- Email: [email protected]
Modelling the hydration of foodstuffs
Simulation Modelling Practice and Theory 13 (2005) 119–128 www.elsevier.com/locate/simpat
Modelling the hydration of foodstuffs A.H. Weerts
a,*
, D.R. Martin b, G. Lian
b,*
, J.R. Melrose
b
a
b
WLjDelft Hydraulics, P.O. Box 177, 2600 MH Delft, The Netherlands Unilever R&D Colworth, Sharnbrook, Bedford MK44 1LQ, Bedfordshire, United Kingdom
Received 13 December 2002; received in revised form 3 August 2004; accepted 1 September 2004
Abstract The rehydration kinetics of dried foodstuffs is of critical importance to their sensory properties and delivery of flavour and functional molecules. Based on the dynamics of capillary flow in partially saturated porous media, a finite element model is developed to predict the infiltration of water into dried food products taking into account temperature effects. The finite element model is based on the mixed form of the mass conservation equation. The constitutive relationships of water retention and hydraulic conductivity are adopted from the fields of hydrology and soil science. The transfer properties of water in the porous medium depend on the moisture content and the microstructure. This is contrast to the constant transfer properties often used in the heat and mass transfer models developed for foods. Rehydration of green tea as a function of temperature has been simulated and results are compared with NMR measurements. There is good agreement. 2004 Elsevier B.V. All rights reserved. Keywords: Rehydration; Porous media; Mass transfer; NMR measurements; Biomaterials
1. Introduction Dried foodstuffs often need to be rehydrated before they are consumed. It is desirable that these foodstuffs hydrate as fast as possible and show adequate structural *
Corresponding authors. E-mail addresses: [email protected] (A.H. Weerts), [email protected] (G. Lian).
1569-190X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2004.09.001
120
A.H. Weerts et al. / Simulation Modelling Practice and Theory 13 (2005) 119–128
and chemical characteristics. Information about water absorption as a function of temperature of those food materials is critically important to their shelf life and product usage. Rehydration of food materials also has an important impact on their nutritional and sensorial properties. A number of studies have been reported to model the hydration kinetics of foodstuffs and different types of models have been used. Two main approaches can be identified. One approach uses the empirical and semi-empirical models like for instance the Peleg and the Weibull equation [18–20,25,30,31]. The other approach employs diffusive models based on FickÕs second law of diffusion [13,26–28,33]. Despite numerous studies using FickÕs law to model liquid water transport in porous foodstuffs, the liquid water movement cannot be simply defined as a diffusion process. Hydration occurs by capillary flow, driven by an energy potential gradient, rather than by diffusion. Some studies using capillary flow approach to model hydration and/or drying of foodstuffs have been reported recently [9,15,17,23,35]. However, the capillary flow approach is still not widely used. The objective of this work is to show the feasibility of modelling the rehydration process of foodstuffs using the capillary flow approach applying constitutive relationships often used in soil science and hydrology. In particular, the effect of temperature is modelled. We first demonstrate that the effect of temperature on hydration can be directly taken into account in the constitutive properties. A finite element model is then proposed. As a model system of porous foodstuff, we use dried green tea leaf material (Sencha). Hydration behavior of green tea at various temperatures is investigated using time domain NMR. Consequently we compare our model predictions with the experimental data of green leaf tea hydration. For a range of temperatures studied, we find good agreement between the model prediction and the experimental data. More details of this work can be found in [37,38]. 2. Theory 2.1. Capillary flow model For fluid flow in a capillary body, we can write the continuity equation as follows [3] r qw J þ qw
oh ¼0 ot
ð1Þ
where qw is the density of water (1000 kg/m3), J is the volumetric mass flux and h is the volumetric moisture (water) content. This equation is a mass balance statement. The first term describes the divergence of the mass flux in the control volume and the second term describes the change of mass in that volume in time. Normally in capillary flow theory the flux of water is considered to be governed by DarcyÕs law: J ¼
kk r ðrP c qw gÞ lw
ð2Þ
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121
where k is the intrinsic permeability (m2) of the porous medium, kr is the relative permeability, and lw (Pa s) is the dynamic viscosity of water, g is the gravity (9.81 m2/s) and Pc is commonly referred to as the capillary pressure, capillary suction or matrix potential. The concepts of volume averages over a representative elementary volume (REV) [39] are assumed to apply. DarcyÕs law provides a relationship between the volumetric flux in Eq. (1) and the pressure in the fluid. In Eq. (2) the relative permeability is introduced. The relative permeability function is assumed to be a non-linear function of saturation. It ranges from 0 when water is not present, to 1 when all pores are filled with water. The relative permeability function and the saturation function will be discussed in the section Constitutive Relationships. The capillary pressure should be thought of as the energy potential of the water and is negative for partially saturated porous media. The flux in DarcyÕs law is proportional to the gradient of the capillary potential. Eq. (1) can be expanded as oh kk r kk r ¼r rP c r q g ð3Þ ot lw lw w to obtain the RichardÕs equation [11]. By combining the capillary potential with some constants we can obtain the pressure head going from Pa to m. Introducing the conductivity, K (m/s) K¼
kk r qw g lw
ð4Þ
we derive the following more convenient form of the RichardÕs equation in partially saturated porous media oh oK ¼ r ðKðhÞrhÞ ot oz
ð5Þ
