Page 244 - Modelling in Transport Phenomena A Conceptual Approach

224 CHAPTER 7. UNSTEADY-STATE MACROSCOPIC BALANCES

for this case to get

Compare Eq. (6) with Eq. (4.5-6) and show that the friction loss per unit mass,
E,,, for pipe flow is given by
jj -- 2f L(V)*
v- D (7)

7.7 A cylindrical tank, 5m in diameter, discharges through a mild steel pipe
system (E = 4.6 x m) connected to the tank base as shown in the figure below.
The drain pipe system has an equivalent length of 100 m and a diameter of 23 cm.
The tank is initially filled with water to an elevation of H with respect to the
reference plane.

Reference
Plane

a) Apply the Bernoulli equation, Q. (5) in Problem 7.6, to the region between
planes “1” and “2” and show that

where Le, is the equivalent length of the drain pipe.
b) Consider the tank as a system and show that the application of the unsteady-
state macroscopic mass balance gives

f
dt=-(z)“/$(l+d)$ Le,
4
Analytical integration of Eq. (2) is possible only if the friction factor f is constant.
e) At any instant, note that the pressure drop in the drain pipe system is equal
to pg(h - H*). Use Eqs. (4.518)-(4.520) to determine f as a function of liquid
height in the tank. Take H* = 1 m, H = 4 m and the final value of h as 1.5 m.