Page 490 - Bird R.B. Transport phenomena

470 Chapter 15 Macroscopic Balances for Nonisothermal Systems

(a) Steady-state behavior for t < 0. When the time derivatives in Eqs. 15.5-12 and 13 are set
equal to zero and the equations added, we get for t < 0, where T = T :
10
}
wC T w + ЬТ тах
p
T = 7 (15.5.14)
20
wC + b
p
Then from Eq. 15.5-13 we can get the initial temperature of the coil
(b) Steady-state behavior for t -> <». When similar operations are performed with 7^ = T ,
loc
we get
wC T }
p
T = — ^ — (15.5-16)
2x
wC + b
p
and
^ m a x

for the final temperature of the coil.
(c) Unsteady state behavior for t > 0. It is convenient to define dimensionless variables
using the steady-state quantities for t < 0 and t —> <»:
T - T
2oo
S = — 2 =7— = dimensionless liquid temperature (15.5-18)
2
ho ~ hoc
T — T
© c = -=r ^f— - dimensionless coil temperature (15.5-19)
r = - ^ - = dimensionless time (15.5-20)
PC V
P
In addition we define three dimensionless parameters:

R = pC V/p C V = ratio of thermal capacities (15.5-21)
p c pc c
F = wC /UA = flow-rate parameter (15.5-22)
p
b/UA = controller parameter (15.5-23)
In terms of these quantities, the unsteady-state balances in Eqs. 15.5-12 and 13 become (after
considerable manipulation):
J/ГЛ
-
—1 = (l + F)@ + (1 - В)в с (15.5-24)
2
dr
^p- = R(® - 0 ) (15.5-25)
2
C
ат
elimination of <d between this pair of equations gives a single second-order linear ordinary
c
differential equation for the exit liquid temperature as a function of time:
— ^ + (1 4- R + F) - ^ + R(B + F)© = 0 (15.5-26)
2
dr UT
This equation has the same form as that obtained for the damped manometer in Eq. 7.7-21
(see also Eq. С1-7). The general solution is then of the form of Eq. 7.7-23 or 24:
©2 = C exp (m r) + C_ exp (m_r) (m Ф mj (15.5-27)
+
+
+
@ 2 = Q exp тт + С т exp тт (m+ = m_ = m) (15.5-28)
2