where h is the pressure head (m), or pressure in equivalent water column. Eq. (5) is called the mixed form [7] of the unsaturated flow equation because of the presence of both h and h. This mixed form has been implemented in various finite difference and finite element models for single and multiphase transport [6,7,34] including solute transport [29] in porous media. It can be also easily extended to include heat transfer [16]. Moreover, its advantages have not been realised in coupled heat and mass transfer in variably saturated porous media where normally the potential form is used [8,14,32]. The potential form suffers from a lack of mass conservation. The mixed form has the advantage of being perfectly mass conservative when implemented numerically. Readers are directed to [7] for an in depth discussion of the mixed versus the potential form and its advantages. 2.2. Constitutive relationships Eq. (5) obtained by combining Eqs. (1) and (2) describes the flow condition of the system. There are several unknowns in this equation. In order to close the system
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constitutive relations that relate the unknowns must be specified. These constitutive relationships can be written in a variety of ways that result in different variables becoming the independent variables for the system. In this section the constitutive relationships will be specified. They are the relative water permeability-saturation and the capillary potential-saturation relations. Heat and mass transfer models developed for foods often assume simple FickÕs diffusion theory applies. As a result, empirically-fitted weak constitutive relationships are often used. Here, the constitutive relationships adopted are physically-based and characteristic to porous media. We first introduce the water retention curves most often used in soil physics and hydrology. The water retention curve describes capillary potential of a porous medium as a function of moisture content and temperature. One of the commonly used equations in soil science is given by [36] h 1 ¼ hs ½1þ j ahjn m
ð6Þ
where h is the pressure head (m), h is the volumetric water content (m3/m3), hs (m3/ m3) is the saturated water content assumed to be equal to the porosity /, a (m1) and n (>1) are two parameters defining the shape of the water activity curve, and m = 1 1/n. This functional form is favoured by many numerical modellers as it is smooth, making it easier to handle numerically. The water retention curve can be related to the water activity curve [4,10] by using the Kelvin equation: h¼
RT lnðAw Þ qw gM w
ð7Þ
where Aw is the water activity, R is the gas constant and Mw is the molecular mass of water. The value of R/Mw is 0.461 MPa/K. The relative permeability function is a function of water content. The function is usually empirically determined. To avoid the necessity of independently measuring a relative permeability-saturation relationship, the function form is often specified in terms of the parameters of the capillary pressure-saturation relationship. For example van Genuchten applied the theory of [22], to Eq. (6) to derive 0:5 h kr ¼ 1 hs
1=m !m !2 h 1 hs
ð8Þ
The viscosity of water depends on temperature. The relationship may be approximately described as follows [32] lw ¼ 661:2ðT 229Þ
1:562
103
where T is the temperature in Kelvin.
ð9Þ
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3. Material and methods 3.1. Leaf tea hydration data Isothermal hydration experiments have been performed on green tea using Time Domain NMR, which is a well established technique for the measurement of rehydration kinetics of porous systems such as dried foods [12]. All NMR measurements were made on a Resonance Instruments (RI) RAMAN spectrometer operating at 23 MHz. Sample temperature was controlled via a standard RI temperature unit, calibrated against an external thermocouple and was found to be stable to ±1 C. To ensure a more homogeneous system during the rehydration experiments, a simple stirring device was used enabling measurements to be made while stirring. The stirring device consists of a central plastic rod with perpendicular cross members that fits inside a standard 18 mm OD (outside diameter) NMR tube. The rod was connected to a general lab stirrer running at 30–50 rpm. The motor was clamped to a frame around the magnet as described elsewhere [1]. This work also showed that the action of shear (stirring) has negligible effect on the acquired NMR signal. Up to 200 mg of dry tea leaf was placed in a clean dry tube within the spectrometer and warmed to the appropriate measurement temperature. Between 2 and 2.5 g of pre-warmed water was then added and data acquisition immediately started. Data was acquired by measuring NMR spin–spin relaxation times (T2) as a function of hydration time. These were measured using the CPMG [5,21] pulse sequence with up to 8192 echoes and an interpulse spacing of 200 ls. Normally two transients were recorded with a recycle delay of 1 s. This recycle delay was found to be sufficient to quantify the fast relaxing component but too short for the slower relaxing component. A final scan was therefore recorded with a recycle delay of 20 s in order to quantify the volume of water added. All the resulting CPMG relaxation decays were fitted to a number of discrete exponential components using the RI software WinFit. It was assumed that the total signal intensity was due solely to the added water, and no account was taken of the initial moisture content of the leaf. It was also assumed that the fastest relaxing component was due to the water within the tea leaf [12]. The intensity of this component was therefore corrected to give the mass of water uptake per gram of dry leaf and plotted as a function of time. The hydration of green tea was measured at different temperatures (288, 298, 308, 318, 328, 338, 348 K). 3.2. Tea leaf model From our study on the hydration of black tea as a function of tea leaf size [38] it became apparent that permeability in the horizontal direction (cut-edges) is much greater than that in the vertical direction (bottom-top). Therefore, the tea leaf is modelled as a two-dimensional block of 0.0002 m · 0.0025 m, representing the average thickness and size of cut fragments of tea leaves. The model tea leaf was then meshed to 3200 triangular finite element mesh with 1681 nodes. The initial moisture content at each node was set to 0.012 m3/m3, about 3% of dry weight. The boundary condition is that the capillary potential for all the nodes at the surface to be zero.
