recetor volumetrico,

com espuma ceramica de celula aberta, atraves de simulacoes numericas. Para tal, desenvolveu-se um

modelo de transferencia de calor que permite a simulacao do comportamento do recetor no regime tran-

siente. De referir que foi assumida uma representacao unidimensional do modelo, pois as dimensoes

transversais sao muito maiores que a dimensao longitudinal do fluxo. Relativamente ao metodo radia-

tivo, foi aplicada a aproximacao de dois fluxos para simplificar a equacao de transferencia radiativa.

Com a otimizacao, foi possıvel estimar uma eficiencia maxima de 39.2 % para uma central eletrica,

com uma irradiancia incidente de 1000 kW.m−2 e um caudal massico de 0.71 kg.s−1. Adicionalmente,

constatou-se que das caracterısticas inerentes ao recetor, a porosidade apresenta o maior impacto no

perfil de temperaturas. Quanto a evolucao da temperatura em regime transiente, foi realizada uma

analise do tempo necessario para se atingir o equilıbrio termico. As simulacoes revelaram que o tempo

de equilıbrio e altamente dependente do caudal massico. Por fim, na analise do regime transiente,

foram identificadas tensoes mecanicas de origem termica que podem causar danos ao recetor. Assim,

estrategias para prevenir estes danos sao discutidas.

Palavras-chave: Energia solar com concentracao; recetor volumetrico; espuma ceramica

de celula aberta; modelo de transferencia de calor; aproximacao de dois fluxos; regime transiente.

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Abstract

Concentrating solar power interest in the power generation market has risen over the years. Particu-

larly, central receiver systems (CRS) show great potential in this sector, due to their high concentration

ratios and storage capacity. Since the capital cost of CRS plants is still elevated, there is a need to

enhance the performance of these systems.

This thesis’s primary goals are to analyse and optimize the ceramic open-cell foam (OCF) volumet-

ric receiver thermal performance in an open loop system, through numerical simulations. To do that,

a heat transfer model was developed to predict the transient behaviour of the OCF receiver. As the

cross-section dimensions of OCF panels are usually much larger than the flow direction length, a unidi-

mensional representation is assumed, and the two-flux approximation was used to simplify the radiative

transfer equation.

A maximum system efficiency of 39.2 % is estimated for an incident irradiance of 1000 kW.m−2 and a

mass flow rate of 0.71 kg.s−1. In addition, simulation results show that porosity is the absorber’s intrinsic

property with the greatest impact on both receiver and air temperature profile. Moreover, a study of the

transient evolution of the receiver’s temperature was developed, mainly focusing on the time required

for the system to reach thermal equilibrium after a perturbation. Simulations have demonstrated that

equilibrium time is highly dependent on the mass flow rate. Finally, sources of thermal stress have

been identified during the transient analysis, which can damage the OCF absorber. Hence, strategies

to prevent this effect are presented.

Keywords: Concentrating solar power; volumetric receiver; ceramic open-cell foam; heat trans-

fer model; two-flux approximation; transient analysis.

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Literature review 7

2.1 Receivers in Solar Towers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Solid particle receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Liquid receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.3 Gas receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Volumetric receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Metallic absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Ceramic absorbers: Monolithic honeycomb . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Ceramic absorbers: Open-cell foam . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Radiative models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Heat transfer model 19

3.1 Geometry of an OCF absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Transient heat transfer equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Internal nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.3 Thermal evolution of the receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Radiative heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 Radiative transfer equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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3.3.2 Two-flux approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.3 Collimated radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.4 Application of the two-flux approximation . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Parameters of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.1 Convective heat transfer coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.2 Optical correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.3 Temperature correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.4 Fluid pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Courant-Friedrichs-Lewy condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.7 Absorber performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Simulation results 41

4.1 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Model comparison with a similar work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Steady state analysis: Absorber’s properties . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3.1 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3.2 Particle diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.3 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3.4 Dispertion ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 Steady state analysis: External conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4.1 Thermal behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4.3 Equilibrium time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 Transient analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5.1 Start-up and shutdown response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5.2 Cloudy weather response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5.3 Thermal stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Conclusions 61

5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Bibliography 63

A Finite-divided-difference formulas 71

A.1 Forward finite-divided-difference formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.2 Backward finite-divided-difference formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.3 Centered finite-divided-difference formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 72

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B Auxiliary calculations 73

B.1 Transient heat transfer: Interior nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

B.2 Transient heat transfer: Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 74

B.2.1 Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B.2.2 Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

B.3 Radiative heat transfer: Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 76

C Thomas algorithm 77

D Numerical implementation of the algorithm 79

E Detailed results of the simulation 91

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List of Tables

1.1 Characteristics of the CSP technology families. . . . . . . . . . . . . . . . . . . . . . . . . 2

3.1 Values of j for each cell geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1 Conditions for the model validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Standard conditions of simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

E.1 Detailed results of simulations performed in the validation. . . . . . . . . . . . . . . . . . . 92

E.2 Energy balance of simulations performed in the validation. . . . . . . . . . . . . . . . . . . 93

E.3 Detailed results of steady state simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 94

E.4 Energy balance of steady state simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 95

E.5 Summary of transient simulation results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

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List of Figures

1.1 Flow diagram of a CSP plant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Main CSP technology families (adapted from [15]). . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Free-falling particle receiver solar tower with embedded storage and heat exchanger. . . . 8

2.2 Falling-film cavity receivers (adapted from [21]). . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Volumetric effect compared with the tubular receiver response. . . . . . . . . . . . . . . . 10

2.4 CRS plant with an open volumetric gas receiver. . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Open loop metallic absorber (from [37]) and closed loop metallic absorber application on

the REFOS project (from [38]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Monolithic honeycomb structure (adapted from [40]) and principle of operation (from [10]). 13

2.7 Open-cell foam (from [47]) and its application on the DIAPR 30-50 project (adapted from [48]). 15

2.8 Radiative transfer equation methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 OCF receiver schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Intensity of radiation between parallel plates. . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Incident radiation on several models in a cold medium simulation. . . . . . . . . . . . . . . 29

3.4 Cell geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Particle’s minimum diameter as a function of the porosity. . . . . . . . . . . . . . . . . . . 37

4.1 Simulation results of the model compared with the experimental data of Pitz-Paal et al. [42]. 42

4.2 Steady state results of the model compared with the work of Kribus et al. [54]. . . . . . . . 44

4.3 Simulation results along the receiver for different solid thermal conductivities. . . . . . . . 46

4.4 Simulation results along the receiver for different particle diameters. . . . . . . . . . . . . 47

4.5 Temperature profile along the absorber for different porosities. . . . . . . . . . . . . . . . . 48

4.6 Simulation results along the absorber for different dispersion ratios. . . . . . . . . . . . . . 49

4.7 Temperature profile along the receiver for different ratios of incident energy per kilogram

of air (Qsup/m) and for different incident irradiances (Ginc). . . . . . . . . . . . . . . . . . 51

4.8 Overall results of the simulations (as a function of the incident energy per kilogram of air). 52

4.9 Overall results of the simulations (as a function of the mass flow rate). . . . . . . . . . . . 53

4.10 Equilibrium time (teq) as a function of the mass flow rate (m) and as a function of the

porosity (ϕ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.11 Start-up and shutdown comparison of the receiver response. . . . . . . . . . . . . . . . . 56

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4.12 Transient thermal response of the receiver inlet (Ts,0) and fluid outlet (Tf,out) to a tempo-

rary decrease of incident irradiance (Ginc), for different time intervals of the cloud passage

(∆tlo). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.13 Transient response for a shutdown procedure, with Ginc,up = 800 kW.m−2 and m =

0.5 kg.s−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.14 Equilibrium time required for the system to recover its normal operation after the cloud

passage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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Nomenclature

Greek symbols

α Absorptance.

αp Specific surface area [m−1].

αsf Specific interfacial area [m−1].

β Extinction coefficient [m−1].

ε Emittance.

ζ Dispersion ratio.

ηcycle Power cycle efficiency.

ηsystem System overall efficiency.

ηthermal Receiver thermal efficiency.

θ Angle [rad, o].

κ Absorption coefficient [m−1].

λ Fourier number.

µ Dynamic viscosity [Pa.s].

µθ Function of the cosine.

ν Kinematic viscosity [m2.s−1].

ρ Density [kg.m−3].

σ Stefan-Boltzmann constant [W.m−2.K−4].

σsc Scattering coefficient [m−1].

σth Thermal stress [Pa].

τ Optical distance.

Φ Scattering phase function.

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φ Biot number.

ϕ Porosity.

Ω Solid angle [rad].

ω Single scattering albedo.

Latin symbols

A Cross section area [m2].

Ap Particle’s area [m2].

a Coefficient of thermal expansion [K−1].

C Courant number.

Cg,2D Linear-focus geometric concentration ratio.

Cg,3D Point-focus geometric concentration ratio.

c Light speed [m.s−1].

cP Specific heat at constant pressure [J.kg−1.K−1].

ct1, ct2 Time counters (of the algorithm).

dh Pore hydraulic diameter [m].

dp Particle’s diameter [m].

E Young’s modulus [Pa].

G Incident radiation [W.m−2].

Ginc Incident irradiance [W.m−2].

h Volumetric heat transfer coefficient [W.m−3.K−1].

hsf Interfacial heat transfer coefficient [W.m−2.K−1].

I Intensity of radiation [W.m−2].

Ib Blackbody intensity [W.m−2].

J Radiosity [W.m−2].

k Thermal conductivity [W.m−1.K−1].

K1 Viscous permeability coefficient [m2].

K2 Inertial permeability coefficient [m].

Lr Receiver’s length [m].

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Loss Thermal loss factor.

m Mass flow rate [kg.s−1].

n Last node of the receiver.

nt1, nt2 Time counter limit (of the algorithm).

Nu Nusselt number.

Pr Prandtl number.

p Pressure [Pa].

Q Power [W].

q Heat flux [W.m2].

Re Reynolds number.

Rf Specific gas constant [J.kg−1.K−1].

s Arbitrary distance [m].

T Temperature [K].

t Time [s].

u,v Darcy velocity [m.s−1].

Vp Particle’s volume [m3].

x Length [m].

y Width [m].

z Height [m].

Subscripts

amb Ambient air.

backsc Backscattering.

C Low temperature reservoir.

c Collimated.

cell Within a cell.

conv Convective.

d Diffuse.

eq Equilibrium.

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f Fluid (air).

H High temperature reservoir.

h Heliostat field.

i Internal node.

in Receiver’s inlet.

k Arbitrary direction.

L Receiver’s end.

l Local.

lo Lower value.

max Maximum.

min Minimum.

n Last node.

out Receiver’s outlet.

pr Printed.

R Radiative.

rad Emitted.

refl Reflected.

s Solid (receiver).

sky Sky.

sun Sun.

sup Supplied.

up Upper value.

v Volume.

w Wall (in front of the receiver’s outlet).

Superscripts

+ Forward.

− Backward.

t Time instant.

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Abbreviations

BC Brayton cycle.

CC Combined cycle.

CFD Computational Fluid Dynamics.

CFL Courant-Friedrichs-Lewy.

CPC Compound parabolic concentrator.

CRS Central receiver system.

CSP Concentrating solar power.

DAR Direct absorption receiver.

DLR German Aerospace Center.

DNI Direct normal irradiance.

DOM Discrete ordinate method.

HiTRec High temperature receiver.

HRSG Heat recovery steam generator.

HTF Heat transfer fluid.

LFR Linear Fresnel reflectors.

MH Monolithic honeycomb.

OCF Open-cell foam.

RC Rankine cycle.

RPC Reticulated porous ceramics.

reSiC Recrystallized silicon carbide.

RTE Radiative transfer equation.

sCO2 Supercritical carbon dioxide.

SiC Silicon carbide.

SiSiC Siliconized silicon carbide.

TES Thermal energy storage.

TSA Technology Program Solar Air Receiver.

VA Volumetric absorber.

WIS Weizmann Institute of Science.

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Chapter 1

Introduction

In this chapter, the concentrating solar power interest is presented in the current context of decar-

bonization of the global economy, due to climate changes. The advantages and principle of operation

of this technology are briefly described. Moreover, the four known families of this kind are presented,

where the central receiver system is highlighted as the most promising technology.

1.1 Motivation

Climate changes occur due greenhouse gas emissions, which mainly results from burning fossil fuels

for electricity, thermal energy and transportation. Since demand for energy is expected to increase and

policies to reduce these emissions are being undertaken, the expansion of renewable energy influence

has been observed over the years in the power generation market [2]. One technology whose interest

has risen over the years is concentrating solar power (CSP), mainly due to the flexibility of its transformed

energy. In fact, from 2012 to 2013, its installed capacity had the fastest growth rate of all renewable tech-

nologies [3, 4]. CSP plants can store heat in a thermal energy storage (TES) system to later convert it

into electricity, which can be dispatched to the grid at any time [5]. In these systems, only the beam solar

radiation can be concentrated. Typically, a CSP plant project requires at least 2000 kWh.m−2 of direct

Figure 1.1: Flow diagram of a CSP plant (from [1]).

1

normal irradiance (DNI) per year to be economic viable currently. With lower DNI, solar photovoltaic

technology becomes a better alternative, since it uses both direct and diffuse irradiance [5, 6].

Commercial CSP plants were first deployed in 1984 in the United States. However, from 1991 to

2005, no plants were built due to a drop in oil and gas prices, and it was only in 2006 that the market

re-emerged [3, 7]. By the end of 2018, global CSP installed capacity reached 6069 MW [8].

The principle of operation of a CSP plant is presented in Figure 1.1. A set of mirrors, which track

the movement of the Sun, focus the beam solar radiation on the receiver. Those rays can be focused

on a receiver, where the Sun’s radiation is converted to thermal energy [5]. The heat attained from the

concentrated radiation in the receiver is then delivered to a heat transfer fluid (HTF). For commercial

applications, synthetic oils, molten salts, steam and air are the most common HTF used, but other fluids

are being studied [1, 9]. The heated fluid can be directly applied in a power generation cycle, or can

go to a TES system to be used later. The fluid heated in the solar receiver can transfer its energy to

the power cycle’s working fluid (in a heat exchanger), or can be directly used as the working fluid. The

latter statement depends on the CSP technology used, the selected HTF and power cycle [1, 3, 10]. If

necessary, a fossil-fuel back-up system can be added to the CSP plant to guarantee capacity and thus

enhancing performance [3, 5].

With the operation explained, the CSP types can now be analysed. The heliostat field can focus

the solar radiation linearly or in a single point. Concentration ratios for a linear-focus (Cg,2D) and for a

point-focus (Cg,3D) concentrators are given by equation (1.1):

Cg,2D =sin θhelsin θsun

, (1.1a)

Cg,3D =sin2 θhel

sin2 θsun, (1.1b)

where θsun = 4.65 mrad is the half-angle of the sun and θhel is the angular spread of the concentrated

radiation incident on the receiver, which ideally is 90 o [11]. Additionally, the solar receiver can either

be a stationary device, independent of the mirror field (fixed receiver), or can move together with the

focusing system (mobile receiver) [5]. Figure 1.2 and Table 1.1 show the main four CSP technologies,

which result from the possible combinations of the focus type with the receiver’s mobility [1, 5, 6, 12]:

• Parabolic troughs consists on parallel rows of reflectors curved in one dimension (typically 100 m

Table 1.1: Characteristics of the CSP technology families (from [3, 5, 6, 13, 14]).

CSP technology Parabolic trough LFR Parabolic dish CRS

Solar Collector Linear Focus Linear Focus Point Focus Point FocusSolar Receiver Mobile Fixed Mobile FixedConcentration ratio 70− 80 > 60 > 1300 > 1000

Typical capacity [MW] 10− 300 10− 200 0.01− 0.025 10− 200

Current efficiency [%] 15− 16 8− 10 20− 25 16− 17

2

(a) Parabolic trough (b) Linear Fresnel reflector

(c) Parabolic dish (d) Central receiver system

Figure 1.2: Main CSP technology families (adapted from [15]).

long or more, with 5 to 6 m of curved aperture). The linear focus property of this system grants

a simpler tracking of the sun (usually a horizontal N-S axis with E-W tracking), and the mobile

receiver feature allows the system to collect more energy (when compared to a fixed receiver). It

has been commercially proven using synthetic oils as HTF, and plants with molten salts are under

demonstration;

• Linear Fresnel reflectors (LFR) uses long rows of flat (or slightly curved) mirrors that reflect the

sun’s rays onto a downward-facing tubular receiver. The existent fixed receiver enables the trans-

port of collected heat to the power block more easily. LFRs are best suited to use direct steam

generation, making storage capacity more difficult to implement. Plants with molten salts are also

under demonstration;

• Parabolic dish (or Stirling dish) is a parabolic dish-shaped concentrator that reflects solar irradiance

onto a receiver at the focal point of the dish, which have an independent Stirling engine embedded

(or a micro-turbine). The point focus receiver allows the use of a two-axis tracking, and hence, a

higher temperature on the receiver. The low available capacity of each dish is the downside of this

technology;

• Central receiver system (CRS), also known as solar tower, consists on a field of heliostats (computer-

3

controlled mirrors with a two-axis tracking system) that focus the sun’s rays in a fixed receiver

placed at the top of a tower. Most of the current commercial plants use molten salts as the HTF,

but other fluids with higher operating temperatures are being considered in future plants.

Of all the CSP types presented, the central receiver system has the greatest potential to become the

preferred technology [3, 6]. The high concentration ratio results in higher temperatures (up to 1000 oC)

and thus in a better efficiency of the power block. This technology has also the flexibility to be imple-

mented in existing fossil fuel plants for hybrid operation, and offers the best TES capacity of all CSP

types [5, 6, 13]. Additionally, the cost of all the central receiver subsystems (heliostat field, solar re-

ceiver and power block) is expected to reduce over the years due to technology innovations. All of

these factors have the potential to make electricity from solar towers cheaper than other CSP technolo-

gies [1, 3, 6, 13]. The numerous developments that are taking place in CRS plants have created an

interest in current innovations on this sector, as well as provided an opportunity to enhance knowledge

about CSP technologies. Those reasons have motivated the presented work.

1.2 Objectives

Since the capital cost of a solar tower is still high, numerical simulations have been increasingly used

as a tool for planning and optimizing CRS plants. With this approach, each subsystem can be tested

and improved individually, so that the power plant overall efficiency is enhanced [16, 17]. In light of this

trend, the purpose of this thesis is to create an one dimensional heat transfer model of an open-cell

foam (OCF) absorber in transient regime. This absorber is a type of volumetric receiver that can be

installed in open loop solar tower power plants, using air as the HTF. So far the literature doesn’t report

any one-dimensional transient model applied to OCF receivers. The development of this model is fully

justified due to the particular geometry of this type of receiver and constitutes a new tool, suitable for the

transient behaviour analysis of OCF absorbers [16].

With this work, the thermal performance of the receiver in steady state can be improved, whether it

is on its intrinsic properties or on the imposed external conditions. Additionally, the absorber’s transient

behaviour is evaluated in order to assess possible problems that may arise during its operation. Regard-

less of the analysis, the model must be flexible enough so that a wide range of these receivers (with

different compositions and properties) can be simulated.

1.3 Methodology

The developed work presents a numerical model for an open-cell foam absorber’s energy distribution

in time and space. This receiver consists of a porous structure, where an air flow is imposed in the lon-

gitudinal direction. Due to the volumetric effect, local thermal non-equilibrium is assumed in this system.

This means that the solid structure and the fluid are treated separately, where the energy expressions

of both media are coupled by a convective term [16]. Moreover, one knows that the dimensions of the

4

cross section of the receiver’s panels are typically much larger than the flow direction length. For this

reason, it is possible to apply a unidimensional representation to the developed work, along with the

two-flux approximation as the radiative method.

The receiver is then discretized in the direction of the fluid flow, where each node contains detailed

information about its thermal properties. This implies that every thermal related quantity of both media

must be discretized as well. Regarding evolution in time, the explicit method was first considered.

However, it was soon rejected, since a complex stability criteria was required for the non-linear equations

of the model [18]. Therefore, the implicit method was used to advance in time. This method is more

difficult to implement than the explicit form, but it has the advantage of being unconditionally stable

numerically [19]. Nevertheless, physical instabilities can arise if the Courant-Friedrichs-Lewy condition

is ignored [20].

The application of the transient heat transfer model is then divided in three steps. First, the model is

validated against experimental data and also compared with a simulation work. Then, an optimization of

the receiver’s performance is carried out by varying the intrinsic properties of the absorber, as well as by

changing the receiver’s surrounding conditions. Lastly, a study more focused on the transient evolution

of the open-cell foam absorber is accomplished. With all those simulations performed, a full assessment

of the model is completed.

1.4 Thesis Outline

This thesis is divided in five parts. The current chapter shows an overview of concentrating solar

power technology, where the principle of operation of each type is explained. Chapter 2 provides a re-

vision on the state-of-the-art receivers for solar towers, with special emphasis on volumetric receivers.

In addition, a literature review on radiative models is also presented. In the third chapter, the heat trans-

fer model is presented. A detailed discussion of all simulation results is given in Chapter 4, where the

validation of the model with experimental data is performed as well. Finally, the fifth chapter summa-

rizes the advantages and limitations of the proposed model. Additionally, this last chapter shows some

suggestions for future studies using the developed model.

5

6

Chapter 2

Literature review

In this chapter, a revision on documented state-of-the-art absorbers of CRS is presented, in which

the receivers are categorized in respect to the state of the heat transfer substance. A special attention is

given to volumetric receivers, where its principle of operation and implementation is evaluated in detail,

both in metallic and in ceramic absorbers. Finally, a brief discussion on radiative models (applied to

volumetric absorbers) is presented, which have highlighted some works where radiative methods are

used on open-cell foam absorbers.

2.1 Receivers in Solar Towers

All of the solar tower’s subsystems are currently being optimized in R&D activities [3]. Regarding

solar receivers, several designs of this component have been tested over the years in order to improve

its performance. Ho and Iverson [21] have suggested dividing these state-of the-art designs in three

categories, according to the state of the heat transfer substance.

2.1.1 Solid particle receivers

Solid particle receivers can work in a wider range of temperatures when compared to other receivers.

During the 1980s, the free-falling particle receiver was the first proposed absorber of this kind. As shown

in Figure 2.1, sand-like ceramic particles are lifted by an elevator to the top of the tower and released

through an opening. This creates a thin curtain of particles that fall through a cavity receiver, which is

directly irradiated by concentrated sunlight. The desired outlet temperature is achieved by controlling

the particles’ mass flow with a slide gate. These high temperature particles are then stored to later

heat a secondary working fluid for the power cycle [21–23]. Although these particles are cheaper than

molten salts, they present some disadvantages, such as higher convective heat losses (due to entrained

airflow), low residence time on the receiver and particle spillage. A thermal efficiency of 50 % was

reached by this system in 2010 [24].

Other designs of solid particle receivers have been studied to solve some of the issues stated above.

For instance, the obstructed falling particle receiver has a porous structure or obstacles which slows the

7

downward velocity of particles, thus increasing their residence time in the receiver. Another example is

the centrifugal receiver for solid particles which uses the rotational velocity of the kiln to achieve a better

control of their residence time. Indirect methods for heating solid particles, such as the gravity-driven

particle flow, avoid particle loss and reduce convective loss. Particles in a enclosure fall around an array

of tubes, whose interior surfaces are irradiated by concentrated sunlight. Fluidization of solid particles

has also been considered [23].

2.1.2 Liquid receivers

Most of the CRS commercial plants use liquid-based receivers, due to their high heat-transfer rates

and high specific heat. Molten salts are currently the most common HTF used, but fluids with higher

boiling points and lower melting points are being studied as well [3, 23, 25].

One type of liquid receivers is the falling-film receiver, in which a film of HTF flows down an inclined

wall (due to gravity only) while it is heated by concentrated sunlight. Figure 2.2 presents two methods

that have been proposed for this kind of receivers. The first one is the direct absorption receiver (DAR),

where the curtain of fluid is irradiated directly at an exposed external wall or at an internal cavity. With

this approach, the outlet fluid temperature reaches the highest value of all liquid receivers, but the ex-

posed HTF will eventually pick up some impurities. Alternatively, one can use indirect-exposure receiver

designs, where the irradiated external walls transfer the heat to the film of fluid that slides inside the

internal cavity walls. In both presented methods, issues with the film stability have yet to be further in-

vestigated [21]. Theoretical predictions show a thermal efficiency of 94 % for the DAR, while the indirect

approach can potentially reach values greater than 80 % with design improvements [26, 27].

