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Model of Heat and Mass Transfer with Moving Boundary during
Roasting of Meat in Convection-Oven
A.H. Feyissa* 1, 2, J. Adler-Nissen 1 and K.V. Gernaey 2
1 National Food Institute, Food production Engineering, Technical University of Denmark
2 Department of Chemical and Biochemical Engineering, Technical University of Denmark
*Corresponding author: Søltofts Plads, 2800, Kgs. Lyngby, DK (e-mail: [email protected])
Abstract: A 2D mathematical model of coupled heat and mass transfer describing oven roasting of meat was formulated from first principles. The current formulation of model equations incorporates the effect of shrinkage phenomena and water holding capacity. The model equations are based on conservation of mass and energy. The pressure driven transport of water in meat is expressed using Darcy’s equation. The arbitrary Lagrangian–Eulerian ( ALE) method was implemented to capture the moving boundary (product-air interface) during the roasting process. The model equations were solved using the Finite Element Method (Multiphysics®^ version 3.5). The state variables (temperature and water content) were predicted. The effect shrinkage on both predictions was evaluated.
Keywords: Coupled heat and mass transfer; Evaporation; Moving boundary; Multiphyics; Shrinkage.
1. Introduction
Roasting in a convection oven is a common way of frying whole meat in households, in professional kitchens and in the ready-meal industry. Mass and heat transfer play an important role in the roasting process. It is essential that their interaction and mechanisms are well understood to allow for better control and optimisation of the roasting process. The effect of shrinkage on meat roasting is often neglected due to the complexity of the process [1]-[4]. However, it is necessary to incorporate such effects into a heat and mass transfer model of meat roasting, because shrinkage is considerable (7-19 % on a area basis [5], and 11- 20.3 % on diameter basis [6]) and plays a key role in the water transport during the roasting process) [7]. Several researchers have formulated different
hypotheses to model mass transfer during roasting, mostly from the perspective of diffusion [1]-[3] while disagreements are often seen with regard to other types of water transport mechanisms [8]-[10]. Purely diffusion based models do not adequately describe the moisture transport phenomena during meat cooking because the effects of water binding capacity and shrinkage phenomena are not considered. These are, however, main driving mechanisms for the exudation of water during the cooking or roasting of meat, and some of the early studies on this topic agree with this fact [5],[8]-[10]. Roasting of meat causes the muscle protein to denature, resulting in a decrease in water holding capacity and leading to shrinkage of the protein network. Shrinkage of the network ultimately induces a pressure gradient inside meat muscle. The excess pressure induces a transport of water inside the meat [11], and in the end leads to water loss from the meat. Most of the published work on the modelling of mass and heat transfer during meat roasting does not at all consider shrinkage, and thus the governing model equations were typically solved using a fixed boundary, where the evaporation interface and the material boundary remain the same for the entire roasting period [1]-[3],[11]. Usually, the reason for making such assumptions is that model equations become considerably simpler and thus easier to solve. However, the model based on such fixed boundary assumptions may not be valid for meat that is heated above the denaturation temperature, where the meat shrinks considerably, loses water and changes its dimensions. When temperatures exceed the denaturation temperature, shrinkage phenomena should therefore be taken into account in the heat and mass transfer model, in order to successfully describe heat and water transport inside the meat product. Therefore the objective of this work is to develop a model of
Excerpt from the Proceedings of the COMSOL Conference 2009 Milan
heat and mass transfer by taking into account the shrinkage effect (moving boundary and pressure driven transport) and ultimately to describe and predict heat and mass transfer processes for meat roasting in a convection oven.
