Conv Supercritical

Overview

Supercritical convection in tubes models forced heat transfer when fluid pressure is above the critical point and properties vary sharply with temperature, especially near the pseudocritical region. In this regime, small changes in state can cause large shifts in density, heat capacity, and transport properties, so conventional single-phase correlations can become unreliable. The functions in this category provide empirical Nusselt-number models for turbulent upward internal flow at supercritical conditions, most often for water-cooled channels used in power and process equipment. For background on the thermodynamic regime, see supercritical fluid.

The shared framework is dimensionless internal convection with correction factors for property variation across the wall and bulk fluid states. All correlations estimate Nu from combinations of Reynolds and Prandtl numbers, typically in power-law form, while some add multiplicative terms such as (\rho_w/\rho_b)^a, (\mu_w/\mu_b)^b, or heat-capacity modifiers to capture near-critical behavior. In design studies, this set is usually applied as a model family: engineers compare several correlations across the same operating window to quantify uncertainty and detect sensitivity to pseudocritical crossing, high heat flux, or deteriorated heat-transfer conditions.

Implementation is based on the Python ht library, specifically ht.conv_supercritical. The ht package provides vetted thermal-engineering correlations with consistent APIs, which makes these methods easy to use in spreadsheets, calculators, and scripted parameter sweeps.

NU_BRINGER_SMITH, NU_GORBAN, NU_MCADAMS, and NU_SHITSMAN are compact forms that rely mainly on Re and one or two Pr definitions. These are practical for baseline estimates, quick screening, and cross-checks when detailed wall/bulk property sets are not yet available. NU_GORBAN is historically important but often treated cautiously in modern comparisons, while NU_MCADAMS remains a simple reference-like form derived from classic turbulent-tube behavior. NU_BRINGER_SMITH and NU_SHITSMAN are likewise useful as lightweight bounds in early-stage studies.

NU_BISHOP, NU_SWENSON, NU_MOKRY, NU_XU, NU_ZHU, NU_GUPTA, NU_ORNATSKY, and NU_PETUKHOV explicitly account for wall-to-bulk property ratios and are typically preferred when temperature gradients are strong. These tools differ in which corrections they emphasize (density, viscosity, conductivity, or mixed Prandtl treatments), but all target improved fidelity in supercritical pipe-flow prediction. They are commonly used in thermal-hydraulic rating for heated channels, especially when evaluating normal versus deteriorated heat-transfer trends under changing mass flux and heat flux.

NU_JACKSON, NU_YAMAGATA, NU_KITOH, NU_GRIEM, NU_KRASN_PROTO, NU_KRASNOSH_PROTO, and NU_KRASNOSHCHEKOV add more structured handling of pseudocritical and heat-capacity effects. NU_JACKSON and NU_YAMAGATA incorporate regime-dependent exponents or factors tied to T_b, T_w, and T_{pc} behavior; NU_KITOH and NU_GRIEM include additional sensitivity to enthalpy and flux-related conditions. NU_KRASN_PROTO and NU_KRASNOSH_PROTO represent two category entries for the Krasnoshchekov-Protopopov style formulation, while NU_KRASNOSHCHEKOV applies a related but distinct corrected baseline form. Together, these functions are most useful for higher-fidelity comparative modeling when property data and operating-state definitions are well characterized.

NU_BISHOP

This function estimates the internal convective heat transfer Nusselt number for turbulent upward flow at supercritical pressure using the Bishop correlation. It uses Reynolds and Prandtl numbers, and can optionally apply density-ratio and thermal-entry corrections when wall and bulk densities and geometry inputs are provided.

Nu = C\,Re^{m}Pr^{n}\,\Phi_{\rho}\,\Phi_{entry}

Excel Usage

=NU_BISHOP(Re, Pr, rho_w, rho_b, D, x)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk properties and averaged heat capacity (-).
  • rho_w (float, optional, default: null): Density at wall temperature (kg/m^3).
  • rho_b (float, optional, default: null): Density at bulk temperature (kg/m^3).
  • D (float, optional, default: null): Tube diameter (m).
  • x (float, optional, default: null): Axial distance along the tube (m).

Returns (float): Nusselt number with wall fluid properties (-).

Example 1: Bishop correlation example

Inputs:

Re Pr rho_w rho_b D x
100000 1.2 330 290 0.01 1.2

Excel formula:

=NU_BISHOP(100000, 1.2, 330, 290, 0.01, 1.2)

Expected output:

265.362

Example 2: Bishop correlation bulk properties only

Inputs:

Re Pr
80000 0.9

Excel formula:

=NU_BISHOP(80000, 0.9)

Expected output:

166.506

Example 3: Bishop correlation with density ratio

Inputs:

Re Pr rho_w rho_b
150000 1.4 350 300

Excel formula:

=NU_BISHOP(150000, 1.4, 350, 300)

Expected output:

419.337

Example 4: Bishop correlation with entry correction

Inputs:

Re Pr rho_w rho_b D x
250000 1.1 400 320 0.02 2

Excel formula:

=NU_BISHOP(250000, 1.1, 400, 320, 0.02, 2)

Expected output:

597.428

Python Code

Show Code 
from ht.conv_supercritical import Nu_Bishop as ht_Nu_Bishop

def Nu_Bishop(Re, Pr, rho_w=None, rho_b=None, D=None, x=None):
    """
    Calculate Nusselt number for supercritical pipe flow using the Bishop correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk properties and averaged heat capacity (-).
        rho_w (float, optional): Density at wall temperature (kg/m^3). Default is None.
        rho_b (float, optional): Density at bulk temperature (kg/m^3). Default is None.
        D (float, optional): Tube diameter (m). Default is None.
        x (float, optional): Axial distance along the tube (m). Default is None.

    Returns:
        float: Nusselt number with wall fluid properties (-).
    """
    try:
        return ht_Nu_Bishop(Re=Re, Pr=Pr, rho_w=rho_w, rho_b=rho_b, D=D, x=x)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk properties and averaged heat capacity (-).
Density at wall temperature (kg/m^3).
Density at bulk temperature (kg/m^3).
Tube diameter (m).
Axial distance along the tube (m).

NU_BRINGER_SMITH

This function computes the Nusselt number for turbulent internal flow near supercritical conditions using the Bringer-Smith empirical correlation. The model relates heat transfer to Reynolds and Prandtl numbers evaluated at the reference and wall-property states.

Nu = C\,Re^{m}Pr^{n}

Excel Usage

=NU_BRINGER_SMITH(Re, Pr)
  • Re (float, required): Reynolds number with reference properties (-).
  • Pr (float, required): Prandtl number with wall properties (-).

Returns (float): Nusselt number with reference temperature properties (-).