Another type is the tubular liquid receiver, which has already been commercially proven. Like the

name suggests, it consists on a set of thin hollow tubes whose external walls are exposed to concen-

trated radiation. The heat is then transferred to the interior of each tube, where the working fluid flows.

The tubular design can either be an external receiver or a cavity absorber. In addition, pipes can be

Figure 2.1: Free-falling particle receiver solar tower with embedded storage and heat exchanger(from [21]).

8

arranged in order that the HTF passes through the receiver multiple times, thus allowing the absorption

of more energy. Although this system needs to be pre-heated in the morning, it has the advantage of

avoiding any source of HTF spillage, as well as tolerating high pressures of the working fluid [21]. Ex-

perimental results from 2010 show a thermal efficiency of 86.2 % for this kind of receiver [28]. Further

improvements are expected with the research of tubular receiver materials that can withstand higher

temperatures and better resist to the working fluid’s corrosion effect [23].

2.1.3 Gas receivers

Although gases have a low thermal conductivity, receivers of this kind can achieve temperatures

over 1000 oC. Air has been the most chosen working fluid for these systems, but applications using

supercritical carbon dioxide (sCO2) are being developed [21, 23].

One absorber of this kind is the small particle air receiver, where an aerosol of carbon nano particles

and air is heated in a pressurized cavity. The cumulative area of the particles absorbs the concentrated

radiation and transfers the energy to the air, which undergoes a Brayton cycle. This method is able

to reach high air temperatures without damaging the receiver, however, it requires a system that can

maintain the aerosol [21]. Theoretical results show that the efficiency of this receiver can be as high as

90 % [29].

Similar to liquid receivers, tubular receivers were also implemented for gases, where the heated

HTF can be directly used in the power block (in a Brayton cycle). Considering air as the working fluid,

liquid-metal heat pipes were proposed due to their high heat transfer coefficient, which can provide

a more compact heat exchange of the concentrated radiation to the gas. High air temperatures and

pressures can be attained for this kind of receiver, but large convective and radiative heat losses are

also expected. For this particular design, a thermal efficiency of 85 % was theoretically obtained, with an

outlet HTF temperature of 815 oC [21, 30]. Other designs of tubular gas receivers were developed using

(a) Direct-exposure absorption receiver. (b) Indirect-exposure absorption receiver.

Figure 2.2: Falling-film cavity receivers (adapted from [21]).

9

sCO2, such as positioning the tubes in a blade structure, to increase light traping. Procedures using this

HTF are able of reaching thermal efficiencies of 50 % using higher temperatures. The downside is that

materials which accommodate the high pressure sCO2 are more expensive [23].

An innovative type of technology is the microchannel tube receiver, that consists on a set of panels

with small tubes where the working fluid passes through. The main advantage of this system is that it

has a higher surface area between the receiver and the HTF when compared to other receivers, thus

increasing heat transfer to the gas. Modules of this receiver can be added to suit higher capacities,

although construction difficulties and increase in cost may arise. Thermal efficiencies of 90 % were

reported for this system [23].

2.2 Volumetric receivers

The gas receiver system left to address is the volumetric air absorber (VA), which first appeared

in the 1980s as a cheaper alternative that had better thermal properties than tubular receivers. Its

working principle is quite simple. A porous material is placed in a volume inside the receiver, that is

irradiated by concentrated sunlight. The radiation is absorbed throughout the porous structure, heating

this material. The attained heat is then transferred to the flowing air. Finally, a blower in this system

makes the air flow through the porous media, where the latter transfers its heat to the passing fluid by

forced convection [10].

The ideal temperature distribution of this receiver is shown on the bottom right of Figure 2.3. Cold air

enters the structure from the irradiated side and cools the receiver in that zone, thus reducing radiative

heat losses. The air gets heated as it crosses the structure, reaching the end of the receiver theoretically

Figure 2.3: Volumetric effect (right) compared with the tubular receiver response (left) (from [31]).

10

with no temperature difference between them. The so called volumetric effect is attained when the inlet

receiver temperature is lower than the outlet temperature. This is an advantage compared to tubular

receivers, whose outer walls must be hotter to provide the same outlet air temperature, which leads to

higher radiative losses. Additionally, volumetric receivers can withstand larger solar fluxes than tubular

receivers, meaning that VAs can achieve the same power with a smaller aperture area [31]. However,

flow instabilities can arise when high solar fluxes (above 2000 kW.m−2) are applied to the volumetric

receiver, which will shorten the material’s durability [32].

CRS plants using volumetric air receivers have been under development over the last years. Up to

date, two possible designs for a CRS plant using volumetric air receivers have been created: closed loop

pressurized system and open loop atmospheric system [10, 23]. Closed loop volumetric receivers use

pressurized air as the working fluid for the power block, which can either be a Brayton Cycle (BC) or a

Combined Cycle (CC). For the latter, cycle hybridization has been proposed, where the solar contribution

would preheat the air that flowed from the compressor to the combustion chamber of the gas turbine.

This application grants a higher efficiency for this cycle, as well as a wider power range [9, 33, 34]. On

the other hand, open loop receivers use atmospheric air to heat a secondary working fluid. The heated

air generates steam in a heat recovery steam generator (HRSG) to power a Rankine Cycle (RC), as

seen in Figure 2.4. Additionally, an air return system can be incorporated within this system, as a way

to recycle colder air that left the heat exchanger and to cool the receiver as well [9, 35, 36].

Volumetric receivers are typically employed when working fluid temperatures above 800 oC are re-

quired, so the porous structure must endure those high temperatures as well. Metallic porous receivers

are used in applications where the maximum outlet air temperature is below 1000 oC. If higher temper-

atures are needed, one must use ceramic absorbers instead. The most common ceramic materials are

silicon carbide (SiC) and siliconized silicon carbide (SiSiC), which can heat the air up to 1500 oC and

1200 oC respectively [10, 21, 23]. The following sections provide a description of each VA family, as well

as its performance on implemented projects.

2.2.1 Metallic absorbers

In this kind of receivers, the passing air is heated by a metal wire mesh. These absorbers have been

applied in several projects which require lower outlet temperatures. In some cases, they were used to

Figure 2.4: CRS plant with an open volumetric gas receiver (from [31]).

11

test the power plant operation before scaling the project to higher temperatures (which require ceramic

absorbers) [10].

Metallic receivers have been applied in open and closed loop power plants. Starting with the former,

Phoebus-TSA (Technology Program Solar Air Receiver) is the reference project of this type. Designed

by the Swiss SOTEL Consortium, the 2.5 MWth metallic receiver was able to reach in 1993 an efficiency

of 85 % at 700 oC. The heated air was then capable to produce steam at 480− 540 oC and 35− 150 bar in

the HRSG. Although this absorber was considered a success, a larger plant using the TSA concept was

never funded, due to the outlet temperature restriction [9]. Nevertheless, Capuano et al. [37] presented in

a recent study an innovative design of a titanium-aluminium alloy receiver, as exhibited in Figure 2.5(a).

With manufacturing improvements, this technology has the potential to surpass the current state-of-the-

art receiver for open loop cycles.

Regarding absorbers in closed loop cycles, REFOS is the reference project, which was developed

by DLR (German Aerospace Center). Figure 2.5(b) shows a representation of the REFOS absorber’s

operation, where the incident radiation is not only focused by the heliostat field but also by a secondary

concentrator. The Incoloy 600 receiver is placed on the convex side of the domed quartz window, where

pressurized air flows through [38]. Design conditions predicted an outlet temperature of 800 oC, an

operating pressure of 15 bar and a thermal efficiency of 80 % for a 350 kWth thermal power module. Tests

in 1996 yield a thermal efficiency of 67 % at 800 oC, which was subsequently raised due to improvements

on the secondary concentrator [9, 38]. The REFOS receiver scheme was later applied in the first phase

of the SOLGATE project [9, 10, 39].

2.2.2 Ceramic absorbers: Monolithic honeycomb

Monolithic honeycomb (MH) receivers are a group of ceramic absorbers that are typically employed

in open loop systems [10]. In this structure, atmospheric air gets heated as it crosses the array of

parallel channels, as presented in Figure 2.6(a). Two main manufacturing processes are known for

(a) Metallic absorber concept. (b) REFOS receiver scheme.

Figure 2.5: Open loop metallic absorber (from [37]) and closed loop metallic absorber application on theREFOS project (from [38]).

12

these receivers: by infiltration, that forms siliconized silicon carbide; or by recrystallization, which gives

recrystallized silicon carbide (reSiC) [41].

Most of the first monolithic honeycomb absorbers showed cracks on its structure after being tested.

One of their main problems was the tensile stress caused by flow instabilities. This issue was first

observed for VAs in 1995, during the Catrec 2 tests. In light of these events, the High Temperature

Receiver (HiTRec) technology was designed to minimize these instabilities, using techniques learned

from tests of former VA [36, 42]. Its principle of operation is revealed in Figure 2.6(b). A set of ceramic

absorbers modules, with less than 0.25 m of diameter each, are mounted on top of a stainless steel

frame. The ceramic absorber is set on a porous cup, making possible for the absorber to move or expand

freely while operating the receiver. Gaps between modules allow the return air of the system to escape

(or be reused) while cooling the stainless steel frame. The gaps also provide an easy replacement of

individual modules, either due to maintenance reasons or to install a new kind of absorber, allowing a

simple way to implement technology innovations on this receiver [36].

In 1998, the 200 kWth HiTRec I was tested, where the goal was to reach a mean outlet temperature

higher than 1000 oC. This receiver consisted on 37 hexagonal modules, making a total aperture area

of 0.491 m2. Each module had a SiSiC cup and a reSiC structure as the absorber material, with an

open porosity of 49.5 %. To cool the structure below the modules, a inner duct using only ambient air

was planned. Tests results showed a maximum outlet temperature of 980 oC associated to an efficiency

of 68 %. Higher temperatures couldn’t be attained due to limitations of the test bed. For an outlet

temperature of 800 oC, thermal efficiencies between 75 % and 80 % were attained. Additionally, a low

temperature difference all over the receiver led to no hot spots. Despite its apparent good operability,

some disadvantages were found. A design error combined with a poor control of the return air caused

deformations the stainless steel, which is unacceptable for larger receivers. In addition, modules had

too much freedom due to an inadequate fixation of those components to the retainer device [36, 43].

In the following years, the 200 KWth HiTRec II was developed to solve the identified issues of its

predecessor. The receiver had 32 hexagonal modules, making a total aperture area of 0.41 m2, The

modules were mass produced using a simplification of the prior receiver, and improvements of the

(a) Monolithic honeycomb samples. (b) HiTRec operating scheme.

Figure 2.6: Monolithic honeycomb structure (adapted from [40]) and principle of operation (from [10]).

13

retainer device were made. In addition, the frame of the receiver was substituted by a steel-nickel alloy

(Incoloy 800) well suited to withstand high temperatures, and whose expansion coefficient is similar to

the SiSiC. A cooling system was also planned with the return air of the heat exchanger. Tests results

showed not only that the structure issue was solved, but also that the return air cool it to temperatures

lower than 500 oC, which meant that cheaper materials could be used in the future for this component [36,

44]. A mean efficiency of 72 % has achieved for a 800 oC outlet air temperature, and 76 % has observed

for a 700 oC outlet air temperature. Efficiencies for HiTRec I in the same conditions were slightly higher,

due to higher radiative losses on the receiver’s sides in the current tests. Nevertheless, HiTRec II solved

the stated problems of its previous version and shows potential for improvements as well. This MH

receiver is still considered to be the current state of the art for open volumetric air receivers in pre-

commercial industrial scale [35–37, 44].

In 2001, a project of two stages was launched: the SOLAIR project, which directly used the HiTRec

technology. The first stage (SOLAIR 200) aimed to design a highly efficient receiver for outlet tempera-

tures over 800 oC in high-flux conditions. The 200 KWth receiver consisted on 36 square-shaped SiSiC

cups, that created an aperture area of 0.62 m2. The modules are mounted on a refractory steel frame,

due to the successful results of HiTRec II [10, 45]. Tests started in 2002 using reSiC as the absorber ma-

terial (configuration 1). Signs of oxidation on the receiver led to the substitution of the top 16 absorbers,

where the material was replaced by SiSiC absorbers (configuration 2). For an outlet air temperature of

800 oC, efficiencies of 75 % and 74 % were achieved respectively for configuration 1 and 2. However, for

a 700 oC outlet temperature, an efficiency of 81 % and 83 % was attained respectively for configuration 1

and 2, meaning that the latter arrangement is well suited for applications below 750 oC. A temperature

difference below 100 oC was observed across the whole absorber for an outlet temperature of 750 oC,

which indicated an easy operability [45].

In 2003, the second stage of the SOLAIR project (SOLAIR 3000) was initiated, where the purpose

was to raise the receiver useful power up to 3 MWth. The 270 square-shaped SiSiC cups with a reSiC

absorber needed to generate stable mean outlet temperatures between 680 oC and 800 oC, as well as

endure temperatures of 1000 oC. Test results showed that this receiver endured an incident power as

high as 2950 kWth. For a solar flux between 370 kW/m2 and 520 kW/m2, and with an outlet temperature

of 750 oC, efficiencies varied from 70 % to 75 %. For the same outlet temperature, a temperature spread

as high as 450 oC was recorded across the whole absorber [10].

With the success of the SOLAIR 3000, a solar thermal power tower plant was planned in 2006. The

solar tower Julich was built to be the first pre-commercial power plant using the open volumetric receiver

concept. The 55 m tower uses the HiTRec technology, where 1080 modules form an overall aperture

area of 23 m2. Combined with a thermal energy storage system, an outlet air temperature of 700 oC

produces steam at 500 oC and 100 bar, in order to drive a 1.5 MWe turbine generator. Solar electricity

is being delivered to the grid since 2009, but research programs are still being done to promote this

receiver to a fully commercial application [10, 31, 46].

14

2.2.3 Ceramic absorbers: Open-cell foam

Open-cell foams (OCF), also known as reticulated porous ceramics (RPC), are another family of

ceramic absorbers. Instead of having an array of thin parallel tubes, RPC are composed by open cells

(with various dimensions) randomly distributed, which gives it a foam shape as seen in Figure 2.7(a). To

build a SiC RPC, the polymer foam replication is the most common method applied [41, 49].

These open-cell foams were mostly employed in projects with closed loop power plants. Particularly,

DIAPR (Directly Irradiated Annular Pressurized Receiver) is the benchmark absorber for closed loop

systems with a OCF, which was designed by the Weizmann Institute of Science (WIS) and Rotem Indus-

tries [10]. Figure 2.7(b) presents a scheme of the implemented RPC receiver on the DIAPR 30-50. In

this design, the Pythagoras alumina–silica OCF is arranged in a configuration called the Porcupine. This

absorber arrangement has been tested individually and within the DIAPR in a broad range of tempera-

tures and irradiances, which the absorber endured without presenting any sort of damage. In addition,

DIAPR 30-50 contains a fused-silica window that withstood pressures above 50 bar. Tests also revealed

that this window isn’t sensitive to contamination agents (such as dirt), since it hadn’t generated hot

spots on this component. Lastly, a compound parabolic concentrator (CPC) was also been included in

the DIAPR design as a secondary concentrator [10, 48].

From 1994 to 1996, the DIAPR was tested for power inputs between 30 KWth to 50 KWth. For solar

irradiances between 3600 kW.m−2 and 5300 kW.m−2 and for air pressures from 17 bar to 20 bar, it was

attained an outlet temperature as high as 1200 oC, which was the largest temperature ever recorded

for a VA at the time. Under these conditions, absorber efficiencies from 70 % to 80 % were reported as

well[10, 48].

Due to the accomplishments of the DIAPR 30-50, some design improvements were suggested to

optimize the absorber performance. As a result, the DIAPR multistage concept was proposed in 1996 to

reduce thermal losses. The new DIAPR design took into account the incident irradiance distribution, by

(a) Open-cell foam sample. (b) DIAPR 30-50 receiver scheme.

Figure 2.7: Open-cell foam (from [47]) and its application on the DIAPR 30-50 project (adaptedfrom [48]).

15

installing preheaters around the main high temperature module. In addition, the main receiver (presented

in Figure 2.7(b)) has suffered some modifications. One significant change was the inclusion of a hot air

inlet that comes from the preheaters, which allowed the air to be gradually heated in two stages [10,

34, 48]. Tests in 1998 have estimated a solar flux between 2500 kW.m−2 and 4000 kW.m−2 in the main

receiver module, as well as predicted an incident irradiance between 850 kW.m−2 and 1400 kW.m−2

in the preheaters. A maximum outlet temperature of 1000 oC was achieved, which has supplied power

between 30 kWth to 60 kWth, with operating pressures from 16 bar to 19 bar. Additionally, the preheaters

delivered air to the main module, with temperatures ranging from 650 oC to 750 oC. Unfortunately, the

receiver efficiency wasn’t determined for this project, since the inlet flux detector wasn’t functional during

these tests [10, 34].

Reticulated porous ceramics were also used in the SOLGATE pilot plant. In this project, a solar-

hybrid power system was built in order to drive a gas turbine in a Brayton cycle. To that end, the

REFOS design (seen in Figure 2.5(b)) was selected as the solar receiver to heat the pressurized air.

However, the second phase of the SOLGATE project required outlet air temperatures of 1000 oC, which

was unattainable with the REFOS metallic receiver. Therefore, that component was replaced by a SiC

OCF absorber in order to achieve the demanded high temperatures. In addition, a window cooling

technology had to be implemented to prevent glass overheating. Tests results in 2003 showed, for a

770 W.m−2 DNI, a 960 oC outlet air temperature and an absolute operating pressure of 5 bar. In these

conditions, efficiencies ranging from 75 % to 90 % were recorded, which leads to a solar fraction of 70 %

in the combustion chamber of the solar-hybrid power plant [10, 39].

Open-cell foam applications were performed at laboratory scale as well. Fend and Pitz-Paal et al.

have tested small RPC and MH samples in a open loop systems, to assess their thermal perfor-

mance [31, 42, 50]. Results have shown that MH receivers can reach higher outlet air temperatures

than OCF absorbers under the same conditions, and thus larger efficiencies are attained for the former

family of ceramic absorbers [41]. Additionally, pressure drops in RPCs have been studied in recent

works [49, 51].

2.3 Radiative models

In volumetric receivers, the incident radiation is absorbed throughout the absorber [31]. The ra-

diative intensity along a porous medium can be described by the radiative transfer equation (RTE). If

one chooses to simulate the thermal behaviour of VAs, the RTE must be included in the overall energy

conservation calculation. Nowadays, solving the analytical RTE for the stated purpose is still a time-

consuming process, even if supercomputers are used [52]. Instead of taking an analytical approach,

works in literature use radiative models to approximate the RTE to a more computable expression.

Avila-Marin et al. [16] has grouped the radiative methods (used to simulate the VA performance)

in several categories, as shown in Figure 2.8. Regarding porous mediums, two simulations types are

considered: one-dimensional models and Computational Fluid Dynamics (CFD) models. The former

model only analyses property variations along the flow direction, which greatly reduces the computation

16

time. On the other hand, CFD models evaluates those property changes on a two-dimensional or three-

dimensional scale, where the flow direction is included. Information about the receiver’s cross section

behaviour is only given by the latter representation, so CFD models are more used than one-dimensional

models [16].

Regarding radiative methods, many have been used in the literature to describe the radiative propa-

gation in open-cell foams. Mey et al. [53] has compared the Rosseland approximation, P1 method and

the two-flux approximation with the Monte Carlo method, which is considered to be the most accurate

approach. In an one-dimensional representation, the two-flux approximation yielded the best results.

Kribus et al. [54] has chosen this approximation to perform a parametric study for OCF absorbers. In the

following year, the same authors have extended this study by using the Discrete ordinate method (DOM),

which is a generalization of the two-flux approximation [55]. Finally, Wu et al. [56] has also performed a

similar investigation on a two-dimensional model, using the P1 method. With the same simulation model

and radiative method, a transient analysis was accomplished as well [57].

Figure 2.8: Radiative transfer equation methods (from [16]).

17

18

Chapter 3

Heat transfer model

In this chapter, the one dimensional transient heat transfer model of an open-cell foam absorber is

presented. One starts by describing the energy equations along the absorber, together with its boundary

conditions. The thermal evolution in time of the system is obtained after the discretization of the model.

Moreover, the general RTE is attained step by step, as well as the two-flux approximation that is applied

to the model. The remaining related thermal parameters of the receiver and fluid are included, and a brief

explanation on physical instabilities is incorporated. Finally, the implemented algorithm is presented,

followed by an analysis on the treatment of the absorber’s performance.

3.1 Geometry of an OCF absorber

The proposed model describes the thermal behaviour of an open-cell foam, which has a porous

structure. Therefore, two different media must be considered: the solid structure itself (SiC) and the fluid

inside the receiver (air) that flows at a given velocity v imposed by a blower placed after the receiver’s

outlet. The porosity ϕ defines the fraction of the absorber’s volume that is occupied by void space [58].

In order to simplify the model, the receiver is defined as a rectangular prism with cross section A

and length Lr, as shown in Figure 3.1. This absorber can be divided in n + 1 nodes, where the nodes

0 and n are at the boundaries of the receiver, while the remaining are located inside the absorber. As

suggested in Figure 3.1, each node is separated by ∆x = Lrn . Note that the presented dimensions of

this figure are distorted, since the length Lr (x direction) is typically much smaller than the cross section

dimensions (y and z directions).

Figure 3.1: OCF Receiver schematic.

19

3.2 Transient heat transfer equations

In the absence of local thermal equilibrium, the general heat transfer equations are presented for the

solid and fluid phase respectively [52, 58]:

(1− ϕ)(ρcP )s∂Ts∂t

= (1− ϕ)∇.(ks∇Ts)−∇qR,s + h(Tf − Ts) , (3.1a)

ϕ(ρcP )f∂Tf∂t

+ (ρcP )fvf .∇Tf = ϕ∇.(kf∇Tf )−∇qR,f + h(Ts − Tf ) , (3.1b)

where ρ is the density of the substance, cP the specific heat at constant pressure, T the temperature, k

the thermal conductivity, qR the radiative heat flux and h the volumetric heat transfer coefficient. Note

that ∂∂t is the partial derivative in respect to time and ∇ the divergence (or the gradient). The indexes s

and f refer to the solid and fluid media respectively.

The right side of expression (3.1a) contains a conduction term, a radiative term and a convective

term, respectively. On the left side, it only has an energy storage term. In expression (3.1b), the same

terms are applied for the fluid. However, it appears an additional term in the left side, which represents

the advection imposed to the fluid.

Some assumptions are made in the model. First, one considers that the foam is homogeneous

throughout the absorber. Furthermore, the y and z directions are considered to be much larger when

compared to the x direction. Hence, the heat transfer model can have a unidimensional representation

in the direction of the fluid flow (x direction). The radiative heat transfer in the fluid phase has a negli-

gible effect, so it can be discarded from equation (3.1b). Finally, the gradient of thermal conductivity is

considered to be negligible in the fluid phase and null in the solid phase. Therefore, the partial derivative

will only affect the temperature gradient. With this proposed assumptions, the general equations are

rewritten:

(1− ϕ)(ρcP )s∂Ts∂t

= (1− ϕ)ks∂2Ts∂x2

− dqRdx

+ h(Tf − Ts) , (3.2a)

ϕ(ρcP )f∂Tf∂t

+ (ρcP )fuf∂Tf∂x

= ϕkf∂2Tf∂x2

+ h(Ts − Tf ) . (3.2b)

where qR and uf are respectively the radiative heat flux and the fluid velocity in the x direction. Ex-

pression (3.2) determines the solid and fluid thermal behaviour at a given time and space. Since these

equations have partial derivatives, one way to solve them is by using the finite difference method.

The purpose of the finite differences is to approximate a given partial derivative to a determined

finite-divided-difference formula. These formulas are derived from the Taylor series expansion, and it’s

accuracy increases with the number of terms used. There are 3 types of approximations available: a

forward difference, a backward difference and a centered difference. Appendix A shows two formulas

for each approximation type (the simplest and one with an additional term of the Taylor series) and for

each derivative order. Note that, for the same order and formula, the centered difference has double the

20

order of accuracy of the forward and backward differences [59].

