An interior wall of a cold furnace, initially at 0°C, is suddenly exposed to a radiant flux of 15 kW/m2 when the furnace is brought on line. The outer surface of the wall is exposed to ambient air at 20°C through a heat transfer coefficient of 10 W/(m2 K). The wall is 20 cm thick and is made of expanded perlite (k = 0.10 W/(mK), ? = 0.03 × 10–5 m2/s) sandwiched between two sheets of oxidized steel. Determine how long after startup will the inner (hot) sheet metal surface get hot enough so that reradiation becomes significant.

Question

GIVEN
• Furnace wall suddenly exposed to radiant heat flux
FIND
(a) How long before reradiation from the heated wall becomes significant.
ASSUMPTIONS
(a) Reradiation becomes significant when the reradiated flux from the exposed wall exceeds 10% of
the incident radiant flux. (b) The oxidized surface of the exposed wall is black.

Answer 1

See the figure to the right for the arrangement of control volumes and nodes and symbol definitions.

The nodes are located at



and the time steps are given by



For the half control volume at i = 1, the explicit form of the energy balance is



For the half control volume at i = N, the explicit form of the energy balance is



For all the interior nodes, i = 2, 3, 4, ... N – 1, the energy balance is



Since the exposed wall is black, the reradiated flux from the hot wall is ?TN4 and the criterion we seek is



For the given values of problem parameters, this equates to TN = 130.3°C. Since we have chosen an explicit method, we can use the marching procedure as described Also, the time step ?t is restricted. After setting up the computer program to step through the time steps, the energy balance on nodes i = 1, 2, and N were checked by hand to insure that the code was correct. Then several runs were made with various values of N and ?t to find how large N and how small ?t must be to get an accurate solution. The table below summarizes these runs