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.
You might also like to view...
Electron Speed: Given that the speed of an electron in the ground state of hydrogen is 2190 km/s, what is the speed of an electron in the n = 4 orbit of hydrogen?
A. 547 km/s B. 31,000 km/s C. 139 km/s D. 178 km/s E. 2180 km/s
Into the gap between the plates of a parallel plate capacitor of capacitance 
a slab of metal is inserted halfway between the plates filling one fourth of the gap between the plates. What is the resulting new capacitance?
A.
B.
C.
D.
E.
A copper sphere that is 10 cm in diameter at an atmospheric pressure of 1.01 × 105 N/m2 is put into a chamber where the pressure is 1.0 × 106 N/m2
What is the change in the diameter of the sphere caused by this change in pressure? The bulk modulus for copper is 1.4 × 1011 Pa. A) -7.1 × 10-5 cm B) -2.1 × 10-5 cm C) -7.1 × 10-4 cm D) -7.1 × 10-6 cm E) -2.1 × 10-4 cm
A blacksmith drops a 550 °F piece of iron into a vat of 75 °F water in order to cool it to 100 °F. How many kilograms of water are needed per kilogram of iron? Assume all the thermal energy from the iron is transferred to the water and none of the water evaporates. The specific heats of water and iron are 4186 J/kg/°C and 448 J/kg/°C, respectively
a. 9.3 kg b. 0.78 kg c. 0.52 kg d. 1.9 kg