Page 482 - Bird R.B. Transport phenomena

462 Chapter 15 Macroscopic Balances for Nonisothermal Systems

in which / is the local friction factor, and R h is the local value of the mean hydraulic ra-
dius. In most applications we omit the dW term, since work is usually done at isolated
points along the flow path. The term dW would be needed in tubes with extensible walls,
magnetically driven flows, or systems with transport by rotating screws.

The d°Form of the Total Energy Balance
If we write Eq. 15.1-3 in differential form, we have (with flat velocity profiles)
2
d(\v ) + gdh + dH = dQ + dW (15.4-3)
Then using Eq. 9.8-7 for dH and Eq. 14.1-8 for dQ we get

vdv + gdh + C dT +\v- г ( | Ю \dp = Ul0C ^ T dl + dW (15.4-4)
p
in which U, is the local overall heat transfer coefficient, Z is the corresponding local
oc
conduit perimeter, and AT is the local temperature difference between the fluids inside
and outside of the conduit.
The examples that follow illustrate applications of Eqs. 15.4-2 and 15.4-4.

EXAMPLE 15.4-1 It is desired to describe the performance of the simple double-pipe heat exchanger shown in
Fig. 15.4-1 in terms of the heat transfer coefficients of the two streams and the thermal resis-
Parallel- or Counter- tance of the pipe wall. The exchanger consists essentially of two coaxial pipes with one fluid
Flow Heat Exchangers stream flowing through the inner pipe and another in the annular space; heat is transferred
across the wall of the inner pipe. Both streams may flow in the same direction, as indicated in
the figure, but normally it is more efficient to use counter flow—that is, to reverse the direc-
tion of one stream so that either w h or w c is negative. Steady-state turbulent flow may be as-
sumed, and the heat losses to the surroundings may be neglected. Assume further that the
local overall heat transfer coefficient is constant along the exchanger.

SOLUTION (a) Macroscopic energy balance for each stream as a whole. We designate quantities refer-
ring to the hot stream with a subscript h and the cold stream with subscript с The steady-
state energy balance in Eq. 15.1-3 becomes, for negligible changes in kinetic and potential
energy,
w m 2 -H ) = Q (15.4-5)
h h h] H
w (H - H ) = Q (15.4-6)
c c2 cl c

Cold stream in
T Plane 2

Hot stream in Jt— Hot stream out

I— dl I

Plane 1 Cold stream out
T = T c2
Fig. 15.4-1. A double-pipe heat exchanger.