Consider transient heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes \(0,1,2,3\), and 4 with a uniform nodal spacing of \(\Delta x\). The wall is initially at a specified temperature. The temperature at the right boundary (node 4) is specified. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary node 0 for the case of combined convection, radiation, and heat flux at the left boundary with an emissivity of \(\varepsilon\), convection coefficient of \(h\), ambient temperature of \(T_{\infty}\), surrounding temperature of \(T_{\text {surr }}\), and uniform heat flux of \(\dot{q}_{0}\) toward the wall. Also, obtain the finite difference formulation for the total amount of heat transfer at the right boundary for the first 20 time steps.

Step by step solution

01

Write the energy balance equation for Node 0

The energy balance equation for Node 0 can be written as: Energy balance: \(q_{\text{in}} - q_{\text{out}} = \textrm{Accumulated energy} + \textrm{Generated energy}\) For Node 0, the energy balance equation can be written as: \[\dot{q}_{0} + q_{\text{conv}} + q_{\text{rad}} - q_{\text{cond}} = \rho c_{p} V\frac{\Delta T_{i}}{\Delta t} + g\], Where \(i=0\) and \(g\) is the heat generation.

02

Write the expressions for heat transfer and accumulated energy

Express the heat transfer terms as follows: 1. Conduction heat transfer: \[q_{\text{cond}} = k\frac{T_{1} - T_{0}}{\Delta x}\], 2. Convection heat transfer: \[q_{\text{conv}} = h(T_{0} - T_{\infty})\], 3. Radiation heat transfer: \[q_{\text{rad}} = \varepsilon \sigma_{SB}(T_{0}^4 - T_{\text{surr}}^4)\], 4. Accumulated energy in Node 0: \[\textrm{Accumulated energy} = \rho c_{p} V\frac{\Delta T_{i}}{\Delta t}\], with \(i=0\).

03

Substitute heat transfer terms in the energy balance equation

Now, replace the heat transfer terms in the energy balance equation: \[\dot{q}_{0} + h(T_{0} - T_{\infty}) + \varepsilon \sigma_SB(T_0^4 - T_{\text{surr}}^4) - k\frac{T_{1} - T_{0}}{\Delta x} = \rho c_p V\frac{\Delta T_0}{\Delta t} + g\]

04

Rearrange the equation to get the finite difference equation for Node 0

Rearrange the equation to get the finite difference equation for \(\Delta T_0\): \[\Delta T_0 = \frac{\Delta t}{\rho c_p V}(k\frac{T_{1} - T_{0}}{\Delta x} - h(T_{0} - T_{\infty}) - \varepsilon \sigma_SB(T_0^4 - T_{\text{surr}}^4) - \dot{q}_{0} + g)\]

05

Calculate the total heat transfer at the right boundary for the first 20 time steps

For the first 20 time steps (n=1 to 20): 1. Calculate the updated \(\Delta T_0\) using the given boundary conditions and the finite difference equation obtained in Step 4, 2. Update the temperature values for all nodes, 3. Calculate the heat transfer at the right boundary (Node 4) for each time step using the updated temperature values, 4. Sum up the heat transfer for each time step to obtain the total heat transfer at the right boundary for the first 20 time steps. By following the above steps, the explicit finite difference formulation for Node 0 and the heat transfer for the first 20 time steps at the right boundary can be determined.

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