Boiling Flow

Overview

Flow boiling describes heat transfer in internal passages where liquid and vapor coexist, typically under forced convection and strong phase-change effects. In design practice, flow boiling models are used to estimate wall heat transfer coefficients, assess operating envelopes, and avoid dryout or unstable thermal behavior. Because boiling coefficients can vary by orders of magnitude across quality, mass flux, and channel size, correlation-based screening is a core part of thermal engineering workflows for evaporators, boilers, and electronics cooling loops.

Core Concepts: The category combines nucleate boiling, convective boiling, and two-phase scaling into practical prediction tools. Many models use a composite structure where a nucleate contribution and a liquid-convective contribution are blended through suppression/enhancement factors, while others are tuned for mini/microchannels using dimensionless groups such as Reynolds, Boiling, and Weber numbers. A useful framing is to separate direct coefficient prediction from support calculations: h_{tp} = f(\text{quality},\; \text{mass flux},\; \text{properties},\; q\;\text{or}\;\Delta T_e), \qquad ext{auxiliary terms} = g(x, \rho, \mu, f_D, \ldots). This helps practitioners choose whether they need a primary heat-transfer estimate, a two-phase intermediate parameter, or a helper equation for an implicit solve.

Implementation: These functions are implemented from the ht library, specifically ht.boiling_flow. The ht package aggregates literature correlations for process and thermal engineering with consistent units and callable APIs, making it well suited for calculator-style evaluation, early-stage sizing, and sensitivity studies.

Composite convective+nucleate formulations include CHEN_BENNETT, CHEN_EDELSTEIN, and LIU_WINTERTON. These methods are commonly selected for saturated tube-flow cases where both forced convection and bubble-driven transfer are important, and where input data is rich enough to evaluate fluid properties, quality, and operating state. CHEN_BENNETT and CHEN_EDELSTEIN follow the classic additive framework with empirical suppression/enhancement factors, while LIU_WINTERTON uses a combined form that balances convective and nucleate terms differently. In practice, engineers often compare these three to bracket expected performance before detailed experimental calibration.

Nucleate and channel-focused empirical correlations are provided by COOPER, FORSTER_ZUBER, LAZAREK_BLACK, LI_WU, SUN_MISHIMA, and YUN_HEO_KIM. COOPER and FORSTER_ZUBER are foundational nucleate-boiling relations and frequently appear as components inside broader flow-boiling models, while LAZAREK_BLACK, LI_WU, SUN_MISHIMA, and YUN_HEO_KIM target mini/microchannel behavior with stronger dependence on compact-geometry scaling. These are useful for refrigerant circuits, compact evaporators, and high heat-flux thermal management where channel size and phase distribution strongly influence h_{tp}. Together, they provide a practical range from classical pool/flow nucleate behavior to modern microscale evaporation fitting.

Two-phase auxiliary and mechanistic tools include LOCKHART_XTT, THOME, TO_SOLVE_Q_THOME, TURBULENT_DITTUS, and TURBULENT_GNIEL. LOCKHART_XTT supplies the Lockhart–Martinelli parameter used as an intermediate variable in many two-phase closure relations, while TURBULENT_DITTUS and TURBULENT_GNIEL provide single-phase turbulent convection baselines often embedded within boiling frameworks. THOME offers a more detailed slug/film microchannel model for direct coefficient prediction, and TO_SOLVE_Q_THOME supports the implicit residual calculation needed when solving for heat flux from a specified wall superheat. This combination supports both quick empirical screening and deeper iterative modeling in the same workflow.

CHEN_BENNETT

This function estimates the two-phase flow boiling heat transfer coefficient using the Chen-Bennett correlation for saturated flow in tubes. It combines a convective contribution with a nucleate-boiling contribution through suppression and enhancement factors.

The combined form is:

h_{tp} = S h_{nb} + F h_{sp,l}

where h_{sp,l} is commonly modeled with a Dittus-Boelter style relation and h_{nb} with a Forster-Zuber style relation. The approach is widely used for engineering estimation of boiling heat transfer in internal flows.

Excel Usage

=CHEN_BENNETT(m, x, D, rhol, rhog, mul, mug, kl, Cpl, Hvap, sigma, dPsat, Te)
  • m (float, required): Mass flow rate (kg/s).
  • x (float, required): Quality at the tube interval (dimensionless).
  • D (float, required): Tube diameter (m).
  • rhol (float, required): Liquid density (kg/m^3).
  • rhog (float, required): Gas density (kg/m^3).
  • mul (float, required): Liquid viscosity (Pa*s).
  • mug (float, required): Gas viscosity (Pa*s).
  • kl (float, required): Liquid thermal conductivity (W/m/K).
  • Cpl (float, required): Liquid heat capacity (J/kg/K).
  • Hvap (float, required): Heat of vaporization (J/kg).
  • sigma (float, required): Surface tension (N/m).
  • dPsat (float, required): Saturation pressure difference (Pa).
  • Te (float, required): Excess wall temperature (K).

Returns (float): Heat transfer coefficient (W/m^2/K), or an error message if invalid.