The finite difference method is used in the model to solve partial derivatives in time and space.

Therefore, one needs to choose between an explicit or implicit form. In the explicit form, an unknown

temperature of a node at a time t+∆t is determined only by known nodal temperatures at the preceding

time t. On the other hand, in the implicit form, an unknown temperature at a time t + ∆t is partially (or

fully) determined by nodal temperatures at a time t + ∆t. The latter form requires that the equations of

all nodes at a time t+ ∆t are solved simultaneously, while the former doesn’t [19].

The explicit form was first considered for this model, due to its simplicity. However, this method was

soon discarded, because of the stability criteria. For a given ∆x, there is a maximum time step (∆t)

that guarantees stability, and it must hold for all nodes [19]. There are some methods to find a stability

condition, such as the Von Neumann analysis, but they only work for linear systems [60, 61]. Due

to the emission terms present in the model, linearity can’t be achieved, so those methods are invalid.

Since methods to non-linear systems are too complex to apply, the explicit form becomes unreliable [18].

Therefore, the implicit finite difference method is applied to this model. Although it’s more complex to

solve the system of equations, this form is unconditionally stable numerically, which means that ∆x and

∆t can be chosen independently [19]. However, one must note that physical instabilities may occur if

the Courant-Friedrichs-Lewy condition is ignored [20].

3.2.1 Internal nodes

The presented model applies the implicit finite differences method. As seen in Figure 3.1, the re-

ceiver is divided in discrete nodes (separated by a distance ∆x between them), where the thermal

properties are calculated. Time also has discrete values, which means that the properties of each node

are computed in a finite number of time instants (that are separated between each other by a time step

∆t).

Equation (3.2) is assigned to each internal node (from 1 to n − 1), and using the finite differences,

the temperature of the respective node can be predicted for the next time instant. To do so, one must

first choose which approximation to use for each partial derivative:

• The energy storage terms contain a partial derivative with respect to time. As previously stated,

the equations intend to determine the temperature of the node at the next time step. Therefore, the

appropriate approximation for this partial derivative is the forward finite-divided-difference formula;

• The conduction terms have a spacial second order partial derivative. Conduction implies an in-

teraction between adjacent nodes, so the applied approximation is the centered finite-divided-

difference formula;

• The term with the imposed advection contains a spacial first order partial derivative. In this phe-

nomena, the fluid is carried at a velocity u, and flows from node ’0’ to node ’n’. The air that reaches

a node at each time step comes from a previous node, so the backward finite-divided-difference

formula should be used to approximate this partial derivative.

21

The finite differences can now be applied to expression (3.2) using the proper approximations:

(1− ϕ)(ρcP )s,iT t+1s,i − T ts,i

∆t= (1− ϕ)ks,i

T t+1s,i+1 − 2T t+1

s,i + T t+1s,i−1

∆x2− dqR

dx

∣∣∣∣i

+ hi(T tf,i − T ts,i

), (3.3a)

ϕ(ρcP )f,iT t+1f,i − T tf,i

∆t+ (ρcP )f,iuf,i

T t+1f,i − T

t+1f,i−1

∆x= ϕkf,i

T t+1f,i+1 − 2T t+1

f,i + T t+1f,i−1

∆x2+ hi

(T ts,i − T tf,i

),

(3.3b)

where the subscript i denotes the number of the current node in study and the superscript t the given

time instant. After manipulating expression (3.3), one gets:

−λs,iT t+1s,i−1 +

(1 + 2λs,i

)T t+1s,i −λs,iT

t+1s,i+1 =

(1−λs,i

φs,i1− ϕ

)T ts,i−λs,i

∆x2

(1− ϕ)ks,i

dqRdx

∣∣∣∣i

+λs,iφs,i

1− ϕT tf,i .

(3.4a)

−(λf,i+uf,i

∆t

ϕ∆x

)T t+1f,i−1 +

(1+2λf,i+uf,i

∆t

ϕ∆x

)T t+1f,i −λf,iT

t+1f,i+1 =

(1−λf,i

φf,iϕ

)T tf,i+λf,i

φf,iϕT ts,i .

(3.4b)

where λ and φ are the Fourier and Biot numbers, respectively. The calculations can be seen step-by-step

in Appendix B.1.

3.2.2 Boundary conditions

In order to close the sets of equations, one must find the appropriate energy balance for each bound-

ary of the receiver. Starting with the entrance, the energy balance of the solid phase is shown in expres-

sion (3.5):

(1− ϕ)αGinc − (1− ϕ)qrad,in − (1− ϕ)qconv,in = −(1− ϕ)ks,0∂T

∂x

∣∣∣∣0

, (3.5)

where α is the absorptance of the receiver and Ginc the incident irradiance on the aperture. qconv,in and

qrad,in are, respectively, the radiative and convective heat fluxes at the receiver’s inlet surface:

qconv,in = hin(T t+1s,0 − T

t+1f,in

), (3.6a)

qrad,in = εσ

[(T ts,0

)4 − (T tsky)4] , (3.6b)

where hin is the inlet convective heat transfer coefficient, Tf,in the temperature of the inlet air close to

the receiver’s surface, ε is the receiver’s emittance, σ the Stefan-Boltzmann constant and Tsky the sky

temperature [19, 62]. Note that in expression (3.6b) the sky is considered to be a blackbody [62].

The left side of equation (3.5) includes an irradiance absorption term, as well as a radiative and a

convective loss terms. The right side includes only a conduction term. The conduction at the boundary

22

occurs in one side only, since there is no solid material at the other one.

Regarding the fluid phase, the energy balance at the receiver’s inlet is given by expression (3.7):

− (cP )f,0 m(T t+1f,0 − T

t+1f,in

)−Aϕkf,0

∂T

∂x

∣∣∣∣0−

= −Aϕkf,0∂T

∂x

∣∣∣∣0+

, (3.7)

where m is the mass flow rate of the fluid [63]:

m = ρf,iAuf,i . (3.8)

The left side of equation (3.7) includes an advection term and a conduction term, and the right side

a conduction term as well. Note that, unlike the solid phase, conduction at the boundary in the fluid

phase occurs in both sides (just like in interior nodes). This happens because air exists not only inside

the material but outside too.

One can now look at the receiver’s outlet. The energy balance of the solid phase is shown in expres-

sion (3.9):

− (1− ϕ)ks,n∂T

∂x

∣∣∣∣n

= (1− ϕ)qconv,out + (1− ϕ)qrad,out . (3.9)

As before, qconv,out and qrad,out are respectively the convective and radiative heat fluxes at the receiver’s

outlet surface:

qconv,out = hout(T t+1s,n − T t+1

f,out

), (3.10a)

qrad,out = εσ

[(T ts,n

)4 − (T t+1w

)4], (3.10b)

where hout is the outlet convective heat transfer coefficient, Tf,out the temperature of the outlet air and

Tw the temperature of the wall in front of the receiver’s outlet surface.

Similar to the entrance, the right side of equation (3.9) contains a radiative and convective loss terms,

and the left side a conduction term. Note that there isn’t irradiance coming out of the receiver outlet,

since it is considered to extinguish before reaching the end of the absorber.

Finally, expression (3.11) gives the energy balance of the fluid phase at the receiver’s outlet. This

equation contains an advection term and a conduction term on both sides of the equation:

− (cP )f,n m(T t+1f,n − T

t+1f,n−1

)−Aϕkf,n

∂T

∂x

∣∣∣∣n−

= −(cP )f,nm(T t+1f,out − T

t+1f,n

)−Aϕkf,n

∂T

∂x

∣∣∣∣n+

. (3.11)

The computation of all the boundary conditions can be seen in Appendix B.2.

23

3.2.3 Thermal evolution of the receiver

The receiver’s temperature at the next time step can be calculated using both systems of equations

formed, where each phase has an associated matrix. Since the matrices are tridiagonal, they can be

solved using the Thomas algorithm (see Appendix C) [59, 60, 64].

The expressions (3.4a), (B.7) and (B.13) found for the solid phase form the matrix equation (3.12):

hin +ks,0∆x −ks,0∆x 0

−λs,1 1 + 2λs,1 −λs,1 0

. . . . . . . . .

0 −λs,n−1 1 + 2λs,n−1 −λs,n−1

0 −ks,n∆xks,n∆x + hout

·

T t+1s,0

T t+1s,1

...

T t+1s,n−1

T t+1s,n

=

S0

S1

...

Sn−1

Sn

, (3.12)

where

S0 = αGinc + hinTt+1f,in + εσ

[(T tsky

)4 − (T ts,0)4] , (3.13a)

Si =

(1− λs,i

φs,i1− ϕ

)T ts,i − λs,i

∆x2

(1− ϕ)ks,i

dqRdx

∣∣∣∣i

+ λs,iφs,i

1− ϕT tf,i 1, 2, . . . , n− 1 , (3.13b)

Sn = houtTtf,n + εσ

[(T tf,n

)4 − (T ts,n)4] . (3.13c)

The expressions (3.4b), (B.10) and (B.16) found for the fluid phase form the matrix equation (3.14):

2FA,0 − ϕkf,0∆x −ϕkf,0∆x 0

−FB,1 FC,1 −λf,1 0

. . . . . . . . .

0 −FB,n−1 FC,n−1 −λf,n−1

0 −FA,n 2FA,n

·

T t+1f,0

T t+1f,1

...

T t+1f,n−1

T t+1f,n

=

F0

F1

...

Fn−1

Fn

, (3.14)

where

FA,i = (ρcP )f,iuf,i +ϕkf,i∆x

, (3.15a)

FB,i = λf,i + uf,i∆t

ϕ∆x, (3.15b)

FC,i = 1 + 2λf,i + uf,i∆t

ϕ∆x, (3.15c)

F0 = FA,0Tt+1f,in , (3.15d)

Fi =

(1− λf,i

φf,iϕ

)T tf,i + λf,i

φf,iϕT ts,i 1, 2, . . . , n− 1 , (3.15e)

Fn = FA,nTtf,n . (3.15f)

24

3.3 Radiative heat transfer

In equation (3.2a), the radiative term (−dqRdx ) represents the amount of radiative energy that is stored

inside an unit volume of the receiver. Due to its importance in the heat transfer model, a detailed

explanation on how to attain this quantity is presented. Note that this demonstration is mostly based

on the information gathered by Modest [52]. Nevertheless, some approximations were added to this

section. All of the following quantities are considered to be independent of the radiation wavelength.

Additionally, the porous medium is considered to be a gray body, meaning that α = ε.

3.3.1 Radiative transfer equation

Consider a light beam (with intensity of radiation I) travelling at a given direction s through a partici-

pating medium. This beam will gain and lose energy along its trajectory, at a rate given by the medium’s

properties.

Starting with the losses, there are two phenomena that diminish the intensity of the beam. One of

them is the absorption of radiation, where a fraction of the beam’s energy is absorbed by the medium

itself. The magnitude of absorbed energy, (dI)abs, is proportional to the distance travelled by the beam

(ds) as well as to its incident intensity of radiation:

(dI)abs = −κIds , (3.16)

where κ is the absorption coefficient. The other effect is the out-scattering of radiation. This occurs

when the medium can redirect a portion of the incident radiation to another direction sk, thus diminishing

the intensity along s. The magnitude of scattered intensity, (dI)abs, is proportional to the same quantities

as equation (3.16):

(dI)sca = −σscIds , (3.17)

where σsc is the scattering coefficient. Note that the coefficients in equations (3.16) and (3.17) define

the extinction coefficient and optical distance (β and τ , respectively):

β = κ+ σsc . (3.18a)

τ =

∫ s

0

βds . (3.18b)

As for the gains, there are two factors that increase the intensity of the beam. The first one is when

the beam can receive energy due to emission from the participating medium. If local thermal equilibrium

is assumed, the emitted intensity is equal to the blackbody intensity (Ib). The emitted intensity per unit

length to the beam is given by expression (3.19):

(dI)emi = κIbds . (3.19)

25

The second effect is the in-scattering of radiation. In this case, the medium is able to redirect a portion

of radiation from sk to s, which is the opposite effect of out-scattering. The magnitude of in-scattering

intensity is shown in equation (3.20):

(dI)sca(s) = dsσsc4π

∫4π

I(sk)Φ(sk, s)dΩk , (3.20)

where Ω is the solid angle and Φ the scattering phase function, that is, the probability of a ray deflect

from sk to s. If the amount of energy is equally scattered to all directions, one gets isotropic scattering

(given by Φ = 1 in equation (3.20)).

Using the effects from equations (3.16), (3.17), (3.19) and (3.20), an expression for the radiative

energy balance can be assembled:

I(s+ ds, s, t+ dt)− I(s, s, t) = κIb(s, t)ds− κI(s, s, t)ds− σscI(s, s, t)ds+ dsσsc4π

∫4π

I(sk)Φ(sk, s)dΩk ,

(3.21)

where:

I(s+ ds, s, t+ dt) = I(s, s, t) + dt∂I

∂t+ ds

∂I

∂s. (3.22)

Since the radiation travels at light speed (c), the relation between length and time is ds = cdt. If one

divides expression (3.21) by ds and insert the definition of β, one attains the radiative transfer equation:

1

c

∂I

∂t+∂I

∂s= κIb − βI +

σsc4π

∫4π

I(sk)Φ(sk, s)dΩk . (3.23)

Equation (3.23) can be simplified when one analyses the heat transfer model scope. Since light

speed is much larger when compared to the length and time scales under study, the partial derivative in

respect to time can be neglected. Therefore, expression (3.23) becomes:

dI

ds= s.∇I = κIb − βI +

σsc4π

∫4π

I(sk)Φ(sk, s)dΩk . (3.24)

Equation (3.24) can be also expressed as a function of optical distance:

dI

dτ= (1− ω)Ib − I +

ω

4π

∫4π

I(sk)Φ(sk, s)dΩk , (3.25)

where ω is the single scattering albedo:

ω =σscβ. (3.26)

3.3.2 Two-flux approximation

With the radiative transfer equations deduced, one must find a unidimensional model that solves it.

One starts by assuming the simplest case, which is only solving the propagation of the diffuse radiation

26

in the participating medium.

The model is assembled assuming that the participating medium is bounded by two parallel plates,

as shown in Figure 3.2. The plates A1 and A2 are located, respectively, at τ = 0 and τ = τL, and are

assumed to be isothermal and isotropic. Additionally, it is assumed that the intensity of radiation that

leaves each plate is directionally dependent, but only on θ. Finally, the properties of the medium can

only change along the τ coordinate. With all these considerations, and using the geometric relation

τs = τcos θ , expression (3.25) becomes:

dI

dτs= cos θ

dI

dτ= (1− ω)Ib − I +

ω

2

∫ π

0

I(τ, θk)Φ(θ, θk) sin θkdθk , (3.27)

which can be rewritten as a function of the cosine (µθ = cos θ), so that:

µθdI

dτ= (1− ω)Ib − I +

ω

2

∫ 1

−1

I(τ, µθ,k)Φ(µθ, µθ,k)dµθ,k . (3.28)

The intensities I+ and I− are also shown in Figure 3.2 (a) and (b) respectively, and are applied at any

point of the medium. I+ is limited to all directions available for A1 (that is, is limited to 0 < µθ < 1).

Analogously, I− is limited to all directions available for A2 (in other words, is limited to −1 < µθ < 0).

Based on these quantities, one unidimensional radiative model that can be employed is the Schuster-

Schwarzschild approximation, (also known as the two-flux approximation), which accurately describes

the propagation of diffuse radiation in the participating medium [53]. This model divides the intensity

of radiation in two hemispheres: an upper (or forward), I+; and a lower (or backward), I−. In each

point, the intensity of radiation within each hemisphere is isotropic. However, the intensity between

hemispheres at the same point may differ. The intensity of radiation in the two-flux approximation is then

given by expression (3.29):

I(τ, µθ) =

I−(τ), −1 < µθ < 0

I+(τ), 0 < µθ < 1

. (3.29)

Using equation (3.29), and assuming isotropic scattering (Φ = 1), expression (3.28) simplifies to:

Figure 3.2: Intensity of radiation between parallel plates on the upward direction (a) and downwarddirection (b) (from [52]).

27

µθdI

dτ= (1− ω)Ib(τ)− I(τ, µθ) +

ω

2(I+(τ) + I−(τ)) . (3.30)

Note that, in a porous medium, the internal emission propagates only in the void space (and not through

the solid). Therefore, the blackbody emission term in equation (3.30) must be multiplied by the poros-

ity [65]. By also applying the substitution dτ = βdx, one gets expression (3.31):

µθdI

dx= β(1− ω)ϕIb(x)− βI(x, µθ) + β

ω

2(I+(x) + I−(x)) . (3.31)

Integrating equation (3.31) in each hemisphere gives expression (3.32):

1

2

dI+

dx= β(1− ω)ϕIb − βI+ + β

ω

2(I+ + I−) , (3.32a)

− 1

2

dI−

dx= β(1− ω)ϕIb − βI− + β

ω

2(I+ + I−) . (3.32b)

Two new quantities are now defined. The first is the diffuse incident radiation (Gd), which represents

the total intensity of diffuse radiation on all directions at a given point. The second quantity is the diffuse

radiative heat flux (qd), which is the balance between incident and outgoing diffuse intensity at a given

point. Since intensity I has only dependence in θ, and using equation (3.29), one gets expression (3.33):

Gd(x) = 2π

∫ π

0

I(x, θ) sin θdθ = 2π

∫ 1

−1

I(x, µθ)dµθ = 2π(I+(x) + I−(x)) . (3.33a)

qd(x) = 2π

∫ π

0

I(x, θ) cos θ sin θdθ = 2π

∫ 1

−1

I(x, µθ)µθdµθ = π(I+(x)− I−(x)) . (3.33b)

With equations (3.32) and (3.33), one finally arrives to a set of differential equations, which are only

dependent on Gd and qd:

dGddx

= −4βqd , (3.34a)

dqddx

= κ(4ϕπIb −Gd) . (3.34b)

Expression (3.34) solves the diffuse radiative transfer inside the receiver. In order to close the system

of equations, one must find a set of boundary conditions to the exterior surfaces. A possible solution is

to attain an expression for I+(x) and I−(x) by manipulating equation (3.33):

I+(x) =Gd(x) + 2qd(x)

4π, (3.35a)

I−(x) =Gd(x)− 2qd(x)

4π. (3.35b)

28

By substituting I+(0) and I−(Lr) for a radiosity J in expressions (3.35a) and (3.35b) respectively, one

obtains the boundary conditions for the two-flux approximation:

I+(0) =J1

π↔ Gd(0) + 2qd(0) = 4J1 , (3.36a)

I−(Lr) =J2

π↔ Gd(Lr)− 2qd(Lr) = 4J2 . (3.36b)

3.3.3 Collimated radiation

The analysis in the previous section only considers the presence of diffuse radiation. However, in a

volumetric receiver, one must also account the collimated radiation.

As stated before, the beam solar radiation is focused on the absorber by a field of heliostats. Due

to this process, solar radiation reaches the receiver as collimated radiation. Since the absorber has a

porous structure, some of the collimated radiation propagates through the participating medium. This

kind of behaviour is unique among solar tower receivers, because collimated energy is partially absorbed

inside the receiver (in opposition to be fully absorbed at the inlet wall) [10, 21, 23].

The inclusion of the collimated term in the two-flux approximation makes the radiative model closer

to reality. Figure 3.3 demonstrates the importance of this parameter. In this figure, the incident radiation

is plotted along the receiver for several models, in a set of conditions where the scattering effect is the

dominant phenomenon. It is assumed that the curve closer to reality is the Monte-Carlo simulation, due

to the experimental results of Mey et al. [53]. With that in mind, one can observe that the collimated

two-flux approximation follows the behaviour of Monte-Carlo simulation, while the diffused two-flux ap-

proximation curve doesn’t. The collimated radiation has a strong impact on the model, and thus it should

be added in the radiative model.

Figure 3.3: Incident radiation on several models in a cold medium simulation (from [55]).

29

The diffuse and collimated radiation must be treated separately [55, 57]. Starting with the latter, its

intensity decreases along its trajectory, as described by equation (3.37). Just like the internal emission,

collimated radiation can only propagate in the void space, so it must be multiplied by ϕ.

Gc(x) = ϕ(1− ζ)Gince−βx , (3.37)

where ζ is the dispersion ratio, which is the fraction of the collimated radiation that is scattered at the

receiver’s entrance. Note that in a real solar tower system, the field of heliostats is usually surrounding

the receiver (at an angle θhel). In this scenario, the radiation that reaches the receiver is a spread

of rays (with angle θhel), as opposed to an array of parallel rays [66]. For this reason, ζ is placed in

equation (3.37) as a correction factor, so that one can treat the incoming radiation as collimated.

Regarding the diffuse radiation, one must add in expression (3.31) the contribution of the collimated

radiation that is scattered along its trajectory [55]:

µθdI

dx= β(1− ω)ϕIb − βI + β

ω

2(I+ + I−) +

βωϕ(1− ζ)Gince−βx

4π. (3.38)

By propagating the effect of the collimated term in expression (3.38), one arrives at a new set of differ-

ential equations:

dGddx

= −4βqd , (3.39a)

dqddx

= κ(4ϕπIb −Gd) + σscϕ(1− ζ)Gince−βx . (3.39b)

Note that equation (3.39a) hasn’t suffered any changes when compared to expression (3.34), while

equation (3.39b) has a new collimated component. However, the boundary conditions remain the same,

since they were deduced using only the definitions of Gd and qd.

3.3.4 Application of the two-flux approximation

The two-flux approximation is now applied to the heat transfer model, in order to find an expression fordqRdx

∣∣i. Before proceeding, one has to compute the diffuse radiation along the receiver. Some quantities

still need to be defined, mainly the blackbody radiation (Ib) and the radiosities J1 and J2, which are

presented in expression (3.40):

Ib =σT 4

s

π, (3.40a)

J1 = ϕ

[ζGinc + εσ

(T ts,0

)4], (3.40b)

J2 = ϕεσ(T ts,n

)4. (3.40c)

30

Notice that the fraction of collimated radiation scattered at the entrance becomes diffuse, so it is added

to equation (3.40b). Note also that some effects were neglected from the radiosities, particularly the

backscattering in J2. It was assumed that those had a smaller contribution to the radiosity in comparison

to the collimated and emitted radiation.

The diffuse radiation can now be attained. To do so, equation (3.39b) is differentiated in order to x:

d2Gddx2

= −4βdqddx

= −4βκ(4ϕπIb −Gd)− 4βσscϕ(1− ζ)Gince−βx , (3.41)

where dqdx was replaced by expression (3.39b). Equation (3.41) can now be discretized (as seen in

Figure 3.1), so that the finite difference method can be performed. Just as before, the centered finite-

divided-difference formula is applied to the second order derivative:

Gd,i−1 − 2Gd,i +Gd,i+1

∆x2= −4βκ

[4ϕσ

(T ts,i)4 −Gd,i]− 4βσscGc,i , (3.42)

where Ib is substituted by equation (3.40a). Note that Gc,i is the collimated radiation at a node i:

Gc,i = ϕ(1− ζ)Gince−iβ∆x . (3.43)

Manipulating expression (3.42) leads to:

−Gd,i−1 + (2 + 4βκ∆x2)Gd,i −Gd,i+1 = 16βκ∆x2ϕσ(T ts,i)4

+ 4βσsc∆x2Gc,i . (3.44)

In order to close this set of equations, one can apply the boundary conditions previously defined

by expression (3.36). The auxiliary calculations performed for the boundary conditions can be seen in

Appendix B.3. With equations (3.44) and (B.21), it is possible to build a matrix to solve Gd:

1 + 2β∆x −1 0

−1 2 + 4βκ∆x2 −1 0

. . . . . . . . .

0 −1 2 + 4βκ∆x2 −1

0 −1 1 + 2β∆x

·

Gd,0

Gd,1...

Gd,n−1

Gd,n

=

R0

R1

...