Nomenclature C Moisture content (wet basis) (kg /kg)
Greek letters
Ceq Water holding capacity at equilibrium (kg /kg)
β Shrinkage coefficient
c (^) p Specific heat (J/(kg. oC) ρ Density (kg/m^3 ) D Diffusion coefficient (m^2 /s)
μw Viscosity (kg/(m.s)) E Elastic modulus(N/m^2 ) (^) ∇ Gradient(1/m)
f Fraction of energy used for evaporation (J/kg) H Latent heat of vaporization (J/kg)
Subscripts
h Heat^ transfer coefficient (W/(m2 oC)
av Average
K Permeability (m^2 ) eq Equilibrium
k Thermal^ conductivity (W/(m. oC))
c Carbohydrate
m Mass (kg) d Solid P Pressure (Pa) evp evaporation q Heat flux (W/m^2 f Fat T Temperature (oC) i Component t Time (s) m Meat
Tσ Sigmoidal temperature(oC)
p Protein
R Radius (m) w Water y (^) i Mass fraction of component i (kg/kg)
0 Initial value
Z Length (m) oven Oven
V Volume (m^3 ) s Surface
v Interface velocity (m/s) r Radial direction u velocity of water (m/s) z Length direction
2. Mathematical Model of Heat and Mass
Transfer
2.1 Process Descriptions and Problem Formulation
The product (meat) is heated in a convection oven by circulating hot air at 175 oC. Heat is supplied to the product surface by convective
heat transfer. The heat is transferred from the surface the product to the center of the product mainly by conduction. Meanwhile, moisture is transported within the product via convection and diffusion processes, and moves from the inside of the product to its surface. With increase in temperature, muscle protein denatures, leading to a decrease in its water holding capacity and shrinkage of the protein network. The shrinkage of network ultimately induces a pressure gradient inside the meat muscle and excess water is expelled to the surface by convection phenomena. Simultaneously, liquid water is evaporated at the product surface and diffuses to the surrounding fluid (hot air). As the meat sample shrinks the interface or the surface at which water is evaporated changes with time. The most important mechanisms occurring during the convection oven roasting process are described in Fig.1, as shown below.
Figure 1: A schematic representation of coupled heat and mass transfer accompanied by shrinkage and evaporation processes
2.2 Assumptions: In this study the following basic assumptions are made to formulate the governing coupled mass and heat transfer equations for a cylindrical body of meat: a) Fat transport is negligible (lean meat is considered having less than 2% fat) b) The crust is thin (this is observed when inspecting a cut through the cooked meat) and does not hinder transport of water to the surface. Evaporation therefore takes place at the surface (moving interface) c) No internal heat generation and no chemical reaction. d) Dissolved matter lost with water can be
meat, with an additional consideration for the effect of pore formation, the following theoretical expressions are formulated. The volume of a cylindrical meat sample at any given time is expressed in terms of the initial volume ( Vo ) and volume of water lost ( Vw,l ) as V = V 0 − β Vw , l (13)
The coefficient β is used to describe the effect of pore formation during roasting process. For shrinkage, the value of β varies between 0 and 1. If β is 1, there is no pore formation (i.e. the volume of water removed is equal to the volume deformation) and if β = 0, then there is no shrinkage (i.e the volume water lost is entirely replaced by air and no deformation occurs). The fraction (1-β) is the fraction of the volume of water removed from the meat during roasting that is replaced by pore space (filled with air). For minced meat, this value is roughly estimated (for a mass loss of 15%, the corresponding pore formation is 3%) to be around 0.2, and in that case β = 0.8 [6]. For isotropic shrinkage [16], Eq. (13) can be re- written as:
R Z
V
V
Z
V
V
R
V
V
V V
wl wl
wl
2
1 / 3
0
0
2 / 3
0
2 0
0
0
π β β π
β
From (14) the expressions for Z and R are given as: 1 / 3
0
0
V
V
Z Z wl
1 / 3
0
0
V
V
R R wl
Differentiating (15) and (16) with respect to time, the interface velocity components can be obtained as:
z wl^ ( V^ wl ) dt
d V
V V
Z dt
dZ v ,
, 1 3
2 / 3
0 0
0
− ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ = =− −
β β (17)
r wl^ ( V^ wl ) dt
d V
V
V
R
dt
dR v ,
2 / 3
0 0
0
−
⎟⎟ ⎠
Vw,l can be expressed as function of water content as in Eq. (19) : ( ) ( ) ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ −
− −
−
−
av
av w w
w l d C
C C
m X X V C C V 1 1
1 , 0
0 00 0 0 ρ
ρ ρ (^) (19)
and the rate change of V (^) w,l is given by (20) : ( ) dt
dC C
V C dt
dV (^) av w av
wl
2 00 0 1
, 1 1 ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ −
=− − ρ
ρ (20)
at the sample center (boundary r = 0) v (^) r = (^0) (21)
3. Numerical Method
The above model equations (system of partial differential equations) describing coupled heat and mass transfer in convection roasting of meat were solved using the finite element software, COMSOL Multiphyics ®version3.5. A 2D cylindrical geometry of dimensions (radius of 20 mm and length of 54 mm) was built in COMSOL for numerical simulations. The coupled partial differential equations for heat and mass transfer along with the boundary condition were solved using the Chemical Engineering module (transient heat transfer and transient mass transfer) and the moving mesh module (ALE). The incorporation of ALE gives the ability to track the position of the product-air interface. The input parameter values and the algebraic expressions in the model are given in table 1.
