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CFD simulation of coupled heat and mass transfer through porous foods during vacuum cooling process
International Journal of Refrigeration 26 (2003) 19–27 www.elsevier.com/locate/ijrefrig
CFD simulation of coupled heat and mass transfer through porous foods during vacuum cooling process Da-Wen Sun*, Zehua Hu FRCFT Group, Department of Agricultural and Food Engineering, University College Dublin, National University of Ireland, Earlsfort Terrace, Dublin 2, Ireland Received 30 May 2001; received in revised form 15 May 2002; accepted 7 June 2002
Abstract A numerical simulation by using a computational fluid dynamics (CFD) code is carried out to predict heat and mass transfer during vacuum cooling of porous foods on the basis of mathematical models of unsteady heat and mass transfer. The simulations allow the simultaneous prediction of temperature distribution, weight loss and moisture content of the meats at low saturation pressure throughout the chilling process. The simulations are also capable of accounting for the effects of the dependent variables such as pressure, temperature, density and water content, thermal shrinkage, and anisotropy of the food. The model is verified by vacuum cooling of cooked meats with cylindrical shape within an experimental vacuum cooler. A data file for pressure history was created from the experimental pressure values, which were applied in the simulations as the boundary condition of the surface temperature. # 2002 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Food; Porous medium; Cooling; Vacuum; Heat transfer; Mass transfer; Simulation; CFD
Produits alimentaires poreux : simulation par dynamique des fluides nume´rise´e des transferts de chaleur et de masse lors du refroidissement sous vide Mots cle´s : Produit alimentaire ; Milieu poreux ; Refroidissement ; Vide ; Transfert de chaleur ; Transfert de masse ; Simulation ; Dynamique des fluides nume´rise´e
1. Introduction Vacuum cooling is an established technique for rapid cooling processing, which has been proven to be one of the most efficient methods available [1–4]. This technology has been traditionally applied in precooling treatment for leafy vegetables such as lettuce [5–9]. In recent years, vacuum cooling technology has attracted much attention and its application has been extended to precooling of mushrooms
* Corresponding author. Tel.: +353-1-716-5528; fax: +3531-475-2119. Website: www.ucd.ie/refrig. E-mail address: [email protected] (D.-W. Sun).
[10], cut flowers [11], sauces [12], meat products [13–16], and fish [17]. With the constant increase in its applications available, vacuum cooling has attracted significant interest in its wider use for cooling process in the food industry since more strict guidelines of cook–chill systems [18] take effect to ensure food safety. However there are still important issues to be resolved, in particular the mechanism of coupled heat and mass transfer during the process. Heat and moisture transfers in wet porous solid foods at low saturation pressure are coupled in a complicated way. In order to model the coupled heat and moisture transfer phenomena during vacuum cooling, sophisticated numerical methods, such as computational fluid dynamics (CFD) is needed [2]. Numerical modelling of
0140-7007/03/$20.00 # 2002 Elsevier Science Ltd and IIR. All rights reserved. PII: S0140-7007(02)00038-5
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Nomenclature A C cp D d F f hm hfg k L M n p Q q R0 s T t V vg W w
Area (m2) Molar concentration (kmol m3) Specific heat (kJ kg1 K-1) Diffusivity (m2 s1) Diameter of pores (m) Mass transfer rate (kg s1) Inner volumetric mass generation rate (kg s1 m3) Surface mass transfer coefficient (kg kPa1 m2 s1) Latent heat of vaporisation of water (kJ kg1) Mass transfer coefficient (kg kPa1 m1 s1) Length (m) Molecular weight (kg kmol1) Normal of surface Partial pressure (kPa) Heat flux (kW) Heat flux per unit area (kW m2) Universal gas constant (=8.314 kJ kmol1 K1) Inner volumetric heat source generation rate (kJ s1 m3) Temperature (K) Time (s) Volume (m3) Specific volume of saturated vapour (m3 kg1) Weight (kg) Weight loss rate per unit weight (kg kg1 s1)
meat chilling has been conducted in the past [19–22]. Hu and Sun [23–27] simulated the air-blast cooling of cooked meats using CFD package. Previously mathematical modelling of vacuum cooling was conducted for liquid and solid foods [28,29]. The theoretical modelling of heat and mass transfer in wet porous media in the presence of evaporation–condensation was also conducted by using the homogenisation method [30]. The aim of current work is to develop validated CFD models for vacuum cooling of porous foods to allow simultaneous prediction of temperature profile and weight loss to provide better understanding of the mechanism of cooling and aid in the improvement of design and operation.
