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International Journal of Heat and Mass Transfer 51 (2008) 4379–4391

An analytical solution for the coupled heat and mass transfer duringthe freezing of high-water content materials

Mariela C. Olguın a, Viviana O. Salvadori b,c,Rodolfo H. Mascheroni b,c, Domingo A. Tarzia d,e,*

a Depto. Matematica, Fac. de Ciencias Exactas, Ingenierıa y Agrimensura, UNR, Rosario, Argentinab CIDCA, Fac. de Ciencias Exactas, UNLP, CONICET, La Plata, Argentina

c MODIAL, Depto. Ing. Quımica, Fac. de Ingenierıa, UNLP, La Plata, Argentinad Depto. Matematica, Fac. Ciencias Empresariales, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina

e Consejo Nacional de Investigaciones Cientıficas y Tecnicas, CONICET, Argentina

Received 28 May 2007; received in revised form 25 January 2008Available online 24 April 2008

Abstract

The coupled problem of heat and mass transfer during the solidification of high-water content materials like soils, foods, tissues andphase-change materials is developed. Assuming quasi-steady heat conduction in the frozen region, the system leads to a set of coupledordinary differential equations. The model takes into account the influence of material characteristics and process variables on theadvance of the freezing and sublimation fronts, temperature and water vapour profiles and weight loss. It was validated against the ana-lytical solution of the freezing (without surface ice sublimation) of a semi-infinite medium and was extensively used to perform a para-metric study.� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Ice sublimation takes place from the surface of high-water content systems like moist soils, aqueous solutions,vegetable or animal tissues and foods that freeze uncoveredor without an impervious and tight packaging material.The rate of both phenomena (solidification and sublima-tion) is determined both by material characteristics (mainlycomposition, structure, shape and size) and cooling condi-tions (temperature, humidity and rate of the media thatsurrounds the phase change material). The sublimationprocess, in spite of its magnitude being much less than thatof freezing process, determines fundamental features of thefinal quality for foods and influences on the structure andutility of frozen tissues.

0017-9310/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2008.01.031

* Corresponding author. Address: Depto. Matematica, Fac. CienciasEmpresariales, Universidad Austral, Paraguay 1950, S2000FZF Rosario,Argentina. Tel.: +54 341 522 3093; fax: +54 341 522 3001.

E-mail address: Domingo.Tarzia@fce.austral.edu.ar (D.A. Tarzia).

Modelling of these simultaneous processes is very diffi-cult owed to the coupling of the heat and mass transfer bal-ances, the existence of two moving phase-change frontsthat advance with very different rates and to the involvedphysical properties which are, in most cases, variable withtemperature and water content.

The process with only-solidification (with no sublima-tion) has been extensively studied [1–3]. A large bibliogra-phy on free and moving boundary problems for the heatdiffusion equation is given by Tarzia [4]. The system hasbeen modelled both by analytical procedures [2,5], andwith numeric methods [1,2,6]. Due to the non-linearity ofthe problem, solutions usually involve mathematical diffi-culties and analytical methods can only be developed foridealized systems or for those with plain structure and con-stant properties. Numerical methods, instead, allow to con-sider the structural and compositional implicanciesthrough detailed models [1,7], and to produce a data setof predicted data from which to develop approximate solu-tions which only enable to predict process times [1,8,9]. The

Nomenclature

C mass concentration of water, kg/m3

Cp specific heat, J/(kg K)Def effective diffusion coefficient of water, m2/sh heat transfer coefficient, W/(m2 K)k thermal conductivity, W/(m K)Km mass transfer coefficient, m/sLs sublimation heat of water, J/kgLf solidification heat of water, J/kgm geometrical index (=0 for slabs, =1 for infinite

cylinders, =2 for spheres)mf frozen mass by unit volume, kg/m3

ms sublimated mass by unit volume, kg/m3

M molecular weight, kg/molRg universal gas constant, (=8.31 J/(mol K))s position of the moving front, mt time, s

T temperature, Kx spatial coordinate, m

Greek symbols

e porosityq density, kg/m3

Subscripts

a water vapor in the aird dehydrated phasef frozen phasei initial (unfrozen material)if initial freezingo sublimation front temperatures surrounding mediumva water vapour in air within the material

4380 M.C. Olguın et al. / International Journal of Heat and Mass Transfer 51 (2008) 4379–4391

process with only sublimation of the already-frozen systemhas been extensively studied for freeze-drying of food andpharmaceutical materials [10,11].

