Page 245 - Bird R.B. Transport phenomena

Problems 229

(с) Combine the results of (a) and (b). Solve the resulting differential equation with an appro-
priate initial condition to get the liquid level z as a function of t. From this get the efflux time

efflux (7B.10-3)
List all the assumptions that have been made and discuss how serious they are. How could
these assumptions be avoided?
(d) Rework part (b) by choosing plane 1 to be stationary and slightly below the liquid surface
at time t. It is understood that the liquid surface does not go below plane 1 during the differ-
ential time interval dt over which the unsteady mechanical energy balance is made. With this
choice of plane 1 the derivative d<3> /dt is zero and there is no work term W . Furthermore
tot
m
the conditions at plane 1 are very nearly those at the liquid surface. Then with the pseudo-
steady-state approximation that the derivative dK /dt is approximately zero and the neglect
tot
of the viscous loss term, the mechanical energy balance, with w l = zv takes the form
2/
0 - \{v\ - vl) - h ) (7B.10-4)
2
7B.11 Disintegration of wood chips (Fig. 7B.11). In the manufacture of paper pulp the cellulose
fibers of wood chips are freed from the lignin binder by heating in alkaline solutions under
pressure in large cylindrical tanks called digesters. At the end of the "cooking" period, a
small port in one end of the digester is opened, and the slurry of softened wood chips is al-
lowed to blow against an impact plate to complete the breakup of the chips and the separa-
tion of the fibers. Estimate the velocity of the discharging stream and the additional force on
the impact plate shortly after the discharge begins. Frictional effects inside the digester, and
the small kinetic energy of the fluid inside the tank, may be neglected. (Note: See Problem
7B.10 for two different methods for selecting the entrance and exit planes.)
Answer: 2810 lb /s (or 1275 kg/s); 10,900 \h (or 48,500 N)
m f

Fig. 7B.11. Pulp digester.

7B.12 Criterion for Vapor-Free Flow in a Pipeline. To ensure that a pipeline is completely liquid-
filled, it is necessary that p > p vap at every point. Apply this criterion to the system in Fig. 7.5-1,
by using mechanical energy balances over appropriate portions of the system.