Numerical solution of multi-dimensional unsteady-state conduction

Two-Dimensional Transient Conduction

Control volume for two-dimensional heat conduction

The energy equation for a two-dimensional transient conduction problem with temperature-dependent thermal conductivity and internal heat source can be written as

$\frac{\partial }{{\partial x}}\left( {k\frac{{\partial T}}{{\partial x}}} \right) + \frac{\partial }{{\partial y}}\left( {k\frac{{\partial T}}{{\partial y}}} \right) + S = \rho c\frac{{\partial T}}{{\partial t}} \qquad \qquad(1)$

To discretize eq. (1), integrating eq. (1) with respect to t in the interval of (t,t + t) and over the control volume P (shaded area in Fig. 1) gives (Tao, 2001)

$\begin{array}{l} \int_t^{t + \Delta t} {\int_s^n {\int_w^e {\frac{\partial }{{\partial x}}\left( {k\frac{{\partial T}}{{\partial x}}} \right)dxdydt} } } + \int_t^{t + \Delta t} {\int_e^w {\int_s^n {\frac{\partial }{{\partial y}}\left( {k\frac{{\partial T}}{{\partial y}}} \right)dydxdt} } } \\ + \int_t^{t + \Delta t} {\int_s^n {\int_w^e {({S_C} + {S_P}T)dxdydt} } } = \int_s^n {\int_e^w {\int_t^{t + \Delta t} {\rho c\frac{{\partial T}}{{\partial t}}dt} dxdy} } \\ \end{array}$

Assuming the heat fluxes are uniform on all faces of the control volume and employing a fully-implicit scheme, the above equation becomes

$\begin{array}{l} \left[ {{k_e}\frac{{{T_E} - {T_P}}}{{{{(\delta x)}_e}}} - {k_w}\frac{{{T_P} - {T_W}}}{{{{(\delta x)}_w}}}} \right]\Delta y\Delta t + \left[ {{k_n}\frac{{{T_N} - {T_P}}}{{{{(\delta y)}_n}}} - {k_s}\frac{{{T_P} - {T_S}}}{{{{(\delta y)}_s}}}} \right]\Delta x\Delta t \\ + ({S_C} + {S_P}T)\Delta x\Delta y\Delta t = {(\rho c)_P}({T_P} - T_P^0)\Delta x\Delta y \\ \end{array}$

which can be rearranged to obtain (Tao, 2001)

${a_P}{T_P} = {a_E}{T_E} + {a_W}{T_W} + {a_N}{T_N} + {a_S}{T_S} + b \qquad \qquad(2)$

where

${a_E} = \frac{{{k_e}\Delta y}}{{{{(\delta x)}_e}}},{\rm{ }}{a_W} = \frac{{{k_w}\Delta y}}{{{{(\delta x)}_w}}},{\rm{ }}{a_N} = \frac{{{k_n}\Delta x}}{{{{(\delta y)}_n}}},{\rm{ }}{a_S} = \frac{{{k_s}\Delta x}}{{{{(\delta y)}_s}}} \qquad \qquad(3)$

${a_P} = {a_E} + {a_W} + {a_N} + {a_S} + a_P^0 - {S_P}\Delta x\Delta y \qquad \qquad(4)$

$b = a_P^0T_P^0 + {S_C}\Delta x\Delta y \qquad \qquad(5)$

$a_P^0 = \frac{{{{(\rho c)}_P}\Delta x\Delta y}}{{\Delta t}} \qquad \qquad(6)$

Three-Dimensional Transient Conduction

The discretized equation for a three-dimensional transient conduction problem with temperature-dependent thermal conductivity and an internal heat source can be obtained by integrating the energy equation with respect to t in the interval of (t,t + t) and over the three-dimensional control volume P (formed by considering two additional neighbors at top, T, and bottom, B). The final form of the discretized equation is (Patankar, 1980)

${a_P}{T_P} = {a_E}{T_E} + {a_W}{T_W} + {a_N}{T_N} + {a_S}{T_S} + {a_T}{T_T} + {a_B}{T_B} + b \qquad \qquad(7)$

where

$\begin{array}{l} {a_E} = \frac{{{k_e}\Delta y\Delta z}}{{{{(\delta x)}_e}}},{\rm{ }}{a_W} = \frac{{{k_w}\Delta y\Delta z}}{{{{(\delta x)}_w}}},{\rm{ }}{a_N} = \frac{{{k_n}\Delta x\Delta z}}{{{{(\delta y)}_n}}} \\ {a_S} = \frac{{{k_s}\Delta x\Delta z}}{{{{(\delta y)}_s}}},{\rm{ }}{a_T} = \frac{{{k_t}\Delta x\Delta y}}{{{{(\delta z)}_t}}},{\rm{ }}{a_B} = \frac{{{k_b}\Delta x\Delta y}}{{{{(\delta z)}_b}}} \\ \end{array} \qquad \qquad(8)$
${a_P} = {a_E} + {a_W} + {a_N} + {a_S} + {a_T} + {a_B} + a_P^0 - {S_P}\Delta x\Delta y\Delta z \qquad \qquad(9)$

$b = a_P^0T_P^0 + {S_C}\Delta x\Delta y\Delta z \qquad \qquad(10)$

$a_P^0 = \frac{{{{(\rho c)}_P}\Delta x\Delta y\Delta z}}{{\Delta t}} \qquad \qquad(11)$

References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC.

Tao, W.Q., 2001, Numerical Heat Transfer, 2nd Ed., Xi’an Jiaotong University Press, Xi’an, China (in Chinese).