INFLUENCE OF BOILER SWING RATE ON DYNAMICS OF THERMAL AND FLOW CHARACTERISTICS IN WATER CIRCULATION...

Computational Thermal Sciences, 3 (6): 483–500 (2011)

INFLUENCE OF BOILER SWING RATE ON DYNAMICSOF THERMAL AND FLOW CHARACTERISTICS INWATER CIRCULATION BOILERS

M. A. Habib,∗ I. Al-Zaharnah,1 T. Ayinde,1 S. Al-Anizi,2 & Y. Al-Awwad2

1Mechanical Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran31261, Saudi Arabia2Consulting Services Department, Saudi Aramco, Dhahran, Saudi Arabia

∗Address all correspondence to M. A. Habib E-mail: [email protected]

Tube overheating may cause tube failure resulting in an unscheduled boiler shutdown that may interrupt plant opera-tion. The impact of this problem is not only due to the cost of replacing defective parts but also due to the frequent needof system shutdown and the possible imminent safety hazards. This paper provides an investigation of the influence ofrapid rise in steam flow rate (swing rate) on the thermal and flow characteristics of the riser tubes in natural circula-tion water tube boilers. A thermal model for the prediction of possible tube overheating was developed. The developedmodel incorporates a nonlinear state space dynamic model which captures the important physical interactions of themain variables of steam generation in drum boilers. The system under consideration includes the drum, the riser, anddowncomer as its major components. A numerical scheme for the solution of the governing differential equations wasestablished. The dynamic response of the system’s state variables due to rapid rises in steam flow rate was investigated.The results show that the rapid rise in the steam flow rate results in decrease in the pressure and an initial increase inthe steam quality which is followed by a decrease in the steam quality. The riser temperature increases partly due to theincrease in the steam temperature and partly due to the dynamic influence resulting from a decrease in the heat transfercoefficient. The present calculations of the water level in the drum provide good comparison with those in the literature.

KEY WORDS: water circulation in boilers, boiler swing rate, thermal and flow characteristics

1. INTRODUCTION

Steam generators that are commonly used in many in-dustrial processes may experience rapid and dynamicchanges in the steam demand. These changes result inrapid variations in drum pressure and water levels in thedrum. The changes in steam demand are normally metby corresponding changes in feedwater flow rates andfiring rates. Consequently, heat flux along the riser anddowncomer tubes are increased and may result in boilertube overheating. This overheating is highly influencedby the fuel flow rate which, in turn, changes to meet theload fluctuations and the drop in drum pressure. The rapidchanges in swing rates may also result in “carry over” or“carry under” phenomena. The carry over” phenomenonmay result in highly wet steam passing into the super-heater while the carry under phenomenon may lead to

steam passing into the downcomer tubes. This may re-sult in disturbance of the water circulation in the boilertube circuits. The boiler tubes in natural circulation boil-ers may suffer tube burnout because of rapid changesin boiler operating variables such as drum pressure andsteam-water ratio. These may result from high load swingrates that are made to meet changes in steam demand.Tube overheating and high stresses may cause tube failureresulting in unscheduled boiler shutdown that may inter-rupt plant operation. The impact of this problem is notonly due to the cost of replacing defective parts but alsodue to the frequent need of system shutdown and the pos-sible imminent safety hazards.

The early work of Davidenko and Rushchinskiy (1975)investigated the numerical solution of a set of partialdifferential equations representing a flow in a straightchannel of a boiler. A single-phase flow was consid-

1940–2503/11/$35.00 c⃝ 2011 by Begell House, Inc. 483

484 Habib et al.

NOMENCLATURE

A cross sectional area of the tube (m2) mr flow rate out of the risers (kg/s)

Ad drum area at normal operating level (m2) ms mass flow rate of steam exiting the boiler

BO boiling number, dimensionless to the superheater and turbine (kg/s)

CO convection number, dimensionless msd steam flow rate through the

∆T Tmetal − Tsat (◦C) liquid surface in the drum (kg/s)

Fr Froude number, dimensionless Nu Nusselt number (dimensionless)

G mass flux (kg/s·m2) P drum pressure (kPa)

h specific enthalpy (kJ/kg), Eqs. (10)–(21) Q heat flow rate to the risers (kW)

h heat transfer coefficient (kW/m2K), Eq. (22) q” heat flux (kW/m2)

hfg specific enthalpy of evaporation, Re Reynolds number (dimensionless)

hfg = hg − hf (kJ/kg) td residence time of the steam in thehw specific enthalpy of saturated drum (s)

liquid water (kJ/kg) Tm metal temperature (◦C)

hg specific enthalpy of saturated Ts Saturated steam temperature (◦C)

water vapor (kJ/kg) u Specific internal energy (kJ/kg)

hfw specific enthalpy of feed water (kJ/kg) ug specific internal energy of steam (kJ/kg)

hs specific enthalpy of steam leaving uw specific internal energy of water (kJ/kg)

the boiler (kJ/kg) UV velocity of the liquid phase inside

k dimensionless friction coefficient the riser tube (m/s) volume (m3)

in the downcomer-riser loop Vd drum volume (m3)

K thermal conductivity (kW/mK) Vdc downcomer volume (m3)

L drum water level (m) Vr riser volume (m3)

Lw level variations caused by changes of Vsd volume of steam under the liquid level (m3)

the amount of water in the drum (m) V 0sd volume of steam in the drum in the

Ls Level variation caused by the hypothetical situation when there is no

steam in the drum (m) condensation of steam in the drum (m3)

Mf mass of saturated liquid in Vst total volume of steam in the system (m3)

water walls and drum (kg) Vt total volume of the drum, downcomer,

Mg mass of saturated vapor in and risers;Vt = Vst + Vwt (m3)

water walls and drum (kg) Vwd volume of water under the liquid level (m3)

Mr total riser mass (kg) Vwt total volume of water in the system (m3)

Mt total mass of metal (including vw specific volume of saturated liquid

tubes and drum) (kg) water (m3/kg)

mcd condensation flow on the drum (kg/s) vs specific volume of saturated vapor

mdc downcomer flow rate (kg/s) water (m3/kg)

mfw mass flow rate of feedwater supplied x mass fraction of steam in the

to the drum (kg/s) flow, dimensionless

Computational Thermal Sciences

Influence of Boiler Swing Rate 485

NOMENCLATURE (Continued)

Subscripts td residence time of the steam in the drumc condensation fw feedwaterd drumdc downcomer Superscriptsm metal average values saturated steamsat saturated Greek Symbolsw saturated liquid α volume fraction of steam in the flowfg difference between vapor and β parameter in empirical formula

liquid ρ densityr riser ρs density of saturated steamt total ρfw density of feedwater

ered and simplified empirical expression of a standardheat transfer coefficient was employed. De Mello (1991)has demonstrated the validity of simplified boiler mod-els that have previously been used to represent steam tur-bine mechanical power response including boiler pres-sure. Boiler response characteristics derived from the ba-sic energy balance, mass balance, and volume balance re-lations using physical boiler parameters were comparedwith those obtained from two other simplified models,one which matched both the steady state and transientopen loop boiler response characteristics, and a simplermodel which matches the initial open loop response. Itwas shown that both simplified models yield acceptableresults of the boiler response including pressure controls.

