A 3-m-long steel rod (k = 43 W/(mK), ? = 1.17 × 10–5 m2/s) is initially at 20°C and insulated completely except for its end faces. One end is suddenly exposed to the flow of combustion gases at 1000°C through a heat transfer coefficient of 250 W/(m2 K) and the other end is held at 20°C. How long does it take for the exposed end to reach 700°C? How much energy will the rod have absorbed if it is circular in cross section and has a diameter of 3 cm?
GIVEN
• Steel rod with one end at fixed temperature and the other end exposed to combustion gases
FIND
(a) Time required for the exposed face to reach 700°C (b) Heat input to the rod
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 temperature is constant so
For the half control volume at i = N, the explicit form of the energy balance is
Solving for TN, m + 1
For all the interior nodes, i = 2, 3, 4, ... N – 1, the energy balance is
Solving for Ti, m + 1
The heat input to the rod after any time step m is given by
The factor of 1/2 is because the control volume at i = N is ?x/2 in width. Since we have chosen an explicit method, we can use the marching procedure as described in Section 4.3.1. 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. The 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
N ?tmax ?t tfinal Qinput (s) (s) (s) (J/m2) 11 3846 10.0 5990 503.72 11 3846 1.0 5993 503.79 21 962 10.0 6350 490.33 41 240 10.0 6440 487.50 81 60 10.0 6460 486.69 81 60 5.0 6455 486.37 Since there is little change between the last 3 runs, the solution is that 6455 seconds are required for the exposed face to reach 700°C and the heat input to the rod is 486.4 joules.
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