2. Materials and methods 2.1. Mathematical model During vacuum cooling, when the vacuum chamber pressure is reduced to the saturation vapour pressure, the
wl l r # r $
Weight loss percentage (w/w%) Linear thermal expanding coefficient (K1) Thermal conductivity (kW m1 K1) Stefan-Boltzman constant for radiation (=5.67 1011 kW m2 K4) Density (kg m3) Volumetric flux per unit area (m3 s1 m2) Viscosity (kPa s) Emissivity of product for radiation (dimensionless); Moisture content (w/w, dimensionless) The porosity of the meat (dimensionless) The tortuosity factor (dimensionless) Ratio in a direction to the average factor (dimensionless)
Subscripts 0 Initial a Average conv Convection evap Evaporation l Liquid m Meat r Radiation sur Surface v Vapour vc Vacuum chamber w Wall x X axis y Y axis z Z axis
water within the meat evaporates and the latent heat of evaporation is removed from the meat. Vapour/water phase changes i.e., vaporisation–condensation condition can be determined by examining local pressure. Therefore, saturation vapour pressure data are very useful for determining the relationship between boiling point and pressure. In the following analysis, the simultaneous processes of heat and mass transfer are examined based on the following assumptions: heat transfer due to convection is insignificant as little air remaining in the chamber as convection media, gravity and physicochemical interactions between the solid and the fluids are also ignored. Vacuum cooling of cooked meat can be treated as vapour transports through the vapour-filled pores of the porous medium. Since significant pressure differences exist as a result of the evaporation of liquid water, and the partial water vapour pressure is higher for hot region than for cool region, the vapour transport within the porous solid is mainly influenced by the structure of the solid and the pressure difference exists during vacuum cooling. The heat transfer equation is given as
D.-W. Sun, Z. Hu / International Journal of Refrigeration 26 (2003) 19–27
@T @ @T @ @T @ @T ¼ l þ l þ l þ sv cp @t @x @x @y @y @z @z
ð1Þ
ð2Þ
and the boundary conditions are as follows: on the x, y and z symmetries, l
@T ¼0 @n
ð3Þ
on the surfaces, l
@T ¼ qevap;sur þ qr @n
p R0 T
ð4Þ
fv ¼
@2 py " M @2 px @2 pz Dvx 2 þ Dvy 2 þ Dvz 2 R0 T @x @y @z
Qr 4 ¼ "r Tsur Tw4 A
@2 py " MD @2 px @2 p z þ þ fv ¼ R0 T x @x2 y @y2 z @z2
sv ¼ hfg fv
ð10Þ
and the boundary conditions are as follows: on the x, y and z symmetries, ð11Þ
on the surfaces, p ¼ pvc
ð12Þ
ð5Þ
where fv is the inner volumetric vapour generation rate for the control volume. The latent heat of vaporisation of water, hfg, which can be calculated by the following formula, is based on data from literature [31]: hfg ¼ 2500:8 ðT 273:15Þ 2:422449
ð9bÞ
where x, y and z are the direction factors for x, y and z directions of the meat fibre. The initial condition for Eq. (9) is
@p ¼0 @n
In Eq. (1) the inner volumetric evaporation heat source generation rate sv is,
ð9aÞ
or,
p ¼ psat;0 ; T ¼ T0 where qr is heat loss by radiation on the surface of the product, which is calculated based on the Stefan–Boltzmann law: qr ¼
ð8Þ
then the inner volumetric mass generation rate is
The initial condition for Eq. (1) is t ¼ 0; T ¼ T0
C¼
21
ð6Þ
The Fick’s second law of diffusion can be used to describe mass transfer phenomenon [32]: @C @ @C ¼ D ð7aÞ @t @n @n For mass transfer through the porous medium, Eq. (7a) can be rewritten as @C @ @C " @ @C ¼ Deff ¼ D ð7bÞ @t @n @n @n @n Since concentration is proportional to the partial pressure, when considering gas permeation through a solid, concentration can be normally expressed as the equilibrium relationship in terms of partial pressure [33]. Noting that for an ideal gas the concentration units can be converted to partial pressure units as follows:
During vacuum cooling, a temperature gradient exists within the food product and the local partial pressure in the presence of the evaporated liquid water depends on the local temperature. The partial water vapour pressure is higher for a hot region than for cool region, and in this case, the vapour transport within the porous food is mainly as a result of the pressure difference. The saturation vapour pressure of water at any point within the solid (psat) is independent of the volume and depends only upon the local temperature of the meat, which can be approximated using the Antoine equation [33,34] according to the vapour property data [35]: 3990:5 ð13Þ 103 psat exp 23:4795 T þ 233:833 The vapour transport within the meat can be also treated as vapour movement through the porous medium [4], which can be expressed as a hydrodynamic transport of vapour in the pores with inner vapour generation rate [36], fv ¼
"
@ @p @ @p @ @p kvx þ kvy þ kvz @x @x @y @y @z @z ð14aÞ
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and the boundary conditions are as follows: on the x, y and z symmetries,
or, 2 " @ p @2 p @2 p fv ¼ kv þ þ x @x2 y @y2 z @z2
ð14bÞ
where kv is the mass transfer coefficient of vapour in the vapour-filled pores, which is given by kv ¼ v
d2a 32
pM R0 T
ð16Þ
ð17Þ pd2
@$l " @2 $l @2 $l @2 $l ¼ Dl l þ þ @t x @x2 y @y2 z @z2
@$l ¼ hm ðPsur Pv Þ @n
Consideration is also given to the effect of thermal shrinkage in the simulation. Given that the thermal linear expanding coefficient of the sample can be calculated based on published data [16], for cooked meat, is determined as 0.0003 K-1, therefore,
Comparing Eq. (17) with Eq. (9b), it is clear that 32a in Eq. (17) is equivalent to D in Eq. (9b). In Eq. (17), is the porosity for cooked meat, which is 0.05 [16]; is a tortuosity factor, which can vary from about 1.5 to 5.0. A correlation of tortuosity verses the porosity of various unconsolidated porous media gives the following approximate values of for different values of : =0.2, =2.0; =0.4, =1.75; =0.6, =1.65 [37], and in the current case, the porosity is 0.05, is hence estimated to be 4.3. M is the molecular weight (kg kmol1), for water the molecular weight is 18 kg kmol1, da is the average pore diameter, which is determined to be 0.0025 m [4]; R0 is the universal gas constant, 8.314 kJ kmol1 K-1; T is the temperature and is the viscosity of vapour. Similar to the analysis for vapour transport, using moisture content $ (w/w%) in the equation, the inner volumetric liquid water generation in a cell can be calculated as, fl ¼ l
on the surfaces, Dleff l
combining Eqs. (14b), (15) and (16), the inner vapour volumetric mass generation rate (kg m3 s1) in meat can be rewritten as 2 " Mpd2a @ p @2 p @2 p þ þ fv ¼ 32R0 T x @x2 y @y2 z @z2
ð20Þ
ð15Þ
Assuming the vapour as an ideal gas, v ¼
@$ ¼0 @n
L ¼ L0 ð1 þ TÞ
ð21Þ
ð22Þ
Eq. (22) is then incorporated to calculate the mass transfer more accurately by taking into account thermal shrinkage effect. The effect of the fibre directions of meat muscles on heat and moisture transfer is also taken into account in the simulation. As shown in Fig. 1, it is assumed that the fibres and fibre bundles are arranged in a hexagonal configuration, with the fibres and fibre bundles rather closely packed in the meat. Water and vapour in the direction parallel to the meat fibres can move easily straight along pores between the fibres, therefore, water and vapour moving in a direction perpendicular to the meat fibres has to cross or move around the fibres and fibre bundles. If there are two paths through the system in the perpendicular direction, and the straight path of vapour diffusion is 0, with the short path through the hexagonal system being 1.150; the long path is then 1.330.