A simple nonlinear model derived from first princi-ples for a drum boiler was developed by Astrom and Bell(1988). The model is characterized by a few physical pa-rameters that are easily obtained from construction data.The models were found to capture the major dynamicalbehavior and were validated against experimental data.The models require steam tables for a limited operatingrange. The model is capable of capturing the essence ofthe steam generation in a heated pipe. The same authors(Astrom and Bell, 1993) derived from first principles anonlinear model for steam generation processes. Compar-ison with data from plant experiments indicated that themodel derives the behavior of the system quite well. Thepredicted pressure swing was large in general. The resultsshowed that increasing the metal mass results in a de-

crease in the swing of the pressure. Possible modificationsto the model include dynamics in the model of circulationflow or making a finer subdivision of the risers. In an an-other investigation (Bell and Astrom, 1996) the investiga-tors derived from first principles a nonlinear model for adrum boiler. The model is characterized by a few physi-cal parameters that are easily obtained from constructiondata and steam tables. Comparisons with data from plantexperiments covered a large operating range for a plantat low and high loads. The results of experiments at lowand high loads included changes in fuel flow, feedwater,and steam demand. The agreement of the model resultswith plant data was good. The pressure dynamics pre-dicted agreed with the plant data. The model captured themajor dynamical behavior of the process which is veri-fied by the extensive comparisons with real plant data pre-sented in the paper. A recursive prediction error methodbased on a restricted nonlinear state space ordinary dif-ferential equation model was applied to simulate data ob-tained from a second-order nonlinear simple drum-boilermodel using ideas and models from Wigren (2005), As-trom and Eklund (1972), and Astrom and Eklund (1975).Good agreement was obtained between the simulated dataand an identified first-order nonlinear model. The majorfactors influencing the performance of the fast load re-sponse of fossil-fueled power plants as well as the meth-ods and features that can improve the performance duringload increase and load reduction were discussed by Ter-muehlen and Gartner (1981).

Volume 3, Number 6, 2011

486 Habib et al.

Water circulation control of steam generation is an im-portant problem that must be considered for plant safetyand reliability. Poor control leads to frequent shutdown.Water circulation in natural circulation drum boilers isone of the critical problems in boiler technology. Poorwater circulation may cause tube burnout resulting in un-scheduled boiler shutdown and interrupting plant opera-tion. Such poor circulation may arise from operational-type problems such as rapid changes in boiler load caus-ing rapid changes in the heat flux as a result of rapidchanges in fuel flow rates. During the last few years,some boiler explosions were attributed to poor water cir-culation (Pereyra et al., 1989). As a result, calculationsand measurements of water circulation and other operat-ing parameters, such as steam quality and void fraction,have become more important not only for boiler manu-facturers but also for large industrial establishments. Ad-equate water circulation is necessary to cool tubes thatform the boiler walls. Criteria are required to determinethe potential for tube overheating. These criteria can beapplied using transient circulation modeling and calcu-lations to identify problem areas. Modeling the steamgenerator system including boiler drum, riser, and down-comer tubes is one of the important problems. Many pre-vious investigations were conducted with the objective ofunderstanding the transient behavior of the system. Thedrum-boiler model which runs in real time has been de-veloped by Flynn and O’Malley (1999) and validated us-ing dynamic data recorded on an actual plant. The modelcan be used for dynamic simulation studies in long timeframes, greater than 30 s, in particular where assessmentof deviations of internal parameters, such as steam pres-sure, drum level, and steam temperature outside safetylimits is essential. The authors indicated that the modelcan be useful for predicting performance capability whilealso modeling critical internal variables such as drumlevel and steam temperatures which may cause the unitto trip if safety limits are violated. The convective heattransfer coefficient in the boiler tubes was approximatedto the Dittus-Boelter formula for turbulent fluid flow.

The difficulties associated with model validation werediscussed by Leva et al. (1999). They presented the re-sults of boiler model validation against a lab scale drumboiler setup. The work gave insight of the validation pro-cess. The most critical steps are the collection of sig-nificant data and also the data reconciliation. Inoue etal. (2000) carried out a computational simulation for apower plant equipped with a once-through boiler underchanges in the power system frequency. The plant con-trol system was modeled together with the boiler-turbine

control system considering a large-size plant undergoingchanges in the system frequency. The model was subdi-vided to three models, namely, the boiler steam pressuremodel, the turbine-governor model, and the plant controlsystem model. In the steam pressure model, a number ofnodes were selected including an evaporating node, a su-perheating node, a steam line node, and a reheater node.The pressure at each node was determined using a simpli-fied lumped model that was based on applying the massand energy conservation equations. The model was testedunder different operating conditions and the sample sim-ulation results were found to be close to actual respon-ses.

Motivated by model-based control, Astrom and Bell(2000) developed a nonlinear dynamic model for drumboilers. The model describes the complicated dynamics ofthe drum, downcomer, and riser components. The modelis based on physical parameters of the plant. The goalwas to develop a model which captures the key dynam-ical properties over a wide operating range. The model isbased on physical principles with a small number of pa-rameters. In their work particular attention was given tothe dynamics of the drum level, as it has been recognizedthat drum level control is an important factor in powerplant operation. The model has four states; Two states ac-count for storage of total energy and total mass, one char-acterizes steam quality at the risers output assuming lin-ear variation of the steam quality along the length of therisers, and another for the steam distribution in the drum.The model is augmented by quadratic approximations ofthe steam tables and physical parameters thus derived.The model has been validated against plant data with veryrich excitation that covers a wide operating range. Theseexperiments have given insight into the behavior of thesystem and have guided the modeling effort. The modelis nonlinear and agrees well with experimental data. Themodel gives insight into the behavior of the system andin particular to the complex shrink and swell phenomenaassociated with the drum water level. The model has a ba-sic assumption of taking the metal temperature identicalto the steam saturation temperature.

The work of Changliang et al. (2001) provided a dy-namic model in terms of drum pressure and water vol-ume assuming metal temperature identical to fluid tem-perature. The drum level was given in terms of steam resi-dence time below the water level which is related to steamrise velocity. No expression was given for the steam risevelocity. They assumed it to be obtainable from test datawithout discussing the procedure for doing that. A com-putational model for the thermal-hydraulic simulation of



the start-up period of a once-through boiler equipped withvarious integral separator systems was developed (Dongand Tingkuan, 2001). The model was based on an im-proved analytical-numerical method for solving the con-tinuity, momentum, and coupled energy equations for theflue gas and the working fluid to obtain the transient heattransfer occurring in the different components of the sys-tem. The results obtained from the computational modelwere compared with the measured data of a 600-MWonce-through boiler. The comparison, which included themain steam pressure, the water separator level, the econ-omizer pressure, as well as its inlet and exit temperaturesshowed reasonable agreement.

