A mathematical model for meat cooking

Abstract

We present an accurate two-dimensional mathematical model for steak cooking based on Flory–Rehner theory. The model treats meat as a poroelastic medium saturated with fluid. Heat from cooking induces protein matrix deformation and moisture loss, leading to shrinkage. Numerical simulations indicate good agreement with experimental data. Moreover, this work presents a new and computationally non-expensive method to account for shrinkage.

Appendix A: Discretization

1.1 Equations

We define the derivative operators $D_x$, $D_y$, $D_{xx}$, and $D_{yy}$ on a function of two variables $f_{i,j}$ as

$$\begin{aligned} D_{x}f_{i,j}&= \frac{f_{i+1,j}-f_{i,j}}{h^x_{i+1,j}+h^x_{i,j}} + \frac{f_{i,j}-f_{i-1,j}}{h^x_{i,j}+h^x_{i-1,j}}&D_{y}f_{i,j} = \frac{f_{i,j+1}-f_{i,j}}{h^y_{i,j+1}+h^y_{i,j}} + \frac{f_{i,j}-f_{i,j-1}}{h^y_{i,j}+h^y_{i,j-1}} \\ D_{xx}f_{i,j}&= \frac{2}{h^x_{i,j}}\bigg [\frac{f_{i+1,j}-f_{i,j}}{h^x_{i+1,j}+h^x_{i,j}} - \frac{f_{i,j}-f_{i-1,j}}{h^x_{i,j}+h^x_{i-1,j}}\bigg ]&D_{yy}f_{i,j} = \frac{2}{h^y_{i,j}}\bigg [\frac{f_{i,j+1}-f_{i,j}}{h^y_{i,j+1}+h^y_{i,j}} - \frac{f_{i,j}-f_{i,j-1}}{h^y_{i,j}+h^y_{i,j-1}}\bigg ]. \end{aligned}$$

We obtain the following systems of equations:

• 1. Porosity–fluid velocity system of equations Discretizing the continuity equation (4) yields

  $$\begin{aligned} \phi ^{k+1}_{i,j} = \phi ^k_{i,j} + \Delta t\left[ D_x \{(1-\phi _{i,j})w^{(1)}_{i,j}\}+ D_y \{(1-\phi _{i,j})w^{(2)}_{i,j}\}+\frac{D_\mathrm{w} t_0}{l^2} \left( D_{xx} \{\phi \}+D_{yy}\{\phi \} \right) \right] . \end{aligned}$$

• 2. Temperature–porosity–fluid velocity system of equations Next, we discretize the dimensionless swelling, elastic, and mixing pressures in (5) and (6):

  $$\begin{aligned} \pi _{\mathrm{sw}_{i,j}}^k&=\pi _{el_{i,j}}^k+\pi _{mix_{i,j}}^k\\ \pi _{el_{i,j}}^k&=\beta _\mathrm{el} \bigg (1+\frac{\nu }{\alpha } T_{i,j}^k\bigg ) \bigg [ {\phi _{i,j}^{k}}^{1/3}\phi _0^{2/3}-\frac{1}{2}\phi _{i,j}^k\bigg ], \\ \pi _{mix_{i,j}}^k&=\beta _{\mathrm{mix}} \bigg (1+\frac{\nu }{\alpha } T_{i,j}^k\bigg ) \bigg [\ln \left( 1-\phi _{i,j}^k\right) +\phi _{i,j}^k + \chi _{i,j}^k\left( {\phi _{i,j}^{k}}\right) ^2\bigg ]. \end{aligned}$$

  Similarly, we discretize the Flory–Huggins parameters in (7) and (8)

  $$\begin{aligned} \chi (T,\phi )_{i,j}^k&= \chi _{p_{i,j}}^k - \left( \chi _{p_{i,j}}^k-\chi _0\right) \left( 1-\phi _{i,j}^k\right) ^2,\\ \chi _\mathrm{p}(T)_{i,j}^k&= \chi _\mathrm{{pn}}-\frac{\chi _{\mathrm{pd}}-\chi _\mathrm{{pn}}}{1+A\exp \left[ \gamma \left( T_{i,j}^k\left( T_\mathrm{D}-T_0\right) +T_0-T_\mathrm{e}\right) \right] }. \end{aligned}$$

  We discretize the dimensionless dynamic viscosity of the fluid $\mu (T)$:

  $$\begin{aligned} \mu _{i,j}^k = \frac{2.42\times 10^{-5}}{\mu (T_0)}\left( 10^{\frac{247.8}{T_{i,j}^k(T_\mathrm{D}-T_0)+T_0-140}}\right) . \end{aligned}$$

