Compressible Flow Pressure Relief Valve

Calculate mass flow rate through a gas pressure relief valve with no header for isentropic and isothermal flow conditions.

The pressure relief valve is assumed to exit directly to ambient pressure (no header). If the ambient pressure is less than the critical pressure the flow is critical (Mc = 1 for isentropic flow and Mc = √(1/γ) for isothermal flow). If the ambient pressure is greater than the critical nozzle pressure, the flow is sub critical (M < Mc). Phase changes are ignored. The piping or pressure vessel is assumed to be at stagnation conditions (M = 0).

The nozzle orifice diameter and cross section area can be calculated from API letter designation (API 526 type D to T). For certified API valves the orifice size is an effective size. The combination of valve and nozzle must be tested with the operating fluid, and certified as having a flow rate greater than or equal to the calculated flow rate for the API size. API 526 assumes isentropic nozzle flow with the compressibility factor Z = 1. For other cases use the API size as an actual size with an appropriate discharge coefficient.

For isothermal flow the inlet temperature should be set equal to the estimated isothermal temperature (eg ambient temperature). The stagnation temperature is constant for isentropic flow, and varies with Mach number for isothermal flow. Use the discharge coefficient to compensate for friction losses and as a safety factor.

Use the Result Plot option to plot nozzle pressure versus stagnation pressure and flowtype, nozzle mach number versus stagnation pressure and flow type, mass flow rate versus stagnation pressure and flow type, mass flowrate versus nozzle area and discharge coefficient, or mass flowrate versus nozzle diameter and discharge coefficient.

Tool Input

• fluidtype : Fluid Type
  • γu : User Defined Specific Heat Ratio
  • SGu : User Defined Gas Specific Gravity
• antype : Nozzle Effective Area Type
  • Dnu : User Defined Nozzle Diameter
  • Anu : User Defined Nozzle Area
• cdtype : Discharge Coefficient Type
  • Cdu : User Defined Discharge Coefficient
• zfactype : Factor Type
  • Zu : User Defined Compressibility Factor
• flowtype : Fluid Flow Type
  • Tou : User Defined Isentropic Stagnation Temperature
  • Tiu : User Defined Isothermal Inlet Temperature
• Po : Stagnation Pressure
• Pa : Isothermal Pressure

Tool Output

• γ : Specific Heat Ratio
• ρn : Nozzle Fluid Density
• An : Nozzle Effective Area
• Cd : Nozzle Discharge Coefficient
• Cn : Nozzle Speed Of Sound
• Dn : Nozzle Effective Diameter
• Fn : Nozzle Reaction Force
• Gn : Nozzle Mass Flux
• Mcn : Critical Nozzle Mach Number
• Mn : Nozzle Mach Number
• Pn : Nozzle Pressure
• Rg : Specific Gas Constant
• SG : Gas Specific Gravity
• Tn : Nozzle Temperature
• Ton : Nozzle Stagnation Temperature
• Vn : Nozzle Fluid Velocity
• Z : Fluid Compressibility Factor
• cvg : Convergence Factor (≅ 1)
• mf : Mass Flowrate
• mmg : Gas Molar Mass
• nf : Mole Flowrate

Calculate critical mass flow rate through a pressure relief valve with no header for isentropic and isothermal flow conditions.

The pressure relief valve is assumed to exit directly to ambient pressure (no header). The flow is assumed to be critical (Mc = 1 for isentropic flow and Mc = √(1/γ) for isothermal flow). Phase changes are ignored. The piping or pressure vessel is assumed to be at stagnation conditions (M = 0).

For isothermal flow the inlet temperature should be set equal to the estimated isothermal temperature (eg ambient temperature). The stagnation temperature is constant for isentropic flow, and varies with Mach number for isothermal flow. Use the discharge coefficient to compensate for friction losses and as a safety factor. Check that the vent exit pressure is greater than or equal to ambient pressure. If the exit pressure is less than ambient pressure the flow is subcritical.

Tool Input

• flowtype : Fluid Flow Type
  • Tou : User Defined Isentropic Stagnation Temperature
  • Tiu : User Defined Isothermal Inlet Temperature
• Po : Stagnation Pressure
• γ : Specific heat ratio
• SG : Specific Gravity
• Z : Compressibility Factor
• Cd : Discharge Coefficient
• Dn : Nozzle Diameter

Tool Output

• An : Nozzle Area
• Mn : Nozzle Mach Number
• Pn : Nozzle Pressure
• Rg : Specific Gas Constant
• Tn : Nozzle Temperature
• mf : Nozzle Mass Flowrate
• nf : Nozzle Mole Flowrate

Calculate mass flow rate through a combined steam pressure relief valve and pressure relief header for isentropic and isothermal valve flow, and adiabatic and isothermal header flow.