In the case of freezing with simultaneous ice sublima-tion, published developments are scarce and no analyticalsolution to the coupled problem has been developed. Icesublimation has been surveyed by several authors in differ-ent systems: in freeze-drying [10,12], in geomorphology [13]and in the case of frozen food products several experimen-tal studies have been published: weight losses of lamb [14],tylose and beef [15], potatoes [16] and meat products[17,18]. Additionally there are no experimental data ontemperature and concentration profiles during freezingwith simultaneous ice sublimation and on the rate of freez-ing and dehydration fronts.

On the other hand, theoretical models were proposed todescribe the heat and mass transfer in these processes but,in most of the published works, only semi-empiric ornumerical methods are used. Aguirre Puente and Sukhwal[19] presented a description of the heat and mass balancesto be solved; meanwhile Pham and Willix [14] suggested theuse of simple equations based on drying theory and on theuse of the psychometric chart to calculate weight loss.Campanone [20] and Campanone et al. [21] developed adetailed numerical model that allows to predict tempera-ture and composition profiles, as well as freezing timeand weight loss. Besides, approximate equations weredeveloped to predict freezing time and weight loss in foodfreezing with simultaneous surface dehydration [22,23].

Based on the abovementioned situation, the objectivesof this work were:

� To develop a physical model and to implement therelated quasi-analytical solution that enables us toaccount for the influence of material characteristicsand process variables on the advance of the freezing

and sublimation fronts, temperature and water vapourprofiles and weight loss during the freezing of high-water content materials.� To use the quasi-analytical solution to predict the influ-

ence of material characteristics and freezing conditionson the dependent variables (freezing and sublimationfronts position, weight loss and temperature and compo-sition profiles).

1.1. The physical problem

When high-water content materials like foods, tissues,gels, soils or water solutions of inorganic or organic sub-stances, held in open, permeable or untightly-sealed con-tainers are refrigerated to below their initial solidificationtemperature, two simultaneous physical phenomena takeplace:

� liquid water solidifies (freeze), and� surface ice sublimates.

The rate and extent of these transfers is determined bydifferent factors:

(a) Material characteristics as shape, size, compositionand structure. These last two determine the value ofthose physical properties relevant to heat and masstransfer: heat capacity, density, thermal conductivityand mass diffusivity. In a first instance, all of theseproperties depend on temperature and composition,and the last three of them also on material structure.

(b) Environmental characteristics: Temperature, humid-ity and rate of the refrigerating medium.

(c) Characteristics of the material–environment inter-phase: Mainly type, shape and material of containerand if this is permeable or not to water vapor, as well

M.C. Olguın et al. / International Journal of Heat and Mass Transfer 51 (2008) 4379–4391 4381

as its tightness to material surface (these characteris-tics influence on the resulting heat and mass transfercoefficients and on the ‘‘microclimate” that surroundsmaterial surface).

From a physical point of view, it is usual to considerthese materials as a combination of a solid matrix, an aque-ous phase and a gaseous phase (air and water vapour).

For the description of the freezing process, the materialcan be divided into three zones: unfrozen, frozen and dehy-drated as can be seen in Fig. 1.

Freezing begins from the refrigerated surface/s, at a tem-perature (Tif) lower than that of pure water, due to thepresence of dissolved materials, and continues along anequilibrium line, whose exact analytical shape is not knownfor most materials. Besides, in most cases, not all of theliquid water is accessible to the freezing process, becausepart of it is linked to solutes and structural materials(bound water). Simultaneously, ice sublimation begins atthe frozen surface and a dehydration front penetrates thematerial, whose rate of advance is again determined byall the abovementioned characteristics of the material andenvironmental conditions. Normally this rate is muchlower than that of the freezing front [20].

1.2. The mathematical problem

A complete mathematical model has to solve both, theheat transfer (freezing) and the mass transfer (weight loss)simultaneously.