Rn−1

Rn

, (3.45)

where

R0 = 8β∆xϕ

[ζGinc + εσ

(T ts,0

)4], (3.46a)

Ri = 16βκ∆x2ϕσ(T ts,i)4

+ 4βσsc∆x2Gc,i 1, 2, . . . , n− 1 , (3.46b)

Rn = 8β∆xϕεσ(T ts,n

)4. (3.46c)

The matrix equation (3.45) is also tridiagonal, thus Gd can be computed using Thomas algorithm (see

Appendix C) [59, 60, 64]. With the diffuse incident radiation discovered, one can finally attain an expres-

31

sion for the radiative energy that is stored inside an unit volume in each node [57]:

dqRdx

∣∣∣∣i

= κ[4ϕσ

(T ts,i)4 −Gi] , (3.47)

where G is the total incident radiation:

Gi = Gc,i +Gd,i . (3.48)

3.4 Parameters of the model

3.4.1 Convective heat transfer coefficient

Equation (3.1) contains a volumetric heat transfer coefficient, which represents the amount of heat

exchanged between the solid and fluid phases through convection (in an unit volume). In the literature,

this quantity is generally given by the following expression [58, 68]:

hi = hsf,iαsf , (3.49)

where hsf is the interfacial convective heat transfer coefficient and αsf the interfacial area per unit

volume. Starting with the αsf , it is commonly assumed that the solid phase of the receiver is formed by

spherical particles. In this condition, it is considered that the receiver is formed by particles packed in a

bed [58, 67]. As seen in Figure 3.4, three different cell geometries can be attained: a simple cubic (SC),

a body centered cubic (BCC) and a face centered cubic (FCC). The particle’s area and volume (Ap and

Vp respectively) are given by expression (3.50), where the values of j depend on the cell geometry (as

shown in Table 3.1) [67].

Ap = j4π

(dp2

)2

, (3.50a)

Vp = j4

3π

(dp2

)3

. (3.50b)

With equation (3.50), one can attain the specific surface area of the particles [67]:

αp =

(ApVp

)cell

=6

dp, (3.51)

Figure 3.4: Cell geometry (adapted from [67]).

32

Table 3.1: Values of j for each cell geometry.

Cell Geometry SC BCC FCCj 1 2 4

where the subscript cell states that the values are calculated within a cell. However, it is different from

αsf , which is the surface area presented to the fluid when the particles are packed in a bed. To attain

this parameter, the following correction must be performed [68, 69]:

αsf = (1− ϕ)αp . (3.52)

With this quantity attained, one can also define the hydraulic diameter of a pore [63, 67, 69]:

dh =4

αp

ϕ

(1− ϕ)=

4ϕ

αsf. (3.53)

Regarding the interfacial heat transfer coefficient, it can be determined from the definition of the

Nusselt number (Nu) [67]:

hsf,i =kf,iNuv,i

dp. (3.54)

Note that in equation (3.54), dp is considered to be the characteristic length. To compute the Nusselt

number, it is necessary to apply an empirical correlation that suits the flow conditions. In packed beds,

it follows the formulation given by expression (3.55)[67]:

Nuv,i = o1 + o2Pro3i Reo4i

(dpdhϕ

)o4, (3.55)

where o1, o2, o3 and o4 are fitting parameters of the Nusselt correlation. Pr and Re are respectively the

Prandtl and Reynolds number:

Pri =(ρcP )f,iµf,iρf,ikf,i

, (3.56a)

Rei =uf,idhϕνf,i

, (3.56b)

where µ and ν are respectively the dynamic and kinematic viscosity.

Several solutions for the fitting parameters of equation (3.55) have been proposed [58]. One of the

most known correlations is the one presented by Wakao and Kaguei, which also has the advantage to

be independent of ϕ [58, 67, 70]. However, it doesn’t give a good correlation for low Reynolds numbers

(Re < 100) [70]. For this reason, the work of Kuwahara et al. is chosen for the model, since it yields a

good correlation for all Reynolds numbers and for a large range of porosities (0.2 < ϕ < 0.9) [71, 72]:

Nuv,i =

[2 +

12(1− ϕ)

ϕ

]+√

1− ϕ 3√

Pri

(dpdhϕRei

)0.6

, (3.57)

The volumetric heat transfer coefficient defined in equation (3.49) can be used in all of the nodes

33

inside the receiver. However, at its boundaries, one only considers the interactions on the surface (as

opposed to the volume as before). The heat transfer coefficient for the convection interactions between

the receiver and the exterior is presented in expression (3.58):

hsf,in(out) =kf,0(n)Nul,0(n)

dp, (3.58)

where the Nusselt number is given by equation (3.59):

Nul,0(n) = 2.0696ϕ0.38Re0.4380(n) . (3.59)

Note that expression (3.58) is limited to 0.66 < ϕ < 0.93 [73].

3.4.2 Optical correlations

In Section 3.3, the absorption, scattering and extinction coefficients are presented, as well as the sin-

gle scattering albedo. These optical properties depend on the absorptance, porosity and pore diameter,

which are inherent quantities of the absorber material.

For a silicon carbide absorber, there are many proposed correlations for these properties [74, 75].

However, Zhao and Tang state that these correlations have a limited range of application, due to the

limited experimental data in which they were based on. Using the Monte Carlo method, a more accurate

correlation for the extinction coefficient is then employed [76]:

β =12.64(1− ϕ)0.7

d0.79h

. (3.60)

If one considers that absorbers can be treated as opaque and diffuse structures, the absorption and

scattering coefficients are attained by expression (3.61) [41]:

κ = αβ , (3.61a)

σsc = (1− α)β . (3.61b)

Note that equation (3.61) respects the definition of β presented in Section 3.3. With equations (3.60)

and (3.61b), ω can be calculated from its definition (expression (3.26)).

3.4.3 Temperature correlations

The heat transfer model is expected to run on a wide range of temperatures (from ambient tempera-

ture to temperatures above 1000oC). Some parameters in this model are highly dependent on temper-

ature, and in the stated range, can even vary an order of magnitude. Therefore, a set of correlations to

this quantities must be defined to accommodate these temperature changes.

For the solid receiver, one knows that thermal conductivity, specific heat and density vary with the

temperature. However, this information can only be accessed by the manufacturers’ reports, and they

34

have only published a single value for each parameter. Therefore, it is assumed that these quantities

are independent of the temperature, and one will only use constant values that other articles have

considered.

For the fluid phase, multiple correlations can be found for dry air. Thermal conductivity and specific

heat are given by expression (3.62), which is valid in the range of 100 K < Tf < 1600 K [77, 78]:

kf,i(T tf,i)

= −3.9333×10−4 + 1.0184×10−4T tf,i−4.8574×10−8(T tf,i)2

+ 1.5207×10−11(T tf,i)3. (3.62a)

(cP )f,i(T tf,i

)= 1.0575×103−4.4890×10−1T t

f,i+1.1407×10−3(T tf,i

)2−7.9999×10−7(T tf,i

)3+1.9327×10−10(T t

f,i

)4.

(3.62b)

Air density is attained with equation (3.63), which is found by manipulating the ideal gas law:

ρf,i(T tf,i)

=pf,i

RfT tf,i, (3.63)

where pf is the fluid pressure and Rf = 287 J.kg−1.K−1 the specific gas constant [79]. The dynamic

viscosity is obtained by Sutherland’s equation:

µf,i(T tf,i)

=C1

(T tf,i) 3

2

T tf,i + C2, (3.64)

where C1 = 1.458× 10−6 kg.m−1.s−1.K−12 is a constant applied for air and C2 = 110.4 K the Sutherland

constant [80]. Notice that its temperature range of validity goes from 100 K to 1880 K [81].

With these quantities defined, one can also find an expression for the kinematic viscosity and for the

fluid velocity [81]:

νf,i =µf,iρf,i

, (3.65a)

uf,i =m

Aρf,i(3.65b)

3.4.4 Fluid pressure

In a porous structure, the mass flow rate is produced by a blower, which creates a pressure drop.

This pressure drop can be described by Darcy’s law with the Forchheimer extension:

− dpfdx

=µfK1

uf +ρfK2

u2f , (3.66)

where K1 and K2 are respectively the viscous and inertial permeability coefficients [32, 58]. Using

equations (3.63) and (3.65b), expression (3.66) becomes:

− pfdpf = RfTtf,i

[µf,iK1

m

A+

1

K2

(m

A

)2]dx . (3.67)

35

Integrating expression (3.67) leads to:

p2f,i−1 − p2

f,i

2= RfT

tf,i∆x

[µf,iK1

m

A+

1

K2

(m

A

)2], (3.68)

and rearranging equation (3.68), one gets the fluid pressure at each node:

pf,i =

√√√√p2f,i−1 − 2RfT tf,i∆x

[µf,iK1

m

A+

1

K2

(m

A

)2]. (3.69)

As for the boundary condition of expression (3.69), it is assumed that the fluid at node 0 is at atmospheric

pressure (pf,0 = 101325 Pa).

The only parameters left to be found are the permeability coefficients (K1 and K2), which depend

only on the absorber’s geometry [42]. To find these constants, an alternate expression of equation (3.66)

was derived for the unidimensional flow:

− dpfdx

=E1(1− ϕ)2µf

ϕ3d2p

uf +E2(1− ϕ)ρf

ϕ3dpu2f , (3.70)

where E1 and E2 are shape factors [58, 82]. Note that if one considers E1 = 150 and E2 = 1.75,

expression (3.70) becomes the Ergun’s equation, which is widely used in the literature [49, 58, 83, 84].

The viscous and inertial permeability coefficients can be attained by comparing expression (3.66) with

equation (3.70):

K1 =ϕ3d2

p

E1(1− ϕ)2, (3.71a)

K2 =ϕ3dp

E2(1− ϕ). (3.71b)

There is an issue left to address, which is the possible flow instabilities that can happen in the ab-

sorber, due to the permeability coefficients. In a real absorber, these instabilities arise when a given

pressure drop has the ability to generate multiple mass flows. If that happens, different outlet tempera-

tures can happen for the same quadratic pressure difference. This means that the mass flow rate might

not be constant throughout the absorber, creating a wide temperature spread that can damage the re-

ceiver [32]. Although this issue doesn’t affect the performance of the unidimensional analysis, the model

must be assembled in order to avoid it. According to Becker et al. [32], flow instabilities only arise when:

K1

K2< 1.94× 10−6 m . (3.72)

Since K1 and K2 are only dependent on dp and ϕ, Figure 3.5 shows the minimum particle diameter

for every porosity value. Particle diameters below the minimum value create flow instabilities in a real

volumetric receiver.

36

Figure 3.5: Particle’s minimum diameter as a function of the porosity

3.5 Courant-Friedrichs-Lewy condition

The transient heat transfer model is composed by discrete nodes. In this model, each node can only

interact with adjacent ones, and air flows at each node with a certain velocity. When dealing with this

kind of problems, one must pay attention to the Courant-Friedrichs-Lewy condition (also known as the

CFL condition). This condition states that, for a certain spacing ∆x and velocity uf , there is a maximum

∆t that must be respected in order to prevent loss of information [18, 20]. The CFL condition is given by

expression (3.73):

C =uf∆t

∆x≤ Cmax ↔ ∆t ≤ ∆xCmax

uf(3.73)

where C is the Courant number. Since the CFL condition is a physical limitation, it also affects implicit

time-marching schemes, even though they are unconditionally stable (numerically). If ∆t is too high, ef-

fects that should be felt several nodes beyond are only transmitted to the adjacent node, which generates

loss of physical information and possibly creates instabilities [18].

The objective is to run the transient model at a maximum constant value of ∆t without generating

loss of physical information. However, the fluid velocity changes along the receiver, so one must assume

a critical maximum value for this parameter (uf,max), which diminish ∆t. According to equation (3.65b),

maximizing uf implies that one finds a minimum value for the fluid density (ρf,min). By testing different

mass flows in the range of 100 K < Tf < 1600 K, one concluded that ρf,min = 0.21 kg.m−3.

The time step can only be maximized by manipulating the maximum Courant number (in expres-

sion (3.73)) through a simple procedure. One starts by running the transient model at Cmax = 1, which

is the upper limit for the explicit time-marching scheme. If the model doesn’t become physically unstable,

one can try to increase the maximum Courant number. Otherwise, one must decrease the maximum

Courant number until instabilities disappear. For the heat transfer model, instabilities have appeared for

Cmax = 1, so the Courant number had to be decreased. After testing, one discovered that the maximum

time step is attained with Cmax = 0.1.

37

3.6 Algorithm

The heat transfer model is implemented using the following algorithm:

1. Define some initial conditions to the model:

(a) Absorber properties: Lr, A, ϕ, dp, α, ε, (cP )s, ks, ρs;

(b) External conditions: Ginc, m, ζ, Tamb, Tsky;

(c) Internal data: n, nt1, nt2, Cmax, ρf,min.

2. Compute constants of the model: ∆x, uf,max, ∆t, αp, αsf , dh, K1, K2 κ σsc, β, Gc,i.

3. Calculate Gc for every node.

4. Start the model at t = 0 s (with the time counters ct1 = ct2 = 0), where all the nodes of both media

are at ambient temperature(T ts(f) = Tamb

).

5. Compute Gd and dqRdx for every node.

6. Calculate temperature dependent parameters for every node using its respective correlation: (cP )f ,

kf , µf , pf , ρf , uf νf .

7. Determine the remaining parameters, which are dependent on the quantities computed in step 6:

Re, Pr, Nuv,l, hsf , h hin,out, λs,f , φs,f .

8. Calculate the temperature of every cell (in the solid and fluid phase) at the next time instant t+ 1,

using expressions (3.12) and (3.14) respectively.

9. Update the temperatures of time t (of each node in both media) with the ones attained in step 8(T ts(f) = T t+1

s(f)

).

10. Is ct1 = nt1?

(a) If so, print the solid and fluid temperatures (at nodes 0 and n) and ct2 and proceed to step 11.

(b) If not, increment ct1 and return to step 5.

11. Is ct2 = nt2?

(a) If so, go to step 12.

(b) If not, increment ct2, reset ct1 and return to step 5.

12. Print the solid and fluid temperatures for each node, as well as the radiative properties (Gc, Gd,dqRdx ) and fluid velocity.

The numerical implementation of this algorithm (in C) is presented in Appendix D.

38

3.7 Absorber performance

When the model simulates a long period of time, it will eventually arrive at thermal equilibrium. If this

state is reached, one can calculate the thermal efficiency of the absorber:

ηthermal =Qf

Qsup=m∫ Tf,outTf,in

(cP )fdT

AGinc, (3.74)

where Qsup is the power supplied to the absorber and Qf the power transferred to the fluid [54]. Note

that, at thermal equilibrium, Tf,out = Tf,n.

One can also predict the efficiency of the power block (ηcycle), and thus calculate the system overall

efficiency (ηsystem). For the former, one could assume the Carnot cycle, in which the power cycle

efficiency is maximized. However, the latter represents a limiting case where the heat sources are at

thermal equilibrium with the working fluid (in the power block). This means that heat is exchanged in

an infinitely slow process, which drives the generated electrical power to 0 W. Since the purpose of

the power block is to maximize electrical power, this process can’t be considered [85]. Instead, some

approaches to estimate ηcycle have been suggested in the literature, where its value is lower than in the

Carnot cycle but the extracted power is maximized. Nokinov [86] has derived an efficiency expression

which takes into account a temperature difference that occurs between the high temperature reservoir

and the working fluid. Moreover, Rebhan [85] has proposed an analysis which considers the influence

of friction losses on this efficiency. For this work, the approach done by Nokinov has been implemented,

and so ηcycle and ηsystem are given by expression (3.75):

ηcycle = 1−√TCTH

, (3.75a)

ηsystem = ηthermal ηcycle , (3.75b)

where TC and TH are the low and high reservoir temperatures respectively [86]. Note that either Nokinov

and Rebhan approaches yield a ηcycle close to what is observed in real power plants [85].

Finally, the receiver’s losses can be analysed as well. In this model, the loss factors are the remaining

fraction of supplied power that isn’t transferred to the fluid. Lossrad,in, Lossconv,in and Lossrefl,in are

respectively the radiative, convective and reflected loss factors at the receiver’s inlet. At the outlet,

the remaining collimated radiation that isn’t scattered or absorbed is also treated as a loss (Lossc,out).

Additionally, the radiative and convective losses at the outlet are considered (Lossrad+conv,out), even

though their contribution is small. The remaining fraction of power can be used to estimate the radiative

backscattering losses (Lossbacksc). Note that the latter source of losses has a fairly small accuracy, due

to the numerical calculation. All of the loss factors are attained by equation (3.76):

Loss1 = Lossrad,in =qrad,inGinc

, (3.76a)

Loss2 = Lossconv,in =qconv,inGinc

, (3.76b)

39

Loss3 = Lossrefl,in = (1− ϕ)(1− α) , (3.76c)

Loss4 = Lossc,out =Gc,nGinc

, (3.76d)

Loss5 = Lossrad+conv,out =qrad,out + qconv,out

Ginc, (3.76e)

Loss6 = Lossbacksc = 1− ηthermal −5∑i=1

Lossi . (3.76f)

40

Chapter 4

Simulation results

In this chapter, a full evaluation of the assembled heat transfer model is carried out. Simulations can

be divided in three categories. First, the model is validated against experimental data and compared

with another work in literature. In the second stage, the model is run without perturbations until a steady

state condition is attained. Results of the simulations are then presented for this state, where special

focus is given to the optimization of the receiver’s thermal performance. Note that for the performed

simulations, the steady state is always reached in less than two minutes. Finally, the absorber’s response

is evaluated for different perturbations of the incident irradiance. The purpose of these simulations is to

further investigate the transient effect on the receiver, mainly the equilibrium time and thermal stress.

4.1 Model validation

The heat transfer model must be validated with experimental data, in order to access its accuracy.

However, most of the experimental works presented in literature show little information about this kind

of receivers. Therefore, the experimental results given by Pitz-Paal et al. [42] are the only ones chosen,

since that work is the one that provides most information about the absorber parameters.

Table 4.1 shows the selected conditions necessary to run the model, in which most of the absorber’s

Table 4.1: Conditions for the model validation.

External conditions Absorber parameters Simulation options

Lr [m] 0.046

A [m2] 0.016742

ks [W.m−1.K−1] 12.5

Ginc [kW.m−2] 1300 (cP )s [J.kg−1.K−1] 1244 n 100

Tamb [K] 300 ρs [kg.m−3] 3210 nt1 15000

Tsky [K] 273.15 dp [mm] 0.351 nt2 1000

α 0.9

ϕ 0.782

ζ 0.1

41

characteristics were taken from the article of Pitz-Paal et al.. The only exceptions are the assumed

values of Tamb, Tsky, α and ζ, as well as the specific heat and density of the receiver, which are retrieved

from the work of Wu and Wang [57]. Since the results of Pitz-Paal et al. were gathered at thermal

equilibrium, the values of nt1 and nt2 are chosen so that the model reaches a steady state situation.

Two sets of simulations are performed, which differ from one another at the inlet fluid boundary

condition. In expression (3.5), there is a convection term that transfers energy between the absorber

inlet wall and the air outside the receiver. In equation (3.7), there isn’t a convection term, because one

considers that the energy transferred to the air outside the receiver is lost in that medium. However,

some authors state that the energy transfered to the air by convection is gained by the inlet air that

enters the receiver [54]. If that case is assumed, one must substitute equation (3.7) in the model by the

following expression:

A(1− ϕ)qconv,in − (cP )f,0 m(T t+1f,0 − T

t+1f,in

)−Aϕkf,0

∂T

∂x

∣∣∣∣0−

= −Aϕkf,0∂T

∂x

∣∣∣∣0+

, (4.1)

Both sets of simulations are compared with the work of Pitz-Paal et al. in Figure 4.1. Tables E.1 and E.2

(in Appendix E) present more information about these simulations.

In Figure 4.1(a), the outlet temperature in both simulations increases with the increase of the energy

per kilogram of air (Qsup/m), which is also predicted by Pitz-Paal et al. [42]. However, the increase of

temperature is higher for the simulation that uses equation (4.1), which also leads to higher efficiencies

(as seen in Figure 4.1(b)). It is possible to conclude from Table E.2 that the difference between efficien-

cies of both simulations (with the same Qsup/m) is mostly due to the convection, which is considered a

loss when using equation (3.7) and a gain when using equation (4.1).

One can now compare the simulations with the experimental results of Pitz-Paal et al. [42]. For low

Qsup/m, the experimental data is closer to the simulation that uses equation (4.1), but for high Qsup/m,

it approaches the simulation with equation (3.7). The results indicate that there might be two operating

regimes for the heat transfer. This can be explained by the effect of the wind speed and its relation with

the inlet velocity.

(a) Outlet air temperature (b) Thermal efficiency

Figure 4.1: Simulation results of the model compared with the experimental data of Pitz-Paal et al. [42].

42

To understand this effect, consider a mass of air placed close to the receiver’s aperture, that has

gained energy from the absorber (due to convection). Note that, in a real CRS, the absorber is placed

at the top of a tower, which is exposed to the wind. If the inlet velocity and wind speed have the same

order of magnitude, it is probable that the mass of air will enter the receiver, bringing in air warmed up

by the receiver aperture’s convective losses. In this case, the energy gained by the air will increase the

receiver efficiency. However, if the wind speed is much greater than the inlet velocity, the mass of air

will most likely be carried away by the wind. Unlike the other case, the energy transferred by convection

must be considered has a loss.

One can now look at the inlet velocities shown in Table E.1. The results with low Qsup/m have a

higher inlet velocity, so the energy transferred to the air is more likely to reenter the absorber. For this

case, expression (4.1) is more accurate. On the other hand, the results with high Qsup/m have a lower

inlet velocity. Therefore, the energy transferred to the air is probably lost, and equation (3.7) must be

considered.

One can then assume that the two points with lowest Qsup/m follow expression (4.1) and the re-

maining are given by equation (3.7). If this interpretation is followed, the maximum relative deviation for

the outlet temperature and thermal efficiency is 3.5 % and 5.4 % respectively. This maximum deviation

is given by the point with the highest Qsup/m. For the remaining results, the relative deviation of both

quantities never exceeds 2.5 %. One has concluded that the information about wind velocity (which is

absent from the experimental work) is highly relevant to validate the model.

Since error bars aren’t presented for the experimental data, one can’t quantitatively see if the simu-

lations are within the expected measurement error. However, it is known that the experimental incident

irradiance changes greatly. Pitz-Paal et al. [42] reported an average density flux of 1.3 MW.m−2 (which

has been used in the simulations), but also stated a peak density flux of 1.9 MW.m−2. This fluctua-

tion must likely generate a measurement error larger than the attained relative error in the simulations.

Therefore, one can consider that the simulation results follow the experimental data, and thus the model

is validated.

Note that for the remaining simulations of this chapter, equation (3.7) is always used as the fluid inlet

boundary condition, since the wind speed is considered to be much greater than the inlet velocity.

4.2 Model comparison with a similar work

With the heat transfer model validated, it can be compared against other simulations in literature.

The work of Kribus et al. is chosen for this comparison, since it also uses the two-flux approximation

as the mechanism for the radiative heat transfer [54]. Table 4.2 shows the conditions that Kribus et al.

imposed for this simulation, where the exceptions are applied for the same quantities stated previously.

Figure 4.2 compares the attained temperature along the receiver with the results of Kribus et al. [54].

Regardless of the absorber’s thermal conductivity, the heat transfer model reaches higher temperatures

at the receiver outlet than the published work. Additionally, the fluid temperature of the presented model

has a faster increase as it crosses the absorber. The latter results might be due to the conduction term

43

Table 4.2: Standard conditions of simulations.

External conditions Absorber parameters Simulation options

Lr [m] 0.02

A [m2] 1

Ginc [kW.m−2] 600 ks [W.m−1.K−1] 40 n 100

m [kg.s−1] 0.6 (cP )s [J.kg−1.K−1] 1244 nt1 15000

Tamb [K] 300 ρs [kg.m−3] 3210 nt2 2000

Tsky [K] 273.15 dp [mm] 0.8

α 0.9

ϕ 0.8

ζ 0.1

of the fluid, which is neglected by Kribus et al. [54]. Although thermal conductivity of air has a low value,

its temperature gradient certainly doesn’t, especially when it is closer to the inlet boundary. Thus, it is

advisable to keep the fluid conduction term, since it predicts its thermal behaviour with better precision.

It is also noticeable in Figure 4.2 that the inlet fluid temperature of the model is lower than in the

results of Kribus et al. [54]. This is because of the convective energy transferred to the fluid, which

has been discussed previously. The gain of the convective energy by the inlet fluid is considered in

the published work, and that causes an increase on both solid and fluid temperatures. This is also the

reason why the volumetric effect is more evident in the presented model than in that work. Particularly

in Figure 4.2(b), the latter phenomenon is non-existent in the results of Kribus et al. [54].