Example 1: Bringer-Smith example

Inputs:

Re Pr
100000 1.2

Excel formula:

=NU_BRINGER_SMITH(100000, 1.2)

Expected output:

208.176

Example 2: Bringer-Smith lower Reynolds number

Inputs:

Re Pr
50000 1

Excel formula:

=NU_BRINGER_SMITH(50000, 1)

Expected output:

110.43

Example 3: Bringer-Smith mid Reynolds number

Inputs:

Re Pr
200000 0.8

Excel formula:

=NU_BRINGER_SMITH(200000, 0.8)

Expected output:

284.037

Example 4: Bringer-Smith higher Reynolds number

Inputs:

Re Pr
800000 1.5

Excel formula:

=NU_BRINGER_SMITH(800000, 1.5)

Expected output:

1167.11

Python Code

Show Code 
from ht.conv_supercritical import Nu_Bringer_Smith as ht_Nu_Bringer_Smith

def Nu_Bringer_Smith(Re, Pr):
    """
    Calculate Nusselt number for near-supercritical flow using the Bringer-Smith correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with reference properties (-).
        Pr (float): Prandtl number with wall properties (-).

    Returns:
        float: Nusselt number with reference temperature properties (-).
    """
    try:
        return ht_Nu_Bringer_Smith(Re=Re, Pr=Pr)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with reference properties (-).
Prandtl number with wall properties (-).

NU_GORBAN

This function estimates the Nusselt number for supercritical turbulent pipe flow using the Gorban correlation. It applies a power-law relationship in Reynolds and Prandtl numbers for quick engineering estimation of convective heat transfer.

Nu = C\,Re^{m}Pr^{n}

Excel Usage

=NU_GORBAN(Re, Pr)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk fluid properties (-).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Gorban correlation example

Inputs:

Re Pr
100000 1.2

Excel formula:

=NU_GORBAN(100000, 1.2)

Expected output:

182.537

Example 2: Gorban correlation lower Reynolds number

Inputs:

Re Pr
40000 1.1

Excel formula:

=NU_GORBAN(40000, 1.1)

Expected output:

80.861

Example 3: Gorban correlation mid Reynolds number

Inputs:

Re Pr
200000 0.9

Excel formula:

=NU_GORBAN(200000, 0.9)

Expected output:

352.59

Example 4: Gorban correlation higher Reynolds number

Inputs:

Re Pr
600000 1.4

Excel formula:

=NU_GORBAN(600000, 1.4)

Expected output:

898.779

Python Code

Show Code 
from ht.conv_supercritical import Nu_Gorban as ht_Nu_Gorban

def Nu_Gorban(Re, Pr):
    """
    Calculate Nusselt number for supercritical flow using the Gorban correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk fluid properties (-).

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Gorban(Re=Re, Pr=Pr)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk fluid properties (-).

NU_GRIEM

This function calculates the Nusselt number for turbulent supercritical internal flow with the Griem correlation. The correlation primarily uses Reynolds and Prandtl numbers and can include an enthalpy-based adjustment factor when enthalpy is supplied.

Nu = C\,Re^{m}Pr^{n}\,\omega(H)

Excel Usage

=NU_GRIEM(Re, Pr, H)
  • Re (float, required): Reynolds number as defined by the correlation (-).
  • Pr (float, required): Prandtl number as defined by the correlation (-).
  • H (float, optional, default: null): Enthalpy of water when applicable (J/kg).

Returns (float): Nusselt number for supercritical pipe flow (-).

Example 1: Griem correlation example

Inputs:

Re Pr
100000 1.2

Excel formula:

=NU_GRIEM(100000, 1.2)

Expected output:

275.482

Example 2: Griem correlation lower Reynolds number

Inputs:

Re Pr
60000 1

Excel formula:

=NU_GRIEM(60000, 1)

Expected output:

166.154

Example 3: Griem correlation with enthalpy

Inputs:

Re Pr H
150000 1.4 1600000

Excel formula:

=NU_GRIEM(150000, 1.4, 1600000)

Expected output:

361.133

Example 4: Griem correlation higher Reynolds number

Inputs:

Re Pr
400000 0.9

Excel formula:

=NU_GRIEM(400000, 0.9)

Expected output:

774.82

Python Code

Show Code 
from ht.conv_supercritical import Nu_Griem as ht_Nu_Griem

def Nu_Griem(Re, Pr, H=None):
    """
    Calculate Nusselt number for supercritical flow using the Griem correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number as defined by the correlation (-).
        Pr (float): Prandtl number as defined by the correlation (-).
        H (float, optional): Enthalpy of water when applicable (J/kg). Default is None.

    Returns:
        float: Nusselt number for supercritical pipe flow (-).
    """
    try:
        return ht_Nu_Griem(Re=Re, Pr=Pr, H=H)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number as defined by the correlation (-).
Prandtl number as defined by the correlation (-).
Enthalpy of water when applicable (J/kg).

NU_GUPTA

This function computes the supercritical-flow Nusselt number using the Gupta correlation. It combines Reynolds and Prandtl dependence with optional density and viscosity ratio corrections between wall and bulk fluid states.

Nu = C\,Re^{m}Pr^{n}\,\Phi_{\rho}\,\Phi_{\mu}

Excel Usage

=NU_GUPTA(Re, Pr, rho_w, rho_b, mu_w, mu_b)
  • Re (float, required): Reynolds number with wall properties (-).
  • Pr (float, required): Prandtl number with wall properties (-).
  • rho_w (float, optional, default: null): Density at wall temperature (kg/m^3).
  • rho_b (float, optional, default: null): Density at bulk temperature (kg/m^3).
  • mu_w (float, optional, default: null): Viscosity at wall temperature (Pa*s).
  • mu_b (float, optional, default: null): Viscosity at bulk temperature (Pa*s).

Returns (float): Nusselt number with wall fluid properties (-).

Example 1: Gupta correlation example

Inputs:

Re Pr rho_w rho_b mu_w mu_b
100000 1.2 330 290 0.0008 0.0009

Excel formula:

=NU_GUPTA(100000, 1.2, 330, 290, 0.0008, 0.0009)

Expected output:

186.201

Example 2: Gupta correlation with bulk and wall properties only

Inputs:

Re Pr
80000 1.1

Excel formula:

=NU_GUPTA(80000, 1.1)

Expected output:

144.414

Example 3: Gupta correlation mid Reynolds number

Inputs:

Re Pr rho_w rho_b mu_w mu_b
200000 0.9 360 310 0.0007 0.00085

Excel formula:

=NU_GUPTA(200000, 0.9, 360, 310, 0.0007, 0.00085)

Expected output:

275.904

Example 4: Gupta correlation higher Reynolds number

Inputs:

Re Pr rho_w rho_b mu_w mu_b
500000 1.5 380 320 0.0006 0.00075

Excel formula:

=NU_GUPTA(500000, 1.5, 380, 320, 0.0006, 0.00075)

Expected output:

947.852

Python Code

Show Code 
from ht.conv_supercritical import Nu_Gupta as ht_Nu_Gupta

def Nu_Gupta(Re, Pr, rho_w=None, rho_b=None, mu_w=None, mu_b=None):
    """
    Calculate Nusselt number for supercritical flow using the Gupta correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with wall properties (-).
        Pr (float): Prandtl number with wall properties (-).
        rho_w (float, optional): Density at wall temperature (kg/m^3). Default is None.
        rho_b (float, optional): Density at bulk temperature (kg/m^3). Default is None.
        mu_w (float, optional): Viscosity at wall temperature (Pa*s). Default is None.
        mu_b (float, optional): Viscosity at bulk temperature (Pa*s). Default is None.