Kim and Choi (2005) developed a model for waterlevel dynamics in the drum-riser-downcomer loop of anatural circulation drum-type boiler. The model is basedon basic conservation rules of mass, momentum, andenergy, together with the constitutional equations. Thework provides an investigation of the response of waterlevel dynamics to changes in steam demand and/or heat-ing rate. The results were compared with those of (As-trom and Bell, 2000). Like the model by Astrom and Bell(2000), the assumption of metal temperature being equalto the steam saturation temperature and the linear varia-tion of the steam quality along the riser tubes is employed.Franke et al. (2003) utilized the Modelica software formodeling the thermo-fluid processes of a simple drumboiler during the startup period. The model was basedon mass and energy conservation equations together withproperty relations for liquid and vapor phases. The modelwas applied to a 70 MW coal-fired boiler consideringthe important components such as the furnace, econo-mizer, superheaters, headers, spray water injection, andlong pipes. Using this model, the start-up time could bereduced from 45 min to 30 min.

Tucakovic et al. (2007) presented a method for carry-ing out a thermal-hydraulic simulation and analysis of aforced circulation loop of a large steam boiler in whichthe evaporating tubes in the furnace are of rifled type. Theeffects of rifled tube geometry on the heat transfer pro-cess, thermal margin against burnout, and friction pres-sure drop in the water circulation loop were considered.The coupled thermal processes on the furnace gas sideand the thermal-hydraulics inside the evaporating tubeswere performed for the entire range of the plant operat-ing loads. The paper focused on the hydraulic calculationsof both forced and natural circulation loops. The thermalsafety margin (against burnout) of rifled tubes was pre-dicted and compared with that of smooth tubes for uni-form and variable heat loads on the walls of the furnace.

The wall temperatures of the rifled tubes were safer evenunder conditions of high void fraction of the water/steammixture that prevents the occurrence of critical heat trans-fer conditions. The pressure drop in the rifled tubes wasalso analyzed and compared with that of smooth tubes.The range of operating conditions in which there is noneed of circulation pumping was also investigated.

In their recent work Habib and co-workers (Habib etal., 2010) investigated the dynamic effects of a rapid risein fuel flow rate (heat input) on the thermal and flow char-acteristics of the riser tubes in natural circulation watertube boilers. The results show that the sudden rise in thefiring rate was followed by an increase in the steam qual-ity, which is accompanied by a decrease in the circulationrate as a result of increase in the pressure. Their calcula-tions of the water level in the drum were in good agree-ment with those in the literature.

This review of the previous work clearly demonstratesthe value of dynamic simulation for the investigation anddevelopment of new control strategies and enhanced op-erational procedures of power plants. The present studyaims at developing and utilizing nonlinear modeling, anumerical procedure for the dynamic simulation of nat-ural circulation steam generator system and investigatingthe dynamic response of the pressure in the drum andsteam quality at the exit of the riser tubes to rapid vari-ations in steam demand in order to optimize the steamgenerator operation to prevent the overheating of the risertubes. One of the main factors is to maintain the tem-perature of the tube walls within safe limits. In order todo that it will be crucial to identify the leading factorsand predict the imminence of tube overheating. This isachieved through the development of a nonlinear dynamicmodel for the investigation of influence of changes in op-erating conditions on the response of natural circulation.The study investigates the dynamic response of the pres-sure in the drum, and steam quality at the exit of the risertubes to rapid variations in steam demand. The differentialequations comprising the nonlinear model and govern-ing the flow inside the boiler tubes were formulated. Thesystem considered (Fig. 1) includes the drum, riser, anddowncomer. The problem considered is transient. The re-sponses of the boiler operating parameters to step changesin steam flow rate are investigated. The operating param-eters include steam quality, flow velocity, mass flow rateinside the riser tubes, and heat transfer characteristics ofthe riser tubes. The parameters also include the volumeof water in the system as well as boiling parameters in-side the steam drum. The following is a description of themathematical models considered in the present study.


488 Habib et al.

2. MATHEMATICAL FORMULATION

The boiler under consideration here is of the water-tubenatural circulation type. The main components of theboiler are the steam drum and the downcomer-riser tubeswhich represent the complete water circulation loop.Thermal energy is added to the fluid while flowing in theriser tubes and thus boiling takes place. The following arethe governing equations.

2.1 The Fluid Flow Dynamic Nonlinear Model

The governing equations consist of conservation of massand energy of the total system, the equations governingthe phase change in the drum including the steam and wa-ter volumes inside the drum and the rate of steam conden-sation, and the equations governing the flow circulationin the riser downcomer loop which govern the transportof the mass, energy, and momentum. Thus, a set of fourdifferential nonlinear equations (Astrom and Bell, 2000)representing the time dependence of the state variables ofthe pressure, steam quality, total water volume, and steamvolume in the drum can be presented as follows (deriva-tions are presented in Appendix A):

FIG. 1: The system of the drum and riser and downcomertubes

a11dp

dt+ a12

dVwt

dt= mfw − ms, (1)

a21dp

dt+ a22

dVwt

dt= Q+ mfwhfw − mshs, (2)

a31dp

dt+ a33

dx

dt= Q− xhfgmdc, (3)

a41dp

dt+ a43

dx

dt+ a44

dVsd

dt=

ρs

Td(V o

sd − Vsd)

+hfw − hw

hfgmfw. (4)

These equations can be presented in a matrix form as fol-lows:

a11a120a21a220a310a33a410a43

000a44

dP/dt

dVwt/dt

dx/dt

dVsd/dt

=

mfw − ms

Q+ mfwhfw − mshs

Q− xhfgmdc

ρs

Td(V o

sd − Vsd) +hfw − hw

hfgmfw

. (5)

The first two equations [Eqs. (1) and (2)] present themass and energy conservation for the global system ofdrum, risers, and downcomers. Equation (3) presents theconservation of the mass and energy in the riser tubesand, therefore, captures the steam dynamics in the ris-ers. Equation (4) presents the boiling process in the steamdrum. The model derived coefficients are given by

a11 = Vwt∂ρw∂p

+ Vst∂ρs∂p

, (6)

a12 = ρw − ρs, (7)

a21 = Vwt

(hw

∂ρw∂p

+ ρw∂hw

∂p

)+ Vst

×(hs

∂ρs∂p

+ ρs∂hs

∂p

)− Vt +mmtCp

∂tm∂p

, (8)

a22 = ρwhw − ρshs, (9)



a31=

(ρw

∂hw

∂p− xhfg

∂ρw∂p

)(1− α)Vr

+

[(1− x)hfg

∂ρs∂p

+ ρs∂hs

∂p

]αVr

+ [ρs + x(ρw − ρs)]hfgVr∂α

∂p− Vr

+mmtCp∂tm∂p

, (10)

a33 = [(1− x)ρs + xρw]hfgVr∂α

∂x, (11)

a41= Vsd∂ρs∂p

+1

hfg

(ρsVsd

∂h

∂p+ ρwVwd

∂hw

∂p

− Vsd − Vwd +mmdCp∂tm∂p

)+x(1+β)Vr

×[α∂ρsp

+ (1− α)∂ρw∂p

+ (ρs − ρw)∂α

∂p

], (12)

a43 = x(1 + β)(ρs − ρw)Vr∂α

∂x, (13)

a44 = ρs. (14)