4. Result and Discussion
4.1 Temperature and water content distributions In the meat roasting process, temperature and water content distributions are important factors which determine the quality of the product. The water content distribution is influenced by the temperature distribution. Fig 2a and 2b show simulated spatial temperature and moisture distribution, respectively, for 2D cylindrical meat sample at different times of roasting process ( t = 0, 500, 1000, 1500, 2000, 2500, 3000, and 3500 s). Generally, inside the meat sample, the temperature increases with increase in time, whereas water content and dimensions are decrease with increase in time. From that the figure, a change of dimensions - a moving boundary - can be noticed.
a)
b) Figure 2. a) Temperature distribution, and b) water content distribution at ( t = 0, 500, 1000, 1500, 2500, 2000, 3000, and 3500 s)
Fig. 2a, illustrates the progress of the temperature distribution during meat roasting in a convection oven. Initially, there is a sharp increase in surface temperature because of the large temperature difference between hot air (175 oC) and the meat (13oC). At t = 500 s, the surface of the meat is at a much higher temperature than the inside part of the meat sample, and a large temperature gradient is developed in the region close to the surface,(see Fig. 2a and Fig. 3). When the roasting process proceeds, this large temperature gradient shifts gradually from near the surface to inside of the product. Moreover, its magnitude decreases as a function of time, as the heat energy is slowly penetrating into the centre of the product, thereby raising its temperature (Fig. 3). In the final period of this roasting experiment, at time t = 3000 s, the temperature of the meat is almost
uniform. Fig. 2b, illustrates the progress of the water content distribution within the meat product during the roasting process. The water content distribution changes from being uniform (= initial condition) to a non-uniform profile. The increase in temperature (to the denaturation temperature zone) causes the meat to reduce its water holding capacity and induces shrinkage. The reduction of the water holding capacity and the shrinkage of the meat protein network cause the meat to exudate water to the surface, which is lost by evaporation at the surface. As a result, the water content gradient is developed within the meat, as shown by iso-concentration lines at t = 500 s (Fig. 4). A large water concentration gradient is observed near the surface and the gradient gradually shifts towards the interior of the product (Fig. 2b). The water transport depends upon the material properties (permeability and elastic modulus), the diffusivity coefficient and the pressure gradient.
Figure 3 Temperature profile across cylindrical sample (Z = 0)
Figure 4 .Iso-concentration, C (kg/kg) at t = 500 s
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7. Acknowledgement
The Author would like to thank DTU for a Ph.D. grant under the aegis of Food-DTU.
8. Appendix
Table 1: Parameters values, thermophysical properties and other expression. Value or expression Reference y (^) p 0.2 kg/kg [5] Initial mass fraction
y (^) c 0.02 kg/kg y (^) f 0.03 kg/kg y (^) w 0.75 kg/kg ρf 920 kg/m^3 [17] ρp 1320 kg/m^3 [17]
ρc 1600 kg/m^3 [17] ρ w 1000 kg/m^3 [17] k (^) m 0.47 W/(m. oC) [17] c (^) p,w 4170 J/(kg. oC [17] Hevap 2.3 10 6 J/kg h 33.4 (W/(m2. oC) Measured
K^10
-17-10-19 (^) (raw meat) 10 -17m^2
[18]
T0ven 175 °C Set T 0 13 °C Set C 0 0.75 kg /kg [5][7] β 0.8 [6] D = 2_._ 23 e- 5 exp (- 3382_._ 212 /T ) [16] − log μ (^) w = 0. 0072 T + 2. 8658 Using data [19]
m i i
y
[17]
3
cpm = (1.6 y c + 2 y p + 2 y f + 4.2 yw ).
[17]
( n ( D ))
mx T o
E T E
E
E E
( 1 exp
For whole meat, E (^) 0=12 kpa , E (^) mx =83 kpa at T =80 o^ C; E (^) n =0.3, and E (^) D =
Using data [5]
q evp = fh ( T oven − Ts )