    Returns:
        float: Nusselt number with wall fluid properties (-).
    """
    try:
        return ht_Nu_Gupta(Re=Re, Pr=Pr, rho_w=rho_w, rho_b=rho_b, mu_w=mu_w, mu_b=mu_b)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with wall properties (-).
Prandtl number with wall properties (-).
Density at wall temperature (kg/m^3).
Density at bulk temperature (kg/m^3).
Viscosity at wall temperature (Pa*s).
Viscosity at bulk temperature (Pa*s).

NU_JACKSON

This function estimates the Nusselt number for turbulent supercritical internal convection via the Jackson correlation. It uses Reynolds and Prandtl numbers, with optional correction factors based on density ratio, heat-capacity ratio, and temperature location relative to the pseudocritical point.

Nu = C\,Re^{m}Pr^{n}\,\Phi_{\rho}\,\Phi_{Cp}(T_b,T_w,T_{pc})

Excel Usage

=NU_JACKSON(Re, Pr, rho_w, rho_b, Cp_avg, Cp_b, T_b, T_w, T_pc)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk fluid properties (-).
  • rho_w (float, optional, default: null): Density at wall temperature (kg/m^3).
  • rho_b (float, optional, default: null): Density at bulk temperature (kg/m^3).
  • Cp_avg (float, optional, default: null): Average heat capacity between wall and bulk temperatures (J/kg/K).
  • Cp_b (float, optional, default: null): Heat capacity at bulk temperature (J/kg/K).
  • T_b (float, optional, default: null): Bulk temperature (K).
  • T_w (float, optional, default: null): Wall temperature (K).
  • T_pc (float, optional, default: null): Pseudocritical temperature at pressure (K).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Jackson correlation example

Inputs:

Re Pr
100000 1.2

Excel formula:

=NU_JACKSON(100000, 1.2)

Expected output:

252.372

Example 2: Jackson correlation with property corrections

Inputs:

Re Pr rho_w rho_b Cp_avg Cp_b T_b T_w T_pc
100000 1.2 330 290 2100 2000 650 700 640

Excel formula:

=NU_JACKSON(100000, 1.2, 330, 290, 2100, 2000, 650, 700, 640)

Expected output:

267.743

Example 3: Jackson correlation mid Reynolds number

Inputs:

Re Pr rho_w rho_b Cp_avg Cp_b T_b T_w T_pc
200000 0.9 350 300 2300 2100 620 680 640

Excel formula:

=NU_JACKSON(200000, 0.9, 350, 300, 2300, 2100, 620, 680, 640)

Expected output:

419.564

Example 4: Jackson correlation higher Reynolds number

Inputs:

Re Pr
400000 1.1

Excel formula:

=NU_JACKSON(400000, 1.1)

Expected output:

753.072

Python Code

Show Code 
from ht.conv_supercritical import Nu_Jackson as ht_Nu_Jackson

def Nu_Jackson(Re, Pr, rho_w=None, rho_b=None, Cp_avg=None, Cp_b=None, T_b=None, T_w=None, T_pc=None):
    """
    Calculate Nusselt number for supercritical flow using the Jackson correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk fluid properties (-).
        rho_w (float, optional): Density at wall temperature (kg/m^3). Default is None.
        rho_b (float, optional): Density at bulk temperature (kg/m^3). Default is None.
        Cp_avg (float, optional): Average heat capacity between wall and bulk temperatures (J/kg/K). Default is None.
        Cp_b (float, optional): Heat capacity at bulk temperature (J/kg/K). Default is None.
        T_b (float, optional): Bulk temperature (K). Default is None.
        T_w (float, optional): Wall temperature (K). Default is None.
        T_pc (float, optional): Pseudocritical temperature at pressure (K). Default is None.

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Jackson(
            Re=Re,
            Pr=Pr,
            rho_w=rho_w,
            rho_b=rho_b,
            Cp_avg=Cp_avg,
            Cp_b=Cp_b,
            T_b=T_b,
            T_w=T_w,
            T_pc=T_pc,
        )
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk fluid properties (-).
Density at wall temperature (kg/m^3).
Density at bulk temperature (kg/m^3).
Average heat capacity between wall and bulk temperatures (J/kg/K).
Heat capacity at bulk temperature (J/kg/K).
Bulk temperature (K).
Wall temperature (K).
Pseudocritical temperature at pressure (K).

NU_KITOH

This function calculates the Nusselt number for supercritical turbulent tube flow using the Kitoh correlation. It is based on Reynolds and Prandtl numbers and can optionally modify the Prandtl exponent as a function of enthalpy, mass flux, and wall heat flux.

Nu = C\,Re^{m}Pr^{n(H,G,q)}

Excel Usage

=NU_KITOH(Re, Pr, H, G, q)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk fluid properties (-).
  • H (float, optional, default: null): Enthalpy of water when applicable (J/kg).
  • G (float, optional, default: null): Mass flux of the fluid (kg/m^2/s).
  • q (float, optional, default: null): Heat flux to the wall (W/m^2).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Kitoh correlation example

Inputs:

Re Pr H G q
100000 1.2 1300000 1500 5000000

Excel formula:

=NU_KITOH(100000, 1.2, 1300000, 1500, 5000000)

Expected output:

331.802

Example 2: Kitoh correlation bulk properties only

Inputs:

Re Pr
80000 1.1

Excel formula:

=NU_KITOH(80000, 1.1)

Expected output:

235.656

Example 3: Kitoh correlation mid Reynolds number

Inputs:

Re Pr H G q
200000 0.9 1800000 1200 3000000

Excel formula:

=NU_KITOH(200000, 0.9, 1800000, 1200, 3000000)

Expected output:

570.322

Example 4: Kitoh correlation higher Reynolds number

Inputs:

Re Pr H G q
450000 1.4 2200000 1700 6000000

Excel formula:

=NU_KITOH(450000, 1.4, 2200000, 1700, 6000000)

Expected output:

416.26

Python Code

Show Code 
from ht.conv_supercritical import Nu_Kitoh as ht_Nu_Kitoh

def Nu_Kitoh(Re, Pr, H=None, G=None, q=None):
    """
    Calculate Nusselt number for supercritical flow using the Kitoh correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk fluid properties (-).
        H (float, optional): Enthalpy of water when applicable (J/kg). Default is None.
        G (float, optional): Mass flux of the fluid (kg/m^2/s). Default is None.
        q (float, optional): Heat flux to the wall (W/m^2). Default is None.

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Kitoh(Re=Re, Pr=Pr, H=H, G=G, q=q)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk fluid properties (-).
Enthalpy of water when applicable (J/kg).
Mass flux of the fluid (kg/m^2/s).
Heat flux to the wall (W/m^2).

NU_KRASN_PROTO

This function computes the Nusselt number using the Krasnoshchekov-Protopopov supercritical correlation. It applies Reynolds and Prandtl scaling with optional correction multipliers from heat-capacity, thermal-conductivity, and viscosity ratios between wall and bulk states.