When applying the principles of process simulation andconsidering solid or semi-solid systems (with neither bulkflow nor deformation) at atmospheric pressure, with unidi-rectional heat and mass transfer and variable properties,the following microscopic balances are obtained:

qCp

oTot¼ k

o2Tox2þ ok

oxoToxþ mk

xoTox

ð1Þ

eoCva

ot¼ Def

o2Cva

ox2þ oDef

oxoCva

oxþ mDef

xoCva

oxð2Þ

Normally uniform initial temperature and composition issupposed and convective boundary conditions at the sur-face for both heat and mass transfer are the usual model-ling assumptions.

sf(t)sd(t)

Fig. 1. Simple schem

Phase-change is accounted for in the following way:

� Solidification (freezing) as a freezing front (x = sf(t))located in the point where material temperature reachesthe initial freezing temperature (Tif), determined bymaterial composition. For temperatures lower than Tif

(the zone nearer to the refrigerated surface) the amountof ice formed is determined by an equilibrium line (icecontent vs temperature and water content) specific tothe material.� On the dehydration front (x = sd(t)) we impose Stefan-

like conditions for temperature distribution and vaporconcentration.

The following assumptions are needed for an adequatesimulation of the system:

� Variable properties (Def, Cp, k, q) as a function of com-position, temperature and structure, with three distinctzones: unfrozen, frozen and dehydrated. The possibledifference between densities of the liquid and frozenphases induces negligible volume change.� Vapor diffusion in the dehydrated zone must be consid-

ered through an adequate effective diffusion coefficientDef. This is an important parameter very difficult tomeasure or calculate, as well as the sorption isothermof water on the material at subzero temperatures (thereis an only one set of values in literature, belonging tobeef).� Uniform initial temperature Ti higher than Tif.

The physical description of the freezing-sublimationprocess presented previously, which led to the balancesand initial and boundary conditions presented in this sec-tion has no analytical solution and – up to now – the onlyway to solve these balances was through fully numericalmodels developed in finite differences or finite elements,as previously stated.

2. Developed model

Consider a semi-infinite material with characteristicssimilar to a very dilute gel (whose properties can be sup-posed equal to those of pure water). The system has initial

Dehydrated region

Frozen region

Unfrozen region

e of the system.

4382 M.C. Olguın et al. / International Journal of Heat and Mass Transfer 51 (2008) 4379–4391

uniform temperature equal to Tif and uncovered flat sur-face which at time t = 0 is exposed to the surrounding med-ium (with constant temperature Ts (lower than Tif) andheat and mass transfer coefficients h and Km). We assumethat Ts < T0(t) < Tif, t > 0 where T0(t) is the unknown sub-limation temperature.

The system behaviour is ruled by balances (1) and (2).To calculate the evolution of temperature and water con-tent in time, we will consider the following free boundaryproblem:

Find the temperatures Td = Td(x, t) and Tf = Tf(x, t), theconcentrations Cva = Cva(x, t), the free boundariessd = sd(t) and sf = sf(t) and the temperature T0 = T0(t) atthe sublimation front x = sd(t) which must satisfy thefollowing:

� Differential equations at the dehydrated region:

qdCpd

oT d

ot¼ kd

o2T d

ox2; 0 < x < sdðtÞ; t > 0 ð3Þ

eoCva

ot¼ Def

o2Cva

ox2; 0 < x < sd tð Þ; t > 0 ð4Þ

� Differential equations at the frozen region:

qfCpf

oT f

ot¼ kf

o2T f

ox2; sdðtÞ < x < sfðtÞ; t > 0 ð5Þ

Free boundary conditions at the sublimation frontx = sd(t):

T dðsdðtÞ; tÞ ¼ T fðsdðtÞ; tÞ ¼ T 0ðtÞ; t > 0 ð6Þ

kf

oT fðsdðtÞ; tÞox

� kd

oT dðsdðtÞ; tÞox

¼ Lsms _sdðtÞ; t > 0 ð7Þ

Def

oCvaðsdðtÞ; tÞox

¼ ms _sdðtÞ ð8Þ

CvaðsdðtÞ; tÞ ¼MP satðT ÞRgT 0ðtÞ

¼ Maexp b� c

T 0ðtÞ

� �RgT 0ðtÞ

ð9Þ

where Cva(sd(t), t) is the equilibrium vapor concentration atT0(t) and the saturation pressure Psat(T) is evaluatedaccording to [24].

