Part two: One$-$dimensional convection diffusion problem

Assignment $($please read the problem description before doing the assignment$)$

Develop a finite difference method $($FDM$)$ formula for solving the equation. The procedure

of deriving the FDM formula must be included in the report.

Develop a MATLAB program and use this program to do the simulation and answer the

following questions. Divide the soil depth into $200$ cells when solving the problem

numerically. You must attach the MATLAB program in your report.

If $=0\mathrm{.}0002\frac{m^2}{s},$ find out when the temperature at the depth of $1\mathrm{.}5m$ reaches $25C.$

If the temperature at the depth of $1\mathrm{.}5m$ reaches $25C$ is defined as $T_{1\mathrm{.}5},$ do the simulations for

$$ in the range of $0\mathrm{.}0002\frac{m^2}{s}$ to $0\mathrm{.}001\frac{m^2}{s}$ with an interval of $0\mathrm{.}0001\frac{m^2}{s}$ and discuss how $$

on $T_{1\mathrm{.}5}$ and why.

Problem description

Note: In the paragraph below, the number $j$ stands for the last digit of your student ID$.$ For example,

if your student ID is $2345673,10+j=13.$

The water on the ground flows into the soil through the pore space of the soil $($the space between soil

particles$).$ The flow velocity is very small of $0\mathrm{.}002(1+j)\frac{m}{s}$ and it does not change along the

vertical direction. Initially, the temperature is $(10+j)C$ in the whole soil volume $($including soil and

the water in the soil$).$ The temperature of the water on top of the ground level is $30C$ and remains

constant. The warm water moves into the soil and increases the temperature in the soil.

A coordinate $x$ is defined and its origin is on the ground level and pointing downwards. The

governing equation is the convection diffusion equation

$\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}=-u\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{x}}+\frac{de{l}^{2}T}{del{x}^2}$

where

$T$ is the temperature in the soil

$t$ is the time

$u$ is the water flow velocity

$$ is the heat diffusion coefficient $($the influence of the soil has been considered$).$

Boundary condition is $T=30C$ at the top boundary and $\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{x}}=0$ at the bottom boundary $($at $x=2m).$