Modifications necessary in my numerical solution of the heat conduction equation

Kummi · November 23, 2021, 02:24

Hello Foamers,
Here is my explanation about the numerical solution followed by the query.
Consider a region of interest to be divided into n elements of width $\Delta x$ . The midpoints of these elements are $P_{1},....,P_{n}$ . The temperature at these points are $T_{1},....,T_{n}$ at time t. At increment of time t + $\Delta t$ , the temperature are notified as $\theta_{1},....,\theta_{n}$ .
The 1D equation, unit $W/m^{3}$

Quote:

${\rho c\frac{\delta T}{\delta t}=\frac{\delta}{\delta x}\left ( K\frac{\delta T}{\delta x} \right )+Q\ }$

The approximation in time for above equation,

$\left [\rho c\frac{\delta T}{\delta t} \right ]_{t+\frac{\Delta t}{2}}=\left [\frac{\delta}{\delta x}\left ( K\frac{\delta T}{\delta x} \right ) \right ]_{_{t+\Delta t}}+Q_{t}$

For i=1 to n-1,

$\rho_{i} c_{i}\frac{(\theta_{i}-T_{i})}{\Delta t}=\left ( K_{U,i}\frac{\left ( \theta _{i+1}-\theta _{i} \right )}{\Delta x} - K_{L,i}\frac{\left ( \theta _{i}-\theta _{i-1} \right )}{\Delta x}\right )/\Delta x+Q_{i}$

where $\rho _{i}$ , $c_{i}$ and $Q_{i}$ are the density (kg/m3), specific heat (J/kg K) and heat release rate (W/m3) of the $i^{th}$ element at time t, and $K_{U,i}$ , $K_{L,i}$ are the average conductivities (W/m K) from $P_{i}$ to $P_{i+1}$ and $P_{i+1}$ to $P_{i}$ respectively at time t.
After rearranging, the final form of the equation (for T>=500deg) is,

Quote:

$l_{i}\theta _{i-1}+g_{i}\theta _{i}+u_{i}\theta _{i+1}=z_{i}$
where,
$l_{i} = -D_{L,i}/D_{0}$ ,
$g_{i} = 1 + (D_{U,i}+D_{L,i})/D_{0}$ ,
$u_{i} = -D_{U,i}/D_{0}$ ,
$z_{i} = T_{i}+\Delta tQ_{i}/\rho _{i}c_{i}$ --> (1)

where, $D_{U,i}$ , $D_{L,i}$ and $D_{0}$ are diffusivities defined by,
$D_{U,i} = K_{U,i}/\rho _{i}c_{i}$
$D_{L,i} = K_{L,i}/\rho _{i}c_{i}$
$D_{0}=\Delta x^{2}/\Delta t$
The above EQU.(1) initializes the temperature distribution for dry coal pyrolysis phenomenon.
In case of wet coal pyrolysis, moisture content should be included with the coal. And thereby, I need to modify the above linear equation (when T=100) slightly as,

Quote:

$l_{i} =0$
$g_{i} = 1$
$u_{i} = 0$
$z_{i} = 100$ --> (2)

The modification implies the approximation that the moisture evaporation occurs at T = 100deg, where there is no effect of diffusion ( $l_{i}$ = $u_{i}$ = 0).
In OpenFOAM, I have made my customized solver for dry coal pyrolysis based on EQU.(1) with default settings.
For wet coal pyrolysis, I don't know where and how to make changes for implementing EQU.(2).
Can someone share ideas here.
Thank you

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