Source ModelsVapor flow through holes and pipes

Vapor flow though holes & pipesVapor flow through holes

Steady flow of vapor through pipes

Example

Liquid versus Vapor flowLiquids Incompressible flow

Kinetic energy term is negligible

Physical properties (density) constantVapors Compressible flowEnergy from pressure converted to kinetic energyTemperature, pressure, density all change when going through a hole or down a pipe

Vapor flow though holes & pipesVapor flow through holesThrottling releaseFree ExpansionNon choked or subsonicChoked, critical or sonicSteady flow of vapor through pipesExample

Vapor flow through holesThrottling flowSmall cracks large frictional losesNot much energy due to pressure is converted to kineticModels require detailed information on physical structure of leak

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

Throttling flowFirst law of thermodynamics

Assume Steady stateAdiabaticNegligible potential and Kinetic energy effectsSingle inlet and outletNo shaft work

Throttling flowHence the process is isenthalpic

Consider the temperature as a function of pressure and enthalpy

Throttling flowTake partial

Definition of Joule-Thomsen coefficient

Throttling flowIf isenthalpic then

Integrate out

Most gases have positive Joule-Thomsen coefficient so as pressure drops, temperature drops

Vapor flow though holes & pipesVapor flow through holesThrottling releaseFree ExpansionNon choked or subsonicChoked, critical or sonicSteady flow of vapor through pipesExample

Vapor flow through holesFree Expansion

AssumeNegligible potential (Z=0)

No shaft workWs=0

Vapor flow through holesMechanical Energy Balance

Friction through hole is defined as before

Vapor flow through holesNeed to have density as a function of pressure to solve integral Assume isentropic flow

Vapor flow through holesSubstitute all into MEB and integrateYou end up with velocity as function of several terms

As before, mass flow rate from velocity

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

Vapor flow though holes & pipesVapor flow through holesThrottling releaseFree ExpansionNon choked or subsonicChoked, critical or sonic

Steady flow of vapor through pipesExample

Choked flow through holesAs 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 holesFor 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

Choked flow through holes

GasPchokedMonotonic~1.670.487P0

Diatomic (air)~1.400.528P0

Triatomic~1.320.542P0

Vapor flow though holes & pipesVapor flow through holesSteady flow of vapor through pipesAdiabatic flow of vapor through pipesNon choked flowsChoked flowsIsothermal flow of vapor through pipesNon choked flowsChoked flowsExample

Vapor flow through pipesThere 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)

Vapor flow though holes & pipesVapor flow through holesSteady flow of vapor through pipesAdiabatic flow of vapor through pipesNon choked flowsChoked flowsIsothermal flow of vapor through pipesNon choked flowsChoked flowsExample

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

Adiabatic vapor flow through pipesThe book doesnt 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 pipesFor most problems you knowL length of piped diameter of pipeT1, P1 upstream temperature, pressureP2 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 pipesFind pipe roughness from Table 4-1Determine f from Eq. 4-27

Determine T2 from Eq. 4-51 (trial & error)Calculation G from Eq. 4-52Calculate Reynolds number to verify Eq 4-27 is valid

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

Vapor flow though holes & pipesVapor flow through holesSteady flow of vapor through pipesAdiabatic flow of vapor through pipesIsothermal flow of vapor through pipesNon choked flowsChoked flowsExample

Isothermal non choked flowsFind roughness from Table 4-1Determine f from Eq. 4-27Compute G from Eq. 4-63Double check Reynolds number

For isothermal non choked flow no need for trial and error, nice analytical equations

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

Vapor flow though holes & pipesVapor flow through holes

Steady flow of vapor through pipes

Example