Nu = C\,Re^{m}Pr^{n}\,\Phi_{Cp}\,\Phi_{k}\,\Phi_{\mu}

Excel Usage

=NU_KRASN_PROTO(Re, Pr, Cp_avg, Cp_b, k_w, k_b, mu_w, mu_b)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk fluid properties (-).
  • Cp_avg (float, optional, default: null): Average heat capacity between wall and bulk temperatures (J/kg/K).
  • Cp_b (float, optional, default: null): Heat capacity at bulk temperature (J/kg/K).
  • k_w (float, optional, default: null): Thermal conductivity at wall temperature (W/m/K).
  • k_b (float, optional, default: null): Thermal conductivity at bulk temperature (W/m/K).
  • mu_w (float, optional, default: null): Viscosity at wall temperature (Pa*s).
  • mu_b (float, optional, default: null): Viscosity at bulk temperature (Pa*s).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Krasnoshchekov-Protopopov example

Inputs:

Re Pr Cp_avg Cp_b k_w k_b mu_w mu_b
100000 1.2 330 290 0.62 0.52 0.0008 0.0009

Excel formula:

=NU_KRASN_PROTO(100000, 1.2, 330, 290, 0.62, 0.52, 0.0008, 0.0009)

Expected output:

228.853

Example 2: Krasnoshchekov-Protopopov bulk properties only

Inputs:

Re Pr
80000 1.1

Excel formula:

=NU_KRASN_PROTO(80000, 1.1)

Expected output:

186.725

Example 3: Krasnoshchekov-Protopopov mid Reynolds number

Inputs:

Re Pr Cp_avg Cp_b k_w k_b mu_w mu_b
200000 0.95 2200 2050 0.68 0.58 0.0007 0.00082

Excel formula:

=NU_KRASN_PROTO(200000, 0.95, 2200, 2050, 0.68, 0.58, 0.0007, 0.00082)

Expected output:

336.777

Example 4: Krasnoshchekov-Protopopov higher Reynolds number

Inputs:

Re Pr Cp_avg Cp_b k_w k_b mu_w mu_b
500000 1.4 2500 2200 0.75 0.65 0.0006 0.00075

Excel formula:

=NU_KRASN_PROTO(500000, 1.4, 2500, 2200, 0.75, 0.65, 0.0006, 0.00075)

Expected output:

931.323

Python Code

Show Code 
from ht.conv_supercritical import Nu_Krasnoshchekov_Protopopov as ht_Nu_Krasnoshchekov_Protopopov

def Nu_Krasn_Proto(Re, Pr, Cp_avg=None, Cp_b=None, k_w=None, k_b=None, mu_w=None, mu_b=None):
    """
    Calculate Nusselt number for supercritical flow using the Krasnoshchekov-Protopopov correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk fluid properties (-).
        Cp_avg (float, optional): Average heat capacity between wall and bulk temperatures (J/kg/K). Default is None.
        Cp_b (float, optional): Heat capacity at bulk temperature (J/kg/K). Default is None.
        k_w (float, optional): Thermal conductivity at wall temperature (W/m/K). Default is None.
        k_b (float, optional): Thermal conductivity at bulk temperature (W/m/K). Default is None.
        mu_w (float, optional): Viscosity at wall temperature (Pa*s). Default is None.
        mu_b (float, optional): Viscosity at bulk temperature (Pa*s). Default is None.

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Krasnoshchekov_Protopopov(
            Re=Re,
            Pr=Pr,
            Cp_avg=Cp_avg,
            Cp_b=Cp_b,
            k_w=k_w,
            k_b=k_b,
            mu_w=mu_w,
            mu_b=mu_b,
        )
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk fluid properties (-).
Average heat capacity between wall and bulk temperatures (J/kg/K).
Heat capacity at bulk temperature (J/kg/K).
Thermal conductivity at wall temperature (W/m/K).
Thermal conductivity at bulk temperature (W/m/K).
Viscosity at wall temperature (Pa*s).
Viscosity at bulk temperature (Pa*s).

NU_KRASNOSH_PROTO

This function computes the Nusselt number using the Krasnoshchekov-Protopopov supercritical correlation. It applies Reynolds and Prandtl scaling with optional correction multipliers from heat-capacity, thermal-conductivity, and viscosity ratios between wall and bulk states.

Nu = C\,Re^{m}Pr^{n}\,\Phi_{Cp}\,\Phi_{k}\,\Phi_{\mu}

Excel Usage

=NU_KRASNOSH_PROTO(Re, Pr, Cp_avg, Cp_b, k_w, k_b, mu_w, mu_b)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk fluid properties (-).
  • Cp_avg (float, optional, default: null): Average heat capacity between wall and bulk temperatures (J/kg/K).
  • Cp_b (float, optional, default: null): Heat capacity at bulk temperature (J/kg/K).
  • k_w (float, optional, default: null): Thermal conductivity at wall temperature (W/m/K).
  • k_b (float, optional, default: null): Thermal conductivity at bulk temperature (W/m/K).
  • mu_w (float, optional, default: null): Viscosity at wall temperature (Pa*s).
  • mu_b (float, optional, default: null): Viscosity at bulk temperature (Pa*s).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Krasnoshchekov-Protopopov example

Inputs:

Re Pr Cp_avg Cp_b k_w k_b mu_w mu_b
100000 1.2 330 290 0.62 0.52 0.0008 0.0009

Excel formula:

=NU_KRASNOSH_PROTO(100000, 1.2, 330, 290, 0.62, 0.52, 0.0008, 0.0009)

Expected output:

228.853

Example 2: Krasnoshchekov-Protopopov bulk properties only

Inputs:

Re Pr
80000 1.1

Excel formula:

=NU_KRASNOSH_PROTO(80000, 1.1)

Expected output:

186.725

Example 3: Krasnoshchekov-Protopopov mid Reynolds number

Inputs:

Re Pr Cp_avg Cp_b k_w k_b mu_w mu_b
200000 0.95 2200 2050 0.68 0.58 0.0007 0.00082

Excel formula:

=NU_KRASNOSH_PROTO(200000, 0.95, 2200, 2050, 0.68, 0.58, 0.0007, 0.00082)

Expected output:

336.777

Example 4: Krasnoshchekov-Protopopov higher Reynolds number

Inputs:

Re Pr Cp_avg Cp_b k_w k_b mu_w mu_b
500000 1.4 2500 2200 0.75 0.65 0.0006 0.00075

Excel formula:

=NU_KRASNOSH_PROTO(500000, 1.4, 2500, 2200, 0.75, 0.65, 0.0006, 0.00075)

Expected output:

931.323

Python Code

Show Code 
from ht.conv_supercritical import Nu_Krasnoshchekov_Protopopov as ht_Nu_Krasnoshchekov_Protopopov

def Nu_Krasnosh_Proto(Re, Pr, Cp_avg=None, Cp_b=None, k_w=None, k_b=None, mu_w=None, mu_b=None):
    """
    Calculate Nusselt number for supercritical flow using the Krasnoshchekov-Protopopov correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk fluid properties (-).
        Cp_avg (float, optional): Average heat capacity between wall and bulk temperatures (J/kg/K). Default is None.
        Cp_b (float, optional): Heat capacity at bulk temperature (J/kg/K). Default is None.
        k_w (float, optional): Thermal conductivity at wall temperature (W/m/K). Default is None.
        k_b (float, optional): Thermal conductivity at bulk temperature (W/m/K). Default is None.
        mu_w (float, optional): Viscosity at wall temperature (Pa*s). Default is None.
        mu_b (float, optional): Viscosity at bulk temperature (Pa*s). Default is None.

