Compressible

Overview

Compressible flow describes fluid motion where density changes are significant—typically when gas velocities approach or exceed the speed of sound. Unlike incompressible flow where density remains constant, compressible flow exhibits complex coupling between velocity, pressure, temperature, and density through thermodynamic relationships. This behavior dominates in high-speed aerodynamics, gas pipeline networks, turbomachinery, and rocket nozzles.

The defining characteristic of compressible flow is the Mach number, M = V/c, where V is flow velocity and c is the local speed of sound. When M < 0.3, density changes remain below 5% and incompressible assumptions suffice. As Mach number increases, compressibility effects intensify: shock waves form, flow can become choked (reaching sonic conditions), and thermodynamic processes dictate pressure-temperature relationships.

Implementation: These tools leverage thermodynamic calculations through SciPy for property evaluation and implement classical gas dynamics correlations for pipeline flow. The tools handle both idealized thermodynamic processes (isentropic, isothermal, polytropic) and practical empirical formulas developed for natural gas transmission.

Thermodynamic Processes govern how gases behave during compression, expansion, and flow through varying areas. Isentropic processes assume reversible adiabatic conditions—ideal for analyzing turbomachinery and nozzles where heat transfer is negligible and losses are small. The ISENTROPIC_WORK, ISENTROPIC_T_RISE, and ISENTROPIC_EFF tools quantify compression work, temperature changes, and efficiency conversions for compressors and turbines. Isothermal processes assume constant temperature—applicable to slow compression with heat removal or long pipelines where gas temperature equilibrates with surroundings. The ISOTHERMAL_WORK and ISOTHERMAL_GAS tools handle these conditions. Polytropic processes represent real compression with some heat transfer, captured by the POLYTROPIC_EXP tool which relates efficiency to the polytropic exponent.

Stagnation Properties measure total energy available in a flow by accounting for both static conditions and kinetic energy. P_STAGNATION and T_STAGNATION convert between static and stagnation states, essential for performance analysis of compressors, turbines, and wind tunnels. The STAGNATION_ENERGY quantifies the enthalpy rise from velocity, while T_STAG_IDEAL provides the theoretical temperature rise.

Critical Flow Conditions occur when flow reaches Mach 1, establishing maximum flow rates that cannot increase regardless of further downstream pressure reduction—a phenomenon called choking. The IS_CHOKED_FLOW tool determines whether flow has reached critical conditions based on pressure ratio, while P_CRITICAL_FLOW and T_CRITICAL_FLOW calculate the exact pressure and temperature at the sonic condition. This behavior governs relief valve sizing, nozzle design, and pipeline capacity limits.

Pipeline Flow Formulas provide empirical correlations for natural gas transmission developed from extensive field data. These formulas account for friction, compressibility, and non-ideal gas behavior in long pipelines. WEYMOUTH_FLOW applies to large-diameter, high-pressure transmission lines with turbulent flow. PANHANDLE_A and PANHANDLE_B suit partially turbulent flow in moderate-diameter lines, with Panhandle B providing better accuracy for modern smooth pipes. FRITZSCHE_FLOW, MULLER_FLOW, and IGT_FLOW offer alternative formulations calibrated to specific pipeline conditions and gas properties.

Figure 1 illustrates the relationship between pressure ratio and Mach number in isentropic flow, and the efficiency conversions between isentropic and polytropic processes in compression.

Figure 1: Compressible Flow Fundamentals: (A) Pressure ratio vs. Mach number in isentropic flow (γ=1.4), showing the critical pressure ratio of 0.528 where flow becomes choked. (B) Isentropic vs. polytropic efficiency for compression at varying pressure ratios, demonstrating how real processes deviate from ideal behavior.

FRITZSCHE_FLOW

This function estimates steady natural-gas pipeline flow at reference conditions using the Fritzsche correlation in SI units. It is commonly used when inlet/outlet pressure, pipe geometry, gas specific gravity, and average temperature are known.

Q \propto \left(\frac{P_1^2-P_2^2}{L\,SG^{0.8587}\,T_{avg}}\right)^{0.538} D^{2.69}

Excel Usage

=FRITZSCHE_FLOW(sg, temp_avg, length, diameter, p_inlet, p_outlet, comp_factor, efficiency)
  • sg (float, required): Specific gravity of gas relative to air [-]
  • temp_avg (float, required): Average temperature of gas in pipeline [K]
  • length (float, required): Length of pipe [m]
  • diameter (float, required): Diameter of pipe [m]
  • p_inlet (float, required): Inlet pressure [Pa]
  • p_outlet (float, required): Outlet pressure [Pa]
  • comp_factor (float, optional, default: 1): Average compressibility factor [-]
  • efficiency (float, optional, default: 1): Pipeline efficiency factor [-]

Returns (float): Gas flow rate at reference conditions [m³/s]

Example 1: Standard natural gas pipeline

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.693 277.15 160000 0.34 9000000 2000000

Excel formula:

=FRITZSCHE_FLOW(0.693, 277.15, 160000, 0.34, 9000000, 2000000)

Expected output:

39.4215

Example 2: Short pipeline section

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.65 290 10000 0.5 5000000 4500000

Excel formula:

=FRITZSCHE_FLOW(0.65, 290, 10000, 0.5, 5000000, 4500000)

Expected output:

111.043

Example 3: Pipeline with 92% efficiency

Inputs:

sg temp_avg length diameter p_inlet p_outlet efficiency
0.7 280 50000 0.4 7000000 3000000 0.92

Excel formula:

=FRITZSCHE_FLOW(0.7, 280, 50000, 0.4, 7000000, 3000000, 0.92)

Expected output:

73.0675

Example 4: Large diameter pipeline

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.6 285 100000 1 8000000 4000000

Excel formula:

=FRITZSCHE_FLOW(0.6, 285, 100000, 1, 8000000, 4000000)

Expected output:

754.802

Python Code

Show Code 
from fluids.compressible import Fritzsche as fluids_fritzsche

def fritzsche_flow(sg, temp_avg, length, diameter, p_inlet, p_outlet, comp_factor=1, efficiency=1):
    """
    Calculate gas flow rate using the Fritzsche formula.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.Fritzsche

    This example function is provided as-is without any representation of accuracy.

    Args:
        sg (float): Specific gravity of gas relative to air [-]
        temp_avg (float): Average temperature of gas in pipeline [K]
        length (float): Length of pipe [m]
        diameter (float): Diameter of pipe [m]
        p_inlet (float): Inlet pressure [Pa]
        p_outlet (float): Outlet pressure [Pa]
        comp_factor (float, optional): Average compressibility factor [-] Default is 1.
        efficiency (float, optional): Pipeline efficiency factor [-] Default is 1.

    Returns:
        float: Gas flow rate at reference conditions [m³/s]
    """
    try:
        sg = float(sg)
        temp_avg = float(temp_avg)
        length = float(length)
        diameter = float(diameter)
        p_inlet = float(p_inlet)
        p_outlet = float(p_outlet)
        comp_factor = float(comp_factor)
        efficiency = float(efficiency)

        if sg <= 0:
            return "Error: Invalid input: sg must be positive."
        if temp_avg <= 0:
            return "Error: Invalid input: temp_avg must be positive."
        if length <= 0:
            return "Error: Invalid input: length must be positive."
        if diameter <= 0:
            return "Error: Invalid input: diameter must be positive."
        if p_inlet <= 0:
            return "Error: Invalid input: p_inlet must be positive."
        if p_outlet < 0:
            return "Error: Invalid input: p_outlet must be non-negative."
        if p_outlet >= p_inlet:
            return "Error: Invalid input: p_outlet must be less than p_inlet."
        if comp_factor <= 0:
            return "Error: Invalid input: comp_factor must be positive."
        if efficiency <= 0 or efficiency > 1:
            return "Error: Invalid input: efficiency must be between 0 and 1."

        result = fluids_fritzsche(SG=sg, Tavg=temp_avg, L=length, D=diameter, 
                                   P1=p_inlet, P2=p_outlet, Zavg=comp_factor, E=efficiency)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Specific gravity of gas relative to air [-]
Average temperature of gas in pipeline [K]
Length of pipe [m]
Diameter of pipe [m]
Inlet pressure [Pa]
Outlet pressure [Pa]
Average compressibility factor [-]
Pipeline efficiency factor [-]

IGT_FLOW

This function computes compressible-gas pipeline flow using the IGT empirical correlation with viscosity effects included. It is suitable for steady-state throughput estimation when pressure drop, line geometry, and gas properties are known.

