Page 236 - Bird R.B. Transport phenomena

220 Chapter 7 Macroscopic Balances for Isothermal Flow Systems

Here / is a coordinate running along the axis of the manometer tube, and L is the distance
along this axis from one manometer interface to the other—that is, the total length of the
manometer fluid. The dimensionless coordinate £ is r/R, and S is the cross-sectional area of
the tube.
The change of potential energy with time is given by
CL Г2тг fR
frtot _ d
(pgz)r dr dB dl
dt dt
о Jo Jo
/integral over portion\ гк+нл гк+н и
+
below z = 0, which + pgS I zdz + pgS I z dz
dt J j
\is constant / ° °
dh
(77-16)
The viscous loss term can also be evaluated as follows:
CL flir fR
E =-\ \ (i:Vv)r dr dO dl
v
JQ JQ JO

2
= SLS/JL(V) /R 2 (7.7-17)
Furthermore, the net work done by the surroundings on the system is
W, = (p - p )S(v) (7.7-18)
a
b
Substitution of the above terms into the mechanical energy balance and letting (v) = dh/dt
then gives the differential equation for h(t) as

2
dt 2 ' \ R )dt ' ^ ' " - 1 ' ' ( 7 7 " 1 9 )
P
which is to be solved with the initial conditions that h = 0 and dh/dt = 0 at t = 0. This second-
order, linear, nonhomogeneous equation can be rendered homogeneous by introducing a
new variable к defined by
k = 2h- Va T Pb (7.7-20)
pL
Then the equation for the motion of the manometer liquid is
(7.7-21)

This equation also arises in describing the motion of a mass connected to a spring and dash-
pot as well as the current in an RLC circuit (see Eq. C.I-7).
mt
We now try a solution of the form k = e . Substituting this trial function into Eq. 7.7-21
shows that there are two admissible values for m:
m ± = \[-(6n/pR ) ± \Z(6fL/pR ) - (6g/L)] (7.7-22)
2 2
2
and the solution is
к = C e mJ + C_e m t when m + Ф m_ (7.7-23)
+
к = C e mt + C te mt when m+ = m_ = m (7.7-24)
}
2
with the constants being determined by the initial conditions.