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Krasnoshchekov_Protopopov(
            Re=Re,
            Pr=Pr,
            Cp_avg=Cp_avg,
            Cp_b=Cp_b,
            k_w=k_w,
            k_b=k_b,
            mu_w=mu_w,
            mu_b=mu_b,
        )
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk fluid properties (-).
Average heat capacity between wall and bulk temperatures (J/kg/K).
Heat capacity at bulk temperature (J/kg/K).
Thermal conductivity at wall temperature (W/m/K).
Thermal conductivity at bulk temperature (W/m/K).
Viscosity at wall temperature (Pa*s).
Viscosity at bulk temperature (Pa*s).

NU_KRASNOSHCHEKOV

This function estimates the supercritical convection Nusselt number using the Krasnoshchekov correlation. It combines a base turbulent heat-transfer relation with optional corrections for density and heat-capacity ratios and temperature-dependent pseudocritical behavior.

Nu = Nu_0\,\Phi_{\rho}\,\Phi_{Cp}(T_b,T_w,T_{pc})

Excel Usage

=NU_KRASNOSHCHEKOV(Re, Pr, rho_w, rho_b, Cp_avg, Cp_b, T_b, T_w, T_pc)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk fluid properties (-).
  • rho_w (float, optional, default: null): Density at wall temperature (kg/m^3).
  • rho_b (float, optional, default: null): Density at bulk temperature (kg/m^3).
  • Cp_avg (float, optional, default: null): Average heat capacity between wall and bulk temperatures (J/kg/K).
  • Cp_b (float, optional, default: null): Heat capacity at bulk temperature (J/kg/K).
  • T_b (float, optional, default: null): Bulk temperature (K).
  • T_w (float, optional, default: null): Wall temperature (K).
  • T_pc (float, optional, default: null): Pseudocritical temperature at pressure (K).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Krasnoshchekov correlation example

Inputs:

Re Pr
100000 1.2

Excel formula:

=NU_KRASNOSHCHEKOV(100000, 1.2)

Expected output:

234.829

Example 2: Krasnoshchekov correlation with property corrections

Inputs:

Re Pr rho_w rho_b Cp_avg Cp_b T_b T_w T_pc
120000 1.1 350 300 2100 2000 650 700 640

Excel formula:

=NU_KRASNOSHCHEKOV(120000, 1.1, 350, 300, 2100, 2000, 650, 700, 640)

Expected output:

275.091

Example 3: Krasnoshchekov correlation mid Reynolds number

Inputs:

Re Pr rho_w rho_b Cp_avg Cp_b T_b T_w T_pc
220000 0.95 360 310 2300 2100 620 690 640

Excel formula:

=NU_KRASNOSHCHEKOV(220000, 0.95, 360, 310, 2300, 2100, 620, 690, 640)

Expected output:

413.04

Example 4: Krasnoshchekov correlation higher Reynolds number

Inputs:

Re Pr
300000 0.9

Excel formula:

=NU_KRASNOSHCHEKOV(300000, 0.9)

Expected output:

470.84

Python Code

Show Code 
from ht.conv_supercritical import Nu_Krasnoshchekov as ht_Nu_Krasnoshchekov

def Nu_Krasnoshchekov(Re, Pr, rho_w=None, rho_b=None, Cp_avg=None, Cp_b=None, T_b=None, T_w=None, T_pc=None):
    """
    Calculate Nusselt number for supercritical flow using the Krasnoshchekov correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk fluid properties (-).
        rho_w (float, optional): Density at wall temperature (kg/m^3). Default is None.
        rho_b (float, optional): Density at bulk temperature (kg/m^3). Default is None.
        Cp_avg (float, optional): Average heat capacity between wall and bulk temperatures (J/kg/K). Default is None.
        Cp_b (float, optional): Heat capacity at bulk temperature (J/kg/K). Default is None.
        T_b (float, optional): Bulk temperature (K). Default is None.
        T_w (float, optional): Wall temperature (K). Default is None.
        T_pc (float, optional): Pseudocritical temperature at pressure (K). Default is None.

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Krasnoshchekov(
            Re=Re,
            Pr=Pr,
            rho_w=rho_w,
            rho_b=rho_b,
            Cp_avg=Cp_avg,
            Cp_b=Cp_b,
            T_b=T_b,
            T_w=T_w,
            T_pc=T_pc,
        )
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk fluid properties (-).
Density at wall temperature (kg/m^3).
Density at bulk temperature (kg/m^3).
Average heat capacity between wall and bulk temperatures (J/kg/K).
Heat capacity at bulk temperature (J/kg/K).
Bulk temperature (K).
Wall temperature (K).
Pseudocritical temperature at pressure (K).

NU_MCADAMS

This function calculates a Nusselt number for supercritical turbulent internal flow using the McAdams-style power-law form. It uses Reynolds and Prandtl numbers with no additional property-ratio correction terms.

Nu = C\,Re^{m}Pr^{n}

Excel Usage

=NU_MCADAMS(Re, Pr)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk fluid properties (-).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: McAdams correlation example

Inputs:

Re Pr
100000 1.2

Excel formula:

=NU_MCADAMS(100000, 1.2)

Expected output:

261.384

Example 2: McAdams correlation lower Reynolds number

Inputs:

Re Pr
50000 1

Excel formula:

=NU_MCADAMS(50000, 1)

Expected output:

139.567

Example 3: McAdams correlation mid Reynolds number

Inputs:

Re Pr
200000 0.8

Excel formula:

=NU_MCADAMS(200000, 0.8)

Expected output:

386.96

Example 4: McAdams correlation higher Reynolds number

Inputs:

Re Pr
700000 1.5

Excel formula:

=NU_MCADAMS(700000, 1.5)

Expected output:

1355.57

Python Code

Show Code 
from ht.conv_supercritical import Nu_McAdams as ht_Nu_McAdams

def Nu_McAdams(Re, Pr):
    """
    Calculate Nusselt number for supercritical flow using the McAdams correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk fluid properties (-).

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_McAdams(Re=Re, Pr=Pr)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk fluid properties (-).

NU_MOKRY

This function estimates the Nusselt number for supercritical internal convection by the Mokry correlation. It uses Reynolds and Prandtl numbers and can include a density-ratio correction when wall and bulk densities are provided.