Q \propto \left(\frac{P_1^2-P_2^2}{L\,Z_{avg}\,T_{avg}}\right)^{5/9}\frac{D^{8/3}}{\mu^{1/9}SG^{4/9}}

Excel Usage

=IGT_FLOW(sg, temp_avg, viscosity, length, diameter, p_inlet, p_outlet, comp_factor, efficiency)
  • sg (float, required): Specific gravity of gas relative to air [-]
  • temp_avg (float, required): Average temperature of gas in pipeline [K]
  • viscosity (float, required): Average viscosity of gas [Pa·s]
  • length (float, required): Length of pipe [m]
  • diameter (float, required): Diameter of pipe [m]
  • p_inlet (float, required): Inlet pressure [Pa]
  • p_outlet (float, required): Outlet pressure [Pa]
  • comp_factor (float, optional, default: 1): Average compressibility factor [-]
  • efficiency (float, optional, default: 1): Pipeline efficiency factor [-]

Returns (float): Gas flow rate at reference conditions [m³/s]

Example 1: Standard natural gas pipeline

Inputs:

sg temp_avg viscosity length diameter p_inlet p_outlet
0.693 277.15 0.00001 160000 0.34 9000000 2000000

Excel formula:

=IGT_FLOW(0.693, 277.15, 0.00001, 160000, 0.34, 9000000, 2000000)

Expected output:

48.9235

Example 2: Short pipeline section

Inputs:

sg temp_avg viscosity length diameter p_inlet p_outlet
0.65 290 0.000012 10000 0.5 5000000 4500000

Excel formula:

=IGT_FLOW(0.65, 290, 0.000012, 10000, 0.5, 5000000, 4500000)

Expected output:

133.548

Example 3: Pipeline with 92% efficiency

Inputs:

sg temp_avg viscosity length diameter p_inlet p_outlet efficiency
0.7 280 0.00001 50000 0.4 7000000 3000000 0.92

Excel formula:

=IGT_FLOW(0.7, 280, 0.00001, 50000, 0.4, 7000000, 3000000, 0.92)

Expected output:

91.1455

Example 4: Large diameter pipeline

Inputs:

sg temp_avg viscosity length diameter p_inlet p_outlet
0.6 285 0.000011 100000 1 8000000 4000000

Excel formula:

=IGT_FLOW(0.6, 285, 0.000011, 100000, 1, 8000000, 4000000)

Expected output:

901.065

Python Code

Show Code 
from fluids.compressible import IGT as fluids_igt

def igt_flow(sg, temp_avg, viscosity, length, diameter, p_inlet, p_outlet, comp_factor=1, efficiency=1):
    """
    Calculate gas flow rate using the IGT (Institute of Gas Technology) formula.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.IGT

    This example function is provided as-is without any representation of accuracy.

    Args:
        sg (float): Specific gravity of gas relative to air [-]
        temp_avg (float): Average temperature of gas in pipeline [K]
        viscosity (float): Average viscosity of gas [Pa·s]
        length (float): Length of pipe [m]
        diameter (float): Diameter of pipe [m]
        p_inlet (float): Inlet pressure [Pa]
        p_outlet (float): Outlet pressure [Pa]
        comp_factor (float, optional): Average compressibility factor [-] Default is 1.
        efficiency (float, optional): Pipeline efficiency factor [-] Default is 1.

    Returns:
        float: Gas flow rate at reference conditions [m³/s]
    """
    try:
        sg = float(sg)
        temp_avg = float(temp_avg)
        viscosity = float(viscosity)
        length = float(length)
        diameter = float(diameter)
        p_inlet = float(p_inlet)
        p_outlet = float(p_outlet)
        comp_factor = float(comp_factor)
        efficiency = float(efficiency)

        if sg <= 0:
            return "Error: Invalid input: sg must be positive."
        if temp_avg <= 0:
            return "Error: Invalid input: temp_avg must be positive."
        if viscosity <= 0:
            return "Error: Invalid input: viscosity must be positive."
        if length <= 0:
            return "Error: Invalid input: length must be positive."
        if diameter <= 0:
            return "Error: Invalid input: diameter must be positive."
        if p_inlet <= 0:
            return "Error: Invalid input: p_inlet must be positive."
        if p_outlet < 0:
            return "Error: Invalid input: p_outlet must be non-negative."
        if p_outlet >= p_inlet:
            return "Error: Invalid input: p_outlet must be less than p_inlet."
        if comp_factor <= 0:
            return "Error: Invalid input: comp_factor must be positive."
        if efficiency <= 0 or efficiency > 1:
            return "Error: Invalid input: efficiency must be between 0 and 1."

        result = fluids_igt(SG=sg, Tavg=temp_avg, mu=viscosity, L=length, D=diameter, 
                             P1=p_inlet, P2=p_outlet, Zavg=comp_factor, E=efficiency)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Specific gravity of gas relative to air [-]
Average temperature of gas in pipeline [K]
Average viscosity of gas [Pa·s]
Length of pipe [m]
Diameter of pipe [m]
Inlet pressure [Pa]
Outlet pressure [Pa]
Average compressibility factor [-]
Pipeline efficiency factor [-]

IS_CHOKED_FLOW

This function checks whether downstream pressure is low enough to produce choked flow for a given upstream pressure and isentropic exponent. It returns a boolean indicating whether sonic conditions are reached at the restriction.

\frac{P_2}{P_1}\le \left(\frac{2}{k+1}\right)^{k/(k-1)}

Excel Usage

=IS_CHOKED_FLOW(p_upstream, p_downstream, k_isentropic)
  • p_upstream (float, required): Higher, source pressure [Pa]
  • p_downstream (float, required): Lower, downstream pressure [Pa]
  • k_isentropic (float, required): Isentropic exponent [-]

Returns (bool): True if flow is choked (critical), False otherwise

Example 1: API 520 Example 1 - not choked

Inputs:

p_upstream p_downstream k_isentropic
670000 532000 1.11

Excel formula:

=IS_CHOKED_FLOW(670000, 532000, 1.11)

Expected output:

false

Example 2: API 520 Example 2 - choked flow

Inputs:

p_upstream p_downstream k_isentropic
670000 101000 1.11

Excel formula:

=IS_CHOKED_FLOW(670000, 101000, 1.11)

Expected output:

true

Example 3: Near critical pressure ratio

Inputs:

p_upstream p_downstream k_isentropic
1000000 500000 1.4

Excel formula:

=IS_CHOKED_FLOW(1000000, 500000, 1.4)

Expected output:

true

Example 4: High pressure ratio - definitely choked

Inputs:

p_upstream p_downstream k_isentropic
5000000 100000 1.4

Excel formula:

=IS_CHOKED_FLOW(5000000, 100000, 1.4)

Expected output:

true

Python Code

Show Code 
from fluids.compressible import is_critical_flow as fluids_is_crit

def is_choked_flow(p_upstream, p_downstream, k_isentropic):
    """
    Determine if a flow is choked (critical) based on pressure ratio.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.is_critical_flow

    This example function is provided as-is without any representation of accuracy.

    Args:
        p_upstream (float): Higher, source pressure [Pa]
        p_downstream (float): Lower, downstream pressure [Pa]
        k_isentropic (float): Isentropic exponent [-]

    Returns:
        bool: True if flow is choked (critical), False otherwise
    """
    try:
        p_upstream = float(p_upstream)
        p_downstream = float(p_downstream)
        k_isentropic = float(k_isentropic)

        if p_upstream <= 0:
            return "Error: Invalid input: p_upstream must be positive."
        if p_downstream < 0:
            return "Error: Invalid input: p_downstream must be non-negative."
        if k_isentropic <= 1:
            return "Error: Invalid input: k_isentropic must be greater than 1."
        if p_downstream >= p_upstream:
            return "Error: Invalid input: p_downstream must be less than p_upstream."

        result = fluids_is_crit(P1=p_upstream, P2=p_downstream, k=k_isentropic)
        return bool(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Higher, source pressure [Pa]
Lower, downstream pressure [Pa]
Isentropic exponent [-]

ISENTROPIC_EFF

This function converts polytropic efficiency to isentropic efficiency for a specified pressure ratio and heat-capacity ratio. It is used in compressor performance calculations where one efficiency definition is known but the other is required.

\eta_s=\frac{(P_2/P_1)^{(k-1)/k}-1}{(P_2/P_1)^{(k-1)/(k\eta_p)}-1}

Excel Usage

=ISENTROPIC_EFF(p_inlet, p_outlet, k_isentropic, eta_poly)
  • p_inlet (float, required): Initial pressure of gas [Pa]
  • p_outlet (float, required): Final pressure of gas [Pa]
  • k_isentropic (float, required): Isentropic exponent of the gas (Cp/Cv) [-]
  • eta_poly (float, required): Polytropic efficiency of the process [-]

Returns (float): Isentropic efficiency calculated from polytropic efficiency [-]

Example 1: 10x compression with 78% polytropic efficiency

Inputs:

p_inlet p_outlet k_isentropic eta_poly
100000 1000000 1.4 0.78

Excel formula:

=ISENTROPIC_EFF(100000, 1000000, 1.4, 0.78)

Expected output:

0.702761

Example 2: 5x compression with 85% polytropic efficiency

Inputs:

p_inlet p_outlet k_isentropic eta_poly
100000 500000 1.4 0.85

Excel formula:

=ISENTROPIC_EFF(100000, 500000, 1.4, 0.85)

Expected output:

0.813456

Example 3: High efficiency compressor

Inputs:

p_inlet p_outlet k_isentropic eta_poly
100000 300000 1.3 0.92

Excel formula:

=ISENTROPIC_EFF(100000, 300000, 1.3, 0.92)

Expected output:

0.909473

Example 4: Different gas (lower k)

Inputs:

p_inlet p_outlet k_isentropic eta_poly
100000 400000 1.2 0.8

Excel formula:

=ISENTROPIC_EFF(100000, 400000, 1.2, 0.8)

Expected output:

0.776255

Python Code

Show Code 
from fluids.compressible import isentropic_efficiency as fluids_isen_eff

def isentropic_eff(p_inlet, p_outlet, k_isentropic, eta_poly):
    """
    Convert between isentropic and polytropic efficiency for compression.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.isentropic_efficiency

    This example function is provided as-is without any representation of accuracy.

