Page 437 - Modelling in Transport Phenomena A Conceptual Approach

PROBLEMS 41 7

9.7 The rate of generation per unit volume is sometimes expressed as a function of
temperature rather than position. Consider the transmission of an electric current
through a wire of radius R. If the surface temperature is constant at TR and the
rate of generation per unit volume is given as

a) Show that the governing equation for temperature is given by

dr (rg) +?(l+aT)r=O

b) Use the transformation
u=l+aT (3)
and reduce Eq. (2) to the form

f (rg) +dru=O (4)

where

c) Solve Eq. (4) to get
T+ (ll4 - Jo (fir)
TR + (l/a) - Jo (@ R)
d) What happens to Eq. (6) when & R = 2.4048?

9.8 For laminar flow forced convection in a circular pipe with a constant wall
temperature) the governing equation for temperature, Eq. (9.3-9)) is integrated
over the cross-sectional area of the tube in Section 9.3.1 to obtain Eq. (9.3-18))
i.e.,
dTb
mCp - aDh(T, -Ta) (1)
=
dz
a) Now let us assume that the flow is turbulent. Over a differential volume ele-
ment of thickness Az, as shown in the figure below, write down the inventory rate
equation for energy and show that the result is identical with Eq. (1).