Flow conditions can be calculated for

• Isentropic valve and adiabatic header
• Isentropic valve and isothermal header
• Isothermal valve and isothermal header

The pressure relief valve is assumed to exit directly into the header. If the header inlet pressure is less than or equal to the nozzle critical pressure the nozzle flow is critical (Mc = 1 for isentropic flow and Mc = √(1/γ) for isothermal flow), the header is over sized and the mass flow rate is restricted by the nozzle. The header inlet pressure is calculated so that the header mass flow rate equals the nozzle mass flow rate. An over sized header is recommended.

If the header inlet pressure is greater than the critical nozzle pressure, the nozzle flow is sub critical (M < Mc), the header is under sized and the mass flow rate is restricted by the header (or low stagnation pressure relative to ambient). The mass flow rate is calculated so that the header inlet pressure is equal to the nozzle pressure. The mass flow rate through the nozzle is always equal to the mass flow rate through the header.

Header pressure losses are calculated from the pressure loss factor (fld = fL/D + K). The Darcy friction factor fd is calculated for fully turbulent flow using the rough pipe equation. The header is assumed to be constant diameter. Minor losses can be included by the minor loss K factor or an equivalent length, and should include valves and bends etc. The header entry and exit losses should not be included (the fluid dynamic pressure loss is included in the calculation). If the ambient pressure is less than the critical header pressure the header exit flow is critical. If the ambient pressure is greater than the critical header pressure, the header exit flow is sub critical (M < Mc). The header entry flow is assumed to be sub critical for all flow conditions.

The nozzle orifice diameter and cross section area can be calculated from API letter designation (API 526 type D to T). For API certified valves, the orifice size is an effective size (the actual area is greater than the effective area). The combination of valve and nozzle must be tested with the operating fluid, and certified as having a flow rate greater than or equal to the calculated flow rate for the effective area. For API certified nozzles use the Isentropic nozzle options with the compressibility factor Z = 1. For other cases use the API size as an actual size with an appropriate discharge coefficient.

For isothermal flow the inlet temperature should be set equal to the estimated isothermal temperature (eg ambient temperature). The stagnation temperature is constant for isentropic/adiabatic flow, and varies with Mach number for isothermal flow. Phase changes are ignored. Use the multiple valve options if the header is shared by two or more pressure relief valves. Use the valve and header discharge coefficients to compensate for valve friction losses and as a safety factor. The discharge coefficients can be set independently, or set equal (the header coefficient equals the valve coefficient). Use the steam table to calculate a suitable value for the specific heat ratio γ, and the compressibility factor Z. For super heated steam γ = 1.334 can be used as an estimate.

Use the Result Plot option to plot nozzle, header inlet and header exit pressure versus stagnation pressure, nozzle, header inlet and header exit mach number versus stagnation pressure, mass flow rate versus stagnation pressure and flow type, or mass flow rate versus stagnation pressure and discharge coefficient.

Tool Input

• schdtype : Header Schedule Type
• diamtype : Header Diameter Type
  • ODu : User Defined Header Outside Diameter
  • IDu : User Defined Header Inside Diameter
• wtntype : Wall Thickness Type
  • tnu : User Defined Header Wall Thickness
• rfactype : Header Internal Roughness Type
  • ru : User Defined Surface Roughness
  • rru : User Defined Relative Roughness
• fdtype : Darcy Friction Factor Type
  • fdu : User Defined Darcy Friction Factor
• leqtype : Minor Pressure Loss Type
  • ku : User Defined Minor Loss K Factor
  • lu : User Defined Minor Loss Length
  • lodu : User Defined Minor Loss Diameters (L/ID)
  • fL/Du : User Defined Pressure Loss Factor
• numtype : Number Of PRV's Type
  • Nu : User Defined Number Of PRVs
• fluidtype : Specific Heat Ratio Type
  • γu : User Defined Specific Heat Ratio
• antype : Nozzle Effective Area Type
  • Dnu : User Defined Nozzle Diameter
  • Anu : User Defined Nozzle Area
• cdtypen : Valve Discharge Coefficient Type
  • Cdvu : User Defined Valve Discharge Coefficient
• cdtypeh : Header Discharge Coefficient Type
  • Cdhu : User Defined Header Discharge Coefficient
• zfactype : Factor Type
  • Zu : User Defined Compressibility Factor
• flowtype : Fluid Flow Type
  • Tou : User Defined Isentropic Stagnation Temperature
  • Tiu : User Defined Isothermal Temperature
• Po : Stagnation Pressure
• Pa : Ambient Pressure At Exit
• L : Header Length