    Args:
        p_inlet (float): Initial pressure of gas [Pa]
        p_outlet (float): Final pressure of gas [Pa]
        k_isentropic (float): Isentropic exponent of the gas (Cp/Cv) [-]
        eta_poly (float): Polytropic efficiency of the process [-]

    Returns:
        float: Isentropic efficiency calculated from polytropic efficiency [-]
    """
    try:
        p_inlet = float(p_inlet)
        p_outlet = float(p_outlet)
        k_isentropic = float(k_isentropic)
        eta_poly = float(eta_poly)

        if p_inlet <= 0:
            return "Error: Invalid input: p_inlet must be positive."
        if p_outlet <= 0:
            return "Error: Invalid input: p_outlet must be positive."
        if k_isentropic <= 1:
            return "Error: Invalid input: k_isentropic must be greater than 1."
        if eta_poly <= 0 or eta_poly > 1:
            return "Error: Invalid input: eta_poly must be between 0 and 1."

        result = fluids_isen_eff(P1=p_inlet, P2=p_outlet, k=k_isentropic, eta_p=eta_poly)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Initial pressure of gas [Pa]
Final pressure of gas [Pa]
Isentropic exponent of the gas (Cp/Cv) [-]
Polytropic efficiency of the process [-]

ISENTROPIC_T_RISE

This function returns the outlet temperature for a gas undergoing isentropic or polytropic compression/expansion, optionally corrected by efficiency. It is useful for estimating thermal loading in compressors and expanders.

T_2=T_1\left[1+\frac{1}{\eta}\left(\left(\frac{P_2}{P_1}\right)^{(k-1)/k}-1\right)\right]

Excel Usage

=ISENTROPIC_T_RISE(temp_init, p_inlet, p_outlet, k_isentropic, efficiency)
  • temp_init (float, required): Initial temperature of gas [K]
  • p_inlet (float, required): Initial pressure of gas [Pa]
  • p_outlet (float, required): Final pressure of gas [Pa]
  • k_isentropic (float, required): Isentropic exponent of the gas (Cp/Cv) [-]
  • efficiency (float, optional, default: 1): Isentropic efficiency of the process [-]

Returns (float): Final temperature of gas after compression/expansion [K]

Example 1: Compression with 8x pressure ratio

Inputs:

temp_init p_inlet p_outlet k_isentropic
286.8 54050 432400 1.4

Excel formula:

=ISENTROPIC_T_RISE(286.8, 54050, 432400, 1.4)

Expected output:

519.523

Example 2: Compression with 80% efficiency

Inputs:

temp_init p_inlet p_outlet k_isentropic efficiency
300 100000 500000 1.4 0.8

Excel formula:

=ISENTROPIC_T_RISE(300, 100000, 500000, 1.4, 0.8)

Expected output:

518.932

Example 3: Expansion cooling

Inputs:

temp_init p_inlet p_outlet k_isentropic
500 1000000 100000 1.4

Excel formula:

=ISENTROPIC_T_RISE(500, 1000000, 100000, 1.4)

Expected output:

258.974

Example 4: Small pressure ratio

Inputs:

temp_init p_inlet p_outlet k_isentropic
300 100000 150000 1.3

Excel formula:

=ISENTROPIC_T_RISE(300, 100000, 150000, 1.3)

Expected output:

329.426

Python Code

Show Code 
from fluids.compressible import isentropic_T_rise_compression as fluids_t_rise

def isentropic_t_rise(temp_init, p_inlet, p_outlet, k_isentropic, efficiency=1):
    """
    Calculate the temperature rise for isentropic compression or expansion.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.isentropic_T_rise_compression

    This example function is provided as-is without any representation of accuracy.

    Args:
        temp_init (float): Initial temperature of gas [K]
        p_inlet (float): Initial pressure of gas [Pa]
        p_outlet (float): Final pressure of gas [Pa]
        k_isentropic (float): Isentropic exponent of the gas (Cp/Cv) [-]
        efficiency (float, optional): Isentropic efficiency of the process [-] Default is 1.

    Returns:
        float: Final temperature of gas after compression/expansion [K]
    """
    try:
        temp_init = float(temp_init)
        p_inlet = float(p_inlet)
        p_outlet = float(p_outlet)
        k_isentropic = float(k_isentropic)
        efficiency = float(efficiency)

        if temp_init <= 0:
            return "Error: Invalid input: temp_init must be positive."
        if p_inlet <= 0:
            return "Error: Invalid input: p_inlet must be positive."
        if p_outlet <= 0:
            return "Error: Invalid input: p_outlet must be positive."
        if k_isentropic <= 1:
            return "Error: Invalid input: k_isentropic must be greater than 1."
        if efficiency <= 0 or efficiency > 1:
            return "Error: Invalid input: efficiency must be between 0 and 1."

        result = fluids_t_rise(T1=temp_init, P1=p_inlet, P2=p_outlet, k=k_isentropic, eta=efficiency)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Initial temperature of gas [K]
Initial pressure of gas [Pa]
Final pressure of gas [Pa]
Isentropic exponent of the gas (Cp/Cv) [-]
Isentropic efficiency of the process [-]

ISENTROPIC_WORK

This function computes molar compression/expansion work for an isentropic (or polytropic-style) gas process with optional efficiency correction. Positive values indicate compression work input and negative values indicate expansion.

W=\frac{k}{k-1}ZRT_1\left[\left(\frac{P_2}{P_1}\right)^{(k-1)/k}-1\right]\frac{1}{\eta}

Excel Usage

=ISENTROPIC_WORK(temp_init, k_isentropic, p_inlet, p_outlet, comp_factor, efficiency)
  • temp_init (float, required): Initial temperature of the gas [K]
  • k_isentropic (float, required): Isentropic exponent of the gas (Cp/Cv) [-]
  • p_inlet (float, required): Inlet pressure [Pa]
  • p_outlet (float, required): Outlet pressure [Pa]
  • comp_factor (float, optional, default: 1): Compressibility factor of the gas [-]
  • efficiency (float, optional, default: 1): Isentropic efficiency of the process [-]

Returns (float): Work performed per mole of gas compressed/expanded [J/mol]

Example 1: Compression with 10x pressure ratio

Inputs:

temp_init k_isentropic p_inlet p_outlet
300 1.4 100000 1000000

Excel formula:

=ISENTROPIC_WORK(300, 1.4, 100000, 1000000)

Expected output:

8125.16

Example 2: Compression with 78% efficiency

Inputs:

temp_init k_isentropic p_inlet p_outlet efficiency
300 1.4 100000 1000000 0.78

Excel formula:

=ISENTROPIC_WORK(300, 1.4, 100000, 1000000, 0.78)

Expected output:

10416.9

Example 3: Expansion (negative work)

Inputs:

temp_init k_isentropic p_inlet p_outlet
300 1.4 1000000 100000

Excel formula:

=ISENTROPIC_WORK(300, 1.4, 1000000, 100000)

Expected output:

-4208.41

Example 4: Small pressure ratio compression

Inputs:

temp_init k_isentropic p_inlet p_outlet
350 1.3 100000 200000

Excel formula:

=ISENTROPIC_WORK(350, 1.3, 100000, 200000)

Expected output:

2187.38

Python Code

Show Code 
from fluids.compressible import isentropic_work_compression as fluids_isentropic_work

def isentropic_work(temp_init, k_isentropic, p_inlet, p_outlet, comp_factor=1, efficiency=1):
    """
    Calculate work of compression or expansion for a gas in an isentropic process.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.isentropic_work_compression

    This example function is provided as-is without any representation of accuracy.

    Args:
        temp_init (float): Initial temperature of the gas [K]
        k_isentropic (float): Isentropic exponent of the gas (Cp/Cv) [-]
        p_inlet (float): Inlet pressure [Pa]
        p_outlet (float): Outlet pressure [Pa]
        comp_factor (float, optional): Compressibility factor of the gas [-] Default is 1.
        efficiency (float, optional): Isentropic efficiency of the process [-] Default is 1.