ð18Þ
or can be described as in the form of effective diffusivity: fl ¼ Dl eff l
@2 $l @2 $l @2 $l þ þ x @x2 y @y2 z @z2
ð18aÞ
where Dl is the diffusivity of liquid water within meat, and Dleff is the effective diffusivity, which is 5.83 1010 m2 s1 at 25 C [38]. The initial condition for water transport is t ¼ 0; $ ¼ $0
ð19Þ
Fig. 1. Schematic illustration of the fibres and fibre bundles in meat.
D.-W. Sun, Z. Hu / International Journal of Refrigeration 26 (2003) 19–27
The mass removed rate per unit weight (mass) can be calculated from the inner volumetric mass generation rate for a control volume as follows: wv ¼
fv m0
ð23Þ
fl m0
ð23aÞ
and, wl ¼
in which m0 is the density of meat, which is 1.05 103 kg m3. Then weight loss percentage (wl) for a control volume after a chilling time t is, wl ¼
W ¼ ðwv þ wl Þ t 100% W0
ð24Þ
The density change after chilling for a period t is, 1 ð1 þ TÞ3 sv Dt 2 2 3 3 þ fl t 0 3 T þ 3 T þ T hfg
m ¼
23
and of 2.0 mm along the x-axis (length) of the ham. The time step was chosen to be 1 minute, which had the same interval as the data acquisition system in the experiment. The initial temperature of the ham was assumed to be uniform at 75 C. The thermal conductivity and specific heat were set at 0.442 Wm1K1 and 3.6 kJ kg1 K1, respectively, according to the published data [40]. The vacuum chamber pressure history during vacuum cooling experiment was recorded and shown in Fig. 2, based on which a data file for the pressure history was created and incorporated into the USER FORTRAN of the CFD simulation as the surface pressure boundary conditions. Starting from prediction of the local saturation pressure distribution within the meat based on the local temperatures, the pressure differences in different directions can then be calculated for each cell. Consequently, the mass transfer rates within the meat caused by the pressure gradient can be predicted, and thus the temperature, weight loss, water content within the meat can be simulated simultaneously throughout the chilling process. 2.3. Experimental verification
ð26Þ
Experiment was carried out in a laboratory vacuum cooler to verify the CFD simulation. The vacuum cooler used in this study has the dimension of 1000 mm (L) 500 mm (W) 500 mm (H). The system consists of two vacuum pumps to provide the vacuum pressure in the cooling chamber. The typical chilling process involves the cooling of the meat in the vacuum cooler from 75 to 4 C. The meat sample used in the study was a cylindrical pork ham in a netting package, with a diameter of 150 mm and a length of 380 mm. Temperatures within the meat and pressures in the vacuum chamber cooler were continuously measured and recorded by a data acquisition system equipped with T-type thermocouples and a pressure sensor. LabVIEW software was used in the data acquisition system to record the temperature
CFD simulations were conducted based on the above mathematical analysis. As the cylindrical ham employed in the study was symmetrical, a three-dimensional model of an eighth of cylindrical shaped meat sample was developed. As the convective heat loss of the meat is insignificant comparing with the evaporative heat loss, in order to simplify the model and save computing time, the ‘meat only’ model was developed to focus on predicting the conduction and mass transfer within the meat. By using FORTRAN subroutine USERGRD [39], the meat geometry was meshed with uniform grids of 1.5 mm along the y-axis (thickness) to the surface,
Fig. 2. Pressure boundary conditions throughout the chilling process.
ð25Þ Approximately, Eq. (25) can be rewritten as 1 sv t þ f t 3 T m ¼ l 0 ð1 þ TÞ3 hfg
ð25aÞ
Therefore, $, the change of water content $ (%) of the cell can be calculated from the change of weight loss values: D$ ¼
D m m
2.2. CFD simulation
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and pressure automatically at a set interval of one minute during the experiment. An electric steam convection oven was used for cooking the meat. During cooking, the steam cooking cycle temperature was pre-set to 82 C, when the surface temperature of the meat reached to 75 C, the steam temperature was then readjusted to 75 C and the temperature was maintained until an uniform temperature was achieved within the meat, so as to enable the experimental temperature conditions to match those in the simulation. Immediately after cooking, the meat was placed in the vacuum cooler, and thermocouples were inserted into the meat at different depths so that the temperatures of the meat could be measured at various locations simultaneously during the cooling process. The cooling process then began by first starting one vacuum pump for 40 min and then two pumps for the remaining cooling period. Temperature distribution obtained from experiment indicates that the core temperature drops to 11.9 C after 73 min cooling, while the weight loss of cooked meat sample was 11.1% after the cooling.
3. Results and discussion
rate gradually drops down again towards the end of chilling process. The predicted temperature distribution within the cylindrical meat after a chilling time of 73 min can be visualised in Fig. 4, which indicates that the temperature of the meat after chilling is higher at the core and decreases from the core towards the surface. Fig. 5 compares the experimental core and surface temperature with the predicted results. The core temperature of the cylindrical ham was cooled from 75 to 11.9 C in 73 min with the predicted temperature at 11.4 C in the same cooling period. This shows good agreement between simulation and experimental data. Fig. 6 shows the comparison of the simulated average weight loss profile with the local weight loss at the centre and that on the surface of the meat sample. The simulated result of 9.84% at chilling time of 73 min, respectively, gives a reasonable agreement with the experimental accumulative weight loss of 11.1%. Meanwhile, the predicted moisture content profiles can be seen in Fig. 6, which indicates that both surface and core moisture content decreases with the chilling time. As an evidence of significant weight loss at the surface after chilling, the moisture content on the surface dropped from 74.5 to 43.47% at chilling time of 73 min.
The chilling rate of cooked meat obtained from simulation is shown in Fig. 3, it is clear to note that the chilling rate is the highest at the beginning of cooling process as the product initial temperature is 75 C. As the product temperature gradually decreases with the chilling time, the local saturated pressure of the meat decreased, thus the pressure difference between the core and surface of the meat is dropping, as a result, the mass transfer rate gradually decreases as the chilling time. However, it can be observed that at the chilling time of 40 min the chilling rate suddenly rises. This is caused by starting the second vacuum pump. Afterwards the chilling Fig. 4. Visualising predicted temperature distribution of the cylindrical ham at 73 min chilling time.