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Fig. 1. Field emission scanning electron microscope (FESEM) picture of frozen dried tea leaf.
Fig. 1 shows of a SEM picture of a frozen dried tea leaf. Computer simulations of hydration have been performed at the same temperatures as the measurements of the hydration (288, 298, 308, 318, 328, 338, 348 K). 3.3. Water activity model parameters From the hydration data, we estimate that the saturated water content of the tea leaf is about 0.8 m3/m3 equivalent to moisture content of 300% (d.b.), assuming a dry bulk density of 290 kg/m3 and matrix inert density of ±1450 kg/m3. Water activity curves of tea were taken from [24]. Using the least square method, the data were fitted to Eq. (6) to obtain the values of a and n. Extrapolation and interpolation between these water retention parameters as a function of temperature yielded the water retention curves at the temperatures of the hydration data. 3.4. Hydraulic conductivity From our study on the hydration of black tea as a function of tea leaf size [38] where we tested the hypothesis that permeability in the horizontal direction (cutedges) is much greater than that in the vertical direction (bottom–top), we know that the hydraulic conductivity in the horizontal direction (from the cut-edges of the leaf) is much greater than that in the vertical direction (top–bottom), about 100–250 times. Therefore, in this study we assumed that the hydraulic conductivity in the horizontal directions is 125 times greater than that in the vertical direction. The hydraulic conductivity at saturation is calibrated on the hydration data obtained at 25 C, the only temperature at which the water activity curve was also measured. Hydraulic conductivity at other temperatures was directly predicted by substituting the viscosity equation (9) into (2).
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4. Results and discussion Fig. 2 shows the measured water activity data compared with the fitted water retention equation proposed by [36] at T = 298 K. The model fits the data well. Results at the other temperatures are similar. Fig. 3 shows a comparison of the modelled and measured rehydration data at T = 298 K. The data at this temperature was used to calibrate the values of the hydraulic conductivity. The modelled hydration curve compares well with the measured data. The calibrated value of the hydraulic conductivity at saturation is 1011 m/s. This value of hydraulic conductivity is comparable to the value obtained by [38] for black tea. Fig. 4 shows the predicted hydration of green tea at T = 288 and T = 308, 318, 328, 338, and 348 K. The measured data are also given in the figure for comparison. The predicted hydration curves agree well with the experimental measurements, suggesting that, by using the physically-based constitutive relationships, the effect of temperature can be directly predicted with good accuracy. The only model parameters that varied with temperature are the viscosity and activity (i.e. surface tension) of water. Their variations with temperature are well documented and are given by simple equations. The experimental data of leaf tea hydration showed moderate variations in the equilibrium moisture content (saturated water content or hs). There is no clear trend in terms of temperature effect. In the model, we assumed that the equilibrium moisture content was constant for all temperatures. This has resulted in some under and overpredictions in the equilibrium moisture uptake at some temperatures. However, the differences are very small (<10%). Experimental studies on other foodstuffs
107 measurements mode1
h, pressurehead (m)
106
105
104
103
102
0
0.1
0.2
0.3
θ, volumetric watercontent (m3/m3) Fig. 2. Measured water activity data [24] and fitted water retention function as a function of water content at T = 298 K.
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hydration (gr.water/gr.drytealeaf)
T=298 K
2
1
measurements model 0
0
200
400
600
800
1000
time (s) Fig. 3. Comparison of the modelled (dots) and measured (circles) rehydration data on green tea at T = 298 K as a function of time. Temperature is indicated in the graph.
