1) A nozzle is a device that accelerates fluid flow by varying the cross-sectional area. Nozzles are used in applications like turbines, rockets, and jets. 2) The document discusses governing equations for nozzle flow, including the continuity and energy equations. It also covers isentropic flow assumptions. 3) Nozzle shape is examined, with convergent-divergent nozzles described as having a throat of minimum area, with subsonic flow before and supersonic after. Critical pressure ratio, temperature ratio, velocity, and area are defined as the conditions at the throat where the velocity is sonic. An example problem is presented to demonstrate these concepts.

1. Thermodynamics – 2
Fall Semester, 2014-2015
Lecture – A
Nozzles
Dr. Ahmed Rezk
1

2. Nozzles
• A nozzle is a device used to accelerate a flowing fluid by varying the
cross-sectional area in the direction of flow. The fluid acceleration
comes on the account of a pressure drop along the nozzle.
• Nozzles applications
• Steam and gas turbine
• Rocket engines
• Jet engines
2

3. Governing equations
• Steady flow continuity equation
• Where m is the mass flow rate at inlet section (in), outlet section (out) and
any section (x) along nozzle passage. The mass flow rate can be
determined as:
3
Constantm
xoutin mmm  
m

4. Governing equations
• ρ is the flowing fluid density (kg/m3)
• V is the velocity normal to the flow area (m/s)
• A the cross-sectional area (m2)
• Ʋ fluid specific volume (m3/kg)
4
 skg
v
AV
AVm / 
x
xx
out
outout
in
inin
v
VA
v
VA
v
VA


5. Governing equations
• Steady flow energy equation (per unit mass)
• Apply the following assumption:
• Negligible heat losses (adiabatic).
• No work done on or by the system.
• The nozzle passage is very small and hence the change in the potential energy is
negligible even in vertical nozzle. 5
   inout
inout
inout zzg
VV
hhwq 






 

2
22

6. Governing equations
• The steady flow energy equation can be describe as:
• Applying the steady flow energy equation between section 1 and any
section X along the fluid flow along the nozzle passage.
6
outin
inout
hh
VV


2
22
 XX hhVV  1
2
1
2
2
 XX hhVV  1
2
1 2

7. Nozzle shape
• This part is to find the change in the nozzle cross-sectional area to increase
the fluid flow velocity to the required value.
• Consider a stream of fluid flow of inlet pressure and enthalpy of Pin and hin.
Assume the inlet velocity (Vin) is very small. Now describe the change of
the nozzle area to increase the fluid velocity.
• Apply the steady state momentum equation and replace the velocity with the
corresponding term from energy equation, and hence:
7

8. Nozzle shape
• The relation between the initial and final thermo-physical properties
depends on the thermodynamic process.
• Assume frictionless fluid flow + adiabatic flow = reversible adiabatic
(isentropic) process.
• At any section X, sx=s1=constant.
8
 X
XX
hhV
v
m
A


1
2
1 2
X
XX
v
VA
m 
X
XX
V
v
m
A



9. Nozzle shape
• At any section, by knowing the pressure and constant entropy, other
parameters can be determined.
9

10. Nozzle shape
• The following graph presents the effect of reducing gas pressure and the
influence on the cross-sectional area and flow velocity.
10
• It is observed that the area decreases
initially, hit the minimum at certain point
then increases again.
• The area decreases, when v-j
increases less rapidly than V-jj.
• The area increases, when v-jj
increases more rapidly than V-j.
X
XX
V
v
m
A



11. Nozzle shape (case study)
11
020406080100
FlowPressure (kPa)
Area
Velocity

12. Nozzle shape
• Based on the aforementioned information. That type of nozzle is called a
convergent-divergent nozzle (the following graph).
12

13. Nozzle shape
• The section of minimum area is called the throat of the nozzle.
• The velocity at the throat of a nozzle operating at its designed
pressure ratio is the velocity of sound at the throat conditions.
• The flow up to the throat is sub-sonic; the flow after the throat is
supersonic
13
• The specific volume of a liquid is
constant over a wide pressure range,
and therefore nozzles for liquids are
always convergent.

14. Nozzle critical pressure ratio
• You can design a convergent divergent nozzle where the velocity at
the nozzle through equal the sound velocity.
• The ratio of the pressure at the section of sonic velocity to the
pressure at nozzle inlet is called the critical pressure ratio Z c.
• Solve the energy and momentum equations between the inlet section
and any point along the nozzle passage.
• In most practical applications the inlet velocity is negligible, so the
energy equation can be reduced to.
14
 hhV  12 hhVV  1
2
1 2

15. Nozzle critical pressure ratio
• The enthalpy is usually expressed in kJ/kg. To find the flow velocity in m/s,
we need to convert the enthalpy to be J/kg.
• Substitute into the momentum equation.
15
     m/s72.442000 11 hhhhV 
 hh
v
m
A


172.44

16. Nozzle critical pressure ratio
• Apply for perfect gas (constant specific heats).
• For isentropic process.
16
   












1
1
11
172.44
72.4472.44
T
T
TCp
v
TTCp
v
hh
v
m
A






 








 1
11 P
P
T
T
RTPv 
P
RT
v 

17. Nozzle critical pressure ratio
• Let the pressure ratio (z)
17
1P
P
z 
 

1
11
1
1
172.44 




zTCpPz
zTR
m
A

        121221211
constantconstant
1
constant
1
constant









zzzzzzzzzm
A


18. Nozzle critical pressure ratio
• To find the value of pressure ratio zc, at which the area per mass flow is
minimum, the differentiation of that term should be equal to zero.
18
 
0
constant
12









  
zzdz
d
1
1 1
2
RatioPressureCritical











P
P
z c
c

19. Critical temperature ratio
• The ratio of temperature at the section where the sonic velocity is attained
to the inlet temperature is called the critical temperature ratio.
19
1
2
RatioeTemperaturCritical
1
11 











P
P
T
T cc

20. Critical Velocity
• To find the critical velocity
20
  





 122 1
1
T
T
TCpTTCpV
  ccc
c
cc TRTCpTCp
T
T
TCpV 













 

11
2
1
212 1

21. Example
• Air at 8.6 bar and 190°C expands at a rate of 4.5 kg/s through a convergent
divergent nozzle into a space at 1.03 bar. Assuming that the inlet velocity is
negligible. Determine the through and the exit cross-sectional area of the
nozzle.
21

22. Example
• The critical pressure, temperature, velocity and area = through area.
22
528.0
1
2 1
1










P
Pc KTTc 8.385
1
2
1 



)/(244.0 3
kgm
P
RT
v
c
c
c 
 
 sm
smTRV cc
/343SpeedSonic
/394

 
c
cc
v
VA
m 
 2
00279.0 m
V
vm
A
c
c
c 