Fig. 3. Predicted chilling rates throughout the vacuum cooling process.
Fig. 5. Comparison of predicted surface and core temperature with experimental results.
D.-W. Sun, Z. Hu / International Journal of Refrigeration 26 (2003) 19–27
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Fig. 6. Comparison of predicted core, surface and average weight loss and moisture content profiles throughout the cooling process.
The image shown in Fig. 7 indicates the weight loss profile of the cylindrical ham after a chilling time of 73 min. It is clearly evident that high weight loss encountered in a thin layer near the surface and at both ends of the ham, however, the weight loss for the other part was under 10%. This has been confirmed by the experimental observation. The accuracy of the simulation is affected by various factors. First of all, for simplicity, heat transfer by convection was ignored in the simulations. As the pressure within the chamber was at low pressure (vacuum), little air and vapour remain in the cooler chamber, which act
Fig. 7. Visualising predicted weight loss profile of the cylindrical ham after 73 min vacuum chilling.
as the convective heat transfer media to remove the heat from the product. However, at the very beginning of cooling process, the pressure began to drop from ambient pressure. Convection between the product and the remaining air was considered high at the beginning of the process, therefore. This resulted in error in heat transfer calculation. It was assumed that the geometry of the meat was a regular cylinder with diameter of 150 mm in the CFD model, however, the diameter of the ham was actually not always a constant along its length instead, the diameter of sample slightly reduced towards its two ends. Therefore, the inaccuracy setting of the geometry may result in errors on the predictions, especially underestimate the weight loss. Further errors may exist due to the accuracy of the physical properties of the meat from the literature. A further source of error may be caused by using the average diameter of pores obtained from the published data to calculate the mass transfer coefficient. Furthermore, the initial temperature of the meat was assumed to be completely uniform in the simulation, however it is very difficult to achieve in practice. After cooking, a couple of minutes were needed to remove the sample from the oven, take measurement of weight and insert the thermocouples before chilling commenced. During this period, natural convection occurred between the sample and air at room temperature hence some cooling of the surface occurred, giving a slightly uneven temperature distribution in the sample. Also deviations of inserting depths of the thermocouples into the meat samples may cause errors on the temperature measurements during the chilling.
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Errors may also exist in the setting of the boundary pressure conditions. Leakage of air into the chamber exists, which is difficult to measure. Since the remained air within the cooler produced partial pressure, which caused error in measuring the surface pressure. Also the experimental pressure data were taken every minute and this reading was treated as an average pressure for the boundary condition for each time step in the simulation. There is another discrepancy as the pressure was not a constant during each time step, especially at the beginning of chilling when the pressure dropped rapidly.
4. Conclusions A CFD model was developed to simulate the coupled heat and mass transfer simultaneously in the presence of water evaporation and moisture transport within porous food (cooked meats) during vacuum cooling process. The predicted results showed good agreements with experimental results in terms of the core temperature and weight loss. The model takes into account the coupling effects of the dependent variables such as pressure, temperature, density and water content, viscosity of vapour, thermal shrinkage and anisotropy of the food so that a comprehensive simulation of coupled heat and moisture transfer at low saturation pressure was conducted to improve the prediction accuracy. It is noted that vapour evaporation and transport within the food dominates the heat and mass transfer during vacuum cooling, while the physical properties of the food and the vapour, and the cooling pressure history throughout the processing are also the determining parameters affecting the chilling rate. To achieve more accurate simulation results, it is suggested to take account of anisotropy, thermal shrinkage, and surface radiation effects into the CFD simulation of coupled heat and mass transfer.
Acknowledgements The authors wish to acknowledge the practical assistance provided by Dr Lijun Wang, John Bryan and Tadhg Brosnan. References [1] Brosnan T, Sun Da-Wen. Precooling technique and applications for horticultural products—a review. International Journal of Refrigeration 2001;24(2):154–70. [2] Sun Da-Wen, editor, Advances in food refrigeration. Surrey (UK): Leatherhead Publishing, LFRA Ltd, 2001. pp. 482. [3] Sun Da-Wen, Wang LJ. Heat transfer characteristics of cooked meats using different cooling methods. International Journal of Refrigeration 2000;23(7):508–16.
[4] Wang LJ, Sun Da-Wen. Modelling vacuum cooling process of cooked meat—part 2: mass and heat transfer of cooked meat under vacuum pressure. International Journal of Refrigeration 2002;25(7):861–2. [5] Shewfelt RL, Phillips RD. Seven principles for better quality of refrigerated fruits and vegetables. In: Refrigeration science and technology proceedings. new developments in refrigeration for food safety and quality, Lexington (KY, USA). 1996, p. 231–6. [6] Sullivan GH, Davenport LR, Julian JW. Precooling: key factor for assuring quality in new fresh market vegetable crops. In: Janick, editor. Progress in new crops. Arlington: ASHS Press; 1996. p. 521–4. [7] Tambuna AF, Morishima H, Kawagoe Y. Measurement of evaporation coefficient of water during vacuum cooling of lettuce. In: Yano Nakamura, editor. Developments in food engineering. UK: Chapman and Hall; 1994. p. 328–30. [8] Varszegi T. Vacuum cooling of vegetables. Hungarian Agricultural Engineering 1994;7:67–8. [9] Turk R, Celik E. The effect of vacuum precooling on the half cooling period and quality characteristics of iceberg lettuce. Acta Horticulture Postharvest 1993;343:321–4. [10] Frost CE, Burton KS, Atkey PT. A fresh look at cooling mushrooms. Mushroom Journal 1989;193:23–9. [11] Sun Da-Wen, Brosnan T. Extension of the vase life of cut daffodil flowers by rapid vacuum cooling. International Journal of Refrigeration 1999;22:472–8. [12] Shaevel ML. Manufacture of frozen prepared meals. In: Malett, editor. Frozen food technology. Glasgow: Blackie Academic and Professional; 1993. p. 281. [13] McDonald K, Sun Da-Wen. Vacuum cooling technology for the food processing industries: a review. Journal of Food Engineering 2000;45(2):55–65. [14] McDonald K, Sun Da-Wen. The formation of pores and their effects in a cooked beef product on the efficiency of vacuum cooling. Journal of Food Engineering 2001;47(3): 173–83. [15] McDonald K, Sun Da-Wen, Kenny T. Comparison of the quality of cooked beef products cooled by vacuum cooling and by conventional cooling. LWT—Food Science and Technology 2000;33:21–9. [16] McDonald K, Sun Da-Wen, Lyng J. Effects of vacuum cooling on the thermophysical properties of a cooked beef product. Journal of Food Engineering 2002;52(2):167–76. [17] Everington DW. Vacuum technology for food processing. Food Technology International Europe 1993:71–4. [18] Anon. Guidelines on cook-chill systems in hospitals and catering premises, 1991. Report No 7. Food Safety Advisory Committee. Dublin (Ireland): Government Publications Sale Office; 1991. [19] Daudin JD, Arnaud G. Simulation en deux dimensions de l’aeraulique de deux tunnels de refrigeration des viandes. International Journal of Refrigeration 1995;18(6):403–12. [20] Maroulis ZB, Kiranoudis CT, Marinos-Kouris D. Heat and mass transfer modelling in air drying of foods. Journal of Food Engineering 1995;26:113–30. [21] Kuitche A, Daudin JD. Modelling of temperature and weight loss kinetics during meat chilling for time-variable conditions using an analytical-based method—I. The model and its sensitivity to certain parameters. Journal of Food Engineering 1996;28:55–84.