3
T=288 K
T=308 K
T=318 K
T=328 K
T=338 K
T=348 K
2
hydration(gr.water/gr.drytealeaf)
1 0 3 2 1 0 4 3 2 measurements model
1 0
0
200
400
600
time (s)
800 1000 0
200
400
600
800 1000
time (s)
Fig. 4. Comparison of the model predicted (dots) and measured (circles) rehydration data on green tea at T = 288 and T = 308–348 K as a function of time.
showed that the equilibrium moisture content varied (decreased) as a function of temperature [2,18,27]. From the current hydration data of leaf tea, the effect of temperature on the equilibrium water uptake is limited.
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5. Conclusions Hydration of foods and biomaterials is greatly influenced by temperature. Presented in this paper is a mixed form of the unsaturated flow equation based on the capillary flow in porous media. Using this approach, the hydration of green tea at a range of temperatures has been successfully modelled. The predicted hydration curves agree well with the experimental data of NMR. The hydraulic conductivity of green tea was estimated to be 1011 m/s at 298 K in the horizontal direction (from the cut-edges of the leaves). The hydraulic conductivity in the vertical direction is 8.0 · 1014 m/s. The mixed form capillary flow model presented in this paper directly predicts the effect of temperature on hydration. Increase in the hydration rate with temperature is due to mainly viscosity and surface tension of water contained in the physicallybased constitutive relationships. The current model can easily be applied to model the hydration and dehydration of foodstuffs. Moreover, this approach can easily be extended to include the coupling with heat and solute transport.
References [1] S. Ablett, A. Darke, D. Martin, Time domain NMR studies under controlled shear conditions, in: P.S. Belton, B.P. Hills, G.A. Webb (Eds.), Advances in Mag. Res. in Food Science, R.S.C., 1999. [2] N. Abu-Ghannam, Modelling textural changes during the hydration process of red beans, J. Food Eng. 38 (1998) 341–352. [3] J. Bear, Hydraulics of Groundwater, McGraw-Hill Book Company, New York, 1979. [4] G.S. Campbell, J.D. Jungbauer, S. Shiozawa, R.D. Hungerford, A one-parameter equation for water sorption isotherms of soils, Soil Sci. 156 (1993) 302–305. [5] H.Y. Carr, E.M. Purcell, Effects of diffusion on free precession in nuclear magnetic resonance experiments, Phys. Rev. 91 (1954) 630–638. [6] M.A. Celia, P. Binning, A mass conservative numerical-solution for 2-phase flow in porous-media with application to unsaturated flow, Water Resour. Res. 28 (1992) 2819–2828. [7] M.A. Celia, E.T. Bouloutas, R.L. Zarba, A general mass-conservative numerical-solution for the unsaturated flow equation, Water Resour. Res. 26 (1990) 1483–1496. [8] P.S. Chen, D.C.T. Pei, A mathematical-model of drying processes, Int. J. Heat Mass Transfer 32 (1989) 297–310. [9] H. Feng, J. Tang, R.P. Cavalieri, O.A. Plumb, Heat and mass transport in microwave drying of porous materials in a spouted bed, AIChE J. 47 (2001) 1499–1512. [10] G.W. Gee, M.D. Campbell, G.S. Campbell, J.H. Campbell, Rapid measurement of low soil-water potentials using a water activity meter, Soil Sci. Soc. Am. J. 56 (1992) 1068–1070. [11] D. Hillel, Fundamentals of Soil Physics, Academic, San Diego, CA, 1980. [12] B. Hills, Magnetic Resonance in Food Science, John Wiley and Sons, New York, 1998. [13] K.H. Hsu, A diffusion-model with a concentration-dependent diffusion-coefficient for describing water-movement in legumes during soaking, J. Food Sci. 48 (1983) 618–622, 643. [14] J. Irudayaraj, Y. Wu, Finite-element analysis of coupled heat, mass, and pressure transfer in porous biomaterials, Numer. Heat Transfer Part A––Applications 26 (1994) 337–350. [15] J. Irudayaraj, Y. Wu, Heat and mass transfer coefficients in drying of starch based food systems, J. Food Sci. 64 (1999) 323–327. [16] C.A. Leech, Water Movement in Unsaturated Concrete: Theory, Experiments, Models University of Queensland, PhD Thesis (2003).
