Page 503 - Bird R.B. Transport phenomena

Problems 483

The behavior of this kind of equipment may be simply analyzed by making the following as-
sumptions:
(i) Steady-state conditions exist.
(ii) The overall heat transfer coefficient U and the heat capacities of the two fluids are
constants.
(iii) The shell-fluid temperature T is constant over any cross section perpendicular to the
B
flow direction.
(iv) There is an equal amount of heating area in each tube fluid "pass"—that is, for
streams I and II in the figure.
(a) Show by an energy balance over the portion of the system between planes a and b that
T - T = R(T l l - T\) where R = \w C /w C \ (15C.1-1)
B B2 A A pA B pB
(b) Show that over a differential section of the exchanger, including a total heat exchange sur-
face dA,

~da~2 B ~ A

da 2

r
! ( + 7
~ 2 ^
in which da = {U/w C )dA, and w and C are defined as in Example 15.4-1.
A
pA
pA
A
]
(c) Show that when T\ and T are eliminated between these three equations, a differential
A
equation for the shell fluid can be obtained:
da 2 da 4
in which S(a) = (T - T )/(T — T ). Solve this equation (see Eq. C.l-7) with the boundary
B B2 m B2
conditions
B.C1: atcr = 0, 0 = 1 (15С.1-6)
B.C. 2: at a = (UA /w C ), 0 = 0 (15C.1-7)
pA
T
A
in which A is the total heat-exchange surface of the exchanger.
T
(d) Use the result of part (c) to obtain an expression for dT /da. Eliminate dT /da from this
B
B
expression with the aid of Eq. 15C.1-3 and evaluate the resulting equation at a = 0 to obtain
the following relation for the performance of the exchanger:
T
w C pA
A
in which ^ = (T - T )/{T m - T ).
A2
M
M
(e) Use this result to obtain the following expression for the rate of heat transfer in the ex-
changer:
Q = UA(AT) ln • У (15С.1-9)
in which
_ (^BI ~ T ) ~ (T - Т )
м
A2
B2
~ 1Ш(Г И - Т )/(Т В2 ~ T J]
А2
A