A 1-cm-thick, 1-m-square steel plate is exposed to sunlight and absorbs a solar flux of 800 W/m2. The bottom of the plate is insulated, the edges are maintained at 20°C by water-cooled clamps, and the exposed face is cooled by a convection coefficient of 10 W/(m2 K) to an ambient temperature of 10°C. The plate is polished to minimize reradiation. Determine the temperature distribution in the plate using a node spacing of 20 cm. The thermal conductivity of the steel is 40 W/(m K).
GIVEN
Square plate with water-cooled edges exposed to solar flux
FIND
(a) Temperature distribution in the plate
ASSUMPTIONS
(a) Neglect temperature gradients through the plate thickness
SKETCH
Because of problem symmetry, we need only consider the 6 nodes in 1/8 th of the square plate as
shown in the above sketch. The boundary condition gives us the temperature of nodes 1, 2, and 3 so
we only need to perform a heat balance on nodes 4, 5, and 6. Define the following symbols
Node 4 transfers heat by conduction with nodes 3, 5, and 6, by convection to ambient, and absorbs the
specified solar flux. The energy balance on node 4 is therefore
Node 5 transfers heat by conduction with node 4, by convection to ambient, and absorbs the specified
flux. The energy balance on node 5 is therefore
Node 6 transfers heat by conduction with nodes 4 and 2, by convection to ambient, and absorbs the
specified flux. The energy balance on node 6 is therefore
This set of equations can be solved by iteration. The table below shows the results of Gauss-Seidel
iteration. Iteration 0 is the first guess for the temperature at nodes 4, 5, and 6.
The solution converges after about 8 iterations giving a peak temperature of 67.727°C at node 5.
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