  We discretize the dimensionless momentum equation (9). The x-component gives

  $$\begin{aligned} \left( 1-\phi ^k_{i,j}\right) w^{(1),k}_{i,j} = \frac{-\kappa ^{(11)}}{\mu ^k_{i,j}}D_x\left\{ \pi ^k_{\mathrm{sw}_{i,j}}\right\} , \end{aligned}$$

  while the y-component yields

  $$\begin{aligned} \left( 1-\phi ^k_{i,j}\right) w^{(2),k}_{i,j} = \frac{-\kappa ^{(22)}}{\mu ^k_{i,j}}D_y\left\{ \pi ^k_{\mathrm{sw}_{i,j}}\right\} . \end{aligned}$$

• 3. Fluid velocity–temperature–porosity system of equations Discretizing the left side of the non-dimensional energy balance equation (10) gives

  $$\begin{aligned}&\left( 1-\phi ^k_{i,j}(1-\nu )\right) \left( \frac{T^{k+1}_{i,j}-T^k_{i,j}}{\Delta t}\right) + \left( \alpha +\nu T^k_{i,j}\right) \left( \frac{\phi ^{k+1}_{i,j}-\phi ^k_{i,j}}{\Delta t}\right) \\&\quad \quad \quad \quad + \left( 1-\phi ^k_{i,j}\right) \left[ w^{(1),k}_{i,j}D_x\{T^k_{i,j}\}+w^{(2),k}_{i,j}D_y\{T^k_{i,j}\}\right] \\&\quad \quad \quad \quad -\frac{D_{w}}{D}\bigg (T_{i,j}^k+\frac{\alpha }{\nu }\bigg )\bigg (D_{xx}{\phi _{i,j}^k}+D_{yy}{\phi _{i,j}^k}\bigg ). \end{aligned}$$

  The discretization of the right side of (10) is

  $$\begin{aligned} \nabla \cdot \left( k \nabla T\right)&= \left( \frac{\omega ^2}{\left( \omega (1-\phi )+1\right) ^2}D_x\{\phi _{i,j}\}D_x\{T^k_{i,j}\}+\frac{\omega }{\omega (1-\phi )+1}D_{xx}\{T^k_{i,j}\}\right) \\&\qquad + \Bigg (\left( \omega -1\right) D_y\{\phi _{i,j}\}D_y\{T^k_{i,j}\}+\left( 1-\phi \left( 1-\omega \right) \right) D_{yy}\{T^k_{i,j}\}\Bigg ). \\ \end{aligned}$$

1.2 Discretized boundary conditions

Porosity and energy balance boundary conditions are a coupled nonlinear system. To avoid the computational expense of solving the nonlinear system each time step instead, we solve the full nonlinear system for the first time step only and use the previous time step values to approximate nonlinear and coupled terms. We discretize on a grid of $\theta N$ by N points, initially with uniform spacing.

• 1. The Dirichlet boundary condition for the porosity

  $$\begin{aligned}&\pi _{\mathrm{sw}_{1,j}}^k\left( T_{1,j}^{k-1},\phi _{1,j}^k\right) =\pi _{\mathrm{sw}_{\theta N,j}}^k\left( T_{\theta N,j}^{k-1},\phi _{\theta N,j}^k\right) =\pi _{\mathrm{sw}_{i,1}}^k\left( T_{i,1}^{k-1},\phi _{i,1}^k\right) \\&=\pi _{\mathrm{sw}_{i,N}}^k(T_{i,N}^{k-1},\phi _{i,N}^k)=0. \end{aligned}$$

• 2. Momentum boundary conditions Taking $\pi _\mathrm{sw}=0$ on the boundary, and using the appropriate one-sided derivative on the boundary, we have the following:

  • (a) On the left:

    $$\begin{aligned} \left( 1-\phi ^k_{1,j}\right) w^{(1),k}_{1,j} = \frac{\kappa ^{(11)}}{\mu ^k_{1,j}}\left( \frac{\pi _{\mathrm{sw}_{2,j}}^k}{h^x_{2,j}+h^x_{1,j}}\right) , \qquad \qquad w^{(2),k}_{1,j}=0 \end{aligned}$$

  • (b) On the right:

    $$\begin{aligned} \left( 1-\phi ^k_{\theta N,j}\right) w^{(1),k}_{\theta N,j} = \frac{-\kappa ^{(11)}}{\mu ^k_{\theta N,j}}\left( \frac{\pi _{\mathrm{sw}_{\theta N-1,j}}^k}{h^x_{\theta N-1,j}+h^x_{\theta N,j}}\right) , \quad w^{(2),k}_{\theta N,j}=0 \end{aligned}$$

  • (c) On the top:

    $$\begin{aligned} \left( 1-\phi ^k_{i,1}\right) w^{(2),k}_{i,1} = \frac{\kappa ^{(22)}}{\mu ^k_{i,1}}\left( \frac{\pi _{\mathrm{sw}_{i,2}}^k}{h^y_{i,1}+h^y_{i,2}}\right) , \qquad \quad \qquad w^{(1),k}_{i,1}=0, \end{aligned}$$