Tool Output

• γ : Specific Heat Ratio
• ρe : Header Exit Density
• ρi : Header Inlet Density
• ρn : Nozzle Density
• An : Nozzle Effective Area
• Cdh : Header Discharge Coefficient
• Cdv : Valve Discharge Coefficient
• Ce : Header Exit Speed Of Sound
• Ci : Header Inlet Speed Of Sound
• Cn : Nozzle Speed Of Sound
• Dn : Nozzle Effective Diameter
• Fe : Combined Exit Reaction Force (All PRVs)
• Fn : Nozzle Reaction Force
• Ge : Header Exit Mass Flux
• Gi : Header Inlet Mass Flux
• Gn : Nozzle Mass Flux
• ID : Header Inside Diameter
• Le : Header Eqivalent Length
• Mce : Header Critical Exit Mach Number
• Mci : Header Critical Inlet Mach Number
• Mcn : Nozzle Critical Mach Number
• Me : Header Exit Mach Number
• Mi : Header Inlet Mach Number
• Mn : Nozzle Mach Number
• N : Number Of PRV's Sharing The Header
• Pe : Header Exit Pressure
• Pi : Header Inlet Pressure
• Pn : Nozzle Pressure
• Rg : Specific Gas Constant
• SG : Gas Specific Gravity
• Te : Header Exit Temperature
• Ti : Header Inlet Temperature
• Tn : Nozzle Temperature
• Toe : Header Exit Stagnation Temperature
• Ve : Header Exit Velocity
• Vi : Header Inlet Velocity
• Vn : Nozzle Fluid Velocity
• Z : Fluid Compressibility Factor
• cvg : Convergence Factor (≅ 1)
• fL/D : Pressure Loss Factor Including Minor Losses
• fd : Darcy Friction Factor
• mh : Header Mass Flowrate Per PRV
• mmg : Gas Molar Mass
• mn : Nozzle Mass Flowrate
• mt : Total Mass Flowrate N PRV's
• nn : Nozzle Mole Flowrate
• nt : Total Mole Flow Rate N PRV's
• rr : Surface Roughness Ratio

Calculate gas header frictional flow critical ratios for adiabatic (constant enthalpy) and isothermal (constant temperature) flow.

Critical flow conditions occur when the exit Mach number equals the critical Mach number and the critical exit pressure is greater than or equal to ambient pressure. For adiabatic flow the critical exit Mach number = 1 (eg sonic flow conditions). For isothermal flow the critical exit Mach number = √γ.

Use the Result Plot option to plot critical pressure loss factor versus inlet Mach number and either flow type or specific heat ratio; or pressure ratio, temperature ratio, density ratio, speed of sound ratio, and velocity ratio, versus either Mach number or pressure loss factor, and versus either flow type or specific heat ratio (Fanno lines). The plot range is set from the calculated value of either the inlet Mach number, or the pressure loss factor in the main calculation. Change the main calculation values to change the plot range.

Tool Input

• fluidtype : Fluid Type
  • γu : User Defined Specific Heat Ratio
• fldtype : Pressure Loss Factor Type
  • Mciu : User Defined Inlet Mach Number
  • fL/Du : User Defined Pressure Loss Factor
• flowtype : Fluid Flow Type

Tool Output

• γ : Specific Heat Ratio
• ρi/ρ* : Inlet Density Over Critical Exit Density Ratio
• Ci/C* : Inlet Speed Of Sound Over Critical Exit Speed Of Sound Ratio
• M* : Critical Exit Mach Number
• Mi : Inlet Mach Number
• Pi/P* : Inlet Pressure Over Critical Exit Pressure Ratio
• Ti/T* : Inlet Temperature Over Critical Exit Temperature Ratio
• Vi/V* : Inlet Velocity Over Critical Velocity Ratio
• cvg : Convergence Factor (≅ 1)
• fL/D* : Critical Duct Pressure Loss factor

Calculate gas header critical and subcritical flow ratios for adiabatic and isothermal flow with friction.