    Returns:
        float: Work performed per mole of gas compressed/expanded [J/mol]
    """
    try:
        temp_init = float(temp_init)
        k_isentropic = float(k_isentropic)
        p_inlet = float(p_inlet)
        p_outlet = float(p_outlet)
        comp_factor = float(comp_factor)
        efficiency = float(efficiency)

        if temp_init <= 0:
            return "Error: Invalid input: temp_init must be positive."
        if k_isentropic <= 1:
            return "Error: Invalid input: k_isentropic must be greater than 1."
        if p_inlet <= 0:
            return "Error: Invalid input: p_inlet must be positive."
        if p_outlet <= 0:
            return "Error: Invalid input: p_outlet must be positive."
        if comp_factor <= 0:
            return "Error: Invalid input: comp_factor must be positive."
        if efficiency <= 0 or efficiency > 1:
            return "Error: Invalid input: efficiency must be between 0 and 1."

        result = fluids_isentropic_work(T1=temp_init, k=k_isentropic, P1=p_inlet, P2=p_outlet, Z=comp_factor, eta=efficiency)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Initial temperature of the gas [K]
Isentropic exponent of the gas (Cp/Cv) [-]
Inlet pressure [Pa]
Outlet pressure [Pa]
Compressibility factor of the gas [-]
Isentropic efficiency of the process [-]

ISOTHERMAL_GAS

This function computes mass flow rate for compressible isothermal gas transport in a pipe using pressure-drop, friction, and geometry terms. It is useful for quick steady-state pipeline sizing and operating checks.

\dot m^2=\frac{\left(\frac{\pi D^2}{4}\right)^2\rho(P_1^2-P_2^2)}{P_1\left(f_d\frac{L}{D}+2\ln\frac{P_1}{P_2}\right)}

Excel Usage

=ISOTHERMAL_GAS(rho, fd, p_inlet, p_outlet, length, diameter)
  • rho (float, required): Average density of gas in pipe [kg/m³]
  • fd (float, required): Darcy friction factor [-]
  • p_inlet (float, required): Inlet pressure [Pa]
  • p_outlet (float, required): Outlet pressure [Pa]
  • length (float, required): Length of pipe [m]
  • diameter (float, required): Diameter of pipe [m]

Returns (float): Mass flow rate of gas through pipe [kg/s]

Example 1: Standard gas pipeline flow

Inputs:

rho fd p_inlet p_outlet length diameter
11.3 0.00185 1000000 900000 1000 0.5

Excel formula:

=ISOTHERMAL_GAS(11.3, 0.00185, 1000000, 900000, 1000, 0.5)

Expected output:

145.485

Example 2: High pressure drop case

Inputs:

rho fd p_inlet p_outlet length diameter
10 0.02 2000000 1000000 5000 0.3

Excel formula:

=ISOTHERMAL_GAS(10, 0.02, 2000000, 1000000, 5000, 0.3)

Expected output:

14.9636

Example 3: Long pipeline with low friction

Inputs:

rho fd p_inlet p_outlet length diameter
8.5 0.015 5000000 4000000 50000 0.6

Excel formula:

=ISOTHERMAL_GAS(8.5, 0.015, 5000000, 4000000, 50000, 0.6)

Expected output:

31.2756

Example 4: Small diameter pipe

Inputs:

rho fd p_inlet p_outlet length diameter
15 0.025 800000 700000 100 0.1

Excel formula:

=ISOTHERMAL_GAS(15, 0.025, 800000, 700000, 100, 0.1)

Expected output:

2.62035

Python Code

Show Code 
from fluids.compressible import isothermal_gas as fluids_iso_gas

def isothermal_gas(rho, fd, p_inlet, p_outlet, length, diameter):
    """
    Calculate mass flow rate for isothermal compressible gas flow in a pipe.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.isothermal_gas

    This example function is provided as-is without any representation of accuracy.

    Args:
        rho (float): Average density of gas in pipe [kg/m³]
        fd (float): Darcy friction factor [-]
        p_inlet (float): Inlet pressure [Pa]
        p_outlet (float): Outlet pressure [Pa]
        length (float): Length of pipe [m]
        diameter (float): Diameter of pipe [m]

    Returns:
        float: Mass flow rate of gas through pipe [kg/s]
    """
    try:
        rho = float(rho)
        fd = float(fd)
        p_inlet = float(p_inlet)
        p_outlet = float(p_outlet)
        length = float(length)
        diameter = float(diameter)

        if rho <= 0:
            return "Error: Invalid input: rho must be positive."
        if fd <= 0:
            return "Error: Invalid input: fd must be positive."
        if p_inlet <= 0:
            return "Error: Invalid input: p_inlet must be positive."
        if p_outlet < 0:
            return "Error: Invalid input: p_outlet must be non-negative."
        if p_outlet >= p_inlet:
            return "Error: Invalid input: p_outlet must be less than p_inlet."
        if length <= 0:
            return "Error: Invalid input: length must be positive."
        if diameter <= 0:
            return "Error: Invalid input: diameter must be positive."

        result = fluids_iso_gas(rho=rho, fd=fd, P1=p_inlet, P2=p_outlet, L=length, D=diameter)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Average density of gas in pipe [kg/m³]
Darcy friction factor [-]
Inlet pressure [Pa]
Outlet pressure [Pa]
Length of pipe [m]
Diameter of pipe [m]

ISOTHERMAL_WORK

This function calculates the ideal isothermal molar work for gas compression or expansion at constant temperature with a compressibility correction. It is a lower-bound work estimate for practical multistage compression systems.

W=ZRT\ln\left(\frac{P_2}{P_1}\right)

Excel Usage

=ISOTHERMAL_WORK(p_inlet, p_outlet, temp, comp_factor)
  • p_inlet (float, required): Inlet pressure [Pa]
  • p_outlet (float, required): Outlet pressure [Pa]
  • temp (float, required): Temperature of the gas [K]
  • comp_factor (float, optional, default: 1): Compressibility factor of the gas, [-]

Returns (float): Work performed per mole of gas compressed/expanded [J/mol]

Example 1: Compression with 10x pressure ratio

Inputs:

p_inlet p_outlet temp
100000 1000000 300

Excel formula:

=ISOTHERMAL_WORK(100000, 1000000, 300)

Expected output:

5743.43

Example 2: Expansion (negative work)

Inputs:

p_inlet p_outlet temp
1000000 100000 300

Excel formula:

=ISOTHERMAL_WORK(1000000, 100000, 300)

Expected output:

-5743.43

Example 3: Compression with compressibility factor

Inputs:

p_inlet p_outlet temp comp_factor
100000 1000000 300 0.95

Excel formula:

=ISOTHERMAL_WORK(100000, 1000000, 300, 0.95)

Expected output:

5456.26

Example 4: Small pressure ratio compression

Inputs:

p_inlet p_outlet temp
100000 150000 350

Excel formula:

=ISOTHERMAL_WORK(100000, 150000, 350)

Expected output:

1179.93

Python Code

Show Code 
from fluids.compressible import isothermal_work_compression as fluids_isothermal_work

def isothermal_work(p_inlet, p_outlet, temp, comp_factor=1):
    """
    Calculate work of compression or expansion for a gas in an isothermal process.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.isothermal_work_compression

    This example function is provided as-is without any representation of accuracy.

    Args:
        p_inlet (float): Inlet pressure [Pa]
        p_outlet (float): Outlet pressure [Pa]
        temp (float): Temperature of the gas [K]
        comp_factor (float, optional): Compressibility factor of the gas, [-] Default is 1.

    Returns:
        float: Work performed per mole of gas compressed/expanded [J/mol]
    """
    try:
        p_inlet = float(p_inlet)
        p_outlet = float(p_outlet)
        temp = float(temp)
        comp_factor = float(comp_factor)

        if p_inlet <= 0:
            return "Error: Invalid input: p_inlet must be positive."
        if p_outlet <= 0:
            return "Error: Invalid input: p_outlet must be positive."
        if temp <= 0:
            return "Error: Invalid input: temp must be positive."
        if comp_factor <= 0:
            return "Error: Invalid input: comp_factor must be positive."

        result = fluids_isothermal_work(P1=p_inlet, P2=p_outlet, T=temp, Z=comp_factor)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Inlet pressure [Pa]
Outlet pressure [Pa]
Temperature of the gas [K]
Compressibility factor of the gas, [-]

MULLER_FLOW

This function estimates gas flow in pipelines with the Muller correlation, which includes viscosity and specific-gravity corrections. It is used for steady operating-point calculations from geometry and pressure conditions.

