Page 483 - Bird R.B. Transport phenomena

§15.4 The d-Forms of the Macroscopic Balances 463

Because there is no heat loss to the surroundings, Q h = —Q . For incompressible liquids with
c
a pressure drop that is not too large, or for ideal gases, Eq. 9.8-8 gives for constant C the rela-
p
tion AH = C AT. Hence Eqs. 15.4-5 and 6 can be rewritten as
/7
w C (T h2 - T ) = Q,, (15.4-7)
h
ph
/rt
zv C (T c2 - Г ) = Q c = -Q h (15.4-8)
с1
pc
c
(b) d-form of the macroscopic energy balance. Application of Eq. 15.4-4 to the hot stream
gives
7
^ff'- ^ 05.4-9)
where r 0 is the outside radius of the inner tube, and U o is the overall heat transfer coefficient
based on the radius r 0 (see Eq. 14.1-8).
Rearrangement of Eq. 15.4-9 gives
dT Qirr )dl
h o
З = ^ ^ (15.4-10)
The corresponding equation for the cold stream is
dT c IT Qm )dl
o
(15.4-11)
T w C
~ 4, c pc
Adding Eqs. 15.4-10 and 11 gives a differential equation for the temperature difference of the
two fluids as a function of /:
f (15.4-12)

\w C , f
h
pl
By assuming that L7 is independent of / and integrating from plane 1 to plane 2, we get
0
In (I" ~ lA = U (-\- + ^-)(2тгг )1 (15.4-13)
o о
\T - T J \ j
h2 C2 WhCph C
This expression relates the terminal temperatures to the stream rates and exchanger dimen-
sions, and it can thus be used to describe the performance of the exchanger. However, it is
conventional to rearrange Eq. 15.4-13 by taking advantage ofthe steady-state energy balances
in Eq. 15.4-7 and 8. We solve each of these equations for wC and substitute the results into
p
Eq. 15.4-13 to obtain
Q c = и (2тгг 1)[ —— =-г, (15.4-14)
о
о
\ln [(Г/,2 - T )/(T h] - T )]/
c2
cl
or
Q, = 17оЛ (Т, - Г ) (15.4-15)
о
с 1п
Here Л is the total outer surface of the inner tube, and (T - T ) is the ''logarithmic mean
о h L ]n
temperature difference" between the two streams. Equations 15.4-14 and 15 describe the rate
of heat exchange between the two streams and find wide application in engineering practice.
Note that the stream rates do not appear explicitly in these equations, which are valid for
both parallel-flow and counter-flow exchangers (see Problem 15A.1).
From Eqs. 15.4-10 and 11 we can also get the stream temperatures as functions of / if de-
sired. Considerable care must be used in applying the results of this example to laminar flow,
for which the variation of the overall heat transfer coefficient may be quite large. An example
of a problem with variable U is Problem 15B.1.
o