A long cylindrical rod, 8 cm in diameter, is initially at a uniform temperature of 20°C. At time t = 0, the rod is exposed to an ambient temperature of 400°C through a heat transfer coefficient of 20 W/(m2 K). The thermal conductivity of the rod is 0.8 W/(mK) and the thermal diffusivity is 3 × 10–6 m2/s. Determine how much time will be required for the temperature change at the centerline of the rod to reach 93.68% of its maximum value. Use an explicit difference equation and compare your numerical results with a chart solution.

Question

GIVEN
• Cylindrical rod suddenly exposed to increased ambient temperature
FIND
(a) Time required for the centerline temperature change to reach 93.68% of its maximum value

Answer 1

Since the rod will eventually reach 400°C, the maximum possible temperature change for any part of the rod is 400 – 20 = 380°C. Taking 93.68% of this temperature difference, we need to find the time such that the centerline temperature is 20 + (0.9368 × 380) = 376°C. the radius is given by



and the time is given by



Note that since all gradients with respect to the circumferential direction, ?, are zero, the index j is not needed. Let the convection coefficient be h and ambient temperature be T?. The following sketch shows the control volumes necessary to solve the problem numerically

Referring to the above sketch, the inner surface area per unit length of the shaded control volume at node i = N is



Referring to the above sketch, the inner surface area per unit length of the shaded control volume at

node i = N is



and the outer surface area is



The volume of the control volume per unit length is



The explicit form of the energy balance on the control volume at i = N gives



For the control volume at the centerline node, i = 1, the volume per unit length is



and the surface area per unit length is



The energy balance on this node is





Using a time step, ?t, such that



then the explicit solution can be solved by marching and it should be stable. For N = 10 and ? = 0.5, the centerline temperature is found to exceed 376°C at 994 seconds.

For the chart solution. The Biot number is



We need to find the abscissa in the figure such that



For the Biot number calculated above, the abscissa is



Solving for the time, we find t = 949 seconds, approximately 5% less than the numerical method predicts. Most likely, the difference is due to the precision with which the charts can be read.