� Free boundary conditions at the freezing front x = sf(t):

T fðsfðtÞ; tÞ ¼ T if ; t > 0 ð10Þ

kf

oT fðsfðtÞ; tÞox

¼ mfLf _sfðtÞ; t > 0 ð11Þ

� The convective boundary conditions at the fixed inter-phase x = 0:

kd

oT dð0; tÞox

¼ hðT dð0; tÞ � T sÞ; t > 0 ð12Þ

Def

oCvað0; tÞox

¼ KmðCvað0; tÞ � CaÞ; t > 0 ð13Þ

� The initial conditions at t = 0:

sfð0Þ ¼ sdð0Þ ¼ 0 ð14ÞT ¼ T if for x P 0 ð15Þ

We will solve the system (3)–(15) by using the quasi-steadymethod [2]. In general, it is a good approximation when theStefan number tends to zero, i.e. when the latent heat of thematerial is high with respect to the heat capacity of the so-lid material. This approximation has often been used whenmodelling the freezing of high-water content materials[2,25]. Therefore, the temperatures Tf, Td and the concen-tration Cva are given by the following expressions:

T dðx; tÞ ¼ AðtÞ þ BðtÞx; 0 < x < sdðtÞ; t > 0 ð16ÞCvaðx; tÞ ¼ DðtÞ þ EðtÞx; 0 < x < sdðtÞ; t > 0 ð17ÞT fðx; tÞ ¼ F ðtÞ þ GðtÞx; sdðtÞ < x < sfðtÞ; t > 0 ð18Þ

where A(t), B(t), D(t), E(t), F(t) and G(t) are unknownfunctions to be determined, as well as the free boundariessd (t) and sf(t) and the temperature T0(t). We have nine un-known functions and we must impose nine conditions (Eqs.(6)–(15)). By considering them we obtain the following sys-tem of differential algebraic equations:

kdBðtÞ ¼ hðAðtÞ � T sÞ ð19ÞDef EðtÞ ¼ KmðDðtÞ � CaÞ ð20ÞAðtÞ þ BðtÞsdðtÞ ¼ T 0ðtÞ ð21ÞF ðtÞ þ GðtÞsdðtÞ ¼ T 0ðtÞ ð22Þ

DðtÞ þ EðtÞsdðtÞ ¼ Maexp b� c

T 0ðtÞ

� �RgT 0ðtÞ

ð23Þ

F ðtÞ þ GðtÞsfðtÞ ¼ T if ð24ÞkfGðtÞ � kdBðtÞ ¼ Lsms _sdðtÞ ð25ÞDef EðtÞ ¼ ms _sdðtÞ ð26ÞkfGðtÞ ¼ mf Lf _sfðtÞ ð27Þ

By considering the first six equations of the previous systemwe can obtain, by direct replacement and rearrangement,A(t), B(t), D(t) and E(t) as a function of T0(t) and sd(t),as well as F(t) and G(t) as a function of T0(t), sd(t) andsf(t), given by the following expressions:

AðtÞ ¼T 0ðtÞ þ T s

hkd

sdðtÞ1þ h

kdsdðtÞ

ð28Þ

BðtÞ ¼ hkd

T 0ðtÞ � T s

1þ hkd

sdðtÞð29Þ

DðtÞ ¼Km

DefCasdðtÞ þMa

expðb� cT 0ðtÞÞ

RgT 0ðtÞ

1þ Km

DefsdðtÞ

ð30Þ

EðtÞ ¼ Km

Def

Maexp b� c

T 0ðtÞ

� �RgT 0ðtÞ

� Ca

1þ Km

DefsdðtÞ

ð31Þ

F ðtÞ ¼ T 0ðtÞsfðtÞ � T if sdðtÞsfðtÞ � sdðtÞ

ð32Þ

GðtÞ ¼ T if � T 0ðtÞsfðtÞ � sdðtÞ

ð33Þ

M.C. Olguın et al. / International Journal of Heat and Mass Transfer 51 (2008) 4379–4391 4383

By considering now the last three conditions of the system(28)–(33) and performing mathematical manipulations weobtain the following system of two ordinary differentialequations and one algebraic equation for sd(t), sf(t) andT0(t) given by:

