Modern Engineering Thermodynamics

174 CHAPTER 6: First Law Open System Applications
Table 6.2 The Effect of Height on Potential Energy
Height, Z
Potential Energy, gZ/g c
ft m Btu/lbm kJ/kg
0 0 0 0
1.0 0.30 1.3 × 10 −3 2.9 × 10 −3
10. 3.0 1.3 × 10 −2 2.9 × 10 −2
100. 30.5 1.3 × 10 −1 2.9 × 10 −1
1000. 305 1.3 2.9
There are, of course, exceptions to this rule of thumb. If h in − h out ≈ 0, then the enthalpy term loses its dominance.
In this case, a small specific kinetic energy term may be quite significant to the analysis. A nozzle or a
diffuser is an example of such an exception.
Now consider the specific potential energy term, gZ/g c . Taking g =32.174ft/s 2 and, g c =32.174lbm· ft/(lbf · s 2 ), then
0
1
gZ
= Z g
B 32:174 ft/s 2 C
= ð½ZŠ ftÞ× @
g c g c
32:174 lbm A= ½ZŠ
.ft
ft .lbf
lbm
lbf .s 2
where the symbol [Z] stands for the numerical value of Z in units of feet. Again, converting to Btu/lbm gives
gZ
= ½ZŠ ft
.lbf
×
1Btu
=
g c lbm 778:16 ft.lbf
In the SI units system, g c = 1.0 and is dimensionless, so
gZ
= Z g
9:807 m/s2
= ð½ZŠ mÞ ×
g c g c
1
where 1 m 2 /s 2 =1J/kg=10 −3 kJ/kg.
½ZŠ
778:16
Btu
lbm
= ½ΖŠ × 9:807 J 9:807
= ½ΖŠ
kg 1000
Table 6.2 gives values of specific potential energy for various heights using these equations. Note that, for
systems with normal engineering dimensions, say less than 1000 ft (305 m) high, the specific potential energy
is very small. Consequently, it is common to neglect the effect of a flow stream’s specific potential energy when
the flow stream enters the system less than 1000 ft (305 m) above or below the potential energy baseline of the
system, or gZ/g c < ~1.0 Btu/lbm (2.3 kJ/kg).
There are also exceptions to this rule of thumb. Again, if h in − h out ≈ 0, then flow stream height changes may be
very important in the analysis. A hydroelectric power plant is an example of such an exception.
Deciding whether to neglect factors such as specific kinetic and potential energies is not always an easy task for
the beginner. Self-confidence comes only with experience. However, one more rule of thumb applies to most
textbook thermodynamic problems: If values for velocity and height are not given in the problem statement and are not
among the problem’s unknowns, then you are supposed to neglect the kinetic and potential energy terms in your analysis.
This means that the person who wrote the problem knew that either V in ≈ V out and Z in ≈ Z out or that all the
velocities and heights were relatively small. The only exception to this last rule of thumb is when you know the
mass flow rate _m , the diameter D, or cross-sectional area A, and the fluid density ρ or specific volume v of a
flow stream. With this information you can calculate the flow stream velocity using Eq. (6.1) as
V = _m ρA = _mv
A
kJ
kg
= 4 _mv
πD 2 (6.14)
If you can make this calculation for V, then you might as well use it in your energy rate balance equation, unless
it is so small that you are certain it will not affect the results of your analysis.
6.5 NOZZLES AND DIFFUSERS
Nozzle is the generic name of any device whose primary function is to convert the pressure energy _mpv of an
inlet flow stream into the kinetic energy _mV 2 /2 of an outlet flow stream. Thus, a nozzle is a very simple energy
conversion device. Similarly, diffuser is the generic name of any device whose primary function is to convert the
kinetic energy of an inlet flow stream into the pressure energy of an outlet flow stream. Note that nozzles and