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[17] G. Lian, C.S. Harris, R. Evans, M. Warboys, Coupled heat and moisture transfer during microwave vacuum drying, J. Microwave Power Electromagn. Energy 32 (1997) 34–44. [18] L.T. Lim, J.M. Tang, J.S. He, Moisture sorption characteristics of freeze-dried blueberries, J. Food Sci. 60 (1995) 810–814. [19] M.D. Machado, F.A.R. Oliveira, V. Gekas, R.P. Singh, Kinetics of moisture uptake and solublesolids loss by puffed breakfast cereals immersed in water, Int. J. Food Sci. Technol. 33 (1998) 225–237. [20] M.F. Machado, F.A.R. Oliveira, L.M. Cunha, Effect of milk fat and total solids concentration on the kinetics of moisture uptake by ready-to-eat breakfast cereal, Int. J. Food Sci. Technol. 34 (1999) 47–57. [21] S. Meiboom, D. Gill, Modified spin-echo method for measuring nuclear relaxation times, Rev. Sci. Instr. 69 (1958) 688–691. [22] Y. Mualem, New model for predicting hydraulic conductivity of unsaturated porous-media, Water Resour. Res. 12 (1976) 513–522. [23] H. Ni, A.K. Datta, K.E. Torrance, Moisture transport in intensive microwave heating of biomaterials: a multiphase porous media model, Int. J. Heat Mass Transfer 42 (1999) 1501–1512. [24] P.C. Panchariya, D. Popovic, A.L. Sharma, Modeling of desorption isotherm of black tea, Drying Technol. 19 (2001) 1177–1188. [25] M. Peleg, An empirical-model for the description of moisture sorption curves, J. Food Sci. 53 (1988) 1216–1217, 1219. [26] N. Sanjuan, J.A. Carcel, G. Clemente, A. Mulet, Modelling of the rehydration process of brocolli florets, Eur. Food Res. Technol. 212 (2001) 449–453. [27] N. Sanjuan, S. Simal, A. Mulet, Modelling of brocolli stems rehydration process, J. Food Eng. 42 (1999) 27–31. [28] S. Simal, A. Femenia, P. Llull, C. Rossello, Dehydration of aloe vera: simulation of drying curves and evaluation of functional properties, J. Food Eng. 43 (2000) 109–114. [29] J. Simunek, T. Vogel, M.Th. van Genuchten, The SWMS_2D code for simulating water flow and solute transport in two-dimensional variably saturated media, USSL, ARS, US Dept. of Agric., Riverside, USA, Research Report no. 132, 1994. [30] P.A. Sopade, E.S. Ajisegiri, M.H. Badau, The use of pelegs equation to model water-absorption in some cereal-grains during soaking, J. Food Eng. 15 (1992) 269–283. [31] P.A. Sopade, J.A. Obekpa, Modeling water-absorption in soybean, cowpea and peanuts at 3 temperatures using peleg equation, J. Food Sci. 55 (1990) 1084–1087. [32] H.R. Thomas, S.D. King, Coupled heat and mass-transfer in unsaturated soil––a potential-based solution, Int. J. Numer. Anal. Meth. Geomech. 16 (1992) 757–773. [33] K. Thorvaldsson, H. Janestad, A model for simultaneous heat, water and vapour diffusion, J. Food Eng. 40 (1999) 167–172. [34] A. Tiktak, W. Bouten, Modeling soil–water dynamics in a forested ecosystem. 3. Model description and evaluation of discretization, Hydrol. Processes 6 (1992) 455–465. [35] I.W. Turner, P.G. Jolly, Combined microwave and convective drying of a porous material, Drying Technol. 9 (1991) 1209–1269. [36] M.T. Van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J. 44 (1980) 892–898. [37] A.H. Weerts, G. Lian, D. Martin, Modelling the hydration of food stuffs: temperature effects, AIChE J. 49 (5) (2003) 1334–1339. [38] A.H. Weerts, G. Lian, D. Martin, Modelling of rehydration of porous biomaterials: anisotropy effects, J. Food Sci. 68 (3) (2003) 937–942. [39] S. Whitaker, Simultaneous heat, mass and momentum transport in porous media: a theory of drying, Adv. Heat Transfer 13 (1977) 119–203.
Modelling the hydration of foodstuffs A.H. Weerts
a,*
, D.R. Martin b, G. Lian
b,*
, J.R. Melrose
b
a
b
WLjDelft Hydraulics, P.O. Box 177, 2600 MH Delft, The Netherlands Unilever R&D Colworth, Sharnbrook, Bedford MK44 1LQ, Bedfordshire, United Kingdom
Received 13 December 2002; received in revised form 3 August 2004; accepted 1 September 2004
Abstract The rehydration kinetics of dried foodstuffs is of critical importance to their sensory properties and delivery of flavour and functional molecules. Based on the dynamics of capillary flow in partially saturated porous media, a finite element model is developed to predict the infiltration of water into dried food products taking into account temperature effects. The finite element model is based on the mixed form of the mass conservation equation. The constitutive relationships of water retention and hydraulic conductivity are adopted from the fields of hydrology and soil science. The transfer properties of water in the porous medium depend on the moisture content and the microstructure. This is contrast to the constant transfer properties often used in the heat and mass transfer models developed for foods. Rehydration of green tea as a function of temperature has been simulated and results are compared with NMR measurements. There is good agreement. 2004 Elsevier B.V. All rights reserved. Keywords: Rehydration; Porous media; Mass transfer; NMR measurements; Biomaterials
1. Introduction Dried foodstuffs often need to be rehydrated before they are consumed. It is desirable that these foodstuffs hydrate as fast as possible and show adequate structural *
Corresponding authors. E-mail addresses: [email protected] (A.H. Weerts), [email protected] (G. Lian).
