Page 327 - Bird R.B. Transport phenomena

§10.8 Forced Convection 311

Forced Convection Free Convection
Heat Transfer Heat Transfer

(No fan)

Heat swept to right by forced Heat transported upward by
stream of air heated air that rises
1. The flow patterns are 1. The flow patterns are
determined primarily by determined by the buoyant
some external force force on the heated fluid
2 First, the velocity profiles are 2. The velocity profiles and
found; then they are used to temperature profiles are
find the temperature profiles interdependent
(usual procedure for fluids
with constant physical
properties)
3. The Nusselt number depends 3. The Nusselt number depends
on the Reynolds and Prandtl on the Grashof and Prandtl
numbers (see Chapter 14) numbers (see Chapter 14)

Fig. 10.8-1. A comparison of forced and free convection in non-
isothermal systems.

how this may be done for tube flow by using the shell balance method. We know that
the velocity distribution so obtained is v = 0, v = 0, and
r e
(10.8-1)

This parabolic distribution is valid sufficiently far downstream from the inlet that the en-
trance length has been exceeded.
In this problem, heat is being transported in both the r and the z directions. There-
fore, for the energy balance we use a "washer-shaped" system, which is formed by the
intersection of an annular region of thickness Ar with a slab of thickness Az (see Fig. 10.8-
2). In this problem, we are dealing with a flowing fluid, and therefore all terms in the e
vector will be retained. The various contributions to Eq. 10.1-1 are
Total energy in at r e \ • lirrkz = (2тгге,)\ г Az (10.8-2)
r r
Total energy out at r + Ar е \ ь,. • 2тг(г + Ar)Az = (10.8-3)
г г+
Total energy in at z e \ • 2тггАг (10.8-4)
z z
Total energy out at z + Az ^ZUAZ • 2тггАг (10.8-5)
Work done on fluid by gravity pv g • (10.8-6)
z z
The last contribution is the rate at which work is done on the fluid within the ring by
gravity—that is, the force per unit volume pg z times the volume 2тгг Ar Az multiplied by
the downward velocity of the fluid.