D.-W. Sun, Z. Hu / International Journal of Refrigeration 26 (2003) 19–27 [22] Davey LM, Pham QT. Predicting the dynamics product heat load and weight loss during beef chilling using a multi-region finite difference approach. International Journal of Refrigeration 1997;20(7):470–82. [23] Hu Z, Sun Da-Wen. The temperature distribution of cooked meat joints in an air-blast chiller during cooling process: CFD simulation and experimental verification. Paper presented at the 20th International Congress of Refrigeration, Sydney, Australia, 1999. [24] Hu Z, Sun Da-Wen. Modelling of an experimental airblast freezer using CFD code. In: Advance in refrigeration system, food technologies and cold chain. Paris (France): International Institute of Refrigeration; 2000. p. 395–400. [25] Hu Z, Sun Da-Wen. CFD simulation of heat and moisture transfer for predicting cooling rate and weight loss of cooked ham during air-blast chilling process. Journal of Food Engineering 2000;46(3):189–97. [26] Hu Z, Sun Da-Wen. Effect of fluctuation in inlet airflow temperature on CFD simulation of air-blast chilling process. Journal of Food Engineering 2001;48(4):311–6. [27] Hu Z, Sun Da-Wen. Predicting local surface heat transfer coefficients during air-blast chilling by different turbulent k-E models. International Journal of Refrigeration 2001; 24(7):702–17. [28] Houska M, Bartos P, Hoffman P, Sestak J. Vacuum cooling of foods. Paper presented at the 20th International Congress of Refrigeration, Sydney, Australia, 1999. [29] Houska M, Podloucky S, Zitny R, Gree R, Sestak J, Dostal M, Burfoot D. Mathematical model of the vacuum cooling of liquids. Journal of Food Engineering 1996;29: 339–48.
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CFD simulation of coupled heat and mass transfer through porous foods during vacuum cooling process Da-Wen Sun*, Zehua Hu FRCFT Group, Department of Agricultural and Food Engineering, University College Dublin, National University of Ireland, Earlsfort Terrace, Dublin 2, Ireland Received 30 May 2001; received in revised form 15 May 2002; accepted 7 June 2002
Abstract A numerical simulation by using a computational fluid dynamics (CFD) code is carried out to predict heat and mass transfer during vacuum cooling of porous foods on the basis of mathematical models of unsteady heat and mass transfer. The simulations allow the simultaneous prediction of temperature distribution, weight loss and moisture content of the meats at low saturation pressure throughout the chilling process. The simulations are also capable of accounting for the effects of the dependent variables such as pressure, temperature, density and water content, thermal shrinkage, and anisotropy of the food. The model is verified by vacuum cooling of cooked meats with cylindrical shape within an experimental vacuum cooler. A data file for pressure history was created from the experimental pressure values, which were applied in the simulations as the boundary condition of the surface temperature. # 2002 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Food; Porous medium; Cooling; Vacuum; Heat transfer; Mass transfer; Simulation; CFD
Produits alimentaires poreux : simulation par dynamique des fluides nume´rise´e des transferts de chaleur et de masse lors du refroidissement sous vide Mots cle´s : Produit alimentaire ; Milieu poreux ; Refroidissement ; Vide ; Transfert de chaleur ; Transfert de masse ; Simulation ; Dynamique des fluides nume´rise´e
1. Introduction Vacuum cooling is an established technique for rapid cooling processing, which has been proven to be one of the most efficient methods available [1–4]. This technology has been traditionally applied in precooling treatment for leafy vegetables such as lettuce [5–9]. In recent years, vacuum cooling technology has attracted much attention and its application has been extended to precooling of mushrooms
* Corresponding author. Tel.: +353-1-716-5528; fax: +3531-475-2119. Website: www.ucd.ie/refrig. E-mail address: [email protected] (D.-W. Sun).
[10], cut flowers [11], sauces [12], meat products [13–16], and fish [17]. With the constant increase in its applications available, vacuum cooling has attracted significant interest in its wider use for cooling process in the food industry since more strict guidelines of cook–chill systems [18] take effect to ensure food safety. However there are still important issues to be resolved, in particular the mechanism of coupled heat and mass transfer during the process. Heat and moisture transfers in wet porous solid foods at low saturation pressure are coupled in a complicated way. In order to model the coupled heat and moisture transfer phenomena during vacuum cooling, sophisticated numerical methods, such as computational fluid dynamics (CFD) is needed [2]. Numerical modelling of
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Nomenclature A C cp D d F f hm hfg k L M n p Q q R0 s T t V vg W w
Area (m2) Molar concentration (kmol m3) Specific heat (kJ kg1 K-1) Diffusivity (m2 s1) Diameter of pores (m) Mass transfer rate (kg s1) Inner volumetric mass generation rate (kg s1 m3) Surface mass transfer coefficient (kg kPa1 m2 s1) Latent heat of vaporisation of water (kJ kg1) Mass transfer coefficient (kg kPa1 m1 s1) Length (m) Molecular weight (kg kmol1) Normal of surface Partial pressure (kPa) Heat flux (kW) Heat flux per unit area (kW m2) Universal gas constant (=8.314 kJ kmol1 K1) Inner volumetric heat source generation rate (kJ s1 m3) Temperature (K) Time (s) Volume (m3) Specific volume of saturated vapour (m3 kg1) Weight (kg) Weight loss rate per unit weight (kg kg1 s1)
meat chilling has been conducted in the past [19–22]. Hu and Sun [23–27] simulated the air-blast cooling of cooked meats using CFD package. Previously mathematical modelling of vacuum cooling was conducted for liquid and solid foods [28,29]. The theoretical modelling of heat and mass transfer in wet porous media in the presence of evaporation–condensation was also conducted by using the homogenisation method [30]. The aim of current work is to develop validated CFD models for vacuum cooling of porous foods to allow simultaneous prediction of temperature profile and weight loss to provide better understanding of the mechanism of cooling and aid in the improvement of design and operation.
