Experiments have been performed on the temperature distribution in a homogeneous long cylinder (0.1 m diameter, thermal conductivity of 0.2 W/(m K) with uniform internal heat generation. By dimensional analysis, determine the relation between the steady-state temperature at the center of the cylinder Tc the diameter, the thermal conductivity, and the rate of heat generation. Take the temperature at the surface as you datum. What is the equation for the center temperature if the difference between center and surface temperature is 30°C when the heat generation is 3000 W/m3?

Question

GIVEN

A homogeneous long cylinder with uniform internal heat generation

Diameter (D) = 0.1 m

Thermal conductivity (k) = 0.2 W/(m K)

Difference between surface and center temperature (Tc – Ts) = 30°C

Heat generation rate q = 3000 W/m3



FIND

(a) Relation between center temperature (Tc), diameter (D), thermal conductivity (k), and rate of heat

generation q



(b) Equation for the center temperature for the given data

ASSUMPTIONS

Steady state

One dimensional conduction in the radial direction

SKETCH

Answer 1

(a) The temperature difference is a function of the variable given



having the following primary dimensions



Let the unknown function be represented by



Where A is a dimensionless constant



Summing the exponents of each primary dimension



The given data can now be used to evaluate the unknown constant



The equation for the center temperature is