Source Models

1. Source Models Vapor flow through holes and pipes

2. Vapor flow though holes & pipes • Vapor flow through holes • Steady flow of vapor through pipes • Example

3. Liquids – Incompressible flow Kinetic energy term is negligible Physical properties (density) constant Vapors – Compressible flow Energy from pressure converted to kinetic energy Temperature, pressure, density all change when going through a hole or down a pipe Liquid versus Vapor flow

4. Vapor flow though holes & pipes • Vapor flow through holes • Throttling release • Free Expansion • Non choked or subsonic • Choked, critical or sonic • Steady flow of vapor through pipes • Example

5. Vapor flow through holes • Throttling flow • Small cracks – large frictional loses • Not much energy due to pressure is converted to kinetic • Models require detailed information on physical structure of leak

6. Throttling flow • A throttling device is a valve or crack or porous material with high resistance to flow that results in a large pressure drop.

7. Throttling flow First law of thermodynamics Assume Steady state Adiabatic Negligible potential and Kinetic energy effects Single inlet and outlet No shaft work

8. Throttling flow Hence the process is isenthalpic Consider the temperature as a function of pressure and enthalpy

9. Throttling flow Take partial Definition of Joule-Thomsen coefficient

10. Throttling flow If isenthalpic then Integrate out Most gases have positive Joule-Thomsen coefficient so as pressure drops, temperature drops

11. Vapor flow though holes & pipes • Vapor flow through holes • Throttling release • Free Expansion • Non choked or subsonic • Choked, critical or sonic • Steady flow of vapor through pipes • Example

12. Vapor flow through holes • Free Expansion Assume Negligible potential (ΔZ=0) No shaft work Ws=0

13. Vapor flow through holes • Mechanical Energy Balance • Friction through “hole” is defined as before

14. Vapor flow through holes • Need to have density as a function of pressure to solve integral – Assume isentropic flow

15. Vapor flow through holes • Substitute all into MEB and integrate You end up with velocity as function of several terms As before, mass flow rate from velocity

16. Vapor flow through holes • Design equation for subsonic flow through holes Eq. 4-38

17. Vapor flow though holes & pipes • Vapor flow through holes • Throttling release • Free Expansion • Non choked or subsonic • Choked, critical or sonic • Steady flow of vapor through pipes • Example

18. Choked flow through holes • As you lower the down stream pressure (or increase upstream pressure) the velocity increases until it reaches a critical velocity, the sonic velocity, or speed of sound. • After that the velocity becomes independent of pressure. Downstream conditions no longer have an effect on velocity.

19. Choked flow through holes • For choked, critical or sonic flow • So at choked conditions Eq. 4-40 • For sharp edged orifice C0=0.61, Worst case scenario C0=1.0

20. Choked flow through holes Gas  Pchoked Monotonic ~1.67 0.487P0 Diatomic (air) ~1.40 0.528P0 Triatomic ~1.32 0.542P0

21. Vapor flow though holes & pipes • Vapor flow through holes • Steady flow of vapor through pipes • Adiabatic flow of vapor through pipes • Non choked flows • Choked flows • Isothermal flow of vapor through pipes • Non choked flows • Choked flows • Example

22. Vapor flow through pipes • There are two cases which we can derive (with much work) relationships for flow of vapors through pipes • Adiabatic – which assumes well insulated walls, no energy loss to surroundings • Isothermal – which assumes constant wall temperature (submerged pipe)

23. Vapor flow though holes & pipes • Vapor flow through holes • Steady flow of vapor through pipes • Adiabatic flow of vapor through pipes • Non choked flows • Choked flows • Isothermal flow of vapor through pipes • Non choked flows • Choked flows • Example

24. Adiabatic vapor flow in pipes • For compressible flow it is best to work things out in terms of the Mach number, Ma.

25. Adiabatic vapor flow through pipes • The book doesn’t even attempt to go through the derivations, just gives the equations. • As before, we need to consider both nonchoked and choked flow.

26. Adiabatic vapor flow through pipes • For most problems you know • L – length of pipe • d – diameter of pipe • T1, P1 – upstream temperature, pressure • P2 – downstream pressure • To get mass flow rate Qm (mass/time) from G, mass flux, (mass/area*time) use Qm=G*A

27. Adiabatic non choked flows in pipes • Find pipe roughness from Table 4-1 • Determine f from Eq. 4-27 • Determine T2 from Eq. 4-51 (trial & error) • Calculation G from Eq. 4-52 • Calculate Reynolds number to verify Eq 4-27 is valid

28. Adiabatic Choked flows in pipes • Find roughness from Table 4-1 • Determine f from Eq 4-27 • Determine Ma1 from Eq 4-57 (use 4-46 to get Y1) (usually trial & error) • Determine mass flux, Gchoked Eq. 4-56 • Determine Pchoked from Eq 4-54 • Double check Reynolds number

29. Vapor flow though holes & pipes • Vapor flow through holes • Steady flow of vapor through pipes • Adiabatic flow of vapor through pipes • Isothermal flow of vapor through pipes • Non choked flows • Choked flows • Example

30. Isothermal non choked flows • Find roughness from Table 4-1 • Determine f from Eq. 4-27 • Compute G from Eq. 4-63 • Double check Reynolds number For isothermal non choked flow no need for trial and error, nice analytical equations

31. Isothermal choked flows • Find roughness from Table 4-1 • Find f from Eq. 4-27 • Determine Ma1 from Eq. 4-71 (trial and error) • Determine G from Eq. 4-70 • Double check the Reynolds number

32. Vapor flow though holes & pipes • Vapor flow through holes • Steady flow of vapor through pipes • Example