Basics of Fluid Mechanics, 2014a

7.1. THE FIRST LAW OF THERMODYNAMICS 207
f(G) 2 h d2 h
dt 2 + gh+ 1 2
( ) [ 2 ( ) ] 2 dh
A
f(G) 2 − =0 (7.52)
dt
A e
Defining a new tank emptying parameter, T e ,as
( ) 2
A
T e =
(7.53)
f(G) A e
This parameter represents the characteristics of the tank which controls the emptying
process. Dividing equation (7.52) byf(G) 2 and using this parameter, equation (7.52)
after minor rearrangement transformed to
( d 2 h
h
dt 2 + gA e 2 )
T e A 2 + 1 2
The solution can either of these equations 16
or
⎧ -
−
⎪⎭
-
⎧ -
-
( ) 2 dh
[1 − T e ]=0 (7.54)
dt
√
dh
(k 1 T e − 2 k 1 ) e ln(h) Te +2gh 2
h (Te− 2) f(G)
= t + k 2 (7.55)
dh
√
(k 1 T e − 2 k 1 ) e ln(h) Te +2gh 2
= t + k 2 (7.56)
⎪⎭ h (Te− 2) f(G)
The solution with the positive solution has no physical meaning because the height
cannot increase with time. Thus define function of the height as
⎧ -
f(h) =−
⎪⎭
-
dh
√
(k 1 T e − 2 k 1 ) e ln(h) Te +2gh 2
h (Te− 2) f(G)
The initial condition for this case are: one the height initial is
The initial boundary velocity is
(7.57)
h(0) = h 0 (7.58)
16 A discussion about this equation appear in the mathematical appendix.
dh
dt
=0 (7.59)