Nu = C\,Re^{m}Pr^{n}\,\Phi_{\rho}

Excel Usage

=NU_MOKRY(Re, Pr, rho_w, rho_b)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk properties and averaged heat capacity (-).
  • rho_w (float, optional, default: null): Density at wall temperature (kg/m^3).
  • rho_b (float, optional, default: null): Density at bulk temperature (kg/m^3).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Mokry correlation example

Inputs:

Re Pr rho_w rho_b
100000 1.2 330 290

Excel formula:

=NU_MOKRY(100000, 1.2, 330, 290)

Expected output:

246.116

Example 2: Mokry correlation bulk properties only

Inputs:

Re Pr
80000 1

Excel formula:

=NU_MOKRY(80000, 1)

Expected output:

165.091

Example 3: Mokry correlation mid Reynolds number

Inputs:

Re Pr rho_w rho_b
200000 0.9 360 310

Excel formula:

=NU_MOKRY(200000, 0.9, 360, 310)

Expected output:

382.639

Example 4: Mokry correlation higher Reynolds number

Inputs:

Re Pr rho_w rho_b
500000 1.5 380 320

Excel formula:

=NU_MOKRY(500000, 1.5, 380, 320)

Expected output:

1258.17

Python Code

Show Code 
from ht.conv_supercritical import Nu_Mokry as ht_Nu_Mokry

def Nu_Mokry(Re, Pr, rho_w=None, rho_b=None):
    """
    Calculate Nusselt number for supercritical flow using the Mokry correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk properties and averaged heat capacity (-).
        rho_w (float, optional): Density at wall temperature (kg/m^3). Default is None.
        rho_b (float, optional): Density at bulk temperature (kg/m^3). Default is None.

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Mokry(Re=Re, Pr=Pr, rho_w=rho_w, rho_b=rho_b)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk properties and averaged heat capacity (-).
Density at wall temperature (kg/m^3).
Density at bulk temperature (kg/m^3).

NU_ORNATSKY

This function computes the Nusselt number for supercritical turbulent flow with the Ornatsky correlation. It uses Reynolds number, both bulk and wall Prandtl numbers, and optionally applies a density-ratio correction.

Nu = C\,Re^{m}\,(\min(Pr_b,Pr_w))^{n}\,\Phi_{\rho}

Excel Usage

=NU_ORNATSKY(Re, Pr_b, Pr_w, rho_w, rho_b)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr_b (float, required): Prandtl number with bulk fluid properties (-).
  • Pr_w (float, required): Prandtl number with wall fluid properties (-).
  • rho_w (float, optional, default: null): Density at wall temperature (kg/m^3).
  • rho_b (float, optional, default: null): Density at bulk temperature (kg/m^3).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Ornatsky correlation example

Inputs:

Re Pr_b Pr_w rho_w rho_b
100000 1.2 1.5 330 290

Excel formula:

=NU_ORNATSKY(100000, 1.2, 1.5, 330, 290)

Expected output:

276.635

Example 2: Ornatsky correlation bulk properties only

Inputs:

Re Pr_b Pr_w
80000 1 1.1

Excel formula:

=NU_ORNATSKY(80000, 1, 1.1)

Expected output:

192.398

Example 3: Ornatsky correlation mid Reynolds number

Inputs:

Re Pr_b Pr_w rho_w rho_b
200000 0.9 1.3 360 310

Excel formula:

=NU_ORNATSKY(200000, 0.9, 1.3, 360, 310)

Expected output:

384.971

Example 4: Ornatsky correlation higher Reynolds number

Inputs:

Re Pr_b Pr_w rho_w rho_b
450000 1.4 1.8 380 320

Excel formula:

=NU_ORNATSKY(450000, 1.4, 1.8, 380, 320)

Expected output:

1055.82

Python Code

Show Code 
from ht.conv_supercritical import Nu_Ornatsky as ht_Nu_Ornatsky

def Nu_Ornatsky(Re, Pr_b, Pr_w, rho_w=None, rho_b=None):
    """
    Calculate Nusselt number for supercritical flow using the Ornatsky correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr_b (float): Prandtl number with bulk fluid properties (-).
        Pr_w (float): Prandtl number with wall fluid properties (-).
        rho_w (float, optional): Density at wall temperature (kg/m^3). Default is None.
        rho_b (float, optional): Density at bulk temperature (kg/m^3). Default is None.

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Ornatsky(Re=Re, Pr_b=Pr_b, Pr_w=Pr_w, rho_w=rho_w, rho_b=rho_b)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk fluid properties (-).
Prandtl number with wall fluid properties (-).
Density at wall temperature (kg/m^3).
Density at bulk temperature (kg/m^3).

NU_PETUKHOV

This function estimates the Nusselt number for supercritical turbulent tube flow using the Petukhov correlation structure. It uses Reynolds and Prandtl numbers and can include optional density and viscosity correction terms in the friction-factor-based model.

Nu = f(Re,Pr,\rho_w/\rho_b,\mu_w/\mu_b)

Excel Usage

=NU_PETUKHOV(Re, Pr, rho_w, rho_b, mu_w, mu_b)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk fluid properties (-).
  • rho_w (float, optional, default: null): Density at wall temperature (kg/m^3).
  • rho_b (float, optional, default: null): Density at bulk temperature (kg/m^3).
  • mu_w (float, optional, default: null): Viscosity at wall temperature (Pa*s).
  • mu_b (float, optional, default: null): Viscosity at bulk temperature (Pa*s).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Petukhov correlation example

Inputs:

Re Pr rho_w rho_b mu_w mu_b
100000 1.2 330 290 0.0008 0.0009

Excel formula:

=NU_PETUKHOV(100000, 1.2, 330, 290, 0.0008, 0.0009)

Expected output:

254.826

Example 2: Petukhov correlation bulk properties only

Inputs:

Re Pr
80000 1.1

Excel formula:

=NU_PETUKHOV(80000, 1.1)

Expected output:

197.156

Example 3: Petukhov correlation mid Reynolds number

Inputs:

Re Pr rho_w rho_b mu_w mu_b
200000 0.9 360 310 0.0007 0.00082

Excel formula:

=NU_PETUKHOV(200000, 0.9, 360, 310, 0.0007, 0.00082)

Expected output:

373.626

Example 4: Petukhov correlation higher Reynolds number

Inputs:

Re Pr rho_w rho_b mu_w mu_b
450000 1.4 380 320 0.0006 0.00075

Excel formula:

=NU_PETUKHOV(450000, 1.4, 380, 320, 0.0006, 0.00075)

Expected output:

950.914

Python Code

Show Code 
from ht.conv_supercritical import Nu_Petukhov as ht_Nu_Petukhov

def Nu_Petukhov(Re, Pr, rho_w=None, rho_b=None, mu_w=None, mu_b=None):
    """
    Calculate Nusselt number for supercritical flow using the Petukhov correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk fluid properties (-).
        rho_w (float, optional): Density at wall temperature (kg/m^3). Default is None.
        rho_b (float, optional): Density at bulk temperature (kg/m^3). Default is None.
        mu_w (float, optional): Viscosity at wall temperature (Pa*s). Default is None.
        mu_b (float, optional): Viscosity at bulk temperature (Pa*s). Default is None.

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Petukhov(Re=Re, Pr=Pr, rho_w=rho_w, rho_b=rho_b, mu_w=mu_w, mu_b=mu_b)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk fluid properties (-).
Density at wall temperature (kg/m^3).
Density at bulk temperature (kg/m^3).
Viscosity at wall temperature (Pa*s).
Viscosity at bulk temperature (Pa*s).

NU_SHITSMAN

This function computes the Nusselt number for supercritical turbulent internal convection using the Shitsman correlation. It depends on Reynolds number and the minimum of bulk and wall Prandtl numbers.