Q \propto \left(\frac{P_1^2-P_2^2}{L\,Z_{avg}\,T_{avg}}\right)^{0.575}\frac{D^{2.725}}{\mu^{0.15}SG^{0.425}}

Excel Usage

=MULLER_FLOW(sg, temp_avg, viscosity, length, diameter, p_inlet, p_outlet, comp_factor, efficiency)
  • sg (float, required): Specific gravity of gas relative to air [-]
  • temp_avg (float, required): Average temperature of gas in pipeline [K]
  • viscosity (float, required): Average viscosity of gas [Pa·s]
  • length (float, required): Length of pipe [m]
  • diameter (float, required): Diameter of pipe [m]
  • p_inlet (float, required): Inlet pressure [Pa]
  • p_outlet (float, required): Outlet pressure [Pa]
  • comp_factor (float, optional, default: 1): Average compressibility factor [-]
  • efficiency (float, optional, default: 1): Pipeline efficiency factor [-]

Returns (float): Gas flow rate at reference conditions [m³/s]

Example 1: Standard natural gas pipeline

Inputs:

sg temp_avg viscosity length diameter p_inlet p_outlet
0.693 277.15 0.00001 160000 0.34 9000000 2000000

Excel formula:

=MULLER_FLOW(0.693, 277.15, 0.00001, 160000, 0.34, 9000000, 2000000)

Expected output:

60.458

Example 2: Short pipeline section

Inputs:

sg temp_avg viscosity length diameter p_inlet p_outlet
0.65 290 0.000012 10000 0.5 5000000 4500000

Excel formula:

=MULLER_FLOW(0.65, 290, 0.000012, 10000, 0.5, 5000000, 4500000)

Expected output:

167.198

Example 3: Pipeline with 92% efficiency

Inputs:

sg temp_avg viscosity length diameter p_inlet p_outlet efficiency
0.7 280 0.00001 50000 0.4 7000000 3000000 0.92

Excel formula:

=MULLER_FLOW(0.7, 280, 0.00001, 50000, 0.4, 7000000, 3000000, 0.92)

Expected output:

114.836

Example 4: Large diameter pipeline

Inputs:

sg temp_avg viscosity length diameter p_inlet p_outlet
0.6 285 0.000011 100000 1 8000000 4000000

Excel formula:

=MULLER_FLOW(0.6, 285, 0.000011, 100000, 1, 8000000, 4000000)

Expected output:

1177.44

Python Code

Show Code 
from fluids.compressible import Muller as fluids_muller

def muller_flow(sg, temp_avg, viscosity, length, diameter, p_inlet, p_outlet, comp_factor=1, efficiency=1):
    """
    Calculate gas flow rate using the Muller formula.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.Muller

    This example function is provided as-is without any representation of accuracy.

    Args:
        sg (float): Specific gravity of gas relative to air [-]
        temp_avg (float): Average temperature of gas in pipeline [K]
        viscosity (float): Average viscosity of gas [Pa·s]
        length (float): Length of pipe [m]
        diameter (float): Diameter of pipe [m]
        p_inlet (float): Inlet pressure [Pa]
        p_outlet (float): Outlet pressure [Pa]
        comp_factor (float, optional): Average compressibility factor [-] Default is 1.
        efficiency (float, optional): Pipeline efficiency factor [-] Default is 1.

    Returns:
        float: Gas flow rate at reference conditions [m³/s]
    """
    try:
        sg = float(sg)
        temp_avg = float(temp_avg)
        viscosity = float(viscosity)
        length = float(length)
        diameter = float(diameter)
        p_inlet = float(p_inlet)
        p_outlet = float(p_outlet)
        comp_factor = float(comp_factor)
        efficiency = float(efficiency)

        if sg <= 0:
            return "Error: Invalid input: sg must be positive."
        if temp_avg <= 0:
            return "Error: Invalid input: temp_avg must be positive."
        if viscosity <= 0:
            return "Error: Invalid input: viscosity must be positive."
        if length <= 0:
            return "Error: Invalid input: length must be positive."
        if diameter <= 0:
            return "Error: Invalid input: diameter must be positive."
        if p_inlet <= 0:
            return "Error: Invalid input: p_inlet must be positive."
        if p_outlet < 0:
            return "Error: Invalid input: p_outlet must be non-negative."
        if p_outlet >= p_inlet:
            return "Error: Invalid input: p_outlet must be less than p_inlet."
        if comp_factor <= 0:
            return "Error: Invalid input: comp_factor must be positive."
        if efficiency <= 0 or efficiency > 1:
            return "Error: Invalid input: efficiency must be between 0 and 1."

        result = fluids_muller(SG=sg, Tavg=temp_avg, mu=viscosity, L=length, D=diameter, 
                                P1=p_inlet, P2=p_outlet, Zavg=comp_factor, E=efficiency)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Specific gravity of gas relative to air [-]
Average temperature of gas in pipeline [K]
Average viscosity of gas [Pa·s]
Length of pipe [m]
Diameter of pipe [m]
Inlet pressure [Pa]
Outlet pressure [Pa]
Average compressibility factor [-]
Pipeline efficiency factor [-]

P_CRITICAL_FLOW

This function computes the critical static pressure corresponding to Mach 1 from stagnation pressure for an isentropic gas. It is used to evaluate choking limits in nozzles and compressible restrictions.

\frac{P^*}{P_0}=\left(\frac{2}{k+1}\right)^{k/(k-1)}

Excel Usage

=P_CRITICAL_FLOW(p_stag, k_isentropic)
  • p_stag (float, required): Stagnation pressure of a fluid with Ma=1 [Pa]
  • k_isentropic (float, required): Isentropic exponent [-]

Returns (float): Critical flow pressure at Ma=1 [Pa]

Example 1: Textbook example (P=1.4MPa, k=1.289)

Inputs:

p_stag k_isentropic
1400000 1.289

Excel formula:

=P_CRITICAL_FLOW(1400000, 1.289)

Expected output:

766813

Example 2: Air at standard conditions (k=1.4)

Inputs:

p_stag k_isentropic
500000 1.4

Excel formula:

=P_CRITICAL_FLOW(500000, 1.4)

Expected output:

264141

Example 3: High pressure nozzle

Inputs:

p_stag k_isentropic
5000000 1.3

Excel formula:

=P_CRITICAL_FLOW(5000000, 1.3)

Expected output:

2728640

Example 4: Steam conditions (k=1.135)

Inputs:

p_stag k_isentropic
1000000 1.135

Excel formula:

=P_CRITICAL_FLOW(1000000, 1.135)

Expected output:

577430

Python Code

Show Code 
from fluids.compressible import P_critical_flow as fluids_p_crit

def p_critical_flow(p_stag, k_isentropic):
    """
    Calculate critical flow pressure for a fluid at Mach 1.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.P_critical_flow

    This example function is provided as-is without any representation of accuracy.

    Args:
        p_stag (float): Stagnation pressure of a fluid with Ma=1 [Pa]
        k_isentropic (float): Isentropic exponent [-]

    Returns:
        float: Critical flow pressure at Ma=1 [Pa]
    """
    try:
        p_stag = float(p_stag)
        k_isentropic = float(k_isentropic)

        if p_stag <= 0:
            return "Error: Invalid input: p_stag must be positive."
        if k_isentropic <= 1:
            return "Error: Invalid input: k_isentropic must be greater than 1."

        result = fluids_p_crit(P=p_stag, k=k_isentropic)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Stagnation pressure of a fluid with Ma=1 [Pa]
Isentropic exponent [-]

P_STAGNATION

This function calculates stagnation pressure from static pressure and the static-to-stagnation temperature ratio for an isentropic gas. It is commonly used in compressible-flow diagnostics and nozzle calculations.

\frac{P_0}{P}=\left(\frac{T_0}{T}\right)^{k/(k-1)}

Excel Usage

=P_STAGNATION(pressure, temp, temp_stag, k_isentropic)
  • pressure (float, required): Normal (static) pressure of a fluid [Pa]
  • temp (float, required): Normal (static) temperature of a fluid [K]
  • temp_stag (float, required): Stagnation temperature [K]
  • k_isentropic (float, required): Isentropic exponent [-]

Returns (float): Stagnation pressure [Pa]

Example 1: Textbook example from Cengel

Inputs:

pressure temp temp_stag k_isentropic
54050 255.7 286.8 1.4

Excel formula:

=P_STAGNATION(54050, 255.7, 286.8, 1.4)

Expected output:

80772.8

Example 2: High speed flow

Inputs:

pressure temp temp_stag k_isentropic
100000 250 300 1.4

Excel formula:

=P_STAGNATION(100000, 250, 300, 1.4)

Expected output:

189293

Example 3: Low speed flow (small temperature rise)

Inputs:

pressure temp temp_stag k_isentropic
101325 290 295 1.4

Excel formula:

=P_STAGNATION(101325, 290, 295, 1.4)

Expected output:

107572

Example 4: Different gas (lower k)

Inputs:

pressure temp temp_stag k_isentropic
200000 300 350 1.3

Excel formula:

=P_STAGNATION(200000, 300, 350, 1.3)

Expected output:

390061

Python Code

Show Code 
from fluids.compressible import P_stagnation as fluids_p_stag

def p_stagnation(pressure, temp, temp_stag, k_isentropic):
    """
    Calculate stagnation pressure from static conditions.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.P_stagnation

    This example function is provided as-is without any representation of accuracy.