2.1.1 Equilibrium Conditions

To obtain the initial conditions, the following model equa-tions at equilibrium conditions were solved. These equa-tions were derived from Eqs. (1)–(4) for steady state con-ditions. Specifyingms andp, the system equilibrium con-ditions are given by the following equations:

Equation (1) reduces to

mfw = ms. (15)

The heat addition is calculated from Eq. (2) as

Q = mshs − mfwhfw = xmdchfg. (16)

Also using Eq. (4), one can write

Vsd = V osd −

Td(hw − hfw)

ρshsmfw. (17)

And from Eqs. (3) and (A17),

Q = xhfg

√2ρwAdc(ρw − ρs)gαVr

k, (18)

Vwd = Vwt − Vdc − (1− x)Vr, (19)

Q = xhfgmdc, (20)

mr = mdc. (21)

The thermodynamic propertiesρf , ρg, hf , hg,(∂ρf )/(∂P ), (∂ρg)/(∂P ), (∂hf )/(∂P ), (∂hg)/(∂P ),(∂Ts)/(∂P ), andTs were obtained from the numericalsolution of the equations representing the thermodynamicproperties which are provided in IFC (1967). These equa-tions represent the physical quantities for the differentproperties such as the specific enthalpy, pressure, specificvolume, etc. The whole region of theTs diagram was di-vided into six regions to provide high accuracy for the cal-culated properties. A separate Fortran subroutine was de-veloped in the present work for the thermodynamic prop-erty calculations which is called by the main program.

2.2 The Thermal Model

2.2.1 Model of the Heat Transfer Coefficient

The two-phase (liquid and vapor) flow pattern in a pipedepends on many parameters such as the flow velocity,the quality of the mixture, the properties of the two phases(density, viscosity, and surface tension), and the pipe ge-ometry as well as its orientation. The flow patterns inhorizontal tubes are slightly different due to the effectof gravity. The flow pattern in a given tube can be de-termined from the flow pattern maps in which the re-gion of operation of each flow pattern is indicated (He-witt and Roberts, 1969; Hewitt, 1982). As shown in theprevious works (Hewitt, 1982; Collier, 1972; Hewitt etal., 1994) and many others, the flow patterns that are nor-mally encountered in vertical pipes are bubble flow, slugflow, churn flow, annular flow, and wispy annular flow.Due to the gravitational effects, the flow patterns in hori-zontal tubes may take any of the forms of stratified flow,plug flow, stratified wavy flow, dispersed bubbly flow, an-nular flow, and intermittent flow. The flow pattern can bepredicted for a given flow condition using the semithe-oretical method of Taitel and Dukler (1976) in which aflow pattern map is represented in terms of a number ofdimensionless parameters.

A large number of empirical and semiempirical corre-lations are available to predict heat transfer coefficientsfor the flow of boiling regimes in vertical and horizon-tal tubes. Some of these correlations have been widelytested and are practically used. Some lack validation. Ex-amples of these are the models described in Chen (1966),Shah (1976, 1982), Bjorge et al. (1982), Gungor and Win-terton (1986), Klimenko (1988, 1990), Kandlikar (1990),Steiner and Taborek (1992). A comparison of these mod-els with experimental data was conducted by Darabi et


490 Habib et al.

al. (1995). The model proposed by Kandlikar (1990)was used to calculate the heat transfer coefficient in thepresent water circulation loop. The equation proposed byKandlikar (1990) for the calculation of the heat transfercoefficient includes a convection evaporation term andnucleate boiling term and is expressed as

h = hL[c1COc2(25Frl)

c5 + c3BOc4Fk], (22)

wherehL is the single-phase heat transfer coefficient andit is calculated from the widely used Dittus-Boelter equa-tion (Dittus and Boelter, 1930). This equation is accuratefor flows in long pipes and for application to turbulentflows and, therefore, is suitable for the present calcula-tions where the riser length to diameter ratio is more than100 and the Reynolds number is in excess of 105.

Nu = 0.023(Re)0.8(Pr)0.4 (23)

Where Re is the Reynolds number of the flow inside thetube and Pr is the Prandtl number for water.

Re=G(1− x)Di

µℓ. (24)

Fk is a fluid-dependent parameter (=1 for water).Fr is the Froude number:

Fr =G2

ρ2LgD. (25)

G is the mass fluxm/A andD is the pipe diameter:

G =m

A. (26)

BO is the boiling number:

BO =q′′

Ghfg. (27)

CO is the convection number:

CO =

(1− x

x

)0.8 (ρv

ρl

)0.5

. (28)

x is the steam quality and the constantsc1, c2, c3, c4, c5are given in Table 1.

2.2.2 Wall Temperature Calculations

The following equations can be used to calculate the out-side wall temperature of the riser tube:

qπDo =To − Ti

Rpipeand qπDi =

Ti − Ts

Rconv, (29)

whereq is the heat flux at the outside surface of the tubes,Do is the pipe outer diameter,Ti andTo are the wall tem-peratures at the inner and outer walls of the tube, andTs isthe water saturation temperature.Rconv andRpipe are re-sistances of the inner water film and the pipe wall, respec-tively. These resistances are calculated from the followingrelations:

Rconv =1

πDih, (30)

whereKpipe is the thermal conductivity of the pipe mate-rial andh is the heat transfer coefficient of the water filminside the tube.