2. Materials and methods 2.1. Mathematical model During vacuum cooling, when the vacuum chamber pressure is reduced to the saturation vapour pressure, the
wl l r # r $
Weight loss percentage (w/w%) Linear thermal expanding coefficient (K1) Thermal conductivity (kW m1 K1) Stefan-Boltzman constant for radiation (=5.67 1011 kW m2 K4) Density (kg m3) Volumetric flux per unit area (m3 s1 m2) Viscosity (kPa s) Emissivity of product for radiation (dimensionless); Moisture content (w/w, dimensionless) The porosity of the meat (dimensionless) The tortuosity factor (dimensionless) Ratio in a direction to the average factor (dimensionless)
Subscripts 0 Initial a Average conv Convection evap Evaporation l Liquid m Meat r Radiation sur Surface v Vapour vc Vacuum chamber w Wall x X axis y Y axis z Z axis
water within the meat evaporates and the latent heat of evaporation is removed from the meat. Vapour/water phase changes i.e., vaporisation–condensation condition can be determined by examining local pressure. Therefore, saturation vapour pressure data are very useful for determining the relationship between boiling point and pressure. In the following analysis, the simultaneous processes of heat and mass transfer are examined based on the following assumptions: heat transfer due to convection is insignificant as little air remaining in the chamber as convection media, gravity and physicochemical interactions between the solid and the fluids are also ignored. Vacuum cooling of cooked meat can be treated as vapour transports through the vapour-filled pores of the porous medium. Since significant pressure differences exist as a result of the evaporation of liquid water, and the partial water vapour pressure is higher for hot region than for cool region, the vapour transport within the porous solid is mainly influenced by the structure of the solid and the pressure difference exists during vacuum cooling. The heat transfer equation is given as
D.-W. Sun, Z. Hu / International Journal of Refrigeration 26 (2003) 19–27
@T @ @T @ @T @ @T ¼ l þ l þ l þ sv cp @t @x @x @y @y @z @z
ð1Þ
ð2Þ
and the boundary conditions are as follows: on the x, y and z symmetries, l
@T ¼0 @n
ð3Þ
on the surfaces, l
@T ¼ qevap;sur þ qr @n
p R0 T
ð4Þ
fv ¼
@2 py " M @2 px @2 pz Dvx 2 þ Dvy 2 þ Dvz 2 R0 T @x @y @z
Qr 4 ¼ "r Tsur Tw4 A
@2 py " MD @2 px @2 p z þ þ fv ¼ R0 T x @x2 y @y2 z @z2
sv ¼ hfg fv
ð10Þ
and the boundary conditions are as follows: on the x, y and z symmetries, ð11Þ
on the surfaces, p ¼ pvc
ð12Þ
ð5Þ
where fv is the inner volumetric vapour generation rate for the control volume. The latent heat of vaporisation of water, hfg, which can be calculated by the following formula, is based on data from literature [31]: hfg ¼ 2500:8 ðT 273:15Þ 2:422449
ð9bÞ
where x, y and z are the direction factors for x, y and z directions of the meat fibre. The initial condition for Eq. (9) is
@p ¼0 @n
In Eq. (1) the inner volumetric evaporation heat source generation rate sv is,
ð9aÞ
or,
p ¼ psat;0 ; T ¼ T0 where qr is heat loss by radiation on the surface of the product, which is calculated based on the Stefan–Boltzmann law: qr ¼
ð8Þ
then the inner volumetric mass generation rate is
The initial condition for Eq. (1) is t ¼ 0; T ¼ T0
C¼
21
ð6Þ
The Fick’s second law of diffusion can be used to describe mass transfer phenomenon [32]: @C @ @C ¼ D ð7aÞ @t @n @n For mass transfer through the porous medium, Eq. (7a) can be rewritten as @C @ @C " @ @C ¼ Deff ¼ D ð7bÞ @t @n @n @n @n Since concentration is proportional to the partial pressure, when considering gas permeation through a solid, concentration can be normally expressed as the equilibrium relationship in terms of partial pressure [33]. Noting that for an ideal gas the concentration units can be converted to partial pressure units as follows:
During vacuum cooling, a temperature gradient exists within the food product and the local partial pressure in the presence of the evaporated liquid water depends on the local temperature. The partial water vapour pressure is higher for a hot region than for cool region, and in this case, the vapour transport within the porous food is mainly as a result of the pressure difference. The saturation vapour pressure of water at any point within the solid (psat) is independent of the volume and depends only upon the local temperature of the meat, which can be approximated using the Antoine equation [33,34] according to the vapour property data [35]: 3990:5 ð13Þ 103 psat exp 23:4795 T þ 233:833 The vapour transport within the meat can be also treated as vapour movement through the porous medium [4], which can be expressed as a hydrodynamic transport of vapour in the pores with inner vapour generation rate [36], fv ¼
"
@ @p @ @p @ @p kvx þ kvy þ kvz @x @x @y @y @z @z ð14aÞ
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and the boundary conditions are as follows: on the x, y and z symmetries,
or, 2 " @ p @2 p @2 p fv ¼ kv þ þ x @x2 y @y2 z @z2
ð14bÞ
where kv is the mass transfer coefficient of vapour in the vapour-filled pores, which is given by kv ¼ v
d2a 32
pM R0 T
ð16Þ
ð17Þ pd2
@$l " @2 $l @2 $l @2 $l ¼ Dl l þ þ @t x @x2 y @y2 z @z2
@$l ¼ hm ðPsur Pv Þ @n
Consideration is also given to the effect of thermal shrinkage in the simulation. Given that the thermal linear expanding coefficient of the sample can be calculated based on published data [16], for cooked meat, is determined as 0.0003 K-1, therefore,