    Args:
        pressure (float): Normal (static) pressure of a fluid [Pa]
        temp (float): Normal (static) temperature of a fluid [K]
        temp_stag (float): Stagnation temperature [K]
        k_isentropic (float): Isentropic exponent [-]

    Returns:
        float: Stagnation pressure [Pa]
    """
    try:
        pressure = float(pressure)
        temp = float(temp)
        temp_stag = float(temp_stag)
        k_isentropic = float(k_isentropic)

        if pressure <= 0:
            return "Error: Invalid input: pressure must be positive."
        if temp <= 0:
            return "Error: Invalid input: temp must be positive."
        if temp_stag <= 0:
            return "Error: Invalid input: temp_stag must be positive."
        if k_isentropic <= 1:
            return "Error: Invalid input: k_isentropic must be greater than 1."
        if temp_stag < temp:
            return "Error: Invalid input: temp_stag must be greater than or equal to temp."

        result = fluids_p_stag(P=pressure, T=temp, Tst=temp_stag, k=k_isentropic)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Normal (static) pressure of a fluid [Pa]
Normal (static) temperature of a fluid [K]
Stagnation temperature [K]
Isentropic exponent [-]

PANHANDLE_A

This function calculates natural-gas pipeline flow with the Panhandle A equation in SI form. It applies an empirical power-law relationship between pressure drop, geometry, gas properties, and throughput.

Q \propto \left(\frac{P_1^2-P_2^2}{L\,SG^{0.8539}\,T_{avg}\,Z_{avg}}\right)^{0.5394}D^{2.6182}

Excel Usage

=PANHANDLE_A(sg, temp_avg, length, diameter, p_inlet, p_outlet, comp_factor, efficiency)
  • sg (float, required): Specific gravity of gas relative to air [-]
  • temp_avg (float, required): Average temperature of gas in pipeline [K]
  • length (float, required): Length of pipe [m]
  • diameter (float, required): Diameter of pipe [m]
  • p_inlet (float, required): Inlet pressure [Pa]
  • p_outlet (float, required): Outlet pressure [Pa]
  • comp_factor (float, optional, default: 1): Average compressibility factor [-]
  • efficiency (float, optional, default: 1): Pipeline efficiency factor [-]

Returns (float): Gas flow rate at reference conditions [m³/s]

Example 1: Standard natural gas pipeline

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.693 277.15 160000 0.34 9000000 2000000

Excel formula:

=PANHANDLE_A(0.693, 277.15, 160000, 0.34, 9000000, 2000000)

Expected output:

46.2618

Example 2: Short pipeline section

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.65 290 10000 0.5 5000000 4500000

Excel formula:

=PANHANDLE_A(0.65, 290, 10000, 0.5, 5000000, 4500000)

Expected output:

126.731

Example 3: Pipeline with 92% efficiency

Inputs:

sg temp_avg length diameter p_inlet p_outlet efficiency
0.7 280 50000 0.4 7000000 3000000 0.92

Excel formula:

=PANHANDLE_A(0.7, 280, 50000, 0.4, 7000000, 3000000, 0.92)

Expected output:

84.8114

Example 4: Large diameter pipeline

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.6 285 100000 1 8000000 4000000

Excel formula:

=PANHANDLE_A(0.6, 285, 100000, 1, 8000000, 4000000)

Expected output:

819.552

Python Code

Show Code 
from fluids.compressible import Panhandle_A as fluids_panhandle_a

def panhandle_a(sg, temp_avg, length, diameter, p_inlet, p_outlet, comp_factor=1, efficiency=1):
    """
    Calculate gas flow rate in a pipeline using the Panhandle A formula.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.Panhandle_A

    This example function is provided as-is without any representation of accuracy.

    Args:
        sg (float): Specific gravity of gas relative to air [-]
        temp_avg (float): Average temperature of gas in pipeline [K]
        length (float): Length of pipe [m]
        diameter (float): Diameter of pipe [m]
        p_inlet (float): Inlet pressure [Pa]
        p_outlet (float): Outlet pressure [Pa]
        comp_factor (float, optional): Average compressibility factor [-] Default is 1.
        efficiency (float, optional): Pipeline efficiency factor [-] Default is 1.

    Returns:
        float: Gas flow rate at reference conditions [m³/s]
    """
    try:
        sg = float(sg)
        temp_avg = float(temp_avg)
        length = float(length)
        diameter = float(diameter)
        p_inlet = float(p_inlet)
        p_outlet = float(p_outlet)
        comp_factor = float(comp_factor)
        efficiency = float(efficiency)

        if sg <= 0:
            return "Error: Invalid input: sg must be positive."
        if temp_avg <= 0:
            return "Error: Invalid input: temp_avg must be positive."
        if length <= 0:
            return "Error: Invalid input: length must be positive."
        if diameter <= 0:
            return "Error: Invalid input: diameter must be positive."
        if p_inlet <= 0:
            return "Error: Invalid input: p_inlet must be positive."
        if p_outlet < 0:
            return "Error: Invalid input: p_outlet must be non-negative."
        if p_outlet >= p_inlet:
            return "Error: Invalid input: p_outlet must be less than p_inlet."
        if comp_factor <= 0:
            return "Error: Invalid input: comp_factor must be positive."
        if efficiency <= 0 or efficiency > 1:
            return "Error: Invalid input: efficiency must be between 0 and 1."

        result = fluids_panhandle_a(SG=sg, Tavg=temp_avg, L=length, D=diameter, 
                                     P1=p_inlet, P2=p_outlet, Zavg=comp_factor, E=efficiency)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Specific gravity of gas relative to air [-]
Average temperature of gas in pipeline [K]
Length of pipe [m]
Diameter of pipe [m]
Inlet pressure [Pa]
Outlet pressure [Pa]
Average compressibility factor [-]
Pipeline efficiency factor [-]

PANHANDLE_B

This function computes natural-gas flow through pipelines using the Panhandle B correlation. It is similar to Panhandle A but uses a different empirical fit for pressure-drop and diameter dependence.

Q \propto \left(\frac{P_1^2-P_2^2}{L\,SG^{0.961}\,T_{avg}\,Z_{avg}}\right)^{0.51}D^{2.53}

Excel Usage

=PANHANDLE_B(sg, temp_avg, length, diameter, p_inlet, p_outlet, comp_factor, efficiency)
  • sg (float, required): Specific gravity of gas relative to air [-]
  • temp_avg (float, required): Average temperature of gas in pipeline [K]
  • length (float, required): Length of pipe [m]
  • diameter (float, required): Diameter of pipe [m]
  • p_inlet (float, required): Inlet pressure [Pa]
  • p_outlet (float, required): Outlet pressure [Pa]
  • comp_factor (float, optional, default: 1): Average compressibility factor [-]
  • efficiency (float, optional, default: 1): Pipeline efficiency factor [-]

Returns (float): Gas flow rate at reference conditions [m³/s]

Example 1: Standard natural gas pipeline

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.693 277.15 160000 0.34 9000000 2000000

Excel formula:

=PANHANDLE_B(0.693, 277.15, 160000, 0.34, 9000000, 2000000)

Expected output:

46.0366

Example 2: Short pipeline section

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.65 290 10000 0.5 5000000 4500000

Excel formula:

=PANHANDLE_B(0.65, 290, 10000, 0.5, 5000000, 4500000)

Expected output:

122.337

Example 3: Pipeline with 92% efficiency

Inputs:

sg temp_avg length diameter p_inlet p_outlet efficiency
0.7 280 50000 0.4 7000000 3000000 0.92

Excel formula:

=PANHANDLE_B(0.7, 280, 50000, 0.4, 7000000, 3000000, 0.92)

Expected output:

81.9639

Example 4: Large diameter pipeline

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.6 285 100000 1 8000000 4000000

Excel formula:

=PANHANDLE_B(0.6, 285, 100000, 1, 8000000, 4000000)

Expected output:

745.369

Python Code

Show Code 
from fluids.compressible import Panhandle_B as fluids_panhandle_b

def panhandle_b(sg, temp_avg, length, diameter, p_inlet, p_outlet, comp_factor=1, efficiency=1):
    """
    Calculate gas flow rate in a pipeline using the Panhandle B formula.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.Panhandle_B

    This example function is provided as-is without any representation of accuracy.

    Args:
        sg (float): Specific gravity of gas relative to air [-]
        temp_avg (float): Average temperature of gas in pipeline [K]
        length (float): Length of pipe [m]
        diameter (float): Diameter of pipe [m]
        p_inlet (float): Inlet pressure [Pa]
        p_outlet (float): Outlet pressure [Pa]
        comp_factor (float, optional): Average compressibility factor [-] Default is 1.
        efficiency (float, optional): Pipeline efficiency factor [-] Default is 1.

