A small aluminum sphere of diameter D, initially at a uniform temperature To, is immersed in a liquid whose temperature, T?, varies sinusoidally according to

Question

T? – Tm = A sin (? t) where: Tm = time-averaged temperature of the liquid A = amplitude of the temperature fluctuation ? = frequency of the fluctuations

If the heat transfer coefficient between the fluid in the sphere, ah , is constant and the system may be treated as ‘lumped capacity,’ derive an expression for the sphere temperature as a function of time.

GIVEN

• A small aluminum sphere is immersed in a liquid whose temperature varies sinusoidally

• Diameter of sphere = D

• Liquid temperature variation: T? – Tm = A sin (? t)

• The heat transfer coefficient = ah (constant)

• The system may be treated as a ‘lumped capacity’

FIND

• An expression for the sphere temperature as a function of time

ASSUMPTIONS

• Constant thermal conductivity

SKETCH

Answer 1

Let k = thermal conductivity of sphere

? = density of sphere c = specific heat of sphere An energy balance on the sphere yields

Change in internal energy = heat transfer to liquid





This is a first order, linear, non-homogeneous differential equation. The general solution is the sum of the homogeneous solution and a particular solution. The homogeneous solution is determined by the characteristic equation, found by substituting ? = e?t into the homogeneous equation



The homogeneous solution is ?h = Ce–mt. As a particular solution, try ?p = K cos (? t) + M sin (? t), substituting ?p and its derivative into the energy balance



Therefore, the general solution is