Basics of Fluid Mechanics, 2014a

7.1. THE FIRST LAW OF THERMODYNAMICS 205
The unsteady state term then obtains the form
⎛
∫ ( )
d U
2
ρ
dt V 2 + gy dV ∼
d = ρ ⎝
[U 2
dt 2 + gh
2
⎞
{}}{
hA⎠ (7.38)
] V
The relationship between the boundary velocity to the height (by definition) is
U b = dh
dt
(7.39)
Therefore, the velocity in the z direction 13 is
U z = dh
dt
U e = A dh
A e dt = −U dh
b
dt
(7.40)
(7.41)
Combining all the three components of the velocity (Pythagorean Theorem) as
U 2 ∼ = Ux
2
+ Uy
2
+ Uz
2
(7.42)
U 2 ∼ =
( (π − 2) r
8 h
) 2 (
dh (π − 1) r
+
dt 4 h
) 2
dh
+
dt
( ) 2 dh
(7.43)
dt
U ∼ = dh
dt
f(G)
√ { }} {
((π ) 2 ( ) 2 − 2) r (π − 1) r
+
+1
8 h
4 h
2 (7.44)
It can be noticed that f(G) is a weak function of the height inverse. Analytical solution
of the governing equation is possible including this effect of the height. However, the
mathematical complication are enormous 14 and this effect is assumed negligible and the
function to be constant.
The last term is
∫ 2
U e
2 U e ρdA= U e 2
2 U e ρA e = 1 ( ) 2 dh A
U e ρA e (7.45)
2 dt A e
A
13 A similar point was provided in mass conservation Chapter 5. However, it easy can be proved by
construction the same control volume. The reader is encouraged to do it to get acquainted with this
concept.
14 The solution, not the derivation, is about one page. It must be remembered that is effect extremely
important in the later stages of the emptying of the tank. But in the same vain, some other effects
have to be taken into account which were neglected in construction of this model such as upper surface
shape.