Page 208 - Modelling in Transport Phenomena A Conceptual Approach

188 CHAPTER 7. UNSTEADY-STATE MACROSCOPIC BfiANCB

(v,) = 0.61 /q 15)
to

Since the pressure in the tank is hydrostatic, lAPl N pgh and Eq. (5) becomes
(v0> = 0.61m = 2.7h (6)

Substitution of Eq. (6) into Eq. (2) gives the governing differential equation for
the liquid height in the tank as
(R-
2.7 (2) fi) = dh
(7)
where
a=- &in (8)
2.7A0
Note that the system reaches steady-state when dhldt = 0 at which point the liquid
height, h,, is given by
h, = R2 (9)
Now, it is worthwhile to investigate two cases:
Case (i) Liquid level in the tank increases
At t = 0, the liquid level in the tank is H/2. Therefore, the liquid level increases,
ie., dhJdt > 0 in Eq. (r), if
R2 > H/2 (10)
Rearrangement of Eq. (7) gives

Integration of Eq. (1 1) yields

t=O.74($) [g-&+Rh( R-JH72 )]
R-fi

Equations (9) and (IO) indicate that h, > H/2. When h, > H, steady-state
condition can never be achieved in the tank. The time required to fill the tank, tj,
is

If H/2 < h, < H, then the time, t,, required for the level of the tank to reach
99% of the steudy-state value is