    Returns:
        float: Gas flow rate at reference conditions [m³/s]
    """
    try:
        sg = float(sg)
        temp_avg = float(temp_avg)
        length = float(length)
        diameter = float(diameter)
        p_inlet = float(p_inlet)
        p_outlet = float(p_outlet)
        comp_factor = float(comp_factor)
        efficiency = float(efficiency)

        if sg <= 0:
            return "Error: Invalid input: sg must be positive."
        if temp_avg <= 0:
            return "Error: Invalid input: temp_avg must be positive."
        if length <= 0:
            return "Error: Invalid input: length must be positive."
        if diameter <= 0:
            return "Error: Invalid input: diameter must be positive."
        if p_inlet <= 0:
            return "Error: Invalid input: p_inlet must be positive."
        if p_outlet < 0:
            return "Error: Invalid input: p_outlet must be non-negative."
        if p_outlet >= p_inlet:
            return "Error: Invalid input: p_outlet must be less than p_inlet."
        if comp_factor <= 0:
            return "Error: Invalid input: comp_factor must be positive."
        if efficiency <= 0 or efficiency > 1:
            return "Error: Invalid input: efficiency must be between 0 and 1."

        result = fluids_panhandle_b(SG=sg, Tavg=temp_avg, L=length, D=diameter, 
                                     P1=p_inlet, P2=p_outlet, Zavg=comp_factor, E=efficiency)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Specific gravity of gas relative to air [-]
Average temperature of gas in pipeline [K]
Length of pipe [m]
Diameter of pipe [m]
Inlet pressure [Pa]
Outlet pressure [Pa]
Average compressibility factor [-]
Pipeline efficiency factor [-]

POLYTROPIC_EXP

This function computes the polytropic exponent from isentropic exponent and polytropic efficiency for gas compression modeling. It helps map real compressor behavior into a polytropic process representation.

n=\frac{k\eta_p}{1-k(1-\eta_p)}

Excel Usage

=POLYTROPIC_EXP(k_isentropic, eta_poly)
  • k_isentropic (float, required): Isentropic exponent of the gas (Cp/Cv) [-]
  • eta_poly (float, required): Polytropic efficiency of the process [-]

Returns (float): Polytropic exponent calculated from isentropic exponent and polytropic efficiency [-]

Example 1: Air with 78% polytropic efficiency

Inputs:

k_isentropic eta_poly
1.4 0.78

Excel formula:

=POLYTROPIC_EXP(1.4, 0.78)

Expected output:

1.57803

Example 2: Air with 85% polytropic efficiency

Inputs:

k_isentropic eta_poly
1.4 0.85

Excel formula:

=POLYTROPIC_EXP(1.4, 0.85)

Expected output:

1.50633

Example 3: Methane (k=1.3) with 80% efficiency

Inputs:

k_isentropic eta_poly
1.3 0.8

Excel formula:

=POLYTROPIC_EXP(1.3, 0.8)

Expected output:

1.40541

Example 4: High efficiency at 95%

Inputs:

k_isentropic eta_poly
1.4 0.95

Excel formula:

=POLYTROPIC_EXP(1.4, 0.95)

Expected output:

1.43011

Python Code

Show Code 
from fluids.compressible import polytropic_exponent as fluids_poly_exp

def polytropic_exp(k_isentropic, eta_poly):
    """
    Calculate polytropic exponent or polytropic efficiency for compression.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.polytropic_exponent

    This example function is provided as-is without any representation of accuracy.

    Args:
        k_isentropic (float): Isentropic exponent of the gas (Cp/Cv) [-]
        eta_poly (float): Polytropic efficiency of the process [-]

    Returns:
        float: Polytropic exponent calculated from isentropic exponent and polytropic efficiency [-]
    """
    try:
        k_isentropic = float(k_isentropic)
        eta_poly = float(eta_poly)

        if k_isentropic <= 1:
            return "Error: Invalid input: k_isentropic must be greater than 1."
        if eta_poly <= 0 or eta_poly > 1:
            return "Error: Invalid input: eta_poly must be between 0 and 1."

        result = fluids_poly_exp(k=k_isentropic, eta_p=eta_poly)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Isentropic exponent of the gas (Cp/Cv) [-]
Polytropic efficiency of the process [-]

STAGNATION_ENERGY

This function computes stagnation-energy rise from flow velocity, representing kinetic energy per unit mass available for conversion to enthalpy. It is used in compressible-flow energy balances and total-state calculations.

\Delta h=\frac{V^2}{2}

Excel Usage

=STAGNATION_ENERGY(velocity)
  • velocity (float, required): Velocity of the fluid [m/s]

Returns (float): Increase in enthalpy due to fluid velocity [J/kg]

Example 1: High speed flow (125 m/s)

Inputs:

velocity
125

Excel formula:

=STAGNATION_ENERGY(125)

Expected output:

7812.5

Example 2: Low speed flow (10 m/s)

Inputs:

velocity
10

Excel formula:

=STAGNATION_ENERGY(10)

Expected output:

50

Example 3: Supersonic flow (400 m/s)

Inputs:

velocity
400

Excel formula:

=STAGNATION_ENERGY(400)

Expected output:

80000

Example 4: Subsonic flow (50 m/s)

Inputs:

velocity
50

Excel formula:

=STAGNATION_ENERGY(50)

Expected output:

1250

Python Code

Show Code 
from fluids.compressible import stagnation_energy as fluids_stag_e

def stagnation_energy(velocity):
    """
    Calculate the increase in enthalpy due to fluid velocity.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.stagnation_energy

    This example function is provided as-is without any representation of accuracy.

    Args:
        velocity (float): Velocity of the fluid [m/s]

    Returns:
        float: Increase in enthalpy due to fluid velocity [J/kg]
    """
    try:
        velocity = float(velocity)

        if velocity == 0:
            return 0.0

        result = fluids_stag_e(V=abs(velocity))
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Velocity of the fluid [m/s]

T_CRITICAL_FLOW

This function computes the critical static temperature for Mach 1 from stagnation temperature for an isentropic gas. It is used with nozzle and choking analyses to determine sonic-state thermodynamic conditions.

\frac{T^*}{T_0}=\frac{2}{k+1}

Excel Usage

=T_CRITICAL_FLOW(temp_stag, k_isentropic)
  • temp_stag (float, required): Stagnation temperature of a fluid with Ma=1 [K]
  • k_isentropic (float, required): Isentropic exponent [-]

Returns (float): Critical flow temperature at Ma=1 [K]

Example 1: Textbook example (T=473K, k=1.289)

Inputs:

temp_stag k_isentropic
473 1.289

Excel formula:

=T_CRITICAL_FLOW(473, 1.289)

Expected output:

413.281

Example 2: Air at standard conditions (k=1.4)

Inputs:

temp_stag k_isentropic
300 1.4

Excel formula:

=T_CRITICAL_FLOW(300, 1.4)

Expected output:

250

Example 3: High temperature nozzle

Inputs:

temp_stag k_isentropic
800 1.3

Excel formula:

=T_CRITICAL_FLOW(800, 1.3)

Expected output:

695.652

Example 4: Steam conditions (k=1.135)

Inputs:

temp_stag k_isentropic
500 1.135

Excel formula:

=T_CRITICAL_FLOW(500, 1.135)

Expected output:

468.384

Python Code

Show Code 
from fluids.compressible import T_critical_flow as fluids_t_crit

def t_critical_flow(temp_stag, k_isentropic):
    """
    Calculate critical flow temperature for a fluid at Mach 1.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.T_critical_flow

    This example function is provided as-is without any representation of accuracy.

    Args:
        temp_stag (float): Stagnation temperature of a fluid with Ma=1 [K]
        k_isentropic (float): Isentropic exponent [-]

    Returns:
        float: Critical flow temperature at Ma=1 [K]
    """
    try:
        temp_stag = float(temp_stag)
        k_isentropic = float(k_isentropic)

        if temp_stag <= 0:
            return "Error: Invalid input: temp_stag must be positive."
        if k_isentropic <= 1:
            return "Error: Invalid input: k_isentropic must be greater than 1."

        result = fluids_t_crit(T=temp_stag, k=k_isentropic)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Stagnation temperature of a fluid with Ma=1 [K]
Isentropic exponent [-]

T_STAG_IDEAL

This function returns ideal stagnation temperature from static temperature, velocity, and constant-pressure heat capacity. It models conversion of kinetic energy into thermal energy in adiabatic deceleration.

T_0=T+\frac{V^2}{2C_p}

Excel Usage

=T_STAG_IDEAL(temp, velocity, cp)
  • temp (float, required): Static temperature [K]
  • velocity (float, required): Velocity [m/s]
  • cp (float, required): Heat capacity at constant pressure [J/kg/K]

Returns (float): Stagnation temperature [K]

Example 1: Textbook example from Cengel

Inputs:

temp velocity cp
255.7 250 1005

Excel formula:

=T_STAG_IDEAL(255.7, 250, 1005)

Expected output:

286.795

Example 2: High speed aircraft

Inputs:

temp velocity cp
220 300 1005

Excel formula:

=T_STAG_IDEAL(220, 300, 1005)

Expected output:

264.776

Example 3: Low speed flow

Inputs:

temp velocity cp
300 50 1005

Excel formula:

=T_STAG_IDEAL(300, 50, 1005)

Expected output:

301.244

Example 4: Different gas (helium, high Cp)

Inputs:

temp velocity cp
300 200 5193

Excel formula:

=T_STAG_IDEAL(300, 200, 5193)

Expected output:

303.851

Python Code

Show Code 
from fluids.compressible import T_stagnation_ideal as fluids_t_stag_ideal

def t_stag_ideal(temp, velocity, cp):
    """
    Calculate ideal stagnation temperature from velocity and heat capacity.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.T_stagnation_ideal

    This example function is provided as-is without any representation of accuracy.