1569-190X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2004.09.001
120
A.H. Weerts et al. / Simulation Modelling Practice and Theory 13 (2005) 119–128
and chemical characteristics. Information about water absorption as a function of temperature of those food materials is critically important to their shelf life and product usage. Rehydration of food materials also has an important impact on their nutritional and sensorial properties. A number of studies have been reported to model the hydration kinetics of foodstuffs and different types of models have been used. Two main approaches can be identified. One approach uses the empirical and semi-empirical models like for instance the Peleg and the Weibull equation [18–20,25,30,31]. The other approach employs diffusive models based on FickÕs second law of diffusion [13,26–28,33]. Despite numerous studies using FickÕs law to model liquid water transport in porous foodstuffs, the liquid water movement cannot be simply defined as a diffusion process. Hydration occurs by capillary flow, driven by an energy potential gradient, rather than by diffusion. Some studies using capillary flow approach to model hydration and/or drying of foodstuffs have been reported recently [9,15,17,23,35]. However, the capillary flow approach is still not widely used. The objective of this work is to show the feasibility of modelling the rehydration process of foodstuffs using the capillary flow approach applying constitutive relationships often used in soil science and hydrology. In particular, the effect of temperature is modelled. We first demonstrate that the effect of temperature on hydration can be directly taken into account in the constitutive properties. A finite element model is then proposed. As a model system of porous foodstuff, we use dried green tea leaf material (Sencha). Hydration behavior of green tea at various temperatures is investigated using time domain NMR. Consequently we compare our model predictions with the experimental data of green leaf tea hydration. For a range of temperatures studied, we find good agreement between the model prediction and the experimental data. More details of this work can be found in [37,38]. 2. Theory 2.1. Capillary flow model For fluid flow in a capillary body, we can write the continuity equation as follows [3] r qw J þ qw
oh ¼0 ot
ð1Þ
where qw is the density of water (1000 kg/m3), J is the volumetric mass flux and h is the volumetric moisture (water) content. This equation is a mass balance statement. The first term describes the divergence of the mass flux in the control volume and the second term describes the change of mass in that volume in time. Normally in capillary flow theory the flux of water is considered to be governed by DarcyÕs law: J ¼
kk r ðrP c qw gÞ lw
ð2Þ
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121
where k is the intrinsic permeability (m2) of the porous medium, kr is the relative permeability, and lw (Pa s) is the dynamic viscosity of water, g is the gravity (9.81 m2/s) and Pc is commonly referred to as the capillary pressure, capillary suction or matrix potential. The concepts of volume averages over a representative elementary volume (REV) [39] are assumed to apply. DarcyÕs law provides a relationship between the volumetric flux in Eq. (1) and the pressure in the fluid. In Eq. (2) the relative permeability is introduced. The relative permeability function is assumed to be a non-linear function of saturation. It ranges from 0 when water is not present, to 1 when all pores are filled with water. The relative permeability function and the saturation function will be discussed in the section Constitutive Relationships. The capillary pressure should be thought of as the energy potential of the water and is negative for partially saturated porous media. The flux in DarcyÕs law is proportional to the gradient of the capillary potential. Eq. (1) can be expanded as oh kk r kk r ¼r rP c r q g ð3Þ ot lw lw w to obtain the RichardÕs equation [11]. By combining the capillary potential with some constants we can obtain the pressure head going from Pa to m. Introducing the conductivity, K (m/s) K¼
kk r qw g lw
ð4Þ
we derive the following more convenient form of the RichardÕs equation in partially saturated porous media oh oK ¼ r ðKðhÞrhÞ ot oz
ð5Þ
where h is the pressure head (m), or pressure in equivalent water column. Eq. (5) is called the mixed form [7] of the unsaturated flow equation because of the presence of both h and h. This mixed form has been implemented in various finite difference and finite element models for single and multiphase transport [6,7,34] including solute transport [29] in porous media. It can be also easily extended to include heat transfer [16]. Moreover, its advantages have not been realised in coupled heat and mass transfer in variably saturated porous media where normally the potential form is used [8,14,32]. The potential form suffers from a lack of mass conservation. The mixed form has the advantage of being perfectly mass conservative when implemented numerically. Readers are directed to [7] for an in depth discussion of the mixed versus the potential form and its advantages. 2.2. Constitutive relationships Eq. (5) obtained by combining Eqs. (1) and (2) describes the flow condition of the system. There are several unknowns in this equation. In order to close the system