Comparing Eq. (17) with Eq. (9b), it is clear that 32a in Eq. (17) is equivalent to D in Eq. (9b). In Eq. (17), is the porosity for cooked meat, which is 0.05 [16]; is a tortuosity factor, which can vary from about 1.5 to 5.0. A correlation of tortuosity verses the porosity of various unconsolidated porous media gives the following approximate values of for different values of : =0.2, =2.0; =0.4, =1.75; =0.6, =1.65 [37], and in the current case, the porosity is 0.05, is hence estimated to be 4.3. M is the molecular weight (kg kmol1), for water the molecular weight is 18 kg kmol1, da is the average pore diameter, which is determined to be 0.0025 m [4]; R0 is the universal gas constant, 8.314 kJ kmol1 K-1; T is the temperature and is the viscosity of vapour. Similar to the analysis for vapour transport, using moisture content $ (w/w%) in the equation, the inner volumetric liquid water generation in a cell can be calculated as, fl ¼ l
on the surfaces, Dleff l
combining Eqs. (14b), (15) and (16), the inner vapour volumetric mass generation rate (kg m3 s1) in meat can be rewritten as 2 " Mpd2a @ p @2 p @2 p þ þ fv ¼ 32R0 T x @x2 y @y2 z @z2
ð20Þ
ð15Þ
Assuming the vapour as an ideal gas, v ¼
@$ ¼0 @n
L ¼ L0 ð1 þ TÞ
ð21Þ
ð22Þ
Eq. (22) is then incorporated to calculate the mass transfer more accurately by taking into account thermal shrinkage effect. The effect of the fibre directions of meat muscles on heat and moisture transfer is also taken into account in the simulation. As shown in Fig. 1, it is assumed that the fibres and fibre bundles are arranged in a hexagonal configuration, with the fibres and fibre bundles rather closely packed in the meat. Water and vapour in the direction parallel to the meat fibres can move easily straight along pores between the fibres, therefore, water and vapour moving in a direction perpendicular to the meat fibres has to cross or move around the fibres and fibre bundles. If there are two paths through the system in the perpendicular direction, and the straight path of vapour diffusion is 0, with the short path through the hexagonal system being 1.150; the long path is then 1.330.
ð18Þ
or can be described as in the form of effective diffusivity: fl ¼ Dl eff l
@2 $l @2 $l @2 $l þ þ x @x2 y @y2 z @z2
ð18aÞ
where Dl is the diffusivity of liquid water within meat, and Dleff is the effective diffusivity, which is 5.83 1010 m2 s1 at 25 C [38]. The initial condition for water transport is t ¼ 0; $ ¼ $0
ð19Þ
Fig. 1. Schematic illustration of the fibres and fibre bundles in meat.
D.-W. Sun, Z. Hu / International Journal of Refrigeration 26 (2003) 19–27
The mass removed rate per unit weight (mass) can be calculated from the inner volumetric mass generation rate for a control volume as follows: wv ¼
fv m0
ð23Þ
fl m0
ð23aÞ
and, wl ¼
in which m0 is the density of meat, which is 1.05 103 kg m3. Then weight loss percentage (wl) for a control volume after a chilling time t is, wl ¼
W ¼ ðwv þ wl Þ t 100% W0
ð24Þ
The density change after chilling for a period t is, 1 ð1 þ TÞ3 sv Dt 2 2 3 3 þ fl t 0 3 T þ 3 T þ T hfg
m ¼
23
and of 2.0 mm along the x-axis (length) of the ham. The time step was chosen to be 1 minute, which had the same interval as the data acquisition system in the experiment. The initial temperature of the ham was assumed to be uniform at 75 C. The thermal conductivity and specific heat were set at 0.442 Wm1K1 and 3.6 kJ kg1 K1, respectively, according to the published data [40]. The vacuum chamber pressure history during vacuum cooling experiment was recorded and shown in Fig. 2, based on which a data file for the pressure history was created and incorporated into the USER FORTRAN of the CFD simulation as the surface pressure boundary conditions. Starting from prediction of the local saturation pressure distribution within the meat based on the local temperatures, the pressure differences in different directions can then be calculated for each cell. Consequently, the mass transfer rates within the meat caused by the pressure gradient can be predicted, and thus the temperature, weight loss, water content within the meat can be simulated simultaneously throughout the chilling process. 2.3. Experimental verification
ð26Þ
Experiment was carried out in a laboratory vacuum cooler to verify the CFD simulation. The vacuum cooler used in this study has the dimension of 1000 mm (L) 500 mm (W) 500 mm (H). The system consists of two vacuum pumps to provide the vacuum pressure in the cooling chamber. The typical chilling process involves the cooling of the meat in the vacuum cooler from 75 to 4 C. The meat sample used in the study was a cylindrical pork ham in a netting package, with a diameter of 150 mm and a length of 380 mm. Temperatures within the meat and pressures in the vacuum chamber cooler were continuously measured and recorded by a data acquisition system equipped with T-type thermocouples and a pressure sensor. LabVIEW software was used in the data acquisition system to record the temperature
CFD simulations were conducted based on the above mathematical analysis. As the cylindrical ham employed in the study was symmetrical, a three-dimensional model of an eighth of cylindrical shaped meat sample was developed. As the convective heat loss of the meat is insignificant comparing with the evaporative heat loss, in order to simplify the model and save computing time, the ‘meat only’ model was developed to focus on predicting the conduction and mass transfer within the meat. By using FORTRAN subroutine USERGRD [39], the meat geometry was meshed with uniform grids of 1.5 mm along the y-axis (thickness) to the surface,
Fig. 2. Pressure boundary conditions throughout the chilling process.