Nevertheless, the temperature profiles of the compared simulations are quite similar. In Figure 4.2(a),

the temperature of the solid increases slightly along its length. As for the fluid temperature, it increases

along its path, and reaches the absorber’s outlet with the same temperature of the solid. Regarding

Figure 4.2(b), the fluid temperature profile attained is similar to the previous case. However, the solid

temperature of both compared simulations experiences a significant decrease closer to the inlet, which

is due to the receiver low thermal conductivity.

The simulations of Figures 4.2(a) and 4.2(b) yield a thermal efficiency of 77.8 % and 79.4 % respec-

(a) ks = 40 W.m−1.K−1 (b) ks = 1 W.m−1.K−1

Figure 4.2: Steady state results of the model compared with the work of Kribus et al. [54].

44

tively, which is an increase of 5.8 % and 9.1 % when compared with the results of Kribus et al. [54]. As

for the radiative losses, the difference between the compared results is less than 1 percentage point. It

can then be concluded that the compared simulations predict an approximate thermal behaviour for both

media. Aside from some minor differences, the simulation results are in agreement with this published

work.

4.3 Steady state analysis: Absorber’s properties

Since the transient model has been successfully validated and compared, it can now be used under

different conditions. For the remaining simulations of this chapter, the selected standard conditions

chosen to run the model are presented in Table 4.2. For each simulation, these parameters are expected

to remain constant unless they are mentioned otherwise.

One can start by performing a study on the absorber’s intrinsic properties (and on ζ as well). These

parameters are changed individually, so that its contribution to the model is identified. Tables E.3 and E.4

(in Appendix E) show more detailed information about the steady state results of each set of presented

simulations.

4.3.1 Thermal conductivity

The effect of thermal conductivity on the absorber thermal behaviour was identified by Kribus et al.

[54], and it has also manifested in this model (as seen in the model comparison). Therefore, a proper

discussion on this parameter must be executed.

Figure 4.3(a) illustrates the temperature profile for each selected value of thermal conductivity. Sim-

ulations with a high ks present a similar absorber temperature throughout its length. However, when

the thermal conductivity is lowered, the receiver temperature near the inlet boundary decreases. By

diminishing ks, the conduction term in expression (3.2a) becomes less significant when compared to the

convection term, which leads to a lower temperature of the solid phase.

The absorber thermal conductivity also affects the thermal behaviour of air. A lower receiver temper-

ature decreases the convection term in equation (3.2b), so less energy is transferred to the fluid. This is

why the temperature gradient of air for ks = 1 W.m−1.K−1 is lower than in the rest of the simulations.

Notice that the volumetric effect is attained for all simulations, but it is more evident for low ther-

mal conductivities. Additionally, the outlet temperatures increase slightly as the thermal conductivities

decrease. Therefore, lowering the absorber thermal conductivity lead to an increase of the thermal effi-

ciency. Simulations show that if ks is decreased from 80 W.m−1.K−1 to 1 W.m−1.K−1, thermal efficiency

increases 1.8 percentage points. Regarding the losses of the system, they aren’t significantly affected

by the different thermal conductivities.

Although decreasing ks raises thermal efficiency, one must also consider thermal stress:

σth,i = aE∆Ts,i , (4.2)

45

Figure 4.3: Simulation results along the receiver for different solid thermal conductivities: (a) Tempera-ture profile; (b) Receiver temperature difference.

where a is the coefficient of thermal expansion, E the Young’s modulus and ∆Ts,i = Ts,i+1− Ts,i−1 [87].

As seen in Figure 4.3(b), the variation of temperature is negligible for high thermal conductivities. How-

ever, for low ks, it presents a pronounced fluctuation near the inlet, which leads to a larger thermal stress

in certain areas of the absorber. This might pose as an issue to the receiver. If σth surpasses the elastic

range, the absorber will be permanently deformed or even become fractured, which is an undesirable

situation. Therefore, the choice of the absorber thermal conductivity must be performed with caution.

4.3.2 Particle diameter

The following simulations address the variation of the particle diameter that compose the receiver.

Although this quantity is sometimes not disclosed in literature, it has a great impact on the thermal be-

haviour of both media, as one can see in Figure 4.4(a) [31]. One can immediately see that the solid

temperature decreases as dp gets lower. One of the main reasons is the high values for the backscatter-

ing losses recorded for low particle diameters (in Table E.4), which must be discussed in detail. Although

backscattering losses are estimated rather than calculated, one knows that this quantity can be related

to the contribution of the collimated radiation to the diffuse radiation (due to expression (3.39b)). Fig-

ure 4.4(b) shows the gradient of diffuse radiative heat flux along the receiver. One can observe that dqddxis much lower at the inlet when dp = 0.2 mm. This means that the lowest particle diameter simulation

has the highest backscattered radiation near the inlet. This radiation is most likely to escape from the

receiver’s inlet than in the other simulations, which leads to higher backscattering losses. The small par-

ticle diameter is also the main reason why the backscattering losses are so high in the model validation.

Note that a small increase of dqddx at the receiver outlet has been observed, which might indicate forward

46

Figure 4.4: Simulation results along the receiver for different particle diameters: (a) Temperature profile;(b) Gradient of the diffuse radiative heat flux.

scattering losses. Although the backscattering loss estimation includes forward scattering losses as

well, the latter contribution is negligible when compared to the former.

Other effects can be seen in Figure 4.4(a). As stated before, the absorber’s temperature raises with

the increase of the particle diameter. However, the gradient of air temperature (at the inlet region) is

lower for larger dp. Both these effects are explained by the heat transfer coefficients, which are inversely

proportional to the particle diameter (h ∝ d−1.4p and hsf,in ∝ d−0.562

p ). Therefore, for low diameters,

more heat is exchanged through convection, which decreases the receiver temperature and increases

the air temperature gradient. Note that the low convection at the inlet for simulations with high particle

diameters causes an overheating of the absorber at that region. For that reason, the volumetric effect

becomes less evident as the diameter increases and is lost when dp = 1.4 mm.

Due to the referred effects of the particle diameter, the energy balance has considerable changes

between simulations. Raising the particle diameter from 0.2 mm to 1.4 mm increased the radiative losses

by 7.3 percentage points, but decreased the convective losses by 10.1 percentage points. Nevertheless,

thermal efficiency has raised significantly with the increase of the particle diameter. From dp = 0.2 mm

to dp = 1.4 mm, an increase of 40.5 % has been recorded for this parameter. Notice that the loss of

collimated energy at the outlet has slightly increased with the raise of the particle diameters. This is

because the extinction coefficient diminishes when dp gets larger, so more collimated radiation exits the

receiver through the outlet aperture.

47

4.3.3 Porosity

The next parameter that is analysed is the porosity, which is one of the most important features of a

volumetric receiver. Figure 4.5 shows the temperature profile of both media for porosities between 0.7

and 0.9, which almost describes the total range of validity of this parameter in the model. Similar to the

previous case, the air temperature gradient (at the inlet region) lowers with the increase of the porosity.

As porosity raises, both αsf and hsf decrease, which leads to a diminish of the volumetric convection

term.

If the same logic of the previous case is applied, one would expect that the decrease of convection led

to an overheating of the absorber at the inlet. However, it is seen in Figure 4.5 that the volumetric effect

becomes more apparent as porosity increases. In addition, the receiver’s inlet temperature decreases

when porosity goes from 0.85 to 0.9, which indicates that there is a maximum in the absorber’s inlet

temperature between 0.8 < ϕ < 0.9.

By definition, the increase of porosity lowers the volume occupied by the solid material. This has

two implications. On one hand, less incident irradiance is absorbed at the inlet solid boundary. On

the other hand, raising the porosity leads to a lower extinction coefficient, meaning that less energy

is absorbed along the receiver (even though the available collimated radiation increases). Therefore,

as porosity increases, the absorbed radiation becomes so low that the inlet air is capable to cool the

receiver entrance, even with a low heat transfer coefficient.

The outlet air temperature raises with the increase of porosity, and so does the thermal efficiency.

When ϕ rises from 0.7 to 0.9, thermal efficiency has an increase of 17 % . Note that if one chooses

an absorber with a lower length and with a high porosity, the outlet air might not reach the receiver

temperature, and thus ηthermal might decrease. Regarding convective losses, they are expected to

decrease when raising the porosity, since hsf,in ∝ ϕ−0.058. A rise in porosity from 0.7 to 0.9 reduces

Figure 4.5: Temperature profile along the absorber for different porosities.

48

convective losses by 4.9 percentage points. As for radiative losses, the difference between the highest

and lowest loss is 1.3 %, which is fairly low.

The remaining loss factors are related to the incident irradiance. As stated before, raising the porosity

diminishes β and increases the collimated radiation travelling through the receiver, so collimated losses

increase as well. In particular, the simulation with the highest porosity recorded a value of 2.6 %, which

is much larger than any other. Concerning the reflection loss factor, it decreased 2 percentage points in

the considerate range of ϕ. Since the absorptance remains unchanged, the variation of this parameter

is linear. Finally, there is an increase of the backscattering losses when porosity diminishes, and the

reason is the same as before. Backscattering is particularly large for the simulation with lowest ϕ.

Nevertheless, its contribution is minor when compared to the one observed for the particle diameter.

4.3.4 Dispertion ratio

The dispersion ratio was introduced in equation (3.37) as a correction factor for the angular spread

of rays. This parameter has never been used in similar models, so one must make an analysis of this

quantity [53, 55, 57]. Different simulations have changed the dispersion ratio from 0 (fully collimated at

the inlet) until 1 (fully diffused), with increments of 0.2.

Figure 4.6(b) illustrates the total incident radiation along the receiver, which according to expres-

sion (3.48), is the sum of the collimated and diffuse radiation. As the dispersion ratio rises, more

collimated radiation becomes diffuse, and thus the total incident radiation increases at the receiver’s

inlet. However, G diminishes along the receiver, and this decrease is greater for high values of ζ. Re-

gardless of the dispersion ratio, the total incident radiation of all simulations arrive to a constant value

Figure 4.6: Simulation results along the absorber for different dispersion ratios: (a) Temperature profile;(b) Total incident radiation.

49

of 0.2 MW.m−2, which corresponds to the internal emission of the receiver. This is why the thermal

behaviour at the outlet is almost indistinguishable between simulations, as seen in Figure 4.6(a). From

ζ = 0 to ζ = 1, the outlet fluid temperature lowers 11.5 K, which corresponds to a decrease of 1.7 % on

the receiver thermal efficiency.

Due to the high values of G at the inlet, the temperature of the absorber increases slightly in this

region for high dispersion ratios. When ζ increases from 0 to 1, so does the radiative losses by 1

percentage point. Additionally, the volumetric effect is lost for dispersion ratios equal or greater than 0.8.

Regarding the convective heat transfer, the dispersion ratio doesn’t affect the heat transfer coefficient,

so the temperature profile of all simulations is similar. Nevertheless, the small increase of the receiver’s

temperature at the inlet rises the convection losses by 0.5 percentage points (from ζ = 0 to ζ = 1).

Finally, the variation of the backscattering losses in these simulations is considered negligible, meaning

that this quantity depends mostly on the absorber geometry rather than on the heliostat field distribution

and focus.

4.4 Steady state analysis: External conditions

An analysis on the external parameters is now carried, specifically to the incident irradiance and to

the mass flow rate. Unlike the absorber’s properties, these quantities can be easily adjusted during the

simulation, in order to reach a maximum system efficiency.

The steady state results of each simulation are also presented in Tables E.3 and E.4. In this study,

four different values of the incident irradiance are chosen. For each Ginc, the mass flow rates are chosen

so that the incident energy per kilogram of air ranges from 600 kJ.kg−1 to 1600 kJ.kg−1, with constant

increments of 200 kJ.kg−1. Notice that the selected range of Qsup/m is similar to the experimental works

performed to this kind of receivers [41].

4.4.1 Thermal behaviour

The thermal behaviour of both phases is illustrated in Figure 4.7. It can be observed that for the

same incident irradiance, the absorber’s temperature raises with the increase of the incident energy

per kilogram of air. When Ginc is constant, an increase of Qsup/m corresponds to a decrease of the

mass flow rate, meaning that the useful power extracted lowers as well. This effect raises the losses of

the system, thus leading to a higher temperature of the receiver. Additionally, a lower mass flow rate

decreases the heat transfer coefficient, which also contributes to the low useful power extracted.

However, the gradient of fluid temperature (at the inlet region) increases when Qsup/m becomes

higher. Considering the information above, the temperature profile of air may seem atypical, but it

can be explained. A decrease on m means that less mass of air enters the absorber (per unit of

time), which lowers its heat capacity. Hence, the fluid temperature increases faster along the absorber.

Moreover, higher fluid temperatures raise the volumetric heat transfer coefficient, which contributes to

the mentioned growth. For those reasons, the volumetric effect is more accentuated for lower mass flow

50

(a) Ginc = 500 kW.m−2 (b) Ginc = 600 kW.m−2

(c) Ginc = 800 kW.m−2 (d) Ginc = 1000 kW.m−2

Figure 4.7: Temperature profile along the receiver for different ratios of incident energy per kilogram ofair (Qsup/m) and for different incident irradiances (Ginc).

rates. For this set of absorber’s properties, this phenomenon is lost for m ≥ 1 kg.s−1 and it is sustained

for mass flow rates lower than 0.83 kg.s−1. This observation can be confirmed in Table E.4, where the

mass flow rates are shown explicitly.

For the same incident energy per kilogram of air, the temperature of the absorber increases when

Ginc raises. Although it isn’t immediately evident, it’s also possible to observe that the fluid temperature

gradient diminishes with the increase of the incident irradiance. In order to maintain the same Qsup/m,

the mass flow rate must increase when Ginc raises. Therefore, the same explanation previously made

to m can also be applied to this analysis.

One must also address the convective and radiative losses of all simulations. For the same incident

irradiance, these losses increase as Qsup/m becomes higher, since the difference between the receiver

temperature and the outside temperatures (Tamb and Tsky) raises. Notice that the increase of tempera-

ture difference has a greater contribution to the convective losses than the decrease of the heat transfer

coefficient. Finally, when maintaining the same incident energy per kilogram of air, the amount of lost

energy increases with the increasing Ginc. Nevertheless, the energy that is supplied to the system is

also greater, and so the fraction of lost energy is actually diminished.

51

4.4.2 Efficiency

With the analysis of the thermal behaviour completed, the optimization of the system efficiency can

be performed. To that end, Figure 4.8 presents all of the efficiencies attained in each simulation, along-

side the outlet fluid temperature. Similar to the model validation, these quantities appear as a function

of Qsup/m, since it is the preferred form of showing these results in literature [41].

For every performed simulation, the air temperature matches the absorber’s temperature at the outlet

(with a maximum difference of 0.5 K). Therefore, the same behaviour as before is seen in Figure 4.8(a),

where Tf,out raises with the increase of Qsup/m and Ginc. As a result, when Ginc remains constant,

the thermal efficiency (shown in Figure 4.8(b)) decreases for higher Qsup/m . However, for the same

Qsup/m, this efficiency raises for larger incident irradiances. This effect is related to the increase of the

mass flow rate, which will become clearer in the figure below.

However, the difference between fluid temperatures with the same incident energy per kilogram of

air becomes larger for higher Qsup/m, and this effect is also manifested in the thermal efficiency. For

Qsup/m = 600 kJ.kg−1, the maximum thermal efficiency difference between simulations is 3.1 percent-

age points; whereas for Qsup/m = 1600 kJ.kg−1, that difference can be as high as 9.8 percentage points.

The purpose of showing data in function of Qsup/m is to normalize it. In theory, this allows to

(a) Outlet air temperature (b) Thermal efficiency

(c) Cycle efficiency (d) System efficiency

Figure 4.8: Overall results of the simulations (as a function of the incident energy per kilogram of air).

52

compare the results against other works with the same conditions [41]. Nevertheless, it has been seen

that simulations with the same incident energy per kilogram of air yield contrasting results, which has

produced differences up to 15.3% in ηthermal. According to Gomez-Garcia et al. [41], this is because the

mass flow rate and the incident irradiance have an independent effect on the outlet air temperature and

consequently on the efficiency. Therefore, Figure 4.9 plots the same simulations for different Ginc, but

this time explicitly as a function of m. With this representation, the mass flow and the irradiance can be

treated separately, which leads to a better understanding of the results. For instance, in Figure 4.9(b),

one can observe that ηthermal is actually lower for higher incident irradiances (that have the same mass

flow). Additionally, this graph suggests that the dependence of ηthermal on m is stronger than on Ginc,

which is a difficult conclusion to extract from Figure 4.8(b). Hence, the graphs in function to the mass

will be preferred through the remaining of this work.

With the attained outlet temperature, the efficiency of the power cycle can be estimated. For this

analysis, it is supposed that the heated air maintains its energy until reaching the heat exchanger, so

that TH = Tf,out. One also assumes that the TC = Tamb. With this considerations, one can see

in Figure 4.9(c) that the profile of the cycle efficiency curves are similar to the outlet fluid temperature

curves in Figure 4.9(a). For an increasing mass flow rate, ηcycle decreases and ηthermal raises. Thus, the

overall system efficiency has an optimal operating point for m, that maximizes the production of energy

(a) Outlet air temperature (b) Thermal efficiency

(c) Cycle efficiency (d) System efficiency

Figure 4.9: Overall results of the simulations (as a function of the mass flow rate).

53

for each Ginc. Figure 4.9(d) shows that for a higher Ginc the point of maximum system efficiency occurs

for larger m. Moreover, if the mass flow is kept constant, ηsystem becomes higher as the irradiance

gets larger. For the performed simulations, a maximum system efficiency of 39.2 % is achieved for

Ginc = 1000 kW.m−2 and m ≈ 0.71 kg.s−1.

4.4.3 Equilibrium time

The presented simulations have started with the all the nodes at ambient temperature. At t = 0 s,

the incident irradiance suddenly increases (from 0 kW.m−2 to the desired Ginc), and these simulations

evolved in time until a steady temperature profile in both media is attained. Thus, the time where thermal

equilibrium is reached can be found.

To do that, one needs to know how to compute time in this model. In theory, it is possible to attain

the thermal properties of the receiver at each time step. However, to reduce the simulation time, a

higher time interval between log values was imposed. The calculation of the printed time is then given

by equation 4.3:

tpr = ∆tct2 nt1 . (4.3)

The equilibrium time (teq) can then be obtained if a ”quasi-steady” state is reached. According to Wu

and Wang [57], one arrives at a ”quasi-steady” state when each measured node is able to maintain a

temperature difference (in respect to time) below 15 K for at least five minutes. Using this definition, the

following general procedure is proposed to calculate teq in each simulation:

1. Record the steady state temperatures of nodes 0 and n (of both media).

2. For each node and phase, report the minimum time where the temperature difference (between

steady state temperature and minimum time temperature) is below 15 K.

3. Finally, subtract the maximum attained time to the initial time (where the incident irradiance pertur-

bation has begun).

Figure 4.10(a) shows the equilibrium time calculated for every simulation. For the same absorber’s

properties, one can see that teq is inversely proportional to the mass flow rate. This conclusion isn’t

unexpected (remember that ∆t ∝ m−1). A larger mass flow rate distributes the energy more quickly

along the receiver, which leads to a faster increase of the temperature, specifically at the absorber’s

outlet. Note that the maximum attained time in the procedure has always occurred for the node n, and

this fact also sustains the latter statement.

If another configuration of absorber’s properties is considered, the values of the equilibrium time

change as well (see Table E.3). This change on the equilibrium time is particularly noticeable when the

porosity of the material is shifted, as shown in Figure 4.10(b). From ϕ = 0.7 to ϕ = 0.9, the equilibrium

time diminishes over a minute. The particle diameter and the thermal conductivity also have impact on

teq, but it is lower when compared to the effect that the porosity induces.

54

(a) teq(m) (b) teq(ϕ)

Figure 4.10: Equilibrium time (teq) as a function of the mass flow rate (m) and as a function of theporosity (ϕ)

4.5 Transient analysis

The previous sections have focused on the steady thermal behaviour of the model, where its param-

eters (defined in Table 4.2) remained undisturbed when performing each simulation individually. In this

section, the response (in time) of the receiver’s model is evaluated, by adding some perturbations on the

external conditions while executing the simulations. These perturbations attempt to recreate the start-up

and shutdown procedures, as well as the passage of a cloud above the heliostat field

In order to perform a successful transient simulation (with perturbations), the following method is

proposed [57]. The algorithm previously defined is applied, so that the steady state temperatures (of both

media along the absorber) are recorded. The attained temperatures are used as the initial conditions

of the simulation, as well as the mass flow and incident irradiance applied in the previous step. In a

central receiver system, the latter quantities can be altered while operating with the absorber. However,

the mass flow rate was assumed to be constant while the absorber is operated. Therefore, Ginc is the

only external parameter that is manipulated along the execution of the simulation.

Note that when dealing with the second stage, some modifications must be made to the algorithm.

In step 4, the steady state temperatures are applied to each node instead of Tamb. Additionally, the

incident irradiance perturbations are inserted in the algorithm between step 10 and 11. Once triggered,

the perturbation can either be a step function or a slope (in respect to time). For the former, Ginc

instantly changes to its setpoint, while for the latter the irradiance gradually varies for a period of time

until reaching its desired value.

4.5.1 Start-up and shutdown response

In the previous section, the receiver has been heated from ambient temperature until a steady state

temperature. These simulations can be associated to the daily start-up procedure of the solar tower

system. In the same way, one can also represent the equivalent of the daily shutdown procedure, that

is, when the absorber is cooled from its steady state temperature to Tamb

55

The conditions for the shutdown procedure will be similar to the ones performed for the start-up with

Ginc = 600 kW.m−2. The same incident irradiance is initially applied, which diminishes to 0 kW.m−2.

Note that this decrease is instantaneous, since it was implicitly applied a step change to the start-up

simulations. Additionally, the selected mass flows in each simulation are equal to the ones at the start-

up. This means that the steady state temperatures are the ones already attained in Figure 4.7(b).

The equilibrium time of both start-up and shutdown is illustrated in Figure 4.11. As one can observe,

the shutdown equilibrium time is also inversely proportional to the mass flow. However, it is always

higher than the start-up teq. In addition, the difference between the start-up and shutdown teq becomes

larger as m decreases.

For the shutdown simulation, convection is the main effect to cool down the absorber. With a low

mass flow rate, air takes more time to pass through the receiver, and thus takes more time to decrease

its temperature. On the other hand, the main mechanism to heat the absorber (in the start-up simulation)

is the absorption of incident irradiance, which happens along the receiver instantly. In the end, the

power absorbed in the start-up procedure is always greater than the power lost through convection (in

the shutdown procedure), and this difference is amplified when m decreases. Therefore, the shutdown

procedure requires again more time to cool the receiver, especially for low mass flow rates.

4.5.2 Cloudy weather response

In a solar tower system, the volumetric receiver is heated due to the direct incident irradiance that is

focused by the heliostats. If a cloud passes nearby the solar tower, it can cast a shadow that covers the

heliostat field (partially or even fully), which decreases the available incident irradiance on the receiver.

The thermal response of the absorber is then evaluated in these conditions, for different time intervals

in which the cloud shades the heliostat field (∆tlo).

Figure 4.11: Start-up and shutdown comparison of the receiver response.

56

Two values for the incident irradiance are assumed in each set of simulations: a lower value for when

the heliostat field is covered (Ginc,lo), and an upper value for when it’s not (Ginc,up). When the irradiance

changes between those values, it doesn’t occur instantaneously. Instead, the irradiance is considered

to vary linearly (from Ginc,up to Ginc,lo or vice versa) in a period of 5 s, to better define the passage of a

cloud. Figure 4.12 shows a set of transient simulations for different ∆tlo. Notice that for each simulation,

∆tlo is the time interval in which the incident irradiance is lower than Ginc,up.

When the irradiance diminishes, the convective and radiative losses become much greater than the

absorbed irradiance. Therefore, the receiver inlet temperature decreases abruptly immediately after

the irradiance drop. However, as Ts,0 decreases, the mentioned losses also diminish. Therefore, this

temperature lowers more slowly as time passes, until it reaches thermal equilibrium (as seen in Fig-

ure 4.12(d)). Note that an analogous analysis can be performed for the inlet temperature increase (after

the cloud passes).

