Source Models Vapor flow through holes and pipes

Source Models Vapor flow through holes and pipes

Vapor flow though holes & pipes Vapor flow through holes Steady flow of vapor through pipes Example

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

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

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

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.

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

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

Throttling flow Take partial Definition of Joule-Thomsen coefficient

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

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

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

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

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

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

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

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

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.

Choked flow through holes For choked, critical or sonic flow So at choked conditions Eq. 4 -40 For sharp edged orifice C 0=0. 61, Worst case scenario C 0=1. 0

Choked flow through holes Gas Pchoked Monotonic ~1. 67 0. 487 P 0 Diatomic (air) ~1. 40 0. 528 P 0 Triatomic ~1. 32 0. 542 P 0

Vapor flow though holes & pipes Vapor flow through holes Steady flow of vapor through pipes n Adiabatic flow of vapor through pipes w Non choked flows w Choked flows n Isothermal flow of vapor through pipes w Non choked flows w Choked flows Example

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

Vapor flow though holes & pipes Vapor flow through holes Steady flow of vapor through pipes n Adiabatic flow of vapor through pipes w Non choked flows w Choked flows n Isothermal flow of vapor through pipes w Non choked flows w Choked flows Example

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

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.

Adiabatic vapor flow through pipes For most problems you know n n L – length of pipe d – diameter of pipe T 1, P 1 – upstream temperature, pressure P 2 – downstream pressure To get mass flow rate Qm (mass/time) from G, mass flux, (mass/area*time) use Qm=G*A

Adiabatic non choked flows in pipes 1) Find pipe roughness from Table 4 -1 2) Determine f from Eq. 4 -27 3) Determine T 2 from Eq. 4 -51 (trial & error) 4) Calculation G from Eq. 4 -52 5) Calculate Reynolds number to verify Eq 4 -27 is valid

Adiabatic Choked flows in pipes 1) Find roughness from Table 4 -1 2) Determine f from Eq 4 -27 3) Determine Ma 1 from Eq 4 -57 (use 4 -46 to get Y 1) (usually trial & error) 4) Determine mass flux, Gchoked Eq. 4 -56 5) Determine Pchoked from Eq 4 -54 6) Double check Reynolds number

Vapor flow though holes & pipes Vapor flow through holes Steady flow of vapor through pipes n n Adiabatic flow of vapor through pipes Isothermal flow of vapor through pipes w Non choked flows w Choked flows Example

Isothermal non choked flows 1) 2) 3) 4) 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

Isothermal choked flows 1) Find roughness from Table 4 -1 2) Find f from Eq. 4 -27 3) Determine Ma 1 from Eq. 4 -71 (trial and error) 4) Determine G from Eq. 4 -70 5) Double check the Reynolds number

Vapor flow though holes & pipes Vapor flow through holes Steady flow of vapor through pipes Example