1þ hkf

T s

T if

sfðtÞ þ hsdðtÞ1

kd

� T s

T if kf

� �� T 0ðtÞ

T if

1þ hkf

sfðtÞ þ hsdðtÞ1

kd

� 1

kf

� �� �

¼ LsKm

T if kf

1þ hkf

sfðtÞ� �

ðsfðtÞ � sdðtÞÞ1þ Km

DefsdðtÞ

Maexp b� c

T 0ðtÞ

� �RgT 0ðtÞ

� Ca

24

35 ð34Þ

_sdðtÞ ¼kfT if

msLs

1þ hkf

T s

T ifsfðtÞ þ hsdðtÞ 1

kd� T s

kf T if

� �� T 0ðtÞ

T if1þ h

kfsfðtÞ þ hsdðtÞ 1

kd� 1

kf

� �h i

1þ hkd

sfðtÞ� �

ðsfðtÞ � sdðtÞÞð35Þ

_sfðtÞ ¼kf T if

mf Lf

1� T oðtÞT if

sfðtÞ � sdðtÞð36Þ

sfð0Þ ¼ sdð0Þ ¼ 0 ð37Þ

The system (34)–(37) can be written in an equivalentform given by Eqs. (38)–(41) by defining the followingparameters:

d1 ¼mf Lf

kf T if

; d2 ¼ms

CaKm

; d3 ¼Lsms

hT if

;

d4 ¼Ma

CaRgT if

; d5 ¼mfLf

hT if

; d6 ¼T s

T if

T 0ðtÞ ¼ T if 1� d1ðsfðtÞ � sdðtÞÞ_sfðtÞ½ � ð38Þ

d2 1þ Km

Def

sdðtÞ� �

_sdðtÞ ¼ T ifd4

exp b� cT 0ðtÞ

� �T 0ðtÞ

� 1 ð39Þ

d5 1þ hkd

sdðtÞ þhkf

ðsfðtÞ � sdðtÞÞ� �

_sfðtÞ

¼ 1� d6 þ d3 1þ hkd

sdðtÞ� � T ifd4

exp b� cT 0ðtÞ

� �T 0ðtÞ

� 1

d2 1þ Km

DefsdðtÞ

� � ð40Þ

sdð0Þ ¼ sfð0Þ ¼ 0 ð41Þ

Moreover, functions F(t) and G(t) can be now rewritten asfollows:

F ðtÞ ¼ T 0ðtÞsfðtÞ � T if sdðtÞsfðtÞ � sdðtÞ

¼ T if ½1� d1sfðtÞ_sfðtÞ� ð42Þ

GðtÞ ¼ T if � T 0ðtÞsfðtÞ � sdðtÞ

¼ T ifd1 _sfðtÞ ð43Þ

and, therefore, the temperatures Td(x, t) and Tf(x, t) aregiven by the following expressions:

T dðx; tÞ ¼T 0ðtÞ 1þ h

kdx

� �þ T s

hkdðsdðtÞ � xÞ

1þ hkd

sdðtÞ;

0 < x < sdðtÞ; t > 0 ð44ÞT fðx; tÞ ¼ T if 1� d1 _sfðtÞðsfðtÞ � xÞ½ �;

sdðtÞ < x < sfðtÞ; t > 0 ð45Þ

and the concentration Cva(x, t) is given by:

Cvaðx; tÞ ¼Km

DefCaðsdðtÞ � xÞ þMa

exp b� cT 0ðtÞ

� �T 0ðtÞ

1þ Km

Defx

� �1þ Km

DefsdðtÞ

;

0 < x < sdðtÞ; t > 0 ð46Þ

Moreover, the temperature Td and the concentration Cva

at the fixed surface x = 0 are given by the followingexpressions:

T dð0; tÞ ¼ AðtÞ ¼T 0ðtÞ þ T s

hkd

sdðtÞ1þ h

kdsdðtÞ

ð47Þ

Cvað0; tÞ ¼ Cað1þ d2 _sdðtÞÞ ð48Þ

As previously expressed, the whole system (3)–(15) with theassumptions (16)–(18) (quasi-steady model) has beensolved with a semi-analytical method. As a result, Eqs.(38)–(40) give expressions that relate sd(t), sf(t) and T0(t)with the first derivates of sd(t) and sf(t). The differentialequations (39) and (40) with initial conditions (41) cannotbe solved analytically, then it must be solved approxi-mately by using an explicit scheme of finite differences thatis coupled with conditions (38) which predicts the freezingtemperature T0(t). These computed values for sd(t), sf(t)and T0(t) were then used for predicting the temperaturesTd(x, t), Tf(x, t) and the concentration Cva(x, t) throughthe expressions (44)–(48). The resulting algorithm wascoded in Scilab version 4.1.

3. Results and discussion

3.1. Validation of the model

At a first instance, it is important to validate anyapproximate model against the analytical solution for therelated physical system. Normally this situation is impossi-ble, that is why approximate models are developed, and theonly option is to compare against the exact solution of asimplified problem.

