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PGE 421K, Spring 2010
HOMEWORK 3
100 points
Due M 22 Feb
1.
Knowing the terms in the mass and energy balances can help you figure out
appropriate (i.e. physically reasonable) forms of correlations, design equations, etc.
The chart above appears in a Dowell Engineers Handbook from the 1950s and shows
the volumetric flow rate that must be supplied to operate fluid jets through orifices of
various sizes, as a function of the pressure drop across the jet.
a.
Start with the general form of the energy balance and derive the
relationship between
Q
and
Δ
P
from the energy balance.
State the
assumptions you make in the derivation.
Assume steady state, isentropic, no shaft work, no change in potential energy.
Mass balance:
gG
±²
g³
= G
´
µ¶
− G
´
·¸¹
= 0
G
´
µ¶
= G
´
·¸¹
Steady state energy balance across the jet:
0 = º
´
+ »
´
¼
+ G
´
µ¶
½¾
¿
µ¶
+
À
µ¶
Á
2
+ ÂÃ
µ¶
Ä − G
´
·¸¹
½¾
¿
·¸¹
+
À
·¸¹
Á
2
+ ÂÃ
·¸¹
Ä
20 pts
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View Full Document Combine energy and mass balances:
0 = g
G
±²
³´
µ
±²
− ´
µ
¶·¸
+
¹
±²
º
− ¹
¶·¸
º
2
»
Assume incompressible fluid. Then:
´
µ
±²
− ´
µ
¶·¸
= ¼
½
¾¿
±²
− ¿
¶·¸
À
and
g
G
±²
=
¹
±²
Á
±²
¼
½
=
¹
¶·¸
Á
¶·¸
¼
½
where A
in
, A
out
are cross sectional areas of pipe and orifice. Hence:
¹
¶·¸
= ¹
±²
Á
±²
Á
¶·¸
Note:
Á
¶·¸
< Á
±²
so
¹
¶·¸
> ¹
±²
.
Substituting into the energy balance, we obtain:
¼
½
∆¿ =
1
2
¾¹
¶·¸
º
− ¹
±²
º
À =
1
2
³¹
¶·¸
º
− ¹
¶·¸
º
Â
Á
¶·¸
Á
±²
Ã
º
»
Solving for
¹
¶·¸
:
¹
¶·¸
=
Ä
2¼
½
∆¿
1 − Å
Á
¶·¸
Á
±²
Æ
º
Flow through orifice is Q=u
out
A
out
so we have:
Ç = Á
¶·¸
Ä
2¼
½
∆¿
1 − Å
Á
¶·¸
Á
±²
Æ
º
orifice
b.
Use the result of (a) to explain the shape of the curves in the chart.
That
is, what is the connection between the mathematical form of (a) and
family of curves in the chart.
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This note was uploaded on 02/13/2012 for the course PGE 421K taught by Professor Bryant during the Spring '11 term at University of Texas at Austin.
 Spring '11
 BRYANT

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