  • (d) On the bottom:

    $$\begin{aligned} \left( 1-\phi ^k_{i,N}\right) w^{(2),k}_{i,N} = \frac{-\kappa ^{(22)}}{\mu ^k_{i,N}}\left( \frac{\pi _{\mathrm{sw}_{i,N-1}}^k}{h^y_{i,N}+h^y_{i,N-1}}\right) , \qquad \quad w^{(1),k}_{i,N}=0. \end{aligned}$$

• 3. Temperature boundary conditions$T^{k-1}_{i,j}$ is used as an approximation of $T^k_{i,j}$ in the nonlinear $j_\mathrm{evap}$ term and $T^{k-1}_{i,j}$ is used to approximate to avoid solving the coupled nonlinear system of the energy boundary condition and the porosity boundary condition simultaneously. This approximation is valid for small $\Delta t$.

  • (a) At the left boundary,

    $$\begin{aligned} -\left( \frac{1}{\omega \left( 1-\phi _{1,j}^{k-1}\right) +\phi _{1,j}^{k-1}}\right) \frac{T_{1,j}^k-T_{2,j}^k}{\frac{1}{2}\left( h^{x,k}_{1,j}+h^{x,k}_{2,j}\right) }&= T_{1,j}^k-1+\lambda j_\mathrm{evap}\left( \phi _{1,j}^{k-1},T_{1,j}^{k-1}\right) , \end{aligned}$$

  • (b) At the right boundary,

    $$\begin{aligned}&-\left( \frac{1}{\omega \left( 1-\phi _{\theta N,j}^{k-1}\right) +\phi _{\theta N,j}^{k-1}}\right) \frac{T_{\theta N,j}^k-T_{\theta N -1,j}^k}{\frac{1}{2}\left( h^{x,k}_{\theta N,j}+h^{x,k}_{\theta N-1,j}\right) }\\&\quad = T_{1,j}^k-1+\lambda j_\mathrm{evap}\left( \phi _{\theta N,j}^{k-1},T_{\theta N,j}^{k-1}\right) , \end{aligned}$$

  • (c) At the top boundary,

    $$\begin{aligned} -\left( \frac{1-\phi _{i,1}^{k-1}}{\omega }+\phi _{i,1}^{k-1}\right) \frac{T_{i,1}^k-T_{i,2}^k}{\frac{1}{2} \left( h^{y,k}_{i,1}+h^{y,k}_{i,2}\right) }&= T_{i,1}^k-1+\lambda j_\mathrm{evap}\left( \phi _{i,1}^{k-1},T_{i,1}^{k-1}\right) , \end{aligned}$$

  • (d) At the bottom boundary,

    $$\begin{aligned} -\left( \frac{1-\phi _{i,N}^{k-1}}{\omega }+\phi _{i,N}^{k-1}\right) \frac{T_{i,N}^k-T_{i,N-1}^k}{\frac{1}{2} \left( h^{y,k}_{i,N}+h^{y,k}_{i,N-1}\right) }&= T_{i,N}^k-1+\lambda j_\mathrm{evap}\left( \phi _{i,N}^{k-1},T_{i,N}^{k-1}\right) . \end{aligned}$$

1.3 Pseudo-code

Our code proceeds in the following order:

Require: $\phi ^k$, $T^k$, $w^k$, $h^k$
1. Continuity	$\phi ^{k+1}$ on bulk $\varOmega$$\leftarrow$$\phi ^k$, ${\mathbf {w}}^{k}$, $h^k$
2. Energy balance	$T^{k+1}$ on bulk $\varOmega$$\leftarrow$$\phi ^k$, $T^{k}$, $h^k$, and $\phi ^{k+1}$ on bulk $\varOmega$
3. Shrinkage	$h^{k+1}$ on bulk $\varOmega$$\leftarrow$$\phi ^{k+1}$ on bulk $\varOmega$
4. Coupled nonlinear B.C.s	$T^{k+1}$, $\phi ^{k+1}$ on $\partial \varOmega$$\leftarrow$$T^{k+1}$, $\phi ^{k+1}$, $h^{k+1}$ on bulk $\varOmega$
5. Shrinkage	$h^{k+1}$ on $\partial \varOmega$$\leftarrow$$\phi ^{k+1}$ on $\partial \varOmega$
6. Momentum balance	${\mathbf {w}}^{k+1}$ on $\partial \varOmega$ and bulk $\varOmega$$\leftarrow$$\phi ^{k+1}$, $T^{k+1}$, $h^{k+1}$ on $\partial \varOmega$ and bulk $\varOmega$