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constitutive relations that relate the unknowns must be specified. These constitutive relationships can be written in a variety of ways that result in different variables becoming the independent variables for the system. In this section the constitutive relationships will be specified. They are the relative water permeability-saturation and the capillary potential-saturation relations. Heat and mass transfer models developed for foods often assume simple FickÕs diffusion theory applies. As a result, empirically-fitted weak constitutive relationships are often used. Here, the constitutive relationships adopted are physically-based and characteristic to porous media. We first introduce the water retention curves most often used in soil physics and hydrology. The water retention curve describes capillary potential of a porous medium as a function of moisture content and temperature. One of the commonly used equations in soil science is given by [36] h 1 ¼ hs ½1þ j ahjn m
ð6Þ
where h is the pressure head (m), h is the volumetric water content (m3/m3), hs (m3/ m3) is the saturated water content assumed to be equal to the porosity /, a (m1) and n (>1) are two parameters defining the shape of the water activity curve, and m = 1 1/n. This functional form is favoured by many numerical modellers as it is smooth, making it easier to handle numerically. The water retention curve can be related to the water activity curve [4,10] by using the Kelvin equation: h¼
RT lnðAw Þ qw gM w
ð7Þ
where Aw is the water activity, R is the gas constant and Mw is the molecular mass of water. The value of R/Mw is 0.461 MPa/K. The relative permeability function is a function of water content. The function is usually empirically determined. To avoid the necessity of independently measuring a relative permeability-saturation relationship, the function form is often specified in terms of the parameters of the capillary pressure-saturation relationship. For example van Genuchten applied the theory of [22], to Eq. (6) to derive 0:5 h kr ¼ 1 hs
1=m !m !2 h 1 hs
ð8Þ
The viscosity of water depends on temperature. The relationship may be approximately described as follows [32] lw ¼ 661:2ðT 229Þ
1:562
103
where T is the temperature in Kelvin.
ð9Þ
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3. Material and methods 3.1. Leaf tea hydration data Isothermal hydration experiments have been performed on green tea using Time Domain NMR, which is a well established technique for the measurement of rehydration kinetics of porous systems such as dried foods [12]. All NMR measurements were made on a Resonance Instruments (RI) RAMAN spectrometer operating at 23 MHz. Sample temperature was controlled via a standard RI temperature unit, calibrated against an external thermocouple and was found to be stable to ±1 C. To ensure a more homogeneous system during the rehydration experiments, a simple stirring device was used enabling measurements to be made while stirring. The stirring device consists of a central plastic rod with perpendicular cross members that fits inside a standard 18 mm OD (outside diameter) NMR tube. The rod was connected to a general lab stirrer running at 30–50 rpm. The motor was clamped to a frame around the magnet as described elsewhere [1]. This work also showed that the action of shear (stirring) has negligible effect on the acquired NMR signal. Up to 200 mg of dry tea leaf was placed in a clean dry tube within the spectrometer and warmed to the appropriate measurement temperature. Between 2 and 2.5 g of pre-warmed water was then added and data acquisition immediately started. Data was acquired by measuring NMR spin–spin relaxation times (T2) as a function of hydration time. These were measured using the CPMG [5,21] pulse sequence with up to 8192 echoes and an interpulse spacing of 200 ls. Normally two transients were recorded with a recycle delay of 1 s. This recycle delay was found to be sufficient to quantify the fast relaxing component but too short for the slower relaxing component. A final scan was therefore recorded with a recycle delay of 20 s in order to quantify the volume of water added. All the resulting CPMG relaxation decays were fitted to a number of discrete exponential components using the RI software WinFit. It was assumed that the total signal intensity was due solely to the added water, and no account was taken of the initial moisture content of the leaf. It was also assumed that the fastest relaxing component was due to the water within the tea leaf [12]. The intensity of this component was therefore corrected to give the mass of water uptake per gram of dry leaf and plotted as a function of time. The hydration of green tea was measured at different temperatures (288, 298, 308, 318, 328, 338, 348 K). 3.2. Tea leaf model From our study on the hydration of black tea as a function of tea leaf size [38] it became apparent that permeability in the horizontal direction (cut-edges) is much greater than that in the vertical direction (bottom-top). Therefore, the tea leaf is modelled as a two-dimensional block of 0.0002 m · 0.0025 m, representing the average thickness and size of cut fragments of tea leaves. The model tea leaf was then meshed to 3200 triangular finite element mesh with 1681 nodes. The initial moisture content at each node was set to 0.012 m3/m3, about 3% of dry weight. The boundary condition is that the capillary potential for all the nodes at the surface to be zero.
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Fig. 1. Field emission scanning electron microscope (FESEM) picture of frozen dried tea leaf.