2.3 The Solution Procedure

The present numerical procedure considers a lumped pa-rameter model in which a transient solution is consid-ered for Eqs. (1)–(4). In this case, the pressure is constantalong the riser tubes. The steam quality is taken linearly,changed along the tube, and the solution is performed forthe quality at the exit of the tube. The steam quality is de-fined as zero at the inlet to the riser tube since the heatingprocess starts at this inlet section. Equations (1)–(4) weresolved simultaneously using an explicit method time stepof 0.3 s for a total time 150 s. The coefficients in theseequations were obtained from Eqs. (6)–(14). These equa-tions together with Eqs. (22)–(30) were used to calcu-late the heat transfer coefficient and metal temperature aswell as other main boiler parameters. The operational dataof the simulated system are given in Table 2. The EES(Engineering Equation Solver) was used to perform thepresent calculations. Time step independence tests wereconducted and the results are shown in Fig. 2. The test was

TABLE 1: The thermal model constants (Kandlikar, 1990)

CO≤0.65 CO≥0.65 CommentsC1 1.136 0.6683C2 –0.9 –0.2C3 667.2 1058C4 0.7 0.7

C5 0.3 0.3C5 = 0 for vertical tubes;

C5 = 0 for horizontal tubes if Fr> 0.04



TABLE 2: The operational data of the simulated system Astrom and Bell (2000)

Parameter Value Parameter ValueDrum pressure 8.5 MPa Riser volume 37 m3

Steam flow rate 49.9 kg/s Downcomer volume 11 m3

Heating value 85.9 MW Water surface area 20 m2

Drum volume 40 m3 Feedwater temperature 238.6◦C

FIG. 2: Response of drum pressure to variations in steam flow rates (time step independence)

performed for the changes in drum pressure in responseto step variations in steam flow at three different valuesof time steps. The figure indicates that the model cap-tures the changes in pressure with maximum variations of0.8% as the time step is reduced from 3 s to 1 s and 0.3%as the time step is reduced from 1 s to 0.3 s. The initialvalues were obtained through the solution of Eqs. (15)–(21) and are taken as given in Table 3. Thus, to obtain theinitial conditions at a given state defined by drum pres-sure and steam flow rate, the procedure used to obtain theinitial conditions is given as follows. The method is iter-ative that starts with a guessed value of the steam qual-ity. Then, Eq. (15) [derived from Eq. (1) at steady stateconditions] is used to calculate the water flow rate andEq. (16) [derived from Eq. (2) at steady state conditions]is used to calculate the heat input (firing rate). The com-bined Eqs. (18) and (20) are used to calculate the down-comer flow rate which is the same as the riser flow rateas given by Eq. (21). A new value of the steam quality isobtained from Eq. (20). Equations (17) and (19) are thenused to calculate the volumes of the water and steam inthe drum under the water level.

TABLE 3: The initial data (steady state conditions) of thesimulated system Astrom and Bell (2000)

Parameter Units ValuePressure,P kPa 8500

Energy input,Q MW 85.9Steam mass flow rate,ms kg/s 49.9

Feedwater mass flow rate,mw kg/s 49.9

3. VALIDATION

In order to validate the present nonlinear dynamic model,the responses of the different boiler parameters to stepvariations in steam flow rate were compared with avail-able experimental data. Validation of the model was per-formed through comparison of the present results againstthe available data (Astrom and Bell, 2000). The mea-surements provide steam flow rate variations with time asshown in Fig. 3. The responses of drum pressure to thesevariations were recorded. The results of the present calcu-lations are shown in Fig. 4 for response to step variationsin steam flow rate. The results show that the model cap-


492 Habib et al.

FIG. 3: Experimental data for distribution of steam flow rate (Astrom and Bell, 2000).

FIG. 4: Comparison of numerical results of response of drum pressure to variations in steam flow rate with experi-mental data of Wigren (2005).

tures the variations in pressure. A reasonably good agree-ment is achieved.

4. RESULTS

The present work presents calculations that were per-formed to investigate the influence of step increase insteam flow rate by 10% and by 20% on the different boilerparameters such as the pressure, the steam quality, themass flow rates in the riser and downcomer, the volumesof water and steam, and the metal temperature, as well asthe heat transfer coefficient.

Figure 5 presents the change of the pressure as a resultof sudden increase in steam flow ratems. As the steamflow rate rises, the steam volume inside the drum de-creases and the drum pressure decreases monotonically.The figure shows that the pressure decreases at a con-stant rate ofdP/dt = 2.67 kPa/s as a result of a steprise in the steam flow rate of 10%. For the case of 20%step rise in the steam demand, the rate of pressure in-crease is 4.8 kPa/s. The reduction in pressure is a directresult of increase in steam flow rate out of the drum.Figure 6 presents the response of the mass flow rates inrisers and downcomers for the two cases of step rise in



FIG. 5: Response of drum pressure for step rise in steam demand

FIG. 6: Response of mass flow rates in riser and downcomer for step rise in steam demand

steam flow rate. The riser mass flow exhibits a sudden in-crease as a result of increase in the steam flow rate. Asthe steam flow ratems is increased, a sudden increasefrom the steady state value in the mass flow rate in theriser occurs. As a result, the downcomer flow exhibits agradual increase with time. Thus, the outlet of the riserfirst increases and then decreases to match the flow in thedowncomer. Thus, the difference between the riser anddowncomer mass flow rates,mr − mdc, increases sud-denly causing increase in the mass flow rate into the drumand the volume of steam water in the drum to compensatefor steam flow out of the drum. The difference between

both of the two flow rates diminishes at around 20 s fromthe moment of the step rise in steam flow rate. This is fol-lowed by a continuous gradual increase in the risers’ anddowncomers’ mass flow rates. The case of 20% step riseshows similar trends but with higher magnitudes of flowrates in risers and downcomers. As well, the differencebetween the risers and downcomers at the start of the riseis much higher for the case of 20%. Consistent with theincrease in steam flow rate and the consequent decrease indrum pressure, the steam quality at the exit of the riser in-creases as a result of an increased rate of evaporation. Thesteam quality increases first at a high rate during the first


494 Habib et al.

10 s. This is followed by a less-steep decrease as shownin Fig. 7 as a result of re-increase in the mass flow rate inthe riser tube as given in Fig. 6.

Figure 8 presents the dynamics of steam and watervolumes in the drum under the water level. As shown inFig. 8, the volume of the steam within the liquid (underwater level) in the drum,Vsd, increases as a result of acombined increase in the riser flow rate and the qualitythe at exit of the riser. The increase in the riser flow ratemr and the qualityx causes a high increase in steam flowinto the drum which results in an increase inVsd. The in-crease inVsd is consistent with the calculations of (Kimand Choi, 2005). Figure 8 exhibits an increase in the wa-ter volume in the drum with high rate during the first fewseconds. This is followed by a reduction in the water withtime. The steam inside the drum under the water level ex-

FIG. 7: Response of steam quality for step rise in steamdemand

FIG. 8: Response of steam and water volumes in thedrum for step rise in steam demand

hibits a continuous increase with time. The reduction inthe rate of increase of the steam volume in the drum af-ter 50 s is due to the condensation of steam inside thedrum which causes a new addition in the water volume.Figure 9 indicates that the total rate of condensation de-creases suddenly and then continues to decrease but at alower rate. The sudden drop in the steam condensation isattributed to the decrease in the drum pressure. Figure 10shows that the total volume of steam in the system,Vst,increases. The increase inVst is a result of an increase ofVsd. Consistent with the increase in the total water volumeVwt, Fig. 10 shows a decrease in the total water volumeVwt according to the relationVst+Vwt=V whereV is thetotal volume of the system.