Regarding the outlet fluid, it has been observed in Figure 4.12 that its temperature doesn’t react

immediately to irradiance variations. This occurs because the receiver’s solid structure doesn’t respond

to these changes uniformly, as shown in Figure 4.13(a). During the shutdown procedure, the colder

inlet air receives energy from the absorber, which increases the fluid temperature and decreases the re-

(a) ∆tlo = 10 s (b) ∆tlo = 20 s

(c) ∆tlo = 35 s (d) ∆tlo = 95 s

Figure 4.12: Transient thermal response of the receiver inlet (Ts,0) and fluid outlet (Tf,out) to a temporarydecrease of incident irradiance (Ginc), for different time intervals of the cloud passage (∆tlo).

57

ceiver’s temperature at the inlet region. At the beginning of this procedure, the energy received from the

absorber’s inlet region is such that air reaches the outlet region at thermal equilibrium with the receiver.

One can see in Figure 4.13(a) that the outlet temperature has barely decreased when the shutdown

procedure has started 5 s ago, while the inlet temperature has diminished abruptly. The absorber’s out-

let region can only be cooled when the receiver’s inlet region has significantly lowered its temperature,

which in Figure 4.13(a) is only possible when t > 5 s. This is the reason for the thermal latent response

between the receiver’s edges observed.

Due to this latency, the outlet air temperature isn’t as affected as the receiver inlet temperature for

low ∆tlo. In Figure 4.12(a), a maximum temperature difference of 63.9 K has been recorded for Tf,out

when ∆tlo = 10 s, which is lower than the 122.7 K attained for Ts,0. Nevertheless, the thermal behaviour

at the outlet becomes similar to the inlet thermal behaviour when a higher value of ∆tlo is considered,

since thermal equilibrium has been attained. Once again, note that an analogous study can be done for

the temperature increase.

With the transient results, one can also find the equilibrium times teq,lo and teq,up, which are associ-

ated to the lower and upper irradiances respectively. When the cloud is shading the heliostat field,teq,lo

is the time needed for the system to reach a ”quasi-steady” state; while teq,up is the time required for the

system to recover its normal operation after the passage of the cloud. Since each simulation starts from

a steady state condition, teq,lo is constant in each set of simulations (as seen in Table E.5). However,

when Ginc is increased, the temperature profile of both media might be different. Therefore, Figure 4.14

presents the different upper equilibrium times as a function of the cloud interval time. Notice that in this

figure, the green line corresponds to the simulations performed in Figure 4.12.

When ∆tlo is greater than the lower equilibrium time, a new ”quasi-steady” state is temporarily at-

Figure 4.13: Transient response for a shutdown procedure, with Ginc,up = 800 kW.m−2 and m =0.5 kg.s−1: (a) Temperature profile; (b) Receiver temperature difference.

58

tained in the receiver (when Ginc = Ginc,lo). As a consequence, teq,up becomes constant for higher

cloud interval times. On the other hand, one can see that the upper equilibrium time decreases as ∆tlo

diminishes. This is due to the latency previously identified in the outlet fluid. As a result, a smaller air

temperature drop results in a faster recovery of the absorber’s normal operating conditions. One of the

most interesting results of this work was to see that for low ∆tlo, the recovery time is much larger than

the perturbation itself. For example, in the simulation illustrated in Figure 4.12(a), a ∆tlo = 10 s results

in a upper equilibrium time of 34.4 s.

Figure 4.14 and Table E.5 also compare different sets of simulations. One can see that when the

irradiance difference is maintained (blue and red simulations in Figure 4.14), a higher mass flow rate

results in lower equilibrium times, which is consistent with the previous results. On the other hand, for a

constant m (blue and green simulations in Figure 4.14), a higher irradiance difference generates larger

equilibrium times. This may seem contradictory when compared to the results of Figure 4.10, since one

would expect similar teq for different Ginc differences. However, the simulations performed in Figure 4.10

have the same starting temperature of the receiver and different Ginc,up; while the simulations of this

section have the same Ginc,up but different starting temperatures of the absorber. The variation of the

starting temperatures is responsible for the different teq,up attained.

The blue and green simulations in Figure 4.14 can be compared as an example. Assuming that

∆tlo > teq,lo, the blue simulation has a Tf,out = Tamb when Ginc starts to increase; whereas in the

green simulation, Tf,out = 1000.1 K when Ginc begins to rise. Since the desired setpoint is the same,

the temperature difference in the blue simulation is much greater than in the green, and thus a higher

teq,up is needed.

Figure 4.14: Equilibrium time required for the system to recover its normal operation after the cloudpassage.

59

4.5.3 Thermal stress

As discussed previously, the transient regime of the receiver displays a non-uniform thermal be-

haviour. During the shutdown procedure, represented in Figure 4.13(a), the absorber’s outlet region

temperature is significantly higher than the inlet region temperature. In the majority of time stamps, it

is observed a noticeable increase of temperature along the receiver. Hence, a study on local thermal

stress (σth,i) throughout the absorber can be performed for this situation (using expression 4.2).

Figure 4.13(b) depicts the temperature difference between adjacent nodes along the absorber, for

each time stamp. A maximum ∆Ts,i of almost 15 K is attained for t = 5 s. Additionally, for time stamps

in the range of 5 s ≤ t ≤ 20 s, a ∆Ts,i > 5 K is recorded for most of the receiver’s length. The highlighted

results present significant temperature differences if one takes into account the short distance between

adjacent nodes, which in these simulations is 2∆x = 0.4 mm. Therefore, the observed ∆Ts,i can produce

local thermal stresses along the absorber.

The presented analysis on thermal stress was done for the shutdown procedure, but it can applied to

every situation where a sudden change of the incident irradiance is observed. Thus, a similar behaviour

of ∆Ts,i is expected in all of the simulations performed in this work.

Note that in a real CRS plant, a high thermal gradient along the absorber must be prevented. If the

temperature difference is high enough, the generated stress has the potential to permanently damage

the receiver, and that can ultimately result in the malfunction of the entire power plant. This is a problem

that happens not only to volumetric absorbers but also to other solar tower receivers. In order to avoid

this situation, different philosophies have been employed in these systems to mitigate radiative flux

variations and, consequently, temperature gradients along the absorber [88–90]. A standard procedure

of CRS power plants is to preheat the receiver in the morning, from ambient temperature to a operating

temperature in a gradual manner. An analogous policy is implemented for the shutdown process at the

end of the day. Moreover, the absorber is kept on a standby operation shortly before the solar tower plant

gets shaded by clouds. This standby mode allows the receiver to rapidly resume its normal operation

after the cloud passage, but without provoking an excessive temperature gradient. When all of these

measures are applied, the temperature along the receiver varies more evenly, and thus the durability of

the receiver is greatly increased [88, 89].

60

Chapter 5

Conclusions

The objective of this thesis was to develop a model to study the transient thermal behaviour of an

open-cell foam volumetric receiver. In this work, the model was greatly simplified due to the inclusion of

a unidimensional representation. This was possible because the cross section dimensions of an OCF

absorber are usually much larger than the receiver’s flow length. This has allowed to implement the two-

flux approximation as a simplification of the radiative transfer equation. As the literature hasn’t reported

an one-dimensional transient model of an OCF receiver, the development of this model is relevant in

transient analysis.

As proposed, the presented heat transfer model shows flexibility. It can be used for absorbers with

porosities ranging from 0.66 to 0.9. In addition, it is capable of predicting fluid temperatures between

100 K to 1600 K. Note that the upper temperature boundary is well below the melting point of the material

used in the simulations [10]. Although this model was designed for a silicon carbide absorber, it can be

applied to other materials. To that end, the optical correlation of β must be modified to match the

material’s characteristics.

The initial simulations were focused on the steady state behaviour of the model. In a first analysis,

the receiver’s intrinsic properties have been studied in order to achieve an optimal design. Tests results

have shown that porosity and particle diameter have the greatest impact on the absorber’s thermal

efficiency. After evaluating its properties, the thermal performance of the absorber was tested under

different external conditions. Since the goal of central receiver systems is to maximize power production,

the optimum system efficiency has been attained for each set of simulations with the same incident

irradiance. During this study, it was concluded that the Qsup/m usually found in literature doesn’t offer

a proper normalization of the results. Moreover, in a real solar power plant, both Ginc and m can be

controlled while the receiver is being operated. Hence, a representation depending on the mass flow

rate has been preferred, due to its practical application and to better understand the influence of these

parameters individually.

Regarding the absorber’s transient response, an important parameter is the equilibrium time. It has

been proven in this work that this parameter is strongly dependent on the mass flow rate. Regarding

the absorber’s intrinsic properties, it was verified that porosity has the most significant influence on

61

equilibrium time.

The transient analysis was also performed considering cloudy weather. Since the presented work

hasn’t addressed explicitly the heliostat field, the executed simulations have dealt with the heliostat

shading indirectly. Even though this assumption simplified the weather issue, some effects that occur in

the mirrors have been neglected in this model. For instance, a recent research shows that the impact of

passing clouds can be associated with the optical efficiency of the heliostat field [17]. Nevertheless, this

study has revealed that the normal operation of the solar tower gets immediately disrupted as soon as

the incident radiation lowers, which indicates that the system is highly sensitive to fluctuations of Ginc.

It was also concluded that for short-duration perturbations, the recovery time is much larger than the

perturbation itself, which is one the most relevant results of this work.

The absorber’s sensitivity to the incident irradiance was identified in all of the presented simulations.

It was observed that a sudden change of this parameter creates a temporary temperature difference

along the solid structure, which induces local thermal stresses that can damage the receiver. To prevent

this issue, operational strategies proposed in the literature have been identified in order to extend the

absorber’s lifetime.

5.1 Future Work

During the model validation, it was discussed the influence of the inlet convection term in the fluid

boundary condition. However, this deliberation has only considered the presence or the absence of the

convection term, instead of evaluating the convective energy that re-enters the absorber as a fraction.

Therefore, further work could be investigating the relation between inlet air velocity and wind speed, in

order to assess the ratio of convective power that is lost.

Additionally, one could perform longer simulations of the volumetric receiver model. These simula-

tions should include the operating philosophies proposed above, where a weather predicting tool can

be added to the model as well. Using meteorogical data of a specific site, the simulations could esti-

mate the power produced over a day or even over a year, in order to assess the feasibility of its real

implementation.

62

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70

Appendix A

Finite-divided-difference formulas

When solving numerical problems, a partial derivative can be computed using finite-divided-difference

formulas, which are derived from the Taylor series expansion. The following equations present the for-

ward, backward and centered finite-divided-difference formulas for the first and second derivative. Note

that for each derivative and approximation type, the first and second order formulas are shown.

A.1 Forward finite-divided-difference formulas

First derivative:

f ′(xi) =f(xi+1)− f(xi)

δx. (A.1a)

f ′(xi) =−f(xi+2) + 4f(xi+1)− 3f(xi)

2δx. (A.1b)

Second derivative:

f ′′(xi) =f(xi+2)− 2f(xi+1) + f(xi)

(δx)2. (A.2a)

f ′′(xi) =−f(xi+3) + 4f(xi+2)− 5f(xi+1) + 2f(xi)

(δx)2. (A.2b)

A.2 Backward finite-divided-difference formulas

First derivative:

f ′(xi) =f(xi)− f(xi−1)

δx. (A.3a)

f ′(xi) =3f(xi)− 4f(xi−1) + f(xi−2)

2δx. (A.3b)

71

Second derivative:

f ′′(xi) =f(xi)− 2f(xi−1) + f(xi−2)

(δx)2. (A.4a)

f ′′(xi) =2f(xi)− 5f(xi−1) + 4f(xi−2)− f(xi−3)

(δx)2. (A.4b)

A.3 Centered finite-divided-difference formulas

First derivative:

f ′(xi) =f(xi+1)− f(xi−1)

2δx. (A.5a)

f ′(xi) =−f(xi+2) + 8f(xi+1)− 8f(xi−1) + f(xi−2)

12δx. (A.5b)

Second derivative:

f ′′(xi) =f(xi+1)− 2f(xi) + f(xi−1)

(δx)2. (A.6a)

f ′′(xi) =−f(xi+2) + 16f(xi+1)− 30f(xi) + 16f(xi−1)− f(xi−2)

12(δx)2. (A.6b)

72

Appendix B

Auxiliary calculations

B.1 Transient heat transfer: Interior nodes

After applying the finite-divided-difference formulas to the partial derivatives, the heat transfer equa-

tions of interior nodes become:

(1− ϕ)(ρcP )s,iT t+1s,i − T ts,i

∆t= (1− ϕ)ks,i

T t+1s,i+1 − 2T t+1

s,i + T t+1s,i−1

∆x2− dqR

dx

∣∣∣∣i

+ hi(Ttf,i − T ts,i) , (B.1a)

ϕ(ρcP )f,iT t+1f,i − T tf,i

∆t+ (ρcP )f,iuf,i

T t+1f,i − T

t+1f,i−1

∆x= ϕkf,i

T t+1f,i+1 − 2T t+1

f,i + T t+1f,i−1

∆x2+ hi(T

ts,i − T tf,i) .

(B.1b)

By manipulating expression (B.1), one gets:

T t+1s,i −T

ts,i =

∆t

(ρcP )s,i

ks,i∆x2

[(T t+1s,i+1−2T t+1

s,i +T t+1s,i−1

)− ∆x2

(1− ϕ)ks,i

dqRdx

∣∣∣∣i

+hi∆x

2

(1− ϕ)ks,i

(T tf,i−T ts,i

)], (B.2a)

T t+1f,i −T

tf,i+uf,i

∆t

ϕ

T t+1f,i − T

t+1f,i−1

∆x=

∆t

(ρcP )f,i

kf,i∆x2

[(T t+1f,i+1−2T t+1

f,i +T t+1f,i−1

)+hi∆x

2

ϕkf,i

(T ts,i−T tf,i

)]. (B.2b)

It is useful to define the Fourier number(λs(f),i = ∆t

(ρcP )s(f),i

ks(f),i∆x2

)and Biot number

(φs(f),i = hi∆x

2

ks(f),i

)to

equation (B.2), so that:

T t+1s,i − T

ts,i = λs,i

[(T t+1s,i+1 − 2T t+1

s,i + T t+1s,i−1

)− ∆x2

(1− ϕ)ks,i

dqRdx

∣∣∣∣i

+φs,i

1− ϕ(T tf,i − T ts,i

)], (B.3a)

T t+1f,i − T

tf,i + uf,i

∆t

ϕ∆x

(T t+1f,i − T

t+1f,i−1

)= λf,i

[(T t+1f,i+1 − 2T t+1

f,i + T t+1f,i−1

)+φf,iϕ

(T ts,i − T tf,i

)], (B.3b)

73

and rearranging equation (B.3), one gets:

−λs,iT t+1s,i−1 +

(1 + 2λs,i

)T t+1s,i −λs,iT

t+1s,i+1 =

(1−λs,i

φs,i1− ϕ

)T ts,i−λs,i

∆x2

(1− ϕ)ks,i

dqRdx

∣∣∣∣i

+λs,iφs,i

1− ϕT tf,i ,

(B.4a)

−(λf,i+uf,i

∆t

ϕ∆x

)T t+1f,i−1 +

(1+2λf,i+uf,i

∆t

ϕ∆x

)T t+1f,i −λf,iT

t+1f,i+1 =

(1−λf,i

φf,iϕ

)T tf,i+λf,i

φf,iϕT ts,i .

(B.4b)

B.2 Transient heat transfer: Boundary conditions

B.2.1 Inlet

Solid

(1− ϕ)αGinc − (1− ϕ)qrad,in − (1− ϕ)qconv,in = −(1− ϕ)ks,0∂T

∂x

∣∣∣∣0

. (B.5)

There aren’t any nodes of the receiver before the entrance of the receiver, so the forward finite-divided-

difference formula is used to replace the partial derivative. Substituting also equation (3.6) in expres-

sion (B.5) gives:

αGinc − εσ[(T ts,0

)4 − (T tsky)4]− hin(T t+1s,0 − T

t+1f,in

)= ks,0

T t+1s,0 − T

t+1s,1

∆x, (B.6)

and rearranging equation (B.6), one gets:

(hin +

ks,0∆x

)T t+1s,0 −

ks,0∆x

T t+1s,1 = αGinc + hinT

t+1f,in + εσ

[(T tsky

)4 − (T ts,0)4] . (B.7)

Fluid

− (cP )f,0 m(T t+1f,0 − T

t+1f,in

)−Aϕkf,0

∂T

∂x

∣∣∣∣0−

= −Aϕkf,0∂T

∂x

∣∣∣∣0+

. (B.8)

In expression (B.8), there are two partial derivatives associated to the conduction terms. On the left side,

the backward finite-divided-difference formula is used; and on the right side, the forward finite-divided-

difference formula is employed. By also replacing equation (3.8) in (B.8), one attains:

(ρcP )f,0uf,0(T t+1f,0 − T

t+1f,in

)+ ϕkf,0

T t+1f,0 − T

t+1f,in

∆x= ϕkf,0

T t+1f,1 − T

t+1f,0

∆x, (B.9)

and rearranging equation (B.9) gives:

[(ρcP )f,0uf,0 + 2

ϕkf,0∆x

]T t+1f,0 −

ϕkf,0∆x

T t+1f,1 =

[(ρcP )f,0uf,0 +

ϕkf,0∆x

]T t+1f,in . (B.10)

74

B.2.2 Outlet

Solid

− (1− ϕ)ks,n∂T

∂x

∣∣∣∣n

= (1− ϕ)qconv,out + (1− ϕ)qrad,out . (B.11)

Some assumptions can be made at the receiver’s outlet. First, one can consider that the wall in front

of the absorber is always at thermal equilibrium with the outlet air (T t+1w = T t+1

f,out). Additionally, it is

assumed that the outlet fluid temperature is the temperature of the fluid at node ’n’ in the previous time

step (T t+1f,out = T tf,n). Analogously to the inlet case, there aren’t any nodes of the receiver after the

receiver’s outlet, so the backward finite-divided-difference formula is used to replace the partial deriva-

tive. Considering all this assumptions, and using the definition of equation (3.10), expression (B.11)

becomes:

ks,nT t+1s,n−1 − T t+1

s,n

∆x= hout

(T t+1s,n − T tf,n

)+ εσ

[(T ts,n

)4 − (T tf,n)4] , (B.12)

and rearranging equation (B.12), one gets:

− ks,n∆x

T t+1s,n−1 +

(ks,n∆x

+ hout

)T t+1s,n = houtT

tf,n + εσ

[(T tf,n

)4 − (T ts,n)4] . (B.13)

Fluid

− (cP )f,n m(T t+1f,n − T

t+1f,n−1

)−Aϕkf,n

∂T

∂x

∣∣∣∣n−

= −(cP )f,nm(T t+1f,out − T

t+1f,n

)−Aϕkf,n

∂T

∂x

∣∣∣∣n+

. (B.14)

Similar to the inlet case, expression (B.14) has two partial derivatives. The left one uses the backward

finite-divided-difference formula and the right one employs the forward finite-divided-difference formula.

Considering once again that T t+1f,out = T tf,n, expression (B.14) gives:

(ρcP )f,nuf,n(T t+1f,n − T

t+1f,n−1

)+ ϕkf,n

T t+1f,n − T

t+1f,n−1

∆x= (ρcP )f,nuf,n

(T tf,n − T t+1

f,n

)+ ϕkf,n

T tf,n − Tt+1f,n

∆x,

(B.15)

and rearranging equation (B.15), one gets:

−[(ρcP )f,nuf,n +

ϕkf,n∆x

]T t+1f,n−1 + 2

[(ρcP )f,nuf,n +

ϕkf,n∆x

]T t+1f,n =

[(ρcP )f,nuf,n +

ϕkf,n∆x

]T tf,n . (B.16)

75

B.3 Radiative heat transfer: Boundary conditions

The boundary conditions of the two-flux approximation applied to the diffuse radiation are given by

expression (B.17):

x = 0 : Gd,0 + 2qd,0 = 4J1 , (B.17a)

x = Lr : Gd,n − 2qd,n = 4J2 . (B.17b)

Before solving the boundary conditions, it is useful to rewrite equation (3.39a) to obtain a expression

for qd(x):

qd = − 1

4β

dGddx

. (B.18)

Using equations (B.18), (3.40b) and (3.40c), expression (B.17) becomes:

Gd,0 −1

2β

dGddx

∣∣∣0

= 4ϕ

[ζGinc + εσ

(T ts,0

)4], (B.19a)

Gd,n +1

2β

dGddx

∣∣∣n

= 4ϕεσ(T ts,n

)4. (B.19b)

The forward and backward finite-divided-difference formulas are now applied to equations (B.19a) and (B.19b)

respectively, in order to resolve the first order derivative:

Gd,0 −1

2β

Gd,1 −Gd,0∆x

= 4ϕ

[ζGinc + εσ

(T ts,0

)4], (B.20a)

Gd,n +1

2β

Gd,n −Gd,n−1

∆x= 4ϕεσ

(T ts,n

)4, (B.20b)

and rearranging expression (B.20), one gets:

(1 + 2β∆x)Gd,0 −Gd,1 = 8β∆xϕ

[ζGinc + εσ

(T ts,0

)4], (B.21a)

−Gd,n−1 + (1 + 2β∆x)Gd,n = 8β∆xϕεσ(T ts,n

)4. (B.21b)

76

Appendix C

Thomas algorithm

The Thomas algorithm is a computational method that simplifies the Gaussian elimination for tridiag-

onal systems. These kind of systems can be expressed by:

Mw = b , (C.1)

where:

M =

f0 g0 0 · · · 0

e1 f1 g1. . .

...

0. . . . . . . . . 0

.... . . en−1 fn−1 gn−1

0 · · · 0 en fn

w =

w0

w1

...

wn−1

wn

b =

b0

b1...

bn−1

bn

. (C.2)

This algorithm has three steps. First, one performs a LU decomposition, where M is divided in a

lower and upper matrices:

M = LU =

ψ0 0 · · · · · · 0

e1 ψ1. . . . . .

...

0. . . . . . . . .

......

. . . en−1 ψn−1 0

0 · · · 0 en ψn

·

1 γ0 0 · · · 0

0 1 γ1. . .

......

. . . . . . . . . 0...

. . . . . . 1 γn−1

0 · · · · · · 0 1

, (C.3)

where

ψ0 = f0 , (C.4a)

ψi = fi − eiγi−1, i = 1, 2, . . . , n , (C.4b)

γi =giψi, i = 0, 1, . . . , n− 1 . (C.4c)

77

Then, a forward substitution is applied with the lower matrix:

Lr = b , (C.5)

where:

r0 =b0ψ0

, (C.6a)

ri =bi − eiri−1

ψi, i = 1, 2, . . . , n . (C.6b)

Finally, the solution is attained after performing a backward substitution with the upper matrix:

Uw = r , (C.7)

where:

wn = rn , (C.8a)

wi = ri − γiwi+1, i = n− 1, n− 2, . . . , 0 . (C.8b)

This algorithm is only stable in certain situations. One necessary conditions for numerical stability is

the matrix M to be diagonal dominant:

|f0| > |g0| > 0 , (C.9a)

|fi| ≥ |ei|+ |gi|, eigi 6= 0, i = 1, 2, . . . , n− 1 . (C.9b)

|fn| > |en| > 0 . (C.9c)

In these conditions, the matrix M is considered to be non-singular, and therefore LU decomposition

is possible. When this method cannot be applied, Gaussian elimination with partial pivoting is recom-

mended [64].

78

Appendix D

Numerical implementation of the

algorithm

In Chapter 3, an algorithm to solve the heat transfer model was proposed. The source code (in C) of

its numerical implementation is presented below (where the standard conditions shown in Table 4.2 are

applied).

