Basics of Fluid Mechanics, 2014a

206 CHAPTER 7. ENERGY CONSERVATION
Combining all the terms into equation (7.27) results in
⎛
✁ρ d ⎝
[U 2
dt 2 + gh
2
⎞
{}}{
hA⎠ − 1 2
] V
taking the derivative of first term on l.h.s. results in
d
[U 2 ] [U
dt 2 + gh
2 ]
hA+
2
2 + gh A dh
2 dt − 1 2
( ) 2 ( ) 2 dh A
U e ρA
dt A e ✁ e =0 (7.46)
( ) 2 ( ) 2 dh A
U e A e =0 (7.47)
dt A e
Equation (7.47) can be rearranged and simplified and combined with mass conservation
15 .
Advance material can be skipped
Dividing equation (7.46) byU e A e and utilizing equation (7.40)
d
[U 2 ]
dt 2 + gh
2
hA
[U 2
+
U e A e 2 + gh
2
Notice that U = U b f(G) and thus
f(G) U b
{}}{
U
dU
dt
hA
+ g dh
U e A e 2 dt
] A A e
A U e
{ }} {
A dh
dt −1 2
hA
[U 2 ]
+
U e A e 2 + gh − 1 2 2
Further rearranging to eliminate the “flow rate” transforms to
1
f(G) h dU ( ) 1 dh
U b A
dt ✟U ✟✟✟✟✯ + gh ✓
✓ ✓✼ dt A
+
e A e 2 ✓ U e A e
[
f(G) 2 h d2 h
dt 2 + gh
2 + f(G) 2
2
( dh
dt
[
f(G) 2
2
( dh
dt
) ]
2
+ gh − 1 2 2
End Advance material
Combining the gh terms into one yields
( ) 2 ( ) 2 dh A
✘U dt A ✘ e A e =0
e
(7.48)
( ) 2 ( ) 2 dh A
=0 (7.49)
dt A e
) ]
2
+ gh − 1 2 2
( ) 2 ( ) 2 dh A
=0
dt A e
(7.50)
( ) 2 ( ) 2 dh A
=0 (7.51)
dt A e
15 This part can be skipped to end of ”advanced material”.