In this case the system that has analytical solution is thatof a semi-infinite medium with constant initial temperaturethat freezes (without surface ice sublimation) and constant

4384 M.C. Olguın et al. / International Journal of Heat and Mass Transfer 51 (2008) 4379–4391

surface temperature (very high heat transfer coefficient).This is called the Lame–Clapeyron–Stefan problem andits solution can be found elsewhere [2,26].

To this aim the prediction program was run with a veryhigh heat transfer coefficient h = 3000 W/(m2 K), with avery low mass transfer coefficient Km = 0.000001 m/sand Ts = 263 K, so as to secure getting almost instanta-neously a constant surface temperature and to make neg-ligible the contribution of the dehydrated zone. If theassumptions done during the development of this method

262

264

266

268

270

272

274

0 0.005 0.01 0.015 0

Posit

Tem

pera

ture

(K

)

time

Fig. 3. Temperature profile in the frozen zone. (—-)

0.0E+00

5.0E-03

1.0E-02

1.5E-02

2.0E-02

2.5E-02

3.0E-02

3.5E-02

0 1000 2000 3000 4000

Ti

Fre

ezin

g fr

ont (

m)

Fig. 2. Comparison of the freezing front position provided

are correct, results provided by the model should tendto those of the analytical problem with equal surfacetemperature.

Fig. 2 shows the comparison of the variation of sf(t) withtime for both cases. As can be seen, results coincide.

In the same sense, Fig. 3 presents the predicted temper-ature profiles in the frozen zone Tf(x, t) for different times.Again results are almost perfectly coincident.

We consider that these results are adequate to validatethe developed model.

.02 0.025 0.03 0.035 0.04

ion (m)

Lame–Clapeyron–Stefan solution, (�) this work.

5000 6000 7000 8000 9000 10000

me (s)

Lame-Clapeyron-Stefan solution

This work

by the Lame–Clapeyron–Stefan solution and this work

M.C. Olguın et al. / International Journal of Heat and Mass Transfer 51 (2008) 4379–4391 4385

3.2. Use of the model

The developed model was extensively used to study theinfluence of system parameters and freezing conditions onthe characteristic dependent variables of the system:Tf(x, t), Td(x, t), sd(t), sf(t) and To(t).

Calculations were performed simulating the freezing of ahigh-water content solid material with physical and ther-mal properties of a very dilute agar–agar gel. So, for mostproperties, values of pure water or ice could be used. Forother properties, like those of the dehydrated phase, valuesfor freeze-dried materials were taken as a valid approach

Table 1Physical properties and other parameters

Physical properties Frozen

q (kg/m3) 1000Cp (J/(kg K)) 1900Def (m2/s) –k (W/(m K)) 1.7

Other parameters

e 0.98Ca (kg/m3) 8.29 �h (W/(m2 K)) 10–150Km (m/s) 0.0072–Lf (J/kg) 3.30 �Ls (J/kg) 2.83 �mf (kg/m3) 980ms (kg/m3) 980M (kg/mol) 0.018Parameters of Eq. (9) and subsequents a = 133Tif (K) 273.16Ts (K) 233–268

0.0E+00

1.0E-02

2.0E-02

3.0E-02

4.0E-02

5.0E-02

6.0E-02

0 1000 2000 3000 4000

Ti

Free

zing

fron

t (m

)

h=10h=25h=75h=100h=150

Fig. 4. Influence of the heat transfer coefficient in the freezing front e

[20]. Table 1 presents a list of the values of the main prop-erties used.

A wide range of possible values of external parameterswas covered, so as to have a clear knowledge of theirspecific influence under different combinations of freezingconditions. Heat transfer coefficients in the range of10–150 W/(m2K) were considered. This range is feasibleto find in practical freezing conditions. In the same sense,external temperatures from 233 to 263 K were used (�40to �10 �C). In all cases a total pressure of 1 atm was used.A maximum calculation time of 10,000 s was consideredsufficient to determine the desired system characteristics.

zone Dehydrated zone

2010009.66 � 10�6

0.07

10�5

0.1082105

106

.33, b = 23.986, c = 6139.9094

5000 6000 7000 8000 9000 10000

me (s)

volution, Ts = 243 K. The other parameters are given in Table 1.