Critical flow conditions occur when the exit Mach number equals the critical Mach number and the critical exit pressure is greater than or equal to ambient pressure. For adiabatic flow the critical exit Mach number = 1 (eg sonic flow conditions). For isothermal flow the critical exit Mach number = √γ. Sub critical duct flow occurs when the ambient pressure at exit is greater than the critical exit pressure, and the exit Mach number is less than the critical exit Mach number.

For sub critical flow an equivalent pressure loss factor is calculated by adding an imaginary extension to the duct. The inlet pressure of the extension equals ambient pressure. The exit pressure of the duct plus extension the equals exit pressure of the extension. The exit Mach number of the original duct equals the inlet Mach number of the extension. The equivalent pressure loss factor is the pressure loss factor for the duct and the extension. For critical flow the equivalent pressure loss factor equals the duct pressure loss factor (the added pressure loss factor = 0).

Use the Result Plot option to plot either inlet and exit Mach number versus pressure loss factor, or added and equivalent pressure loss factor versus pressure loss factor.

Tool Input

• fluidtype : Fluid Type
  • γu : User Defined Specific Heat Ratio
• flowtype : Fluid Flow Type
• Pi : Inlet Pressure
• Pa : Amient Pressure
• fL/D : Input Friction Loss Factor

Tool Output

• γ : Specific Heat Ratio
• ρi/ρe : Inlet Density Over Exit Density Ratio
• Ci/Ce : Inlet Speed Of Sound Over Exit Speed Of Sound Ratio
• Mce : Critical Exit Mach Number
• Mci : Critical Inlet Mach Number
• Me : Exit Mach Number
• Mi : Inlet Mach Number
• Pe : Exit Pressure
• Pi/Pe : Inlet Pressure Over Exit Pressure Ratio
• Ti/Te : Inlet Temperature Over Exit Temperature Ratio
• Vi/Ve : Inlet Velocity Over Exit Velocity Ratio
• cvg : Convergence Factor (≅ 1)
• fL/Da : Added Friction Loss Factor
• fL/De : Effective Friction Loss Factor

Calculate mass flow rate through a gas pressure relief header for adiabatic and isothermal flow conditions.

The pressure relief header is assumed to be part of a pressure relief system, and connected to the outlet of a pressure relief valve, relief vent, or a smaller diameter relief header. The header inlet pressure should be set equal to exit pressure of the upstream relief valve, relief vent, or relief header. The header should be sized so that the calculated header mass flowrate is greater than or equal to the valve, or vent mass flowrate. For headers with more than one inlet, the header mass flowrate is divided by the number of inlets. The calculated header flowrate is the maximum possible value. The actual mass flowrate will always be equal to the upstream flowrate and the actual header inlet pressure will be less than or equal to the upstream exit pressure.

For adiabatic flow, the inlet temperature is calculated from the upstream stagnation temperature (the inlet temperature varies with the inlet Mach number). For isothermal flow the inlet temperature should be set equal to the estimated isothermal temperature (eg ambient temperature). The stagnation temperature is constant for adiabatic flow, and varies with Mach number for isothermal flow. Phase changes are ignored.

Header pressure losses are calculated from the pressure loss factor (fld = fL/D + K). The Darcy friction factor f is calculated for fully turbulent flow using the rough pipe equation. The header is assumed to be constant diameter. Minor losses can be included by the minor loss K factor or an equivalent length, and should include valves and bends etc. The header entry and exit losses should not be included (the fluid dynamic pressure loss is included in the calculation). The discharge coefficient can also be used for minor losses, and as a safety factor. If the ambient pressure is less than the critical exit pressure the header exit flow is critical (Mc = 1 for isentropic flow and Mc = √(1/γ) for isothermal flow). If the ambient pressure is greater than the critical exit pressure, the header exit flow is sub critical (M < Mc). The header entry flow is assumed to be sub critical for all flow conditions.

Use the Result Plot option to plot inlet and exit pressure versus stagnation pressure, inlet and exit mach number versus stagnation pressure, or mass flow rate versus stagnation pressure and flow type.