ð25Þ Approximately, Eq. (25) can be rewritten as 1 sv t þ f t 3 T m ¼ l 0 ð1 þ TÞ3 hfg
ð25aÞ
Therefore, $, the change of water content $ (%) of the cell can be calculated from the change of weight loss values: D$ ¼
D m m
2.2. CFD simulation
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and pressure automatically at a set interval of one minute during the experiment. An electric steam convection oven was used for cooking the meat. During cooking, the steam cooking cycle temperature was pre-set to 82 C, when the surface temperature of the meat reached to 75 C, the steam temperature was then readjusted to 75 C and the temperature was maintained until an uniform temperature was achieved within the meat, so as to enable the experimental temperature conditions to match those in the simulation. Immediately after cooking, the meat was placed in the vacuum cooler, and thermocouples were inserted into the meat at different depths so that the temperatures of the meat could be measured at various locations simultaneously during the cooling process. The cooling process then began by first starting one vacuum pump for 40 min and then two pumps for the remaining cooling period. Temperature distribution obtained from experiment indicates that the core temperature drops to 11.9 C after 73 min cooling, while the weight loss of cooked meat sample was 11.1% after the cooling.
3. Results and discussion
rate gradually drops down again towards the end of chilling process. The predicted temperature distribution within the cylindrical meat after a chilling time of 73 min can be visualised in Fig. 4, which indicates that the temperature of the meat after chilling is higher at the core and decreases from the core towards the surface. Fig. 5 compares the experimental core and surface temperature with the predicted results. The core temperature of the cylindrical ham was cooled from 75 to 11.9 C in 73 min with the predicted temperature at 11.4 C in the same cooling period. This shows good agreement between simulation and experimental data. Fig. 6 shows the comparison of the simulated average weight loss profile with the local weight loss at the centre and that on the surface of the meat sample. The simulated result of 9.84% at chilling time of 73 min, respectively, gives a reasonable agreement with the experimental accumulative weight loss of 11.1%. Meanwhile, the predicted moisture content profiles can be seen in Fig. 6, which indicates that both surface and core moisture content decreases with the chilling time. As an evidence of significant weight loss at the surface after chilling, the moisture content on the surface dropped from 74.5 to 43.47% at chilling time of 73 min.
The chilling rate of cooked meat obtained from simulation is shown in Fig. 3, it is clear to note that the chilling rate is the highest at the beginning of cooling process as the product initial temperature is 75 C. As the product temperature gradually decreases with the chilling time, the local saturated pressure of the meat decreased, thus the pressure difference between the core and surface of the meat is dropping, as a result, the mass transfer rate gradually decreases as the chilling time. However, it can be observed that at the chilling time of 40 min the chilling rate suddenly rises. This is caused by starting the second vacuum pump. Afterwards the chilling Fig. 4. Visualising predicted temperature distribution of the cylindrical ham at 73 min chilling time.
Fig. 3. Predicted chilling rates throughout the vacuum cooling process.
Fig. 5. Comparison of predicted surface and core temperature with experimental results.
D.-W. Sun, Z. Hu / International Journal of Refrigeration 26 (2003) 19–27
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Fig. 6. Comparison of predicted core, surface and average weight loss and moisture content profiles throughout the cooling process.
The image shown in Fig. 7 indicates the weight loss profile of the cylindrical ham after a chilling time of 73 min. It is clearly evident that high weight loss encountered in a thin layer near the surface and at both ends of the ham, however, the weight loss for the other part was under 10%. This has been confirmed by the experimental observation. The accuracy of the simulation is affected by various factors. First of all, for simplicity, heat transfer by convection was ignored in the simulations. As the pressure within the chamber was at low pressure (vacuum), little air and vapour remain in the cooler chamber, which act
Fig. 7. Visualising predicted weight loss profile of the cylindrical ham after 73 min vacuum chilling.
as the convective heat transfer media to remove the heat from the product. However, at the very beginning of cooling process, the pressure began to drop from ambient pressure. Convection between the product and the remaining air was considered high at the beginning of the process, therefore. This resulted in error in heat transfer calculation. It was assumed that the geometry of the meat was a regular cylinder with diameter of 150 mm in the CFD model, however, the diameter of the ham was actually not always a constant along its length instead, the diameter of sample slightly reduced towards its two ends. Therefore, the inaccuracy setting of the geometry may result in errors on the predictions, especially underestimate the weight loss. Further errors may exist due to the accuracy of the physical properties of the meat from the literature. A further source of error may be caused by using the average diameter of pores obtained from the published data to calculate the mass transfer coefficient. Furthermore, the initial temperature of the meat was assumed to be completely uniform in the simulation, however it is very difficult to achieve in practice. After cooking, a couple of minutes were needed to remove the sample from the oven, take measurement of weight and insert the thermocouples before chilling commenced. During this period, natural convection occurred between the sample and air at room temperature hence some cooling of the surface occurred, giving a slightly uneven temperature distribution in the sample. Also deviations of inserting depths of the thermocouples into the meat samples may cause errors on the temperature measurements during the chilling.
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Errors may also exist in the setting of the boundary pressure conditions. Leakage of air into the chamber exists, which is difficult to measure. Since the remained air within the cooler produced partial pressure, which caused error in measuring the surface pressure. Also the experimental pressure data were taken every minute and this reading was treated as an average pressure for the boundary condition for each time step in the simulation. There is another discrepancy as the pressure was not a constant during each time step, especially at the beginning of chilling when the pressure dropped rapidly.
4. Conclusions A CFD model was developed to simulate the coupled heat and mass transfer simultaneously in the presence of water evaporation and moisture transport within porous food (cooked meats) during vacuum cooling process. The predicted results showed good agreements with experimental results in terms of the core temperature and weight loss. The model takes into account the coupling effects of the dependent variables such as pressure, temperature, density and water content, viscosity of vapour, thermal shrinkage and anisotropy of the food so that a comprehensive simulation of coupled heat and moisture transfer at low saturation pressure was conducted to improve the prediction accuracy. It is noted that vapour evaporation and transport within the food dominates the heat and mass transfer during vacuum cooling, while the physical properties of the food and the vapour, and the cooling pressure history throughout the processing are also the determining parameters affecting the chilling rate. To achieve more accurate simulation results, it is suggested to take account of anisotropy, thermal shrinkage, and surface radiation effects into the CFD simulation of coupled heat and mass transfer.
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