Nu = C\,Re^{m}(\min(Pr_b,Pr_w))^{n}

Excel Usage

=NU_SHITSMAN(Re, Pr_b, Pr_w)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr_b (float, required): Prandtl number with bulk fluid properties (-).
  • Pr_w (float, required): Prandtl number with wall fluid properties (-).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Shitsman correlation example

Inputs:

Re Pr_b Pr_w
100000 1.2 1.6

Excel formula:

=NU_SHITSMAN(100000, 1.2, 1.6)

Expected output:

266.117

Example 2: Shitsman correlation lower Reynolds number

Inputs:

Re Pr_b Pr_w
60000 1 1.3

Excel formula:

=NU_SHITSMAN(60000, 1, 1.3)

Expected output:

152.844

Example 3: Shitsman correlation mid Reynolds number

Inputs:

Re Pr_b Pr_w
200000 0.9 1.4

Excel formula:

=NU_SHITSMAN(200000, 0.9, 1.4)

Expected output:

368.083

Example 4: Shitsman correlation higher Reynolds number

Inputs:

Re Pr_b Pr_w
400000 1.3 1.7

Excel formula:

=NU_SHITSMAN(400000, 1.3, 1.7)

Expected output:

860.063

Python Code

Show Code 
from ht.conv_supercritical import Nu_Shitsman as ht_Nu_Shitsman

def Nu_Shitsman(Re, Pr_b, Pr_w):
    """
    Calculate Nusselt number for supercritical flow using the Shitsman correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr_b (float): Prandtl number with bulk fluid properties (-).
        Pr_w (float): Prandtl number with wall fluid properties (-).

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Shitsman(Re=Re, Pr_b=Pr_b, Pr_w=Pr_w)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk fluid properties (-).
Prandtl number with wall fluid properties (-).

NU_SWENSON

This function estimates Nusselt number for supercritical internal turbulent flow using the Swenson correlation. It uses Reynolds and Prandtl numbers and optionally incorporates wall-to-bulk density-ratio effects.

Nu = C\,Re^{m}Pr^{n}\,\Phi_{\rho}

Excel Usage

=NU_SWENSON(Re, Pr, rho_w, rho_b)
  • Re (float, required): Reynolds number with wall fluid properties (-).
  • Pr (float, required): Prandtl number with wall properties and averaged heat capacity (-).
  • rho_w (float, optional, default: null): Density at wall temperature (kg/m^3).
  • rho_b (float, optional, default: null): Density at bulk temperature (kg/m^3).

Returns (float): Nusselt number with wall fluid properties (-).

Example 1: Swenson correlation example

Inputs:

Re Pr rho_w rho_b
100000 1.2 330 290

Excel formula:

=NU_SWENSON(100000, 1.2, 330, 290)

Expected output:

217.928

Example 2: Swenson correlation bulk properties only

Inputs:

Re Pr
80000 1

Excel formula:

=NU_SWENSON(80000, 1)

Expected output:

153.945

Example 3: Swenson correlation mid Reynolds number

Inputs:

Re Pr rho_w rho_b
200000 0.9 360 310

Excel formula:

=NU_SWENSON(200000, 0.9, 360, 310)

Expected output:

348.029

Example 4: Swenson correlation higher Reynolds number

Inputs:

Re Pr rho_w rho_b
500000 1.5 380 320

Excel formula:

=NU_SWENSON(500000, 1.5, 380, 320)

Expected output:

1114.67

Python Code

Show Code 
from ht.conv_supercritical import Nu_Swenson as ht_Nu_Swenson

def Nu_Swenson(Re, Pr, rho_w=None, rho_b=None):
    """
    Calculate Nusselt number for supercritical flow using the Swenson correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with wall fluid properties (-).
        Pr (float): Prandtl number with wall properties and averaged heat capacity (-).
        rho_w (float, optional): Density at wall temperature (kg/m^3). Default is None.
        rho_b (float, optional): Density at bulk temperature (kg/m^3). Default is None.

    Returns:
        float: Nusselt number with wall fluid properties (-).
    """
    try:
        return ht_Nu_Swenson(Re=Re, Pr=Pr, rho_w=rho_w, rho_b=rho_b)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with wall fluid properties (-).
Prandtl number with wall properties and averaged heat capacity (-).
Density at wall temperature (kg/m^3).
Density at bulk temperature (kg/m^3).

NU_XU

This function calculates the Nusselt number for supercritical turbulent convection in vertical pipes using the Xu correlation. It applies Reynolds and Prandtl scaling with optional density and viscosity ratio corrections.

Nu = C\,Re^{m}Pr^{n}\,\Phi_{\rho}\,\Phi_{\mu}

Excel Usage

=NU_XU(Re, Pr, rho_w, rho_b, mu_w, mu_b)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk properties and averaged heat capacity (-).
  • rho_w (float, optional, default: null): Density at wall temperature (kg/m^3).
  • rho_b (float, optional, default: null): Density at bulk temperature (kg/m^3).
  • mu_w (float, optional, default: null): Viscosity at wall temperature (Pa*s).
  • mu_b (float, optional, default: null): Viscosity at bulk temperature (Pa*s).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Xu correlation example

Inputs:

Re Pr rho_w rho_b mu_w mu_b
100000 1.2 330 290 0.0008 0.0009

Excel formula:

=NU_XU(100000, 1.2, 330, 290, 0.0008, 0.0009)

Expected output:

289.133

Example 2: Xu correlation bulk properties only

Inputs:

Re Pr
80000 1.1

Excel formula:

=NU_XU(80000, 1.1)

Expected output:

226.556

Example 3: Xu correlation mid Reynolds number

Inputs:

Re Pr rho_w rho_b mu_w mu_b
200000 0.9 360 310 0.0007 0.00082

Excel formula:

=NU_XU(200000, 0.9, 360, 310, 0.0007, 0.00082)

Expected output:

380.01

Example 4: Xu correlation higher Reynolds number

Inputs:

Re Pr rho_w rho_b mu_w mu_b
450000 1.4 380 320 0.0006 0.00075

Excel formula:

=NU_XU(450000, 1.4, 380, 320, 0.0006, 0.00075)

Expected output:

1054.51

Python Code

Show Code 
from ht.conv_supercritical import Nu_Xu as ht_Nu_Xu

def Nu_Xu(Re, Pr, rho_w=None, rho_b=None, mu_w=None, mu_b=None):
    """
    Calculate Nusselt number for supercritical flow using the Xu correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk properties and averaged heat capacity (-).
        rho_w (float, optional): Density at wall temperature (kg/m^3). Default is None.
        rho_b (float, optional): Density at bulk temperature (kg/m^3). Default is None.
        mu_w (float, optional): Viscosity at wall temperature (Pa*s). Default is None.
        mu_b (float, optional): Viscosity at bulk temperature (Pa*s). Default is None.

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Xu(Re=Re, Pr=Pr, rho_w=rho_w, rho_b=rho_b, mu_w=mu_w, mu_b=mu_b)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk properties and averaged heat capacity (-).
Density at wall temperature (kg/m^3).
Density at bulk temperature (kg/m^3).
Viscosity at wall temperature (Pa*s).
Viscosity at bulk temperature (Pa*s).