4386 M.C. Olguın et al. / International Journal of Heat and Mass Transfer 51 (2008) 4379–4391

In Figs. 4 and 5 the evolution of both moving bound-aries is shown, studying the influence of the transfercoefficients h and Km (Km is directly proportional to h).We can see that both sf(t) and sd(t) increase linearly forvery short times, but as the dehydrated layer grows, bothprocesses (freezing and sublimation) continue at steadilylower rates. This is due, as shown in Fig. 1, to the develop-ment of the dehydrated surface layer that has lower k andDef than the frozen zone, so lowering heat and mass trans-fer rates.

245

250

255

260

265

270

275

0 1000 2000 3000 4000 50

Tim

Sub

limat

ion

fron

t tem

pera

ture

(K

)

Fig. 6. Influence of the heat transfer coefficient in the sublimation front

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

3.5E-04

0 1000 2000 3000 4000

Ti

Sub

limat

ion

fron

t (m

)

h=10h=25h=75h=100h=150

Fig. 5. Influence of the heat transfer coefficient in the sublimation fron

Another fact observed from Fig. 4 is that sf(t) is mono-tonely increasing with respect to the parameter h mean-while the same is not valid for sd(t) (Fig. 5). In thesecond case there is a double and opposed influence ofthe increase of h: it means higher sublimation rate by anincrease of Km but, at the same time, a lower driving forcedue to the rapid cooling of the surface that decreases watervapor pressure. At short times the first effect prevails, butwith time the second factor has heavier weight and at suf-ficiently long times higher h implies lower water loss.

00 6000 7000 8000 9000 10000

e (s)

h=10h=25h=75h=100h=150

temperature, Ts = 243 K. The other parameters are given in Table 1.

5000 6000 7000 8000 9000 10000

me (s)

t evolution, Ts = 243 K. The other parameters are given in Table 1.

M.C. Olguın et al. / International Journal of Heat and Mass Transfer 51 (2008) 4379–4391 4387

Another characteristic of the influence of h is that ahigher h means lower sublimation front temperature To.The effect of the increase in h is more remarkable at lowvalues of this parameter, and tends to an asymptote forhigh-h values (Fig. 6).

When we consider Ts as parameter, we obtain that sf(t)is a monotonely decreasing and sd(t) is a monotonelyincreasing function of Ts (Figs. 7 and 8, respectively). Inthe case of sd(t), initially Ts has no traceable influence,

0.0E+00

5.0E-03

1.0E-02

1.5E-02

2.0E-02

2.5E-02

0 1000 2000 3000 4000

Ti

Fre

ezin

g fr

ont (

m)

Ts=233

Ts=243

Ts=253

Ts=263

Fig. 7. Influence of the surrounding temperature in the evolution of the free

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

3.5E-04

0 1000 2000 3000 4000

Ti

Sub

limat

ion

fron

t (m

)

Ts=233

Ts=243

Ts=253

Ts=263

Fig. 8. Influence of the surrounding temperature in the evolution of the sublim

but at higher times there is a clear influence of external tem-perature on its value.

Besides, data given in Figs. 4–8 clearly show that forequal system characteristics and freezing conditions thevalue of sf(t) is – at least – two orders of magnitude higherthan that of sd(t) and that for any real system and normalfreezing times ice sublimation is constrained to a very thinsurface layer. For the examples given in this work the highestvalues of sd(t)attained after 10,000 s were less than 0.3 mm.

5000 6000 7000 8000 9000 10000

me (s)

zing front, h = 10 W/(m2K). The other parameters are given in Table 1.

5000 6000 7000 8000 9000 10000

me (s)

ation front, h = 10 W/(m2K). The other parameters are given in Table 1.

4388 M.C. Olguın et al. / International Journal of Heat and Mass Transfer 51 (2008) 4379–4391

Fig. 9 presents the influence of Ts on To. It shows that alower external temperature always means lower sublima-tion temperature and that its influence increases with time(higher separation between sublimation temperatures fordifferent external temperatures).

Moreover, in Fig. 10, we see the change in the tempera-ture profiles in the dehydrated and frozen zones with timefor a given value of parameters h and Ts. From the figure asharp profile is seen in the dehydrated zone and a moregradual – but always linear – variation in the frozen zone.

