The rate of heat flow per unit length q/L through a hollow cylinder of inside radius ri and outside radius ro is q/L = ( A k ?T)/(ro – ri) where A = 2? (ro – ri)/ln(ro/ri). Determine the per cent error in the rate of heat flow if the arithmetic mean area ? (ro + ri) is used instead of the logarithmic mean area A for ratios of outside to inside diameters (Do/Di) of 1.5, 2.0, and 3.0. Plot the results.

Question

GIVEN

• A hollow cylinder

• Inside radius = ri

• Outside radius = ro

• Heat flow per unit length as given above

FIND

(a) Per cent error in the rate of heat flow if the arithmetic rather than the logarithmic mean area is used for ratios of outside to inside diameters of 1.5, 2.0, and 3.0. (b) Plot the results

ASSUMPTIONS

• Radial conduction only

• Constant thermal conductivity

• Steady state prevails

SKETCH

Answer 1

The rate of heat transfer per unit length using the logarithmic mean area is



The rate of heat transfer per unit length using the arithmetic mean area is



The per cent error is



For a ratio of outside to inside diameters of 1.5



The percent errors for the other diameter ratios can be calculated in a similar manner with

the following results



(b)