The response of heat transfer coefficienth is shown inFig. 11 and the response of the liquid heat transfer coeffi-

FIG. 9: Response of steam condensation for step rise insteam demand

FIG. 10: Response of total water and steam volumes forstep rise in steam demand



FIG. 11: Response of heat transfer coefficient for steprise in steam demand

cienthL is shown in Fig. 12. As shown in Fig. 11,h de-creases sharply during the first few seconds. This is thenfollowed by a reduction in the rate of decrease inh withtime. Despite the fact that the case of 20% exhibits highervalues ofh, the rate of decrease inh is higher than in thecase of 10%. The temperature distributions are shown inFig. 13 and indicate an increase in inner wall temperatureabove the saturation temperature with time. It is noted thatthe rise in temperature due to rise in steam flow rate inthe present case of a general riser tube is not significant.However, this rise in temperature could be significant incases where riser tubes have low flow rates resulting fromhigh pressure losses and/or high heat flux. In such a case,the steam quality can be high causing tube failures. In or-der to explain the behavior of the heat transfer coefficient,the response of the heat transfer coefficient of the liquid,

FIG. 12: Response of liquid heat transfer coefficient forstep rise in steam demand

FIG. 13: Response of the difference between the metaland saturated steam temperatures for step rise in steamdemand

hL (Fig. 12), and the qualityx, (Fig. 7) should be investi-gated. As the pressure decreases,ρw increases and resultsin a decrease in the volume flow rate which, in turn, leadsto reduction in the liquid flow velocity. The heat transfercoefficient of the liquid phasehL is influenced by the flowvelocity which is shown in Fig. 14. The differences be-tween the heat transfer coefficient of the two-phase flowand the liquid phase is due to the boiling number and theconvection number as given by Eqs. (42) and (43). Therate of velocity decreases with time to a minimum valuearound a time of 65 s and then increases. This is consis-tent with the distribution of liquid heat transfer coefficientin Fig. 14. The increase is at a much higher rate for thecase of 20%. This explains the higher rate of decrease as

FIG. 14: Response of liquid flow velocity for step rise insteam demand


496 Habib et al.

shown in Fig. 11 andhL in Fig. 12. Figure 15 exhibitsthe distribution of the void fraction that indicates a rapidincrease in the first few seconds which is then followedby a less-steep increase in its value. As a result, the heattransfer coefficient decreases as shown in Fig. 11 and ex-plains the behavior of the heat transfer coefficienth in thisfigure.

Comparison of the response of water levels is madefor the present results of step rise in steam flow rate of10% with those of Astrom and Bell (2000) and also withthe results of Kim and Choi (2005). Figure 16 provides acomparison of the water level due to liquid and steam con-tents. The present results confirm those of Kim and Choi(2005), however, with higher magnitudes. The present

FIG. 15: Response of steam volume fraction for step risein steam demand

FIG. 16: Response of the drum level for step rise in steamflow rate

calculations and those of Astrom and Bell (2000) and Kimand Choi (2005) present the same water and steam contri-bution to water level in the first period of time at around20 s following the change in the steam flow rate. As thelevel decreases, the results show good agreement with As-trom and Bell (2000). However, the results of Kim andChoi (2005) show a higher rate of decrease in the drumlevel.

5. CONCLUSIONS

A nonlinear dynamic model for the investigation of influ-ence of operating conditions on the response of naturalcirculation in water tube boilers was developed. The dif-ferential equations comprising the nonlinear model andgoverning the flow inside the boiler tubes were formu-lated. A thermal model for the prediction of possible tubeoverheating in water tube boilers was also developed. Thestudy investigates the dynamic response of the boiler op-erating parameters to rapid variations in steam demand.The operating parameters include steam quality, flow ve-locity, mass flow rate inside the riser tubes, and heat trans-fer characteristics of the riser tubes. The parameters alsoinclude the volume of water in the system as well asboiling parameters inside the steam drum. A numericalscheme for the solution of the governing differential equa-tions was established.

The results indicate that increasing the steam flow rateby 10% and by 20% results in a monotonic decrease inthe drum pressure and large variations in the steam qual-ity and water level in the drum. As well, the changes inthe heat transfer coefficient lead to a slight temperatureincrease in the riser metal temperature. The riser temper-ature increases partly due to the increase in the steam tem-perature and partly due to the decrease in the heat transfercoefficient. The calculated response of the water level inthe drum in response to step rise in steam flow rate is inreasonable comparison with those available in the litera-ture.

ACKNOWLEDGMENT

The authors wish to acknowledge the support receivedfrom King Fahd University of Petroleum & Minerals dur-ing this study.

REFERENCES

Astrom, K. J. and Eklund, K., A simplified non-linear model ofa drum boiler-turbine unit,Int. J. Control, vol. 16, no. 1, pp.145–169, 1972.



Astrom, K. J. and Eklund, K., A simple non-linear drum boilermodel,Int. J. Control, vol. 22, no. 5, pp. 739–740, 1975.

Astrom, K. J. and Bell, R. D., Simple drum-boiler models, inProc. of IFAC Power Systems Modelling and Control Appli-cations, Brussels, Belgium, pp. 123–127, 1988.

Astrom, K. J. and Bell, R. D., A nonlinear model for steamgeneration processes,Proc. of IFAC 12th Triennial WorldCongress, Sydney, Australia, pp. 649–652, 1993.

Astrom, K. J. and Bell, R. D., Drum-boiler dynamic,Automat-ica, vol. 36, pp. 363–378, 2000.

Bell, R. D. and Astrom, K. J., A fourth order non-linear modelfor drum-boiler dynamics,Proc. of 13th Triennial WorldCongress, San Francisco, pp. 31–36, 1996.

Bjorge, R. W., Hall, G. R., and Rohsenow, W. M., Correlationof forced convection boiling heat transfer data,Int. J. HeatMass Transfer, vol. 25, pp. 753–757, 1982.

Changliang, L., Jizhen, L., Yuguang, N., and Wiping, L., Non-linear boiler model of 300 MW power unit for system dy-namic performance studies,Proc. of ISIE, Pusan, Korea, pp.1296–1300, 2001.

Chen, J. C., A correlation for boiling heat transfer to saturatedfluids in vertical flow,Ind. Eng. Chem., Proc. Design Dev.,vol. 5, no. 3, pp. 322–339, 1966.

Collier, J. G.,Convective Boiling and Condensation, Maiden-head, UK: McGraw-Hill Book Company (UK) Ltd., 1972.

Darabi, J., Salehi, M., Saeedi, M. H., and Ohadi, M. M., Reviewof available correlations for prediction of flow boiling heattransfer in smooth and augmented tubes,ASHRAE Transac-tions, vol. 101, pp. 225–235, 1995.