NU_YAMAGATA

This function computes the Nusselt number for turbulent supercritical pipe flow using the Yamagata correlation. It uses Reynolds and Prandtl numbers and can apply pseudocritical-property corrections based on Prandtl number at the pseudocritical state, heat-capacity ratio, and the bulk/wall temperature window.

Nu = C\,Re^{m}Pr^{n}\,F(Pr_{pc},Cp_{avg}/Cp_b,T_b,T_w,T_{pc})

Excel Usage

=NU_YAMAGATA(Re, Pr, Pr_pc, Cp_avg, Cp_b, T_b, T_w, T_pc)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk fluid properties (-).
  • Pr_pc (float, optional, default: null): Prandtl number at the pseudocritical temperature (-).
  • Cp_avg (float, optional, default: null): Average heat capacity between wall and bulk temperatures (J/kg/K).
  • Cp_b (float, optional, default: null): Heat capacity at bulk temperature (J/kg/K).
  • T_b (float, optional, default: null): Bulk temperature (K).
  • T_w (float, optional, default: null): Wall temperature (K).
  • T_pc (float, optional, default: null): Pseudocritical temperature at pressure (K).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Yamagata correlation example

Inputs:

Re Pr Pr_pc Cp_avg Cp_b T_b T_w T_pc
100000 1.2 1.5 2080.845 2048.621 650 700 600

Excel formula:

=NU_YAMAGATA(100000, 1.2, 1.5, 2080.845, 2048.621, 650, 700, 600)

Expected output:

292.347

Example 2: Yamagata correlation bulk properties only

Inputs:

Re Pr
80000 1

Excel formula:

=NU_YAMAGATA(80000, 1)

Expected output:

203.004

Example 3: Yamagata correlation mid Reynolds number

Inputs:

Re Pr Pr_pc Cp_avg Cp_b T_b T_w T_pc
200000 0.9 1.2 2200 2100 620 680 640

Excel formula:

=NU_YAMAGATA(200000, 0.9, 1.2, 2200, 2100, 620, 680, 640)

Expected output:

270.924

Example 4: Yamagata correlation higher Reynolds number

Inputs:

Re Pr Pr_pc Cp_avg Cp_b T_b T_w T_pc
450000 1.4 1.3 2400 2200 700 760 680

Excel formula:

=NU_YAMAGATA(450000, 1.4, 1.3, 2400, 2200, 700, 760, 680)

Expected output:

1374.86

Python Code

Show Code 
from ht.conv_supercritical import Nu_Yamagata as ht_Nu_Yamagata

def Nu_Yamagata(Re, Pr, Pr_pc=None, Cp_avg=None, Cp_b=None, T_b=None, T_w=None, T_pc=None):
    """
    Calculate Nusselt number for supercritical flow using the Yamagata correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk fluid properties (-).
        Pr_pc (float, optional): Prandtl number at the pseudocritical temperature (-). Default is None.
        Cp_avg (float, optional): Average heat capacity between wall and bulk temperatures (J/kg/K). Default is None.
        Cp_b (float, optional): Heat capacity at bulk temperature (J/kg/K). Default is None.
        T_b (float, optional): Bulk temperature (K). Default is None.
        T_w (float, optional): Wall temperature (K). Default is None.
        T_pc (float, optional): Pseudocritical temperature at pressure (K). Default is None.

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Yamagata(
            Re=Re,
            Pr=Pr,
            Pr_pc=Pr_pc,
            Cp_avg=Cp_avg,
            Cp_b=Cp_b,
            T_b=T_b,
            T_w=T_w,
            T_pc=T_pc,
        )
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk fluid properties (-).
Prandtl number at the pseudocritical temperature (-).
Average heat capacity between wall and bulk temperatures (J/kg/K).
Heat capacity at bulk temperature (J/kg/K).
Bulk temperature (K).
Wall temperature (K).
Pseudocritical temperature at pressure (K).

NU_ZHU

This function estimates supercritical turbulent-convection Nusselt number with the Zhu correlation. It uses Reynolds and Prandtl numbers and optionally includes density and thermal-conductivity ratio corrections between wall and bulk conditions.

Nu = C\,Re^{m}Pr^{n}\,\Phi_{\rho}\,\Phi_{k}

Excel Usage

=NU_ZHU(Re, Pr, rho_w, rho_b, k_w, k_b)
  • Re (float, required): Reynolds number with bulk fluid properties (-).
  • Pr (float, required): Prandtl number with bulk properties and averaged heat capacity (-).
  • rho_w (float, optional, default: null): Density at wall temperature (kg/m^3).
  • rho_b (float, optional, default: null): Density at bulk temperature (kg/m^3).
  • k_w (float, optional, default: null): Thermal conductivity at wall temperature (W/m/K).
  • k_b (float, optional, default: null): Thermal conductivity at bulk temperature (W/m/K).

Returns (float): Nusselt number with bulk fluid properties (-).

Example 1: Zhu correlation example

Inputs:

Re Pr rho_w rho_b k_w k_b
100000 1.2 330 290 0.63 0.69

Excel formula:

=NU_ZHU(100000, 1.2, 330, 290, 0.63, 0.69)

Expected output:

240.146

Example 2: Zhu correlation bulk properties only

Inputs:

Re Pr
80000 1.1

Excel formula:

=NU_ZHU(80000, 1.1)

Expected output:

186.796

Example 3: Zhu correlation mid Reynolds number

Inputs:

Re Pr rho_w rho_b k_w k_b
200000 0.9 360 310 0.7 0.62

Excel formula:

=NU_ZHU(200000, 0.9, 360, 310, 0.7, 0.62)

Expected output:

398.964

Example 4: Zhu correlation higher Reynolds number

Inputs:

Re Pr rho_w rho_b k_w k_b
450000 1.4 380 320 0.75 0.66

Excel formula:

=NU_ZHU(450000, 1.4, 380, 320, 0.75, 0.66)

Expected output:

1099.64

Python Code

Show Code 
from ht.conv_supercritical import Nu_Zhu as ht_Nu_Zhu

def Nu_Zhu(Re, Pr, rho_w=None, rho_b=None, k_w=None, k_b=None):
    """
    Calculate Nusselt number for supercritical flow using the Zhu correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_supercritical.html

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

    Args:
        Re (float): Reynolds number with bulk fluid properties (-).
        Pr (float): Prandtl number with bulk properties and averaged heat capacity (-).
        rho_w (float, optional): Density at wall temperature (kg/m^3). Default is None.
        rho_b (float, optional): Density at bulk temperature (kg/m^3). Default is None.
        k_w (float, optional): Thermal conductivity at wall temperature (W/m/K). Default is None.
        k_b (float, optional): Thermal conductivity at bulk temperature (W/m/K). Default is None.

    Returns:
        float: Nusselt number with bulk fluid properties (-).
    """
    try:
        return ht_Nu_Zhu(Re=Re, Pr=Pr, rho_w=rho_w, rho_b=rho_b, k_w=k_w, k_b=k_b)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number with bulk fluid properties (-).
Prandtl number with bulk properties and averaged heat capacity (-).
Density at wall temperature (kg/m^3).
Density at bulk temperature (kg/m^3).
Thermal conductivity at wall temperature (W/m/K).
Thermal conductivity at bulk temperature (W/m/K).