Tool Input

• schdtype : Header Schedule Type
• diamtype : Header Diameter Type
  • ODu : User Defined Header Outside Diameter
  • IDu : User Defined Header Inside Diameter
• wtntype : Wall Thickness Type
  • tnu : User Defined Header Wall Thickness
• rfactype : Header Internal Roughness Type
  • ru : User Defined Surface Roughness
  • rru : User Defined Relative Roughness
• fdtype : Darcy Friction Factor Type
  • fdu : User Defined Darcy Friction Factor
• leqtype : Minor Pressure Loss Type
  • ku : User Defined Minor Loss K Factor
  • lu : User Defined Minor Loss Length
  • lodu : User Defined Minor Loss Diameters (L/ID)
  • fL/Du : User Defined Pressure Loss Factor
• numtype : Number Of Inlets Type
  • Nu : User Defined Number Of Inlets
• fluidtype : Fluid Type
  • γu : User Defined Specific Heat Ratio
  • SGu : User Defined Gas Specific Gravity
• zfactype : Compressibility Factor Type
  • Zu : User Defined Compressibility Factor
• cdtype : Discharge Coefficient Type
  • Cdu : User Defined Discharge Coefficient
• flowtype : Flow Type
  • Tou : User Defined Adiabatic Stagnation Temperature
  • Tiu : User Defined Isothermal Inlet Temperature
• Pi : Header Inlet Pressure
• Pa : Ambient Pressure (At Exit)
• L : Header Length

Tool Output

• γ : Specific Heat Ratio
• ρe : Header Exit Density
• ρi : Header Inlet Density
• Cd : Discharge Coefficient
• Ce : Header Exit Speed Of Sound
• Ci : Header Inlet Speed Of Sound
• Fe : Combined Exit Reaction Force (All PRVs)
• Ge : Header Exit Mass Flux
• Gi : Header Inlet Mass Flux
• ID : Header Inside Diameter
• Le : Header Eqivalent Length
• Mce : Header Critical Exit Mach Number
• Mci : Header Critical Inlet Mach Number
• Me : Header Exit Mach Number
• Mi : Header Inlet Mach Number
• N : Number Of PRV's Sharing The Header
• Pe : Header Exit Pressure
• Rg : Specific Gas Constant
• SG : Gas Specific Gravity Relative To Air
• Te : Header Exit Temperature
• Ti : Header Inlet Temperature
• Toe : Exit Stagnation Temperature
• Ve : Header Exit Velocity
• Vi : Header Inlet Velocity
• Z : Compressibility Factor
• cvg : Convergence Factor (≅ 1)
• fL/D : Pressure Loss Factor Including Minor Losses
• fL/Da : Added Friction Loss Factor
• fL/De : Effective Friction Loss Factor
• fd : Darcy Friction Factor
• mf : Mass Flowrate Per PRV
• mmg : Gas Molar Mass
• nf : Mole Flow Rate Per PRV
• rr : Surface Roughness Ratio

Calculate critical mass flow rate through a pressure relief header for adiabatic and isothermal flow conditions.

The pressure relief header is assumed to be part of a pressure relief system, and connected to the outlet of a pressure relief valve, relief vent, or a smaller diameter relief header. The header inlet pressure should be set equal to exit pressure of the upstream relief valve, relief vent, or relief header. The header should be sized so that the calculated header mass flowrate is greater than or equal to the valve, or vent mass flowrate. The calculated header flowrate is the maximum possible value. The actual mass flowrate will always be equal to the upstream flowrate and the actual header inlet pressure will be less than or equal to the upstream exit pressure.

For adiabatic flow, the inlet temperature is calculated from the upstream stagnation temperature (the inlet temperature varies with the inlet Mach number). For isothermal flow the inlet temperature should be set equal to the estimated isothermal temperature (eg ambient temperature). The stagnation temperature is constant for adiabatic flow, and varies with Mach number for isothermal flow. Phase changes are ignored.

Header pressure losses are calculated from the pressure loss factor (fld = fL/D + K). Minor losses can accounted for in either the pressure loss factor, or the discharge coefficient. The header exit flow is assumed to be critical (Mc = 1 for isentropic flow and Mc = √(1/γ) for isothermal flow). The header entry flow is sub critical for all flow conditions. Check that the vent exit pressure is greater than or equal to ambient pressure. If the exit pressure is less than ambient pressure the flow is subcritical.

Tool Input

• flowtype : Flow Type
  • Tou : User Defined Adiabatic Stagnation Temperature
  • Tiu : User Defined Isothermal Inlet Temperature
• Pi : Header Inlet Pressure
• γ : Spefic Heat Ratio
• SG : Specific Gravity
• Z : Compressibility Factor
• Cd : Discharge Coefficient
• ID : Header Internal Diameter
• fL/ID : Pressure Loss factor

Tool Output

• Me : Header Exit Mach Number
• Mi : Header Inlet Mach Number
• Pe : Header Exit Pressure
• Rg : Specific Gas Constant
• Te : Header Exit Temperature
• Ti : Header Inlet Temperature
• mf : Header Mass Flowrate
• nf : Header Mole Flow Rate