Davidenko, K. Y. and Rushchinskiy, V. M., An algorithm fornumerical determination of transient processes in supercriti-cal boilers,Heat Transfer - Sov. Res., vol. 7, no. 5, pp.17–30,1975.

De Mello, F. P., Boiler models for system dynamic performancestudies,IEEE Trans. Power Syst., vol. 6, no. 2, pp. 66–75,1991.

Dittus, F. W. and Boelter, L. M. K., University of California,Berkeley,Publ. Eng., no. 2, p. 443, 1930.

Dong, Y. and Tingkuan, C., HGSSP—A computer program forsimulation of once-through boiler start-up behavior,HeatTransfer Eng., vol. 22, no. 5, pp. 50–60, 2001.

Flynn, M. E. and O’ Malley, M. J., A drum-boiler model for longterm power system dynamic simulation,IEEE Trans. PowerSyst., vol. 14, no. 1, pp. 209–217, 1999.

Franke, R., Rode, M., and Kruger, K., On-line optimization ofdrum boiler startup,Proc. of Third International ModelicaConference, Linkoping, Sweden, November 3–4, 2003, Fritz-son, P., ed., pp. 287–96, 2003.

Gungor, K. E. and Winterton, R. H. S., A general correlation forflow boiling in tubes and annuli,Int. J. Heat Mass Transfer,

vol. 29, pp. 351–358, 1986.

Hewitt, G. F. and Roberts, D. N., Studies of two-phase flow pat-terns by simultaneous x-ray and flash photography, Reportno. AERE-M2159, Atomic Energy Research Establishment,Harwell, UK, 1969.

Hewitt, G. F., Gas-liquid flow, inHandbook of Multiphase Sys-tems, New York: Hemisphere Publishing, 1982.

Hewitt, G. F, Shires, G. L., and Bott, P. R.,Process Heat Trans-fer, CRC Press, Boca Raton, FL, 1994.

Habib, M. A., Emara-Shabaik, H. E., Al-Zaharnah, I., andAyinde, T., A thermal nonlinear dynamic model for watertube drum boilers,Int. J. Energy Res., vol. 34, pp. 20–35,2010.

Inoue, T., Taniguchi, H., and Ikeguchi, Y., A model of fossilfueled plant with once-through boiler for power system fre-quency simulation studies,IEEE Trans. Power Syst., vol. 15,no. 4, pp. 1322–1328, 2000.

International Formulation Committee (IFC), A formulation ofthe Thermodynamic Properties of Ordinary Water Substance1967, The 1967 IFC Formulation for Industrial Use, VereinDeutscher Ingenieure, Dusseldorf, 1967.

Kandlikar, S. G., A general correlation for saturated two-phaseflow boiling heat transfer inside horizontal and vertical tubes,J. Heat Transfer, vol. 112, pp. 219–228, 1990.

Kim, H. and Choi, S., A model on water level dynamics in nat-ural circulation drum-type boilers,Int. Commun. Heat MassTransfer, vol. 32, pp. 786–796, 2005.

Klimenko, V. V., A general correlation for two-phase flow heattransfer,Int. J. Heat Mass Transfer, vol. 31, pp. 541–552,1988.

Klimenko, V. V., A generalized correlation for two-phase forcedflow heat transfer, second assessment,Int. J. Heat MassTransfer, vol. 33, pp. 2073–2088, 1990.

Leva, A., Maffezzoni, C., and Benell, G., Validation of drumboiler models through complete dynamic tests,Control Engi-neering Practice, vol. 7, pp. 11–26, 1999.

Pereyra, E. J., Shah, B. A., and Edwards, L. L., Circulation ratesand tube burnout in natural circulation boilers,Tappi J., vol.72, pp. 143–148, 1989.

Shah, M. M., A new correlation for heat transfer during boilingflow through pipes,ASHRAE Trans., vol. 82, no. 2, pp. 66–86, 1976.

Shah, M. M., Chart correlation for saturated boiling heat trans-fer: Equations for further study,ASHRAE Trans., vol. 88, no.1, pp. 185–196, 1982.

Steiner, D. and Taborek, J., Flow boiling heat transfer in verticaltubes correlated by an asymptotic model,Heat Transfer Eng.,vol. 13, no. 2, pp. 43–69, 1992.

Taitel, Y. and Dukler, A. E., A model for predicting flow regimetransitions in horizontal and near horizontal gas-liquid flow,


498 Habib et al.

AIChe J., vol. 22, pp. 47–55, 1976.

Termuehlen, H. and Gartner, G., Sustained fast turbine-generator load response in fossil-fueled power plants,IEEETrans. Power Appar. Syst., vol. 100, no. 5, pp. 2495–2503,1981.

Tucakovic, D. R., Stevanovic, V. D., Zivanovic, T., Jovovic, A.,and Ivanovic, V. B., Thermal-hydraulic analysis of a steamboiler with rifled evaporating tubes,Appl. Therm. Eng., vol.27, pp. 509–519, 2007.

Wigren, T., Recursive identification based on nonlinear statespace models applied to drum-boiler dynamics with nonlin-ear output equations,Proc. of 2005 American Control Con-ference, Portland, OR, pp. 5066–5072, 2005.

APPENDIX A: THE FLUID FLOW DYNAMICNONLINEAR MODEL

The boiler under consideration here is of the water-tubenatural circulation type (Fig. 1). The main components ofthe boiler are the steam drum, the downcomer-riser tubeswhich represent the complete water circulation loop. Thefollowing are the governing equations.

A.1 Mass Conservation

Considering the system of drum, risers, and downcomers(Fig. 1), the total mass of steam and water in the systemis governed by,

dM

dt= mfw − ms, (A1)

where M represents the mass of steam and water in thesystem and is given as

M = Ms +Mw = ρsVst + ρwVwt, (A2)

whereρs andρw are the density of saturated steam andsaturated water.Vst and Vwt are the total volumes ofsteam and water. The mass is governed by

d

dt[ρsVst + ρwVwt] = mfw − ms. (A3)

The total volume of the drum, downcomer, and risers,Vt,is

Vt = Vst + Vwt = Vd + Vdc + Vr. (A4)

Substituting forVst = Vt − Vwt, thus

d

dt[ρs(Vt − Vwt) + ρwVwt] = mfw − ms. (A5)

Differentiating, thus

Vtdρsdt

+ (ρw − ρs)dVwt

dt+ Vwt

d

dt(ρw − ρs)

= mfw − ms. (A6)

Resubstituting forVt = Vst + Vwt, sinceρw andρs arefunctions of the pressure only, thus(Vst

∂ρs∂p

+ Vwt∂ρw∂p

)dp

dt+(ρw−ρs)

dVwt

dt= mfw−ms.