1 # inc lude <s t d i o . h>

2 # inc lude <s t d l i b . h>

3 # inc lude <s t r i n g . h>

4 # inc lude <t ime . h>

5 # inc lude <math . h>

6

7 / / / / / / / / / / / / / / / / Ob jec t i ve : Temperature p r o f i l e / / / / / / / / / / / / / / /

8 double Ts [101 ] , Tf [ 1 0 1 ] ;

9

10 / / / / / / / / / / / / / / / / / / / / / / Global Var iab les / / / / / / / / / / / / / / / / / / / / / / /

11

12 / / User data

13 long i n t Ginc=600000;

14 double Lr =0.02 , Area=1 , dotm =0.6 , Tamb=300 , Tsky =273.15;

15

16 / / I n t e r n a l data

17 i n t n = 100;

18 double varph i =0.8 , alpha =0.9 , eps i l on =0.9 , dp=0.0008 , zeta =0 .1 ;

19 double sigma=5.67∗pow(10 ,−8) , Tk = 273.15 , c1 = 1.458∗pow(10 ,−6) , c2 = 110.4 , pf0 = 101325;

20 double Cmax=0.1 , rhofmin =0.21 , k1k2min = 1.94∗pow(10 ,−6) ;

21

22 / / Model ’ s constants

23 double dx , alphap , a lphasf , dh , kappa , sigmasc , beta , dt , k1 , k2 , k1k2 , umax ;

24

25 / / Time , t ime counters and space counters

26 long i n t t =0 , n t =15000, taux =0 , ntaux =2000;

79

27 double t r e a l ;

28 i n t i ;

29

30 / / Rad ia t i ve p r o f i l e

31 double Gc[101 ] , Gd[101 ] , Gt [ 101 ] , dq [ 1 0 1 ] ;

32

33 / / Temperature c o r r e l a t i o n s

34 double ks [101 ] , cps [101 ] , rhos [101 ] , as [ 1 0 1 ] ;

35 double k f [ 101 ] , cp f [ 101 ] , r ho f [ 101 ] , muf [ 101 ] , nuf [ 101 ] , p f [ 101 ] , u f [ 1 0 1 ] ;

36 i n t Rf = 287;

37

38 / / Temperature dependent p r o p e r t i e s

39 double Re[101 ] , Pr [ 101 ] , Nu[101 ] , hs f [ 101 ] , h [ 1 0 1 ] ;

40 double lambdas [101 ] , lambdaf [ 101 ] , ph is [ 101 ] , p h i f [ 1 0 1 ] ;

41

42 / / Other temperatures and c o r r e l a t i o n s

43 double T f i n ;

44

45 / / Record o ld values

46 double Tsold [101 ] , T fo ld [ 1 0 1 ] ;

47 double T f i n o l d ;

48

49

50 / / / / / / / / / / / / / / / / / / / / / / Funct ions / / / / / / / / / / / / / / / / / / / / / / /

51

52 / / D i f f use r a d i a t i o n , t o t a l r a d i a t i o n and energy deposi ted

53 vo id Gdqcalc ( )

54

55 / / Arrays

56 double f [ 101 ] , g [101 ] , e [101 ] , b [ 1 0 1 ] ;

57 double ps i [ 101 ] , gama[101 ] , r [ 101 ] , w[ 1 0 1 ] ;

58

59 / / Clear v a r i a b l e s

60 f o r ( i =0; i<=n ; ++ i )

61

62 w[ i ] = 0 ;

63 ps i [ i ] = 0 ;

64 gama [ i ] = 0 ;

65 r [ i ] = 0 ;

66

67

68 / / Set i n i t i a l values

69 f [ 0 ] = 1 + 2∗beta∗dx ;

70 g [ 0 ] = −1;

71 e [ 0 ] = 0 ;

72 b [ 0 ] = 8∗beta∗dx∗ varph i ∗ ( zeta ∗ ( double ) Ginc + eps i l on ∗sigma∗pow( Tsold [ 0 ] , 4 ) ) ;

73 f [ n ] = 1 + 2∗beta∗dx ;

74 g [ n ] = 0 ;

75 e [ n ] = −1;

80

76 b [ n ] = 8∗beta∗dx∗ varph i ∗eps i l on ∗sigma∗pow( Tsold [ n ] , 4 ) ;

77

78 f o r ( i =1; i<=n−1; ++ i )

79

80 f [ i ] = 2 + 4∗beta∗kappa∗dx∗dx ;

81 g [ i ] = −1;

82 e [ i ] = −1;

83 b [ i ] = 16∗beta∗kappa∗dx∗dx∗ varph i ∗sigma∗pow( Tsold [ i ] , 4 ) + 4∗beta∗sigmasc∗dx∗dx∗Gc[ i ] ;

84

85

86 / / LU Decomposit ion

87 ps i [ 0 ] = f [ 0 ] ;

88 gama [ 0 ] = g [ 0 ] / ps i [ 0 ] ;

89

90 f o r ( i =1; i<=n−1; ++ i )

91

92 ps i [ i ] = f [ i ] − e [ i ]∗gama [ i −1];

93 gama [ i ] = g [ i ] / ps i [ i ] ;

94

95 ps i [ n ] = f [ n ] − e [ n ]∗gama [ n−1];

96

97 / / Forward s u b s t i t u t i o n ( Lr = b )

98 r [ 0 ] = b [ 0 ] / ps i [ 0 ] ;

99

100 f o r ( i =1; i<=n ; ++ i )

101

102 r [ i ] = ( b [ i ] − e [ i ]∗ r [ i −1]) / ps i [ i ] ;

103

104

105 / / Backward s u b s t i t u t i o n (Ua = r )

106 w[ n ] = r [ n ] ;

107

108 f o r ( i =n−1; i >=0; −− i )

109

110 w[ i ] = r [ i ] − gama [ i ]∗w[ i + 1 ] ;

111

112

113 / / Ca lcu la te Gd, Gt e dq

114 f o r ( i =0; i<=n;++ i )

115

116 Gd[ i ] = w[ i ] ;

117 Gt [ i ] = Gc [ i ] + Gd[ i ] ;

118 dq [ i ] = kappa∗(4∗ varph i ∗sigma∗pow( Tsold [ i ] , 4 ) − Gt [ i ] ) ;

119

120

121

122 / / F l u i d temperature c o r r e l a t i o n s

123 vo id Tcor rca lc ( )

124

81

125 / / I n l e t

126 k f [ 0 ] = −3.9333∗pow(10 ,−4) + 1.0184∗pow(10 ,−4)∗Tfo ld [ 0 ] − 4.8574∗pow(10 ,−8)∗pow( T fo ld [ 0 ] , 2 )

+ 1.5207∗pow(10 ,−11)∗pow( T fo ld [ 0 ] , 3 ) ;

127 cpf [ 0 ] = 1.0575∗pow(10 ,3 ) − 4.4890∗pow(10 ,−1)∗Tfo ld [ 0 ] + 1.1407∗pow(10 ,−3)∗pow( T fo ld [ 0 ] , 2 ) −

7.9999∗pow(10 ,−7)∗pow( T fo ld [ 0 ] , 3 ) + 1.9327∗pow(10 ,−10)∗pow( T fo ld [ 0 ] , 4 ) ;

128 muf [ 0 ] = ( c1∗pow( T fo ld [ 0 ] , 1 . 5 ) ) / ( T fo ld [ 0 ] + c2 ) ;

129 r ho f [ 0 ] = p f [ 0 ] / ( ( double ) Rf∗Tfo ld [ 0 ] ) ;

130 nuf [ 0 ] = muf [ 0 ] / r ho f [ 0 ] ;

131 uf [ 0 ] = dotm / ( Area∗ r ho f [ 0 ] ) ;

132

133 / / I n t e r i o r nodes

134 f o r ( i =1; i<=n ; ++ i )

135

136 k f [ i ] = −3.9333∗pow(10 ,−4) + 1.0184∗pow(10 ,−4)∗Tfo ld [ i ] − 4.8574∗pow(10 ,−8)∗pow( T fo ld [ i

] , 2 ) + 1.5207∗pow(10 ,−11)∗pow( T fo ld [ i ] , 3 ) ;

137 cpf [ i ] = 1.0575∗pow(10 ,3 ) − 4.4890∗pow(10 ,−1)∗Tfo ld [ i ] + 1.1407∗pow(10 ,−3)∗pow( T fo ld [ i ] , 2 )

− 7.9999∗pow(10 ,−7)∗pow( T fo ld [ i ] , 3 ) + 1.9327∗pow(10 ,−10)∗pow( T fo ld [ i ] , 4 ) ;

138 muf [ i ] = ( c1∗pow( T fo ld [ i ] , 1 . 5 ) ) / ( T fo ld [ i ]+ c2 ) ;

139 pf [ i ] = s q r t (pow( p f [ i −1] ,2)−2∗Rf∗Tfo ld [ i ]∗ dx ∗ ( ( ( muf [ i ]∗dotm ) / ( k1∗Area ) ) + (1 / k2 ) ∗pow( dotm /

Area , 2 ) ) ) ;

140 r ho f [ i ] = p f [ i ] / ( ( double ) Rf∗Tfo ld [ i ] ) ;

141 nuf [ i ] = muf [ i ] / r ho f [ i ] ;

142 uf [ i ] = dotm / ( Area∗ r ho f [ i ] ) ;

143

144

145

146

147 / / Temperature dependent p r o p e r t i e s

148 vo id Tpropcalc ( )

149

150 / / Reynolds and Prand t l number

151 f o r ( i =0; i<=n ; ++ i )

152

153 Re[ i ] = ( u f [ i ]∗dh ) / ( va rph i ∗nuf [ i ] ) ;

154 Pr [ i ] = ( cp f [ i ]∗muf [ i ] ) / k f [ i ] ;

155

156

157 / / Nussel t number and heat t r a n s f e r c o e f f i c i e n t ( i n l e t and o u t l e t )

158 Nu[0]=2.0696∗pow( varphi , 0 . 3 8 ) ∗pow(Re[ 0 ] , 0 . 4 3 8 ) ;

159 Nu[ n]=2.0696∗pow( varphi , 0 . 3 8 ) ∗pow(Re [ n ] , 0 . 4 3 8 ) ;

160 hsf [ 0 ] = ( k f [ 0 ]∗Nu [ 0 ] ) / dp ;

161 hsf [ n ] = ( k f [ n ]∗Nu[ n ] ) / dp ;

162 f o r ( i =1; i<=n−1; ++ i )

163

164 / / Nussel t number and heat t r a n s f e r c o e f f i c i e n t ( i n t e r i o r nodes )

165 Nu[ i ]=2+((12∗(1− varph i ) ) / va rph i ) + s q r t (1−varph i ) ∗pow( Pr [ i ] , ( 1 / 3 ) ) ∗pow ( ( Re [ i ]∗dp∗ varph i ) / dh

, 0 . 6 ) ;

166 hsf [ i ] = ( k f [ i ]∗Nu[ i ] ) / dp ;

167 h [ i ]= hs f [ i ]∗ a lphas f ;

82

168

169 / / Four ie r and B io t number

170 lambdas [ i ] = ( d t∗ks [ i ] ) / ( rhos [ i ]∗ cps [ i ]∗ dx∗dx ) ;

171 phis [ i ] = ( h [ i ]∗ dx∗dx ) / ks [ i ] ;

172 lambdaf [ i ] = ( d t∗ k f [ i ] ) / ( r ho f [ i ]∗ cpf [ i ]∗ dx∗dx ) ;

173 p h i f [ i ] = ( h [ i ]∗ dx∗dx ) / k f [ i ] ;

174

175

176

177

178 / / Temperatures outs ide the rece i ve r

179 vo id Textca lc ( )

180

181 T f i n = Tamb;

182

183

184

185 / / Ca lcu la te the s o l i d phase temperature

186 vo id Tscalc ( )

187

188 / / Arrays

189 double f [ 101 ] , g [101 ] , e [101 ] , b [ 1 0 1 ] ;

190 double ps i [ 101 ] , gama[101 ] , r [ 101 ] , w[ 1 0 1 ] ;

191

192 / / Clear v a r i a b l e s

193 f o r ( i =0; i<=n ; ++ i )

194

195 w[ i ] = 0 ;

196 ps i [ i ] = 0 ;

197 gama [ i ] = 0 ;

198 r [ i ] = 0 ;

199

200

201 / / Set i n i t i a l values

202 f [ 0 ] = hsf [ 0 ] + ( ks [ 0 ] / dx ) ;

203 g [ 0 ] = −(ks [ 0 ] / dx ) ;

204 e [ 0 ] = 0 ;

205 b [ 0 ] = alpha ∗ ( double ) Ginc + hsf [ 0 ]∗ T f i n + eps i l on ∗sigma ∗ (pow( Tsky , 4 ) − pow( Tsold [ 0 ] , 4 ) ) ;

206 f [ n ] = ( ks [ n ] / dx ) + hsf [ n ] ;

207 g [ n ] = 0 ;

208 e [ n ] = −(ks [ n ] / dx ) ;

209 b [ n ] = hsf [ n ]∗ Tfo ld [ n ] + eps i l on ∗sigma ∗ (pow( T fo ld [ n ] , 4 ) − pow( Tsold [ n ] , 4 ) ) ;

210

211 f o r ( i =1; i<=n−1; ++ i )

212

213 f [ i ] = 1 + 2∗ lambdas [ i ] ;

214 g [ i ] = −lambdas [ i ] ;

215 e [ i ] = −lambdas [ i ] ;

216 b [ i ] = (1−(( lambdas [ i ]∗ phis [ i ] ) /(1− varph i ) ) ) ∗Tsold [ i ] − ( ( lambdas [ i ]∗ dx∗dx ) /((1− varph i ) ∗ks

83

[ i ] ) ) ∗dq [ i ] + ( ( lambdas [ i ]∗ phis [ i ] ) /(1− varph i ) ) ∗Tfo ld [ i ] ;

217

218

219 / / LU Decomposit ion

220 ps i [ 0 ] = f [ 0 ] ;

221 gama [ 0 ] = g [ 0 ] / ps i [ 0 ] ;

222

223 f o r ( i =1; i<=n−1; ++ i )

224

225 ps i [ i ] = f [ i ] − e [ i ]∗gama [ i −1];

226 gama [ i ] = g [ i ] / ps i [ i ] ;

227

228 ps i [ n ] = f [ n ] − e [ n ]∗gama [ n−1];

229

230 / / Forward s u b s t i t u t i o n ( Lr = b )

231 r [ 0 ] = b [ 0 ] / ps i [ 0 ] ;

232

233 f o r ( i =1; i<=n ; ++ i )

234

235 r [ i ] = ( b [ i ] − e [ i ]∗ r [ i −1]) / ps i [ i ] ;

236

237

238 / / Backward s u b s t i t u t i o n (Ua = r )

239 w[ n ] = r [ n ] ;

240

241 f o r ( i =n−1; i >=0; −− i )

242

243 w[ i ] = r [ i ] − gama [ i ]∗w[ i + 1 ] ;

244

245

246 / / Ca lcu la te Ts

247 f o r ( i =0; i<=n;++ i )

248

249 Ts [ i ] = w[ i ] ;

250

251

252

253

254 / / Ca lcu la te the f l u i d phase temperature

255 vo id T fca l c ( )

256

257 / / Arrays

258 double f [ 101 ] , g [101 ] , e [101 ] , b [ 1 0 1 ] ;

259 double ps i [ 101 ] , gama[101 ] , r [ 101 ] , w[ 1 0 1 ] ;

260

261 / / Clear v a r i a b l e s

262 f o r ( i =0; i<=n ; ++ i )

263

264 w[ i ] = 0 ;

84

265 ps i [ i ] = 0 ;

266 gama [ i ] = 0 ;

267 r [ i ] = 0 ;

268

269

270 / / Set i n i t i a l values

271 f [ 0 ] = rho f [ 0 ]∗ cpf [ 0 ]∗ uf [ 0 ] + ( (2∗ varph i ∗ k f [ 0 ] ) / dx ) ;

272 g [ 0 ] = −(( va rph i ∗ k f [ 0 ] ) / dx ) ;

273 e [ 0 ] = 0 ;

274 b [ 0 ] = rho f [ 0 ]∗ cpf [ 0 ]∗ uf [ 0 ]∗ T f i n + ( ( va rph i ∗ k f [ 0 ] ) / dx ) ∗T f i n ;

275 f [ n ] = 2∗ r ho f [ n ]∗ cpf [ n ]∗ uf [ n ] + ( (2∗ varph i ∗ k f [ n ] ) / dx ) ;

276 g [ n ] = 0 ;

277 e [ n ] = −( r ho f [ n ]∗ cpf [ n ]∗ uf [ n ] + ( ( va rph i ∗ k f [ n ] ) / dx ) ) ;

278 b [ n ] = rho f [ n ]∗ cpf [ n ]∗ uf [ n ]∗ Tfo ld [ n ] + ( ( va rph i ∗ k f [ n ] ) / dx ) ∗Tfo ld [ n ] ;

279

280 f o r ( i =1; i<=n−1; ++ i )

281

282 f [ i ] = 1 + 2∗ lambdaf [ i ] + ( ( u f [ i ]∗ dt ) / ( va rph i ∗dx ) ) ;

283 g [ i ] = −lambdaf [ i ] ;

284 e [ i ] = −( lambdaf [ i ] + ( ( u f [ i ]∗ dt ) / ( va rph i∗dx ) ) ) ;

285 b [ i ] = (1−(( lambdaf [ i ]∗ p h i f [ i ] ) / va rph i ) ) ∗Tfo ld [ i ] + ( ( lambdaf [ i ]∗ p h i f [ i ] ) / va rph i ) ∗Tsold [ i

] ;

286

287

288 / / LU Decomposit ion

289 ps i [ 0 ] = f [ 0 ] ;

290 gama [ 0 ] = g [ 0 ] / ps i [ 0 ] ;

291

292 f o r ( i =1; i<=n−1; ++ i )

293

294 ps i [ i ] = f [ i ] − e [ i ]∗gama [ i −1];

295 gama [ i ] = g [ i ] / ps i [ i ] ;

296

297 ps i [ n ] = f [ n ] − e [ n ]∗gama [ n−1];

298

299 / / Forward s u b s t i t u t i o n ( Lr = b )

300 r [ 0 ] = b [ 0 ] / ps i [ 0 ] ;

301

302 f o r ( i =1; i<=n ; ++ i )

303

304 r [ i ] = ( b [ i ] − e [ i ]∗ r [ i −1]) / ps i [ i ] ;

305

306

307 / / Backward s u b s t i t u t i o n (Ua = r )

308 w[ n ] = r [ n ] ;

309

310 f o r ( i =n−1; i >=0; −− i )

311

312 w[ i ] = r [ i ] − gama [ i ]∗w[ i + 1 ] ;

85

313

314

315 / / Ca lcu la te Tf

316 f o r ( i =0; i<=n;++ i )

317

318 Tf [ i ] = w[ i ] ;

319

320

321

322 / / / / / / / / / / / / / / / / / / Main f u n c t i o n / / / / / / / / / / / / / / / / / / / / / / / / / /

323

324 i n t main ( i n t argc , char ∗∗argv )

325

326

327 i n t j ;

328 / / F i l e to p r i n t t r a n s i e n t values

329 FILE ∗ t t r a n s i e n t ;

330

331 t t r a n s i e n t = fopen ( ” D a t a t r a ns i e n t . t x t ” , ” wt ” ) ;

332

333 f p r i n t f ( t t r a n s i e n t , ” t r e a l \ t \ t Ts0 \ t \ t Ts100 \ t \ t Ts200 \ t \ t Tf0 \ t \ t Tf100 \ t \ t

Tf200 \n ” ) ;

334

335 p r i n t f ( ” Begin a lgo r i t hm \n ” ) ;

336

337 / / Clear ar rays

338 f o r ( i =0; i<=n ; ++ i )

339

340 Ts [ i ] = 0 ;

341 Tf [ i ] = 0 ;

342 Gc[ i ] = 0 ;

343 ks [ i ] = 0 ;

344 cps [ i ] = 0 ;

345 rhos [ i ] = 0 ;

346 as [ i ] = 0 ;

347 k f [ i ] = 0 ;

348 cpf [ i ] = 0 ;

349 r ho f [ i ] = 0 ;

350 muf [ i ] = 0 ;

351 nuf [ i ] = 0 ;

352 pf [ i ] = 0 ;

353 uf [ i ] = 0 ;

354 Re[ i ] = 0 ;

355 Pr [ i ] = 0 ;

356 Nu[ i ] = 0 ;

357 hsf [ i ] = 0 ;

358 h [ i ] = 0 ;

359 lambdas [ i ] = 0 ;

360 lambdaf [ i ] = 0 ;

86

361 phis [ i ] = 0 ;

362 p h i f [ i ] = 0 ;

363 Gd[ i ] = 0 ;

364 Gt [ i ] = 0 ;

365 dq [ i ] = 0 ;

366 Tsold [ i ] = 0 ;

367 Tfo ld [ i ] = 0 ;

368

369

370 / / Def ine mesh

371 dx = Lr / ( double ) n ;

372 umax = dotm / ( Area∗ rhofmin ) ;

373 dt = ( dx∗Cmax) / umax ;

374 p r i n t f ( ”%f \n ” , d t ) ;

375

376 / / Def ine pore diameter and pe rm eab i l i t y

377 alphap = 6/ dp ;

378 a lphas f = (1−varph i ) ∗alphap ;

379 dh = (4∗ varph i ) / a lphas f ;

380 k1 =(pow( varphi , 3 ) ∗dp∗dp ) / (150∗pow(1−varphi , 2 ) ) ;

381 k2 =(pow( varphi , 3 ) ∗dp ) /(1.75∗(1− varph i ) ) ;

382 k1k2=k1 / k2 ;

383 / / Evaluate p o s s i b i l i t y o f f l ow i n s t a b i l i t i e s

384 i f ( k1k2 >= k1k2min )

385 p r i n t f ( ”K1 / K2 OK\n ” ) ;

386 else

387 p r i n t f ( ”K1 / K2 NOT OK OK\n ” ) ;

388

389 / / Op t i ca l c o r r e l a t i o n

390 beta = (12.64∗pow(1−varphi , 0 . 7 ) ) / ( pow( dh , 0 . 7 9 ) ) ;

391 kappa = alpha∗beta ;

392 sigmasc = (1−alpha ) ∗beta ;

393

394 / / I n l e t pressure

395 pf [ 0 ] = pf0 ;

396

397 / / P rope r t i es o f the absorber

398 f o r ( i =0; i<=n ; ++ i )

399

400 ks [ i ] = 40;

401 cps [ i ] = 1244;

402 rhos [ i ] = 3210;

403

404

405 / / Def ine i n i t i a l cond i t i ons ( temperature p r o f i l e , co l l ima ted r a d i a t i o n and cu r ren t t ime )

406 t r e a l = taux∗nt + d t∗ t ;

407 T f i n = Tamb;

408 T f i n o l d = T f i n ;

409 f o r ( i =0; i<=n ; ++ i )

87

410

411 Gc[ i ] = varph i∗(1−zeta ) ∗Ginc∗exp(− i ∗beta∗dx ) ;

412 Ts [ i ] = Tamb;

413 Tf [ i ] = Tamb;

414 Tsold [ i ] = Ts [ i ] ;

415 Tfo ld [ i ] = Tf [ i ] ;

416

417

418 / / Evo lu t i on i n t ime

419 f o r ( taux =0; taux<=ntaux ; ++taux )

420

421 f o r ( t =1; t<=nt ; ++ t )

422

423 t r e a l = d t ∗ ( taux∗nt + t ) ;

424 / / Ca lcu la te d i f f u s e r a d i a t i o n , t o t a l r a d i a t i o n and energy deposi ted i n the s o l i d

425 Gdqcalc ( ) ;

426

427 / / Ca lcu la te temperature c o r r e l a t i o n s

428 Tcor rca lc ( ) ;

429

430 / / Ca lcu la te temperature dependent p r o p e r t i e s

431 Tpropcalc ( ) ;

432

433 / / Ca lcu la te temperatures outs ide the rece i ve r

434 Textca lc ( ) ;

435

436 / / Ca lcu la te the s o l i d phase temperature

437 Tscalc ( ) ;

438

439 / / Ca lcu la te the f l u i d phase temperature

440 Tfca l c ( ) ;

441

442 / / Store temperatures i n backup v a r i a b l e s

443 f o r ( i =0; i<=n ; ++ i )

444

445 Tsold [ i ] = Ts [ i ] ;

446 Tfo ld [ i ] = Tf [ i ] ;

447

448 T f i n o l d = T f i n ;

449

450

451 / / P r i n t t r a n s i e n t values

452 f p r i n t f ( t t r a n s i e n t , ” %.5 f \ t %.2 f \ t %.2 f \ t %.2 f \ t %.2 f \ t %.2 f \ t %.2 f \n ” , t r e a l , Ts

[ 0 ] , Ts [ 5 0 ] , Ts [100 ] , Tf [ 0 ] , Tf [ 5 0 ] , Tf [ 1 0 0 ] ) ;

453

454

455 f c l o s e ( t t r a n s i e n t ) ;

456

457 / / F i l e to p r i n t f i n a l values ( poss ib l y i n steady−s ta te c o n d i t i o n )

88

458 FILE ∗ t f i n a l ;

459 t f i n a l = fopen ( ” Data steady . t x t ” , ” wt ” ) ;

460

461 / / P r i n t f i n a l values

462 f p r i n t f ( t f i n a l , ” t r e a l \ t \ t i \ t x \ t \ t \ t Ginc \ t \ t Gc \ t \ t Gd \ t G \ t \ t dq \ t \ t p \ t

\ t u \ t \ t Ts \ t \ t Tf \n ” ) ;

463 f o r ( i =0; i<=n ; ++ i )

464

465 f p r i n t f ( t f i n a l , ” %.5 f \ t %d \ t %.5 f \ t %l d \ t %.0 f \ t %.0 f \ t %.0 f \ t %.0 f \ t %.0 f \ t %.2 f \

t %.2 f \ t %.2 f \n ” , t r e a l , i , i ∗dx , Ginc , Gc [ i ] , Gd[ i ] , Gt [ i ] , dq [ i ] , p f [ i ] , u f [ i ] , Ts [ i ] ,

Tf [ i ] ) ;

466

467

468 f c l o s e ( t f i n a l ) ;

469

470 p r i n t f ( ”End a lgo r i t hm \n ” ) ;

471 r e t u r n 0 ;

472

89

90

Appendix E

Detailed results of the simulation

The following appendix presents detailed information about the simulation results in Chapter 4.

