Modelling the Diffusion of Oxygen and Cellular Respiration in Meat

Lasse Sønderskov Hansen, Michael Skytte Petersen, Anders Rømer Gylling & Nathan Hugh Barr

Studenteropgave: Kandidatprojekt

Abstrakt

A better understanding of the oxygen consuming processes in meat can lead to a management of packaging that increases the shelf-life of meat. Previous efforts to model the diffusion of oxygen and cellular respiration in meat have not incorporated a non-linear reaction term. In this report we model the diffusion and consumption of oxygen in meat with the reaction-diffusion equation applying a non-linear reaction term and by: \begin{equation*} \frac{\partial c}{\partial t}= D\frac{\partial^2 c}{\partial x^2}-k_1(1-e^{-k_2c}). \end{equation*} We choose the boundary conditions, $c(0)=c_i$ and $\left.\frac{dc(x)}{dx}\right|_{x_m}=0$, where $x_m$ denotes the distance from the surface to the middle of the meat. By analysing the model in the steady state, we find that the behaviour of the model is governed by the quantity $k_2c_i$, that is the boundary concentration of oxygen at the surface of meat multiplied by a constant governing the non-linear consumption. When $k_2c_i\ll1$, the oxygen profiles are exponential decreasing functions with respect to spatial coordinates. When $k_2c_i\gg1$, the oxygen profiles to the model are constant, with values equal to the boundary concentration, $c_i$. When $k_2c_i=1$, the approximated solution is a second order polynomial found by the perturbation method. We conclude that when the diffusion constant is $D = 2.6 \times 10^{-2} \si{\square\centi\metre\per\hour}$,$k_2c_i=1$, and $k_1<2 \times 10^{-8}\si{\mole\per\cubic\centi\metre\per\hour}$, the perturbation solution is a better solution than a reaction-diffusion equation with a linear reaction term.

UddannelserMatematik, (Bachelor/kandidatuddannelse) Kandidat
SprogEngelsk
Udgivelsesdato21 dec. 2016
Antal sider71
VejledereJesper Schmidt Hansen