265

266

267

268

269

270

271

272

273

274

0 1000 2000 3000 4000 50

Tim

Sub

limat

ion

fron

t tem

pera

ture

(K

)

Ts=233

Ts=243

Ts=253

Ts=263

Fig. 9. Influence of the surrounding temperature in the sublimation

time

sf

time

sf

time

sf

240

245

250

255

260

265

270

275

0 0.01 0.02

Pos

Tem

pera

ture

(K

)

time

sf

Fig. 10. Temperature profiles, in dehydrated and frozen zones, and position oh = 75 W/(m2K), Ts = 243 K. The other parameters are given in Table 1.

Both profiles are each time less steep as moving boundariesadvance towards the interior of the system.

To get a better scope of temperature evolution, Fig. 11presents the influence of h and Ts on the predicted temper-ature profiles at a given time (1000 s) for the dehydratedzone. Plots have the expected trends with lower tempera-tures for higher h and lower Ts, and can be matched toget a complete temperature profile of the system. Plotshows the width and range of the dehydrated zone at thetime of calculation (1000 s) and is equivalent to those of

00 6000 7000 8000 9000 10000

e (s)

front temperature. The other parameters are given in Table 1.

2 4 5

2 5 0

2 5 5

2 6 0

2 6 5

2 7 0

2 7 5

0 0 .0 0 0 1 0 0 0

sd

2 4 5

2 5 0

2 5 5

2 6 0

6 5

2 7 0

2 7 5

0 0 .0 0 0 1 0 0 0 0

sd

0.03 0.04 0.05

ition (m)

2 4 5

2 5 0

2 5 5

2 0

6 5

2 7 0

2 7 5

0 0 . 0 0 1 0 0 0

sd

245

250

255

260

265

270

275

0 0.0003

sd

0.00020.0001

f freezing and sublimation fronts at different times (from 100 to 10,000 s),

0

1

2

x 10-4

240

250

260

240

250

260

270

280

Position (m)Ts (K)

Td (

K)

h = 10

h = 25

h = 75h = 100

h = 150

Fig. 11. Temperature profiles in the dehydrated zone at time = 1000 s, as a function of h and Ts. The other parameters are given in Table 1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035

Position (m)

Vap

our

Con

cent

ratio

n (k

g/m

3 )

time

Fig. 12. Vapour concentration profiles at different times (from 100 to 10,000 s), in dehydrated zone, h = 75 W/(m2K), Ts = 243 K. The other parametersare given in Table 1.

M.C. Olguın et al. / International Journal of Heat and Mass Transfer 51 (2008) 4379–4391 4389

previous figures. A similar plot for the frozen zone can bedeveloped using the present model.

Besides the right limit of the plot in Fig. 11 give thevalue of the sublimation temperature at that time, h andTs.

Related water vapor concentration, we can see in Fig. 12the linear profiles of vapour concentration in the dehy-drated zone, for different times (from 100 to 10,000 s), atspecified h and Ts values. As expected, concentration valuesare lower at higher times due to the related diminution ofT0(t) that, in turn, lowers Cva(sd(t), t).

These results are complemented by those of Fig. 13which shows the influence of h and Ts on the predicted con-centration profiles at a given time (1000 s). The depth andrange of the dehydrated zone at the time of calculation areobviously equal to those of Fig. 11.

4. Conclusions

The developed quasi-analytical model, with few simpli-fying assumptions, enables to easily predict the influenceof material characteristics and freezing conditions on the

0

1

2

x 10-4

240

250

260

0.2

0.4

0.6

0.8

1

1.2

1.4

Position (m)Temperature (K)

Vap

our

conc

entr

atio

n (k

g/m

3 )

h = 10

h = 25

h = 75h = 100

h = 150

Fig. 13. Vapour concentration profiles in the dehydrated zone at time = 1000 s, as a function of h and Ts. The other parameters are given in Table 1.

4390 M.C. Olguın et al. / International Journal of Heat and Mass Transfer 51 (2008) 4379–4391

evolution of both freezing and sublimation fronts, temper-ature in the frozen and dehydrated zones, temperature atthe sublimation front and vapour concentration in thedehydrated layer.

It can be used to optimize the freezing conditions of cer-tain types of materials as well as a base for the developmentof more detailed models that include a freezing range orinitial temperatures different from that of material freezing.

Acknowledgements

This paper has been partially sponsored by CONICETand project ANPCYT-PAV 120–00005 from Argentina.We also thank the anonymous two referees whose detailedcomments helped us to improve the content of the paper.

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