91

TableE

.1:D

etailedresults

ofsimulations

performed

inthe

validation.

InitialconditionsS

imulation

results

Node

0N

oden

Maxim

um

mQinc

mTs

Tf

Ts

Tf

xs

Ts

xf

Tf

uf,0

Qf

ηth

erm

al

[kg.s −

1][k

J.k

g−1]

[K]

[K]

[K]

[K]

[mm

][K

][m

m]

[K]

[m.s −

1][W

][−

]

Using

equation(3.7)

0.0

3478

625.7

3745.3

303.9

710.0

710.0

0.0

0745.3

23.9

2710.4

1.7

914754

0.6

78

0.0

3061

711.1

0782.9

305.1

759.4

759.5

0.0

0782.9

23.4

6760.0

1.5

814621

0.6

72

0.0

2476

879.1

3855.1

308.2

851.7

851.8

0.0

0855.1

22.0

8852.8

1.2

914337

0.6

59

0.0

2463

883.8

1857.1

308.3

854.1

854.2

0.0

0857.1

21.6

2855.3

1.2

814329

0.6

58

0.0

1878

1158.6

8969.1

315.1

990.7

990.8

18.8

6993.2

19.3

2993.2

1.0

013819

0.6

35

0.0

1501

1450.0

11078.8

324.3

1118.0

1118.2

16.5

61123.0

17.0

21123.0

0.8

213247

0.6

09

Using

equation(4.1)

0.0

3478

625.7

3781.9

366.9

757.6

757.6

0.0

0781.9

23.4

6758.1

2.1

616546

0.7

60

0.0

3061

711.1

0825.6

380.2

813.9

814.0

0.0

0825.6

22.0

8814.7

1.9

716448

0.7

56

0.0

2476

879.1

3910.2

409.5

920.0

920.1

20.7

0921.5

21.1

6921.5

1.7

216228

0.7

46

0.0

2463

883.8

1912.5

410.4

922.8

922.9

19.7

8924.3

20.2

4924.3

1.7

116222

0.7

45

0.0

1878

1158.6

81045.5

467.1

1082.4

1082.6

18.4

01086.0

18.8

61086.0

1.4

815800

0.7

26

0.0

1501

1450.0

11177.2

536.4

1233.3

1233.5

15.1

81240.6

15.6

41240.6

1.3

615283

0.7

02

92

Tabl

eE

.2:

Ene

rgy

bala

nce

ofsi

mul

atio

nspe

rform

edin

the

valid

atio

n.

Initi

alco

nditi

ons

Sim

ulat

ion

para

met

ers

Ene

rgy

bala

nce

mQsup

mhsf,in

hsf,out

Gc,n

ηth

erm

al

Loss

1L

oss

2L

oss

3L

oss

4L

oss

5L

oss

6

[kg.s−1]

[kJ.k

g−1]

[W.m

−2.K

−1]

[W.m

−2.K

−1]

[W.m

−2]

[%]

[%]

[%]

[%]

[%]

[%]

[%]

Usi

ngeq

uatio

n(3

.7)

0.0

3478

625.7

31156.7

1782.9

0.0

67.8

1.2

8.6

2.2

0.0

0.0

20.2

0.0

3061

711.1

01096.2

1740.0

0.0

67.2

1.5

8.9

2.2

0.0

0.0

20.3

0.0

2476

879.1

31004.4

1672.6

0.0

65.9

2.1

9.3

2.2

0.0

0.0

20.5

0.0

2463

883.8

11002.2

1671.0

0.0

65.8

2.7

10.3

2.2

0.0

0.0

19.0

0.0

1878

1158.6

8900.6

1591.0

0.0

63.5

3.4

10.1

2.2

0.0

0.0

20.8

0.0

1501

1450.0

1829.2

1529.5

0.0

60.9

5.3

10.8

2.2

0.0

0.0

20.8

Usi

ngeq

uatio

n(4

.1)

0.0

3478

625.7

31279.9

1838.1

0.0

76.0

1.4

0.0

2.2

0.0

0.0

20.3

0.0

3061

711.1

01233.1

1797.2

0.0

75.6

1.8

0.0

2.2

0.0

0.0

20.4

0.0

2476

879.1

31168.5

1733.9

0.0

74.6

2.7

0.0

2.2

0.0

0.0

20.6

0.0

2463

883.8

11167.0

1732.4

0.0

74.5

2.7

0.0

2.2

0.0

0.0

20.6

0.0

1878

1158.6

81108.2

1660.5

0.0

72.6

4.7

0.0

2.2

0.0

0.0

20.5

0.0

1501

1450.0

11077.2

1609.4

0.0

70.2

7.5

0.0

2.2

0.0

0.0

20.1

93

TableE

.3:D

etailedresults

ofsteadystate

simulations

performed

inC

hapter4.Initialconditions

Sim

ulationresults

Node

0N

oden

Maxim

um

Ginc

mQsup

mks

ϕdp

ζTs

Tf

Ts

Tf

xs

Ts

xf

Tf

teq

uf,0

Qf

ηthermal

ηcycle

ηsystem

[kW.m

−2

][kg.s −

1]

[kJ.k

g−

1]

[W.m

−1.K

−1

][−

][m

m]

[−]

[K]

[K]

[K]

[K]

[mm

][K

][m

m]

[K]

[s][m.s −

1]

[W]

[−]

[−]

[−]

500

0.8

3333

600

40

0.8

00.8

0.1

764.6

305.3

765.8

765.9

11.0

766.6

13.0

766.5

46.6

50.7

2403913

0.8

08

0.3

74

0.3

02

500

0.6

2500

800

40

0.8

00.8

0.1

876.3

310.8

891.2

891.4

10.2

892.9

11.4

892.8

63.3

00.5

5389561

0.7

79

0.4

20

0.3

27

500

0.5

0000

1000

40

0.8

00.8

0.1

976.9

318.8

1000.0

1000.1

9.2

1002.9

10.2

1002.9

78.2

50.4

5373144

0.7

46

0.4

52

0.3

38

500

0.4

1667

1200

40

0.8

00.8

0.1

1065.1

329.1

1092.5

1092.7

8.6

1097.2

9.4

1097.1

90.8

70.3

9355395

0.7

11

0.4

76

0.3

38

500

0.3

5714

1400

40

0.8

00.8

0.1

1141.1

341.5

1170.4

1170.7

8.0

1177.1

8.6

1177.1

101.0

80.3

5337133

0.6

74

0.4

94

0.3

33

500

0.3

1250

1600

40

0.8

00.8

0.1

1206.0

355.4

1235.7

1235.9

7.4

1244.5

8.0

1244.4

109.0

70.3

1319063

0.6

38

0.5

07

0.3

24

600

1.0

0000

600

40

0.8

00.8

0.1

780.2

304.1

771.5

771.6

0.0

780.2

13.6

772.1

38.8

10.8

6490879

0.8

18

0.3

76

0.3

08

600

0.7

5000

800

40

0.8

00.8

0.1

892.7

308.3

901.8

902.0

10.4

903.4

12.0

903.3

53.0

90.6

5476371

0.7

94

0.4

23

0.3

36

600

0.6

0000

1000

40

0.8

00.8

0.1

996.4

314.5

1016.8

1017.0

9.6

1019.8

10.8

1019.7

66.1

50.5

3459376

0.7

66

0.4

57

0.3

50

600

0.5

0000

1200

40

0.8

00.8

0.1

1089.5

322.7

1116.6

1116.8

8.8

1121.3

9.8

1121.2

77.6

20.4

6440476

0.7

34

0.4

82

0.3

54

600

0.4

2857

1400

40

0.8

00.8

0.1

1171.2

332.9

1201.9

1202.2

8.2

1208.9

9.0

1208.7

87.1

70.4

0420466

0.7

01

0.5

00

0.3

51

600

0.3

7500

1600

40

0.8

00.8

0.1

1242.2

344.7

1274.3

1274.6

7.8

1283.7

8.2

1283.6

95.0

90.3

7400121

0.6

67

0.5

15

0.3

43

800

1.3

3333

600

40

0.8

00.8

0.1

811.4

302.7

778.5

778.7

0.0

811.4

14.6

779.0

29.0

11.1

4664815

0.8

31

0.3

79

0.3

15

800

1.0

0000

800

40

0.8

00.8

0.1

923.2

305.5

915.3

915.5

0.0

923.2

12.8

916.6

39.7

50.8

7650356

0.8

13

0.4

28

0.3

48

800

0.8

0000

1000

40

0.8

00.8

0.1

1029.2

309.5

1038.8

1039.1

10.0

1041.6

11.6

1041.5

50.1

60.7

0632803

0.7

91

0.4

63

0.3

66

800

0.6

6667

1200

40

0.8

00.8

0.1

1127.3

315.0

1148.8

1149.2

9.2

1153.5

10.4

1153.4

59.7

20.5

9612518

0.7

66

0.4

89

0.3

74

800

0.5

7143

1400

40

0.8

00.8

0.1

1216.4

322.0

1245.4

1245.8

8.8

1252.5

9.6

1252.3

68.1

30.5

2590128

0.7

38

0.5

09

0.3

76

800

0.5

0000

1600

40

0.8

00.8

0.1

1295.9

330.5

1329.2

1329.6

8.0

1339.1

8.8

1338.9

75.2

20.4

7566409

0.7

08

0.5

25

0.3

72

1000

1.6

6667

600

40

0.8

00.8

0.1

843.2

302.0

782.7

782.8

0.0

843.2

15.6

783.1

23.1

71.4

3838582

0.8

39

0.3

81

0.3

19

1000

1.2

5000

800

40

0.8

00.8

0.1

953.3

304.0

923.4

923.7

0.0

953.3

13.6

924.6

31.7

01.0

8824410

0.8

24

0.4

30

0.3

55

1000

1.0

0000

1000

40

0.8

00.8

0.1

1059.0

306.9

1052.4

1052.8

0.0

1059.0

12.2

1054.9

40.1

30.8

7806725

0.8

07

0.4

66

0.3

76

1000

0.8

3333

1200

40

0.8

00.8

0.1

1158.9

310.8

1169.3

1169.8

9.6

1173.9

11.0

1173.6

48.1

60.7

3785772

0.7

86

0.4

94

0.3

88

1000

0.7

1429

1400

40

0.8

00.8

0.1

1251.6

315.8

1273.9

1274.4

9.0

1280.8

10.2

1280.6

55.4

80.6

4761939

0.7

62

0.5

15

0.3

92

1000

0.6

2500

1600

40

0.8

00.8

0.1

1336.2

322.1

1366.2

1366.7

8.6

1376.1

9.4

1375.9

61.8

90.5

7735932

0.7

36

0.5

31

0.3

91

600

0.6

0000

1000

10.8

00.8

0.1

1006.4

313.8

1029.1

1030.5

13.0

1036.8

13.8

1036.7

45.2

60.5

3468649

0.7

81

0.4

60

0.3

60

600

0.6

0000

1000

20

0.8

00.8

0.1

986.2

314.2

1019.5

1019.8

10.2

1023.8

11.2

1023.7

62.3

70.5

3461309

0.7

69

0.4

58

0.3

52

600

0.6

0000

1000

60

0.8

00.8

0.1

1001.1

314.6

1015.7

1015.8

9.4

1017.9

10.4

1017.8

67.7

30.5

3458544

0.7

64

0.4

57

0.3

49

600

0.6

0000

1000

80

0.8

00.8

0.1

1003.7

314.7

1015.0

1015.1

9.2

1016.8

10.4

1016.8

68.4

60.5

3458083

0.7

63

0.4

56

0.3

48

600

0.6

0000

1000

40

0.7

00.8

0.1

959.6

323.2

960.6

960.7

5.2

962.9

5.8

962.9

97.4

40.5

5420844

0.7

01

0.4

41

0.3

09

600

0.6

0000

1000

40

0.7

50.8

0.1

980.1

319.1

987.3

987.4

7.0

990.0

7.8

990.0

82.2

20.5

4439035

0.7

32

0.4

49

0.3

28

600

0.6

0000

1000

40

0.8

50.8

0.1

1000.7

309.6

1049.0

1049.4

13.4

1050.9

15.2

1050.6

49.6

70.5

3481677

0.8

03

0.4

65

0.3

74

600

0.6

0000

1000

40

0.9

00.8

0.1

976.5

305.2

1066.6

1064.7

19.0

1066.7

20.0

1064.7

33.5

00.5

2492304

0.8

21

0.4

69

0.3

85

600

0.6

0000

1000

40

0.8

00.2

0.1

828.7

355.5

847.0

847.0

4.2

849.1

4.2

849.1

57.3

30.6

0344339

0.5

74

0.4

05

0.2

32

600

0.6

0000

1000

40

0.8

00.5

0.1

942.2

328.2

969.8

969.8

7.2

972.8

7.6

972.8

63.4

20.5

6427057

0.7

12

0.4

44

0.3

16

600

0.6

0000

1000

40

0.8

01.1

0.1

1036.8

308.8

1039.6

1040.0

11.2

1042.0

13.6

1041.6

68.0

40.5

2475182

0.7

92

0.4

63

0.3

67

600

0.6

0000

1000

40

0.8

01.4

0.1

1073.5

306.0

1051.9

1052.3

0.0

1073.5

16.8

1052.7

69.7

20.5

2483717

0.8

06

0.4

66

0.3

76

600

0.6

0000

1000

40

0.8

00.8

0993.5

314.4

1018.0

1018.2

9.8

1020.9

10.8

1020.8

66.2

60.5

3460160

0.7

67

0.4

57

0.3

51

600

0.6

0000

1000

40

0.8

00.8

0.2

999.3

314.5

1015.7

1015.9

9.4

1018.7

10.4

1018.6

66.1

50.5

3458592

0.7

64

0.4

57

0.3

49

600

0.6

0000

1000

40

0.8

00.8

0.4

1005.2

314.7

1013.4

1013.6

9.0

1016.5

10.2

1016.4

66.1

50.5

3457018

0.7

62

0.4

56

0.3

47

600

0.6

0000

1000

40

0.8

00.8

0.6

1011.0

314.8

1011.1

1011.3

8.2

1014.3

9.6

1014.2

66.1

50.5

4455445

0.7

59

0.4

55

0.3

46

600

0.6

0000

1000

40

0.8

00.8

0.8

1016.8

315.0

1008.8

1009.0

0.0

1016.8

8.8

1012.1

66.1

50.5

4453865

0.7

56

0.4

55

0.3

44

600

0.6

0000

1000

40

0.8

00.8

11022.6

315.1

1006.5

1006.7

0.0

1022.6

7.8

1010.1

66.1

50.5

4452286

0.7

54

0.4

54

0.3

42

94

Tabl

eE

.4:

Ene

rgy

bala

nce

ofst

eady

stat

esi

mul

atio

nspe

rform

edin

Cha

pter

4.

Initi

alco

nditi

ons

Sim

ulat

ion

para

met

ers

Ene

rgy

bala

nce

Ginc

mQsup

mks

ϕdp

ζhsf,in

hsf,out

Gc,n

ηthermal

Loss

1L

oss

2L

oss

3L

oss

4L

oss

5L

oss

6

[kW.m

−2

][k

g.s

−1

][k

J.k

g−

1]

[W.m

−1.K

−1

][−

][m

m]

[−]

[W.m

−2.K

−1

][W.m

−2.K

−1

][W.m

−2

][%

][%

][%

][%

][%

][%

][%

]

500

0.8

3333

600

40

0.8

00.8

0.1

512.4

816.4

9.3

680.8

3.4

9.5

2.0

0.0

0.0

4.3

500

0.6

2500

800

40

0.8

00.8

0.1

456.2

772.4

9.3

677.9

6.0

10.5

2.0

0.0

0.0

3.6

500

0.5

0000

1000

40

0.8

00.8

0.1

419.4

739.5

9.3

674.6

9.2

11.4

2.0

0.0

0.0

2.8

500

0.4

1667

1200

40

0.8

00.8

0.1

393.9

712.7

9.3

671.1

13.1

12.1

2.0

0.0

0.0

1.8

500

0.3

5714

1400

40

0.8

00.8

0.1

375.6

689.7

9.3

667.4

17.2

12.6

2.0

0.0

0.0

0.7

500

0.3

1250

1600

40

0.8

00.8

0.1

361.9

669.4

9.3

663.8

21.5

13.1

2.0

0.0

0.0

−0.5

600

1.0

0000

600

40

0.8

00.8

0.1

553.8

887.3

11.2

481.8

3.1

8.9

2.0

0.0

0.0

4.2

600

0.7

5000

800

40

0.8

00.8

0.1

491.9

841.3

11.2

479.4

5.4

9.7

2.0

0.0

0.0

3.5

600

0.6

0000

1000

40

0.8

00.8

0.1

450.9

807.4

11.2

476.6

8.3

10.5

2.0

0.0

0.0

2.6

600

0.5

0000

1200

40

0.8

00.8

0.1

422.2

780.3

11.2

473.4

11.9

11.1

2.0

0.0

0.0

1.5

600

0.4

2857

1400

40

0.8

00.8

0.1

401.2

757.5

11.2

470.1

16.0

11.7

2.0

0.0

0.0

0.3

600

0.3

7500

1600

40

0.8

00.8

0.1

385.6

737.4

11.2

466.7

20.2

12.1

2.0

0.0

0.0

−1.0

800

1.3

3333

600

40

0.8

00.8

0.1

626.7

1010.7

14.9

883.1

2.7

8.0

2.0

0.0

0.0

4.1

800

1.0

0000

800

40

0.8

00.8

0.1

555.2

960.9

14.9

881.3

4.6

8.7

2.0

0.0

0.0

3.4

800

0.8

0000

1000

40

0.8

00.8

0.1

507.1

925.3

14.9

879.1

7.1

9.2

2.0

0.0

0.0

2.5

800

0.6

6667

1200

40

0.8

00.8

0.1

472.6

898.0

14.9

876.6

10.3

9.8

2.0

0.0

0.0

1.4

800

0.5

7143

1400

40

0.8

00.8

0.1

447.1

875.8

14.9

873.8

13.9

10.2

2.0

0.0

0.0

0.0

800

0.5

0000

1600

40

0.8

00.8

0.1

427.6

856.8

14.9

870.8

18.0

10.6

2.0

0.0

0.0

−1.4

1000

1.6

6667

600

40

0.8

00.8

0.1

690.2

1117.3

18.7

383.9

2.6

7.5

2.0

0.0

0.0

4.1

1000

1.2

5000

800

40

0.8

00.8

0.1

610.6

1064.0

18.7

382.4

4.2

8.0

2.0

0.0

0.0

3.4

1000

1.0

0000

1000

40

0.8

00.8

0.1

556.6

1026.8

18.7

380.7

6.4

8.4

2.0

0.0

0.0

2.5

1000

0.8

3333

1200

40

0.8

00.8

0.1

517.4

999.2

18.7

378.6

9.2

8.9

2.0

0.0

0.0

1.3

1000

0.7

1429

1400

40

0.8

00.8

0.1

487.9

977.8

18.7

376.2

12.5

9.3

2.0

0.0

0.0

0.0

1000

0.6

2500

1600

40

0.8

00.8

0.1

465.1

960.3

18.7

373.6

16.2

9.6

2.0

0.0

0.0

−1.5

600

0.6

0000

1000

10.8

00.8

0.1

450.4

812.5

11.2

478.1

8.7

10.6

2.0

0.0

0.1

0.5

600

0.6

0000

1000

20

0.8

00.8

0.1

450.7

808.5

11.2

476.9

8.0

10.3

2.0

0.0

0.0

2.8

600

0.6

0000

1000

60

0.8

00.8

0.1

451.0

806.9

11.2

476.4

8.5

10.5

2.0

0.0

0.0

2.5

600

0.6

0000

1000

80

0.8

00.8

0.1

451.1

806.7

11.2

476.3

8.6

10.6

2.0

0.0

0.0

2.5

600

0.6

0000

1000

40

0.7

00.8

0.1

364.2

625.4

0.0

070.1

7.2

12.0

3.0

0.0

0.0

7.7

600

0.6

0000

1000

40

0.7

50.8

0.1

402.2

704.5

0.0

873.2

7.8

11.4

2.5

0.0

0.0

5.1

600

0.6

0000

1000

40

0.8

50.8

0.1

519.0

951.4

653.0

580.3

8.5

9.1

1.5

0.1

0.0

0.5

600

0.6

0000

1000

40

0.9

00.8

0.1

628.6

1169.5

15826.9

582.1

7.7

7.1

1.0

2.6

0.0

−0.5

600

0.6

0000

1000

40

0.8

00.2

0.1

1049.7

1614.9

0.0

057.4

4.0

18.5

2.0

0.0

0.0

18.1

600

0.6

0000

1000

40

0.8

00.5

0.1

600.9

1028.0

0.1

071.2

6.7

12.9

2.0

0.0

0.0

7.3

600

0.6

0000

1000

40

0.8

01.1

0.1

373.3

682.4

117.6

079.2

9.8

9.2

2.0

0.0

0.0

−0.2

600

0.6

0000

1000

40

0.8

01.4

0.1

324.4

599.3

488.4

980.6

11.2

8.4

2.0

0.1

0.0

−2.3

600

0.6

0000

1000

40

0.8

00.8

0450.9

807.8

12.4

876.7

8.2

10.4

2.0

0.0

0.0

2.6

600

0.6

0000

1000

40

0.8

00.8

0.2

451.0

807.0

9.9

976.4

8.4

10.5

2.0

0.0

0.0

2.6

600

0.6

0000

1000

40

0.8

00.8

0.4

451.1

806.1

7.4

976.2

8.6

10.6

2.0

0.0

0.0

2.6

600

0.6

0000

1000

40

0.8

00.8

0.6

451.2

805.2

4.9

975.9

8.8

10.7

2.0

0.0

0.0

2.5

600

0.6

0000

1000

40

0.8

00.8

0.8

451.3

804.3

2.5

075.6

9.0

10.8

2.0

0.0

0.0

2.5

600

0.6

0000

1000

40

0.8

00.8

1451.4

803.5

0.0

075.4

9.3

10.9

2.0

0.0

0.0

2.5

95

Table E.5: Summary of transient simulation results performed in Chapter 4.

Initial conditions Simulation resultsm Ginc,up Ginc,lo ∆tlo teq,lo teq,up

[kg.s−1] [kW.m−2] [kW.m−2] [s] [s] [s]

0.5 800 0 10 104.33 50.780.5 800 0 20 104.33 65.270.5 800 0 35 104.33 72.200.5 800 0 65 104.33 76.360.5 800 0 95 104.33 77.360.5 800 0 125 104.33 77.62

0.8 800 0 10 61.58 38.750.8 800 0 20 61.58 47.640.8 800 0 35 61.58 51.190.8 800 0 65 61.58 52.530.8 800 0 95 61.58 52.680.8 800 0 125 61.58 52.76

0.5 800 500 10 63.76 34.400.5 800 500 20 63.76 47.880.5 800 500 35 63.76 53.900.5 800 500 65 63.76 56.700.5 800 500 95 63.76 57.200.5 800 500 125 63.76 57.20

96