Fig. 1 shows of a SEM picture of a frozen dried tea leaf. Computer simulations of hydration have been performed at the same temperatures as the measurements of the hydration (288, 298, 308, 318, 328, 338, 348 K). 3.3. Water activity model parameters From the hydration data, we estimate that the saturated water content of the tea leaf is about 0.8 m3/m3 equivalent to moisture content of 300% (d.b.), assuming a dry bulk density of 290 kg/m3 and matrix inert density of ±1450 kg/m3. Water activity curves of tea were taken from [24]. Using the least square method, the data were fitted to Eq. (6) to obtain the values of a and n. Extrapolation and interpolation between these water retention parameters as a function of temperature yielded the water retention curves at the temperatures of the hydration data. 3.4. Hydraulic conductivity From our study on the hydration of black tea as a function of tea leaf size [38] where we tested the hypothesis that permeability in the horizontal direction (cutedges) is much greater than that in the vertical direction (bottom–top), we know that the hydraulic conductivity in the horizontal direction (from the cut-edges of the leaf) is much greater than that in the vertical direction (top–bottom), about 100–250 times. Therefore, in this study we assumed that the hydraulic conductivity in the horizontal directions is 125 times greater than that in the vertical direction. The hydraulic conductivity at saturation is calibrated on the hydration data obtained at 25 C, the only temperature at which the water activity curve was also measured. Hydraulic conductivity at other temperatures was directly predicted by substituting the viscosity equation (9) into (2).
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4. Results and discussion Fig. 2 shows the measured water activity data compared with the fitted water retention equation proposed by [36] at T = 298 K. The model fits the data well. Results at the other temperatures are similar. Fig. 3 shows a comparison of the modelled and measured rehydration data at T = 298 K. The data at this temperature was used to calibrate the values of the hydraulic conductivity. The modelled hydration curve compares well with the measured data. The calibrated value of the hydraulic conductivity at saturation is 1011 m/s. This value of hydraulic conductivity is comparable to the value obtained by [38] for black tea. Fig. 4 shows the predicted hydration of green tea at T = 288 and T = 308, 318, 328, 338, and 348 K. The measured data are also given in the figure for comparison. The predicted hydration curves agree well with the experimental measurements, suggesting that, by using the physically-based constitutive relationships, the effect of temperature can be directly predicted with good accuracy. The only model parameters that varied with temperature are the viscosity and activity (i.e. surface tension) of water. Their variations with temperature are well documented and are given by simple equations. The experimental data of leaf tea hydration showed moderate variations in the equilibrium moisture content (saturated water content or hs). There is no clear trend in terms of temperature effect. In the model, we assumed that the equilibrium moisture content was constant for all temperatures. This has resulted in some under and overpredictions in the equilibrium moisture uptake at some temperatures. However, the differences are very small (<10%). Experimental studies on other foodstuffs
107 measurements mode1
h, pressurehead (m)
106
105
104
103
102
0
0.1
0.2
0.3
θ, volumetric watercontent (m3/m3) Fig. 2. Measured water activity data [24] and fitted water retention function as a function of water content at T = 298 K.
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hydration (gr.water/gr.drytealeaf)
T=298 K
2
1
measurements model 0
0
200
400
600
800
1000
time (s) Fig. 3. Comparison of the modelled (dots) and measured (circles) rehydration data on green tea at T = 298 K as a function of time. Temperature is indicated in the graph.
3
T=288 K
T=308 K
T=318 K
T=328 K
T=338 K
T=348 K
2
hydration(gr.water/gr.drytealeaf)
1 0 3 2 1 0 4 3 2 measurements model
1 0
0
200
400
600
time (s)
800 1000 0
200
400
600
800 1000
time (s)
Fig. 4. Comparison of the model predicted (dots) and measured (circles) rehydration data on green tea at T = 288 and T = 308–348 K as a function of time.
showed that the equilibrium moisture content varied (decreased) as a function of temperature [2,18,27]. From the current hydration data of leaf tea, the effect of temperature on the equilibrium water uptake is limited.
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5. Conclusions Hydration of foods and biomaterials is greatly influenced by temperature. Presented in this paper is a mixed form of the unsaturated flow equation based on the capillary flow in porous media. Using this approach, the hydration of green tea at a range of temperatures has been successfully modelled. The predicted hydration curves agree well with the experimental data of NMR. The hydraulic conductivity of green tea was estimated to be 1011 m/s at 298 K in the horizontal direction (from the cut-edges of the leaves). The hydraulic conductivity in the vertical direction is 8.0 · 1014 m/s. The mixed form capillary flow model presented in this paper directly predicts the effect of temperature on hydration. Increase in the hydration rate with temperature is due to mainly viscosity and surface tension of water contained in the physicallybased constitutive relationships. The current model can easily be applied to model the hydration and dehydration of foodstuffs. Moreover, this approach can easily be extended to include the coupling with heat and solute transport.
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