(A7)

A.2 Energy Conservation

The energy balance of the whole system is given by

dU

dt= Q+ mfwhfw − mshs (A8)

whereU represents the total energy of steam, water, andmetal in the system and is given by

U = Msus +Mwuw +MmtCptm. (A9)

Thus

d

dt[ρsusVst + ρwuwVwt +MmtCptm]

= Q+ mfwhfw − mshs. (A10)

Since the internal energy isu = h−p/ρ, the global energybalance can be written as

d

dt[ρshsVst + ρwhwVwt − pVt +MmtCptm]


Substituting forVst = Vt − Vwt in Eq. (A10), thus[Vst

(hs

∂ρs∂p

+ ρs∂hs

∂p

)+ Vwt

(hw

∂ρw∂p

+ ρw∂hw

∂p

)− Vt +MmtCp

∂tm∂p

]dp

dt+ (ρwhw − ρshs)

dVwt

dt


A.3 Momentum Conservation

Considering natural circulation inside the riser-downcomer loop which is driven by the density dif-



ference between the vapor and liquid phases, the mo-mentum balance results in the following equation for thedowncomer and riser flow rates:

dρV

dt+

dρV 2

dy= −dp

dy+ ρg +

d

dyµdV

dy. (A13)

Considering

m = ρV A, (A14)

thus

dm

dt+

d

dx

(m2

ρA

)= −dp

dx+ (ρw − ρs) g

− k

2

d

dx

(m2

ρA

)(A15)

wherek is a dimensionless friction coefficient. Integrat-ing for the loop of riser and downcomer, and neglectingthe inertia force, thus

(Lr+Ldc)dmdc

dt=(ρw−ρs)gαVr−

k

2

m2dc

ρwAdc. (A16)

As indicated by [14], the time constant is very short andtherefore, steady state conditions are considered:

mdc =

√2ρwAdc(ρw − ρs)gαVr

k. (A17)

A.4 The Riser Equations

Considering the riser tubes, the total mass of steam andwater in the riser is governed by

dMr

dt= mdc − mr, (A18)

whereM represents the mass of saturated steam and sat-urated water in the riser tubes and is given as

Mr = ρsαVr + ρw(1− α)Vr, (A19)

whereVr is the volume of the riser tubes andα is thevolume fraction of steam in the riser tubes. Thus, the massin the riser is governed by

d

dt[ρsαVr + ρw(1− α)Vr] = mdc − mr. (A20)

The energy balance of the riser is thus given by

dUr

dt= Q+ mdchw − mr(hw + xhfg), (A21)

wherehfg is the enthalpy of evaporation of water at con-stant pressure such thathfg = hs − hw whereUr rep-resents the total energy of steam, water, and metal in theriser and is given by

Ur =[ρsαVrhs + ρw(1− α)Vrhw − pVr

+MmrCptm]. (A22)

Thus, since the internal energy isu = h − p/ρ, the riserenergy balance can be written as

d

dt[ρsαVrhs + ρw(1− α)Vrhw − pVr +MmrCptm]

= Q+ mdchw − ms(hw + xhfg). (A23)

Multiplying Eq. (A20) by [−(hw + xhc)] and adding toEq. (A23), thus the flow through the risers is given by

(1− x)hfgd

dt(ρsαVr) + ρw (1− α)Vr

dhw

dt

− xhfgd

dt[ρw (1− α)Vr] + ρsαVr

dhs

dt− Vr

dp

dt

+mmrCpdtmdt

= Q− xhfgmdc. (A24)

Differentiating, thus

(1− x)hfgαVr∂ρs∂p

dp

dt+ (1− x)hfgρsVr

×[∂α

∂x

dx

dt+

∂α

∂p

dp

dt

]+ (1− α) ρwVr

∂hw

∂p

dp

dt

− xhfgVr (1− α)∂ρw∂p

dp

dt+ xhfgρwVr

×[∂α

∂x

dx

dt+

∂α

∂p

dp

dt

]+ ρs ¯αVr

∂hs

∂p

dp

dt− Vr

dp

dt

+MmrCpdtmdt

= Q− xhfgmdc. (A25)

Rearranging the terms, thus{(ρw

∂hw

∂p− xhfg

∂ρw∂p

)(1− α)Vr+

[(1−x)hfg

∂ρs∂p

+ ρs∂hs

∂p

]αVr + [ρs + x(ρw − ρs)]hfgVr

∂α

∂p− Vr

+MmrCp∂tm∂p

}dp

dt+

{[(1− x)ρs + xρw]hfgVr

∂α

∂x

}× dx

dt= Q− xhfgmdc. (A26)


500 Habib et al.

A.5 The Governing Equations for the Flow inside theDrum

The mass balance for the steam under the liquid level inthe drum is

d

dt[ρsVsd] = xmr − msd − mcd, (A27)

wheremsd is the steam flow rate through the liquid sur-face in the drum andmcd is the rate of condensation inthe drum that can be expressed as

mcd =(ρw − ρfw)

hcmfw +

1

hc

[ρsVsd

dhs

dt

+ ρwVwddhw

dt−(Vsd+Vwd)

∂p

∂t+MdCp

∂ts∂t

], (A28)

msd =ρs

Td(Vsd − V o

sd) + xmdc + xrβ

× (mdc − mr) . (A29)

The mass flow rate in the downcomer tubes can be calcu-lated from Eq. (A20) as

mr = mdc − Vr∂

∂p[(1− α) ρw + αρs]

dP

dt

+ Vr (ρw − ρs)∂α

∂x

dx

dt, (A30)

which can be presented as

mr = mdc − Vr

[α∂ρs∂p

+ (1− α)∂ρw∂p

+ (ρw − ρs)

× ∂ ¯α

∂p

]dP

dt+ Vr (ρw − ρs)

∂α

∂x

dx

dt. (A31)

Substitutingmr, msd, andmcd into Eq. (A27), thus

ρsdVsd

dt+ Vsd

dρsdt

+1

hfg

[ρsVsd

dhs

dt+ ρwVwd

dhw

dt

− (Vsd + Vwd)dp

dt+mmdCp

dtmdt

]+ x(1 + β)Vr

× d

dt[(1− α)ρw + αρs] =

ρs

Td(V o

sd − Vsd)

+hfw − hw

hfgmfw. (A32)

A.6 The Overall Boiler Dynamic Model

Based on the above analyses [Eqs. (A7), (A12), (A26),and (A32)], the overall fluid dynamics model of the boileris given by the following set of equations:

a11a120a21a220a310a33a410a43

000a44

dP/dtdVwt/dtdx/dtdVsd/dt

=

mfw − ms

Q+ mfwhfw − mshs

Q− xhfgmdc

ρs

Td(V o

sd − Vsd) +hfw − hw

hfgmfw

. (A33)


INFLUENCE OF BOILER SWING RATE ON DYNAMICS OF THERMAL AND FLOW CHARACTERISTICS IN WATER CIRCULATION...

Documents

Transcript of INFLUENCE OF BOILER SWING RATE ON DYNAMICS OF THERMAL AND FLOW CHARACTERISTICS IN WATER CIRCULATION...