Page 234 - Bird R.B. Transport phenomena

218 Chapter 7 Macroscopic Balances for Isothermal Flow Systems

This is to be solved with the two initial conditions:
I.C.I: atf = O, h = H (7.7-6)
I.C. 2: atf = O, ^ = V2^H(R /R) 2 (77-7)
0
The second of these is Torricelli's equation at the initial instant of time.
The second-order differential equation for h can be converted to a first-order equation for
2
the function u(h) by making the change of variable (dh/dt) = u. This gives
hj^-(N-l)u+2gh = 0 (7.7-8)

The solution to this first-order equation can be verified to be 1
N
= Ch ~ l + 2gh/(N - 2) (7.7-9)
u
N
The second initial condition then gives С = —4g/[N(N - 2)H ~ ] for the integration constant;
2
since N » 1, we need not concern ourselves with the special case that N = 2. We can next take
the square root of Eq. 7.7-9 and introduce a dimensionless liquid height r\ = h/H; this gives
4

in which the minus sign must be chosen on physical grounds. This separable, first-order
equation can be integrated from t = 0 to t = f efflux to give

( W - 2 1 H f dr, f2NH MKn П7ЛЛЛ
^ Jo VqQ/NW 1 = V"8" Ф
The function ф(Ы) gives the deviation from the quasi-steady-state solution obtained in Eq.
7.7-3. This function can be evaluated as follows:
, 1 I N - 2 Г 1 dtn
fKn

The integrations can now be performed. When the result is expanded in inverse powers of N,
one finds that

+ o(^) (7.7-13)

Since N = (R/R ) 4 is a very large number, it is evident that the factor ф(Ы) differs only very
o
slightly from unity.
It is instructive now to return to Eq. 7.7-4 and omit the term describing the change in
total kinetic energy with time. If this is done, one obtains exactly the expression for efflux
time in Eq. 7.7-3 (or Eq. 7.7-11, with ф(Ы) = 1. We can therefore conclude that in this type of
problem, the change in kinetic energy with time can safely be neglected.

1
See E. Kamke, Differentialgleichungen: Losungsmethoden und Losungen, Chelsea Publishing
Company, New York (1948), p. 311, #1.94; G. M. Murphy, Ordinary Differential Equations and Their
Solutions, Van Nostrand, Princeton, NJ. (I960), p. 236, #157.