    Args:
        temp (float): Static temperature [K]
        velocity (float): Velocity [m/s]
        cp (float): Heat capacity at constant pressure [J/kg/K]

    Returns:
        float: Stagnation temperature [K]
    """
    try:
        temp = float(temp)
        velocity = float(velocity)
        cp = float(cp)

        if temp <= 0:
            return "Error: Invalid input: temp must be positive."
        if cp <= 0:
            return "Error: Invalid input: cp must be positive."

        result = fluids_t_stag_ideal(T=temp, V=velocity, Cp=cp)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Static temperature [K]
Velocity [m/s]
Heat capacity at constant pressure [J/kg/K]

T_STAGNATION

This function computes stagnation temperature from static temperature and pressure ratio under isentropic assumptions. It is useful when recovering total temperature from measured static flow conditions.

T_0=T\left(\frac{P_0}{P}\right)^{(k-1)/k}

Excel Usage

=T_STAGNATION(temp, pressure, p_stag, k_isentropic)
  • temp (float, required): Normal (static) temperature of a fluid [K]
  • pressure (float, required): Normal (static) pressure of a fluid [Pa]
  • p_stag (float, required): Stagnation pressure [Pa]
  • k_isentropic (float, required): Isentropic exponent [-]

Returns (float): Stagnation temperature [K]

Example 1: Textbook example from Cengel

Inputs:

temp pressure p_stag k_isentropic
286.8 54050 432400 1.4

Excel formula:

=T_STAGNATION(286.8, 54050, 432400, 1.4)

Expected output:

519.523

Example 2: High speed flow

Inputs:

temp pressure p_stag k_isentropic
250 50000 100000 1.4

Excel formula:

=T_STAGNATION(250, 50000, 100000, 1.4)

Expected output:

304.753

Example 3: Low speed flow (small pressure rise)

Inputs:

temp pressure p_stag k_isentropic
300 100000 110000 1.4

Excel formula:

=T_STAGNATION(300, 100000, 110000, 1.4)

Expected output:

308.282

Example 4: Different gas (lower k)

Inputs:

temp pressure p_stag k_isentropic
320 200000 400000 1.3

Excel formula:

=T_STAGNATION(320, 200000, 400000, 1.3)

Expected output:

375.507

Python Code

Show Code 
from fluids.compressible import T_stagnation as fluids_t_stag

def t_stagnation(temp, pressure, p_stag, k_isentropic):
    """
    Calculate stagnation temperature from pressure ratio.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.T_stagnation

    This example function is provided as-is without any representation of accuracy.

    Args:
        temp (float): Normal (static) temperature of a fluid [K]
        pressure (float): Normal (static) pressure of a fluid [Pa]
        p_stag (float): Stagnation pressure [Pa]
        k_isentropic (float): Isentropic exponent [-]

    Returns:
        float: Stagnation temperature [K]
    """
    try:
        temp = float(temp)
        pressure = float(pressure)
        p_stag = float(p_stag)
        k_isentropic = float(k_isentropic)

        if temp <= 0:
            return "Error: Invalid input: temp must be positive."
        if pressure <= 0:
            return "Error: Invalid input: pressure must be positive."
        if p_stag <= 0:
            return "Error: Invalid input: p_stag must be positive."
        if k_isentropic <= 1:
            return "Error: Invalid input: k_isentropic must be greater than 1."
        if p_stag < pressure:
            return "Error: Invalid input: p_stag must be greater than or equal to pressure."

        result = fluids_t_stag(T=temp, P=pressure, Pst=p_stag, k=k_isentropic)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Normal (static) temperature of a fluid [K]
Normal (static) pressure of a fluid [Pa]
Stagnation pressure [Pa]
Isentropic exponent [-]

WEYMOUTH_FLOW

This function estimates compressible-gas pipeline flow with the Weymouth equation under steady-state assumptions. It relates volumetric flow to pressure drop, pipe size, and gas thermophysical corrections.

Q \propto \left(\frac{P_1^2-P_2^2}{L\,SG\,T_{avg}\,Z_{avg}}\right)^{1/2}D^{2.667}

Excel Usage

=WEYMOUTH_FLOW(sg, temp_avg, length, diameter, p_inlet, p_outlet, comp_factor, efficiency)
  • sg (float, required): Specific gravity of gas relative to air [-]
  • temp_avg (float, required): Average temperature of gas in pipeline [K]
  • length (float, required): Length of pipe [m]
  • diameter (float, required): Diameter of pipe [m]
  • p_inlet (float, required): Inlet pressure [Pa]
  • p_outlet (float, required): Outlet pressure [Pa]
  • comp_factor (float, optional, default: 1): Average compressibility factor [-]
  • efficiency (float, optional, default: 1): Pipeline efficiency factor [-]

Returns (float): Gas flow rate at reference conditions [m³/s]

Example 1: Standard natural gas pipeline

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.693 277.15 160000 0.34 9000000 2000000

Excel formula:

=WEYMOUTH_FLOW(0.693, 277.15, 160000, 0.34, 9000000, 2000000)

Expected output:

34.8666

Example 2: Short pipeline section

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.65 290 10000 0.5 5000000 4500000

Excel formula:

=WEYMOUTH_FLOW(0.65, 290, 10000, 0.5, 5000000, 4500000)

Expected output:

97.7999

Example 3: Pipeline with 92% efficiency

Inputs:

sg temp_avg length diameter p_inlet p_outlet efficiency
0.7 280 50000 0.4 7000000 3000000 0.92

Excel formula:

=WEYMOUTH_FLOW(0.7, 280, 50000, 0.4, 7000000, 3000000, 0.92)

Expected output:

63.1529

Example 4: Large diameter pipeline

Inputs:

sg temp_avg length diameter p_inlet p_outlet
0.6 285 100000 1 8000000 4000000

Excel formula:

=WEYMOUTH_FLOW(0.6, 285, 100000, 1, 8000000, 4000000)

Expected output:

655.567

Python Code

Show Code 
from fluids.compressible import Weymouth as fluids_weymouth

def weymouth_flow(sg, temp_avg, length, diameter, p_inlet, p_outlet, comp_factor=1, efficiency=1):
    """
    Calculate gas flow rate in a pipeline using the Weymouth formula.

    See: https://fluids.readthedocs.io/fluids.compressible.html#fluids.compressible.Weymouth

    This example function is provided as-is without any representation of accuracy.

    Args:
        sg (float): Specific gravity of gas relative to air [-]
        temp_avg (float): Average temperature of gas in pipeline [K]
        length (float): Length of pipe [m]
        diameter (float): Diameter of pipe [m]
        p_inlet (float): Inlet pressure [Pa]
        p_outlet (float): Outlet pressure [Pa]
        comp_factor (float, optional): Average compressibility factor [-] Default is 1.
        efficiency (float, optional): Pipeline efficiency factor [-] Default is 1.

    Returns:
        float: Gas flow rate at reference conditions [m³/s]
    """
    try:
        sg = float(sg)
        temp_avg = float(temp_avg)
        length = float(length)
        diameter = float(diameter)
        p_inlet = float(p_inlet)
        p_outlet = float(p_outlet)
        comp_factor = float(comp_factor)
        efficiency = float(efficiency)

        if sg <= 0:
            return "Error: Invalid input: sg must be positive."
        if temp_avg <= 0:
            return "Error: Invalid input: temp_avg must be positive."
        if length <= 0:
            return "Error: Invalid input: length must be positive."
        if diameter <= 0:
            return "Error: Invalid input: diameter must be positive."
        if p_inlet <= 0:
            return "Error: Invalid input: p_inlet must be positive."
        if p_outlet < 0:
            return "Error: Invalid input: p_outlet must be non-negative."
        if p_outlet >= p_inlet:
            return "Error: Invalid input: p_outlet must be less than p_inlet."
        if comp_factor <= 0:
            return "Error: Invalid input: comp_factor must be positive."
        if efficiency <= 0 or efficiency > 1:
            return "Error: Invalid input: efficiency must be between 0 and 1."

        result = fluids_weymouth(SG=sg, Tavg=temp_avg, L=length, D=diameter, 
                                  P1=p_inlet, P2=p_outlet, Zavg=comp_factor, E=efficiency)
        return float(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Specific gravity of gas relative to air [-]
Average temperature of gas in pipeline [K]
Length of pipe [m]
Diameter of pipe [m]
Inlet pressure [Pa]
Outlet pressure [Pa]
Average compressibility factor [-]
Pipeline efficiency factor [-]