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Nozzles - Lecture A

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Ahmed Rezk

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.

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Thermodynamics 2 Fall Semester, 2014-2015 Lecture A Nozzles Dr. Ahmed Rezk 1
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
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 Constantm xoutin mmm 緒 m
Governing equations is the flowing fluid density (kg/m3) V is the velocity normal to the flow area (m/s) A the cross-sectional area (m2) 堂 fluid specific volume (m3/kg) 4 skg v AV AVm /緒 駕 x xx out outout in inin v VA v VA v VA 緒
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 緒 2 22
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
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
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 緒 X XX V v m A
Nozzle shape At any section, by knowing the pressure and constant entropy, other parameters can be determined. 9
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
Nozzle shape (case study) 11 020406080100 FlowPressure (kPa) Area Velocity
Nozzle shape Based on the aforementioned information. That type of nozzle is called a convergent-divergent nozzle (the following graph). 12
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.
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
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 緒 hh v m A 172.44
Nozzle critical pressure ratio Apply for perfect gas (constant specific heats). For isentropic process. 16 件 э 1 1 11 172.44 72.4472.44 T T TCp v TTCp v hh v m A 件 э 件 э 1 11 P P T T RTPv P RT v
Nozzle critical pressure ratio Let the pressure ratio (z) 17 1P P z э э 1 11 1 1 172.44 zTCpPz zTR m A ээээээээээээ 121221211 constantconstant 1 constant 1 constant zzzzzzzzzm A
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 ээ zzdz d 1 1 1 2 RatioPressureCritical 件 э 緒緒 P P z c c
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 緒件 э 緒 P P T T cc
Critical Velocity To find the critical velocity 20 緒 122 1 1 T T TCpTTCpV ccc c cc TRTCpTCp T T TCpV 緒緒 緒件 э 緒 э 11 2 1 212 1
Example Air at 8.6 bar and 190属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
Example The critical pressure, temperature, velocity and area = through area. 22 528.0 1 2 1 1 緒件 э P Pc KTTc 8.385 1 2 1 緒 )/(244.0 3 kgm P RT v c c c 緒 sm smTRV cc /343SpeedSonic /394 緒 c cc v VA m 緒 2 00279.0 m V vm A c c c

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Nozzles - Lecture A

  • 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 Constantm xoutin mmm 緒 m
  • 4. Governing equations is the flowing fluid density (kg/m3) V is the velocity normal to the flow area (m/s) A the cross-sectional area (m2) 堂 fluid specific volume (m3/kg) 4 skg v AV AVm /緒 駕 x xx out outout in inin v VA v VA v VA 緒
  • 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 緒 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 緒 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 緒 hh v m A 172.44
  • 16. Nozzle critical pressure ratio Apply for perfect gas (constant specific heats). For isentropic process. 16 件 э 1 1 11 172.44 72.4472.44 T T TCp v TTCp v hh v m A 件 э 件 э 1 11 P P T T RTPv P RT v
  • 17. Nozzle critical pressure ratio Let the pressure ratio (z) 17 1P P z э э 1 11 1 1 172.44 zTCpPz zTR m A ээээээээээээ 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 ээ zzdz d 1 1 1 2 RatioPressureCritical 件 э 緒緒 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 緒件 э 緒 P P T T cc
  • 20. Critical Velocity To find the critical velocity 20 緒 122 1 1 T T TCpTTCpV ccc c cc TRTCpTCp T T TCpV 緒緒 緒件 э 緒 э 11 2 1 212 1
  • 21. Example Air at 8.6 bar and 190属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 緒件 э P Pc KTTc 8.385 1 2 1 緒 )/(244.0 3 kgm P RT v c c c 緒 sm smTRV cc /343SpeedSonic /394 緒 c cc v VA m 緒 2 00279.0 m V vm A c c c