Example 1: Example from reference

Inputs:

m x D rhol rhog mul mug kl Cpl Hvap sigma dPsat Te
0.106 0.2 0.0212 567 18.09 0.000156 0.00000711 0.086 2730 200000 0.02 100000 3

Excel formula:

=CHEN_BENNETT(0.106, 0.2, 0.0212, 567, 18.09, 0.000156, 0.00000711, 0.086, 2730, 200000, 0.02, 100000, 3)

Expected output:

4938.28

Example 2: Higher quality with larger diameter

Inputs:

m x D rhol rhog mul mug kl Cpl Hvap sigma dPsat Te
0.2 0.4 0.03 900 15 0.0002 0.00001 0.12 3200 180000 0.025 80000 5

Excel formula:

=CHEN_BENNETT(0.2, 0.4, 0.03, 900, 15, 0.0002, 0.00001, 0.12, 3200, 180000, 0.025, 80000, 5)

Expected output:

7946.77

Example 3: Low quality at small diameter

Inputs:

m x D rhol rhog mul mug kl Cpl Hvap sigma dPsat Te
0.05 0.1 0.015 950 8 0.0003 0.000012 0.14 3500 220000 0.03 60000 2

Excel formula:

=CHEN_BENNETT(0.05, 0.1, 0.015, 950, 8, 0.0003, 0.000012, 0.14, 3500, 220000, 0.03, 60000, 2)

Expected output:

7020.18

Example 4: Mid-range inputs

Inputs:

m x D rhol rhog mul mug kl Cpl Hvap sigma dPsat Te
0.12 0.3 0.025 700 12 0.00018 0.000008 0.1 2600 190000 0.018 90000 4

Excel formula:

=CHEN_BENNETT(0.12, 0.3, 0.025, 700, 12, 0.00018, 0.000008, 0.1, 2600, 190000, 0.018, 90000, 4)

Expected output:

5751.45

Python Code

Show Code 
from ht.boiling_flow import Chen_Bennett as ht_Chen_Bennett

def Chen_Bennett(m, x, D, rhol, rhog, mul, mug, kl, Cpl, Hvap, sigma, dPsat, Te):
    """
    Compute the Chen-Bennett boiling heat transfer coefficient.

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

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

    Args:
        m (float): Mass flow rate (kg/s).
        x (float): Quality at the tube interval (dimensionless).
        D (float): Tube diameter (m).
        rhol (float): Liquid density (kg/m^3).
        rhog (float): Gas density (kg/m^3).
        mul (float): Liquid viscosity (Pa*s).
        mug (float): Gas viscosity (Pa*s).
        kl (float): Liquid thermal conductivity (W/m/K).
        Cpl (float): Liquid heat capacity (J/kg/K).
        Hvap (float): Heat of vaporization (J/kg).
        sigma (float): Surface tension (N/m).
        dPsat (float): Saturation pressure difference (Pa).
        Te (float): Excess wall temperature (K).

    Returns:
        float: Heat transfer coefficient (W/m^2/K), or an error message if invalid.
    """
    try:
        return ht_Chen_Bennett(m=m, x=x, D=D, rhol=rhol, rhog=rhog, mul=mul, mug=mug,
            kl=kl, Cpl=Cpl, Hvap=Hvap, sigma=sigma, dPsat=dPsat, Te=Te)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Mass flow rate (kg/s).
Quality at the tube interval (dimensionless).
Tube diameter (m).
Liquid density (kg/m^3).
Gas density (kg/m^3).
Liquid viscosity (Pa*s).
Gas viscosity (Pa*s).
Liquid thermal conductivity (W/m/K).
Liquid heat capacity (J/kg/K).
Heat of vaporization (J/kg).
Surface tension (N/m).
Saturation pressure difference (Pa).
Excess wall temperature (K).

CHEN_EDELSTEIN

This function estimates the two-phase flow boiling heat transfer coefficient using the Chen-Edelstein correlation for saturated tube flow. It blends nucleate-boiling and convective mechanisms using empirical enhancement and suppression factors.

The model structure is:

h_{tp} = S h_{nb} + F h_{sp,l}

where the single-phase liquid term and nucleate-boiling term are combined to represent mixed boiling behavior across a range of flow conditions.

Excel Usage

=CHEN_EDELSTEIN(m, x, D, rhol, rhog, mul, mug, kl, Cpl, Hvap, sigma, dPsat, Te)
  • m (float, required): Mass flow rate (kg/s).
  • x (float, required): Quality at the tube interval (dimensionless).
  • D (float, required): Tube diameter (m).
  • rhol (float, required): Liquid density (kg/m^3).
  • rhog (float, required): Gas density (kg/m^3).
  • mul (float, required): Liquid viscosity (Pa*s).
  • mug (float, required): Gas viscosity (Pa*s).
  • kl (float, required): Liquid thermal conductivity (W/m/K).
  • Cpl (float, required): Liquid heat capacity (J/kg/K).
  • Hvap (float, required): Heat of vaporization (J/kg).
  • sigma (float, required): Surface tension (N/m).
  • dPsat (float, required): Saturation pressure difference (Pa).
  • Te (float, required): Excess wall temperature (K).

Returns (float): Heat transfer coefficient (W/m^2/K), or an error message if invalid.

Example 1: Example from reference

Inputs:

m x D rhol rhog mul mug kl Cpl Hvap sigma dPsat Te
0.106 0.2 0.0212 567 18.09 0.000156 0.00000711 0.086 2730 200000 0.02 100000 3

Excel formula:

=CHEN_EDELSTEIN(0.106, 0.2, 0.0212, 567, 18.09, 0.000156, 0.00000711, 0.086, 2730, 200000, 0.02, 100000, 3)

Expected output:

3289.06

Example 2: Higher quality with larger diameter

Inputs:

m x D rhol rhog mul mug kl Cpl Hvap sigma dPsat Te
0.18 0.45 0.03 820 12 0.00021 0.000011 0.11 3000 210000 0.022 90000 6

Excel formula:

=CHEN_EDELSTEIN(0.18, 0.45, 0.03, 820, 12, 0.00021, 0.000011, 0.11, 3000, 210000, 0.022, 90000, 6)

Expected output:

4517.15

Example 3: Low quality at small diameter

Inputs:

m x D rhol rhog mul mug kl Cpl Hvap sigma dPsat Te
0.08 0.15 0.018 930 7.5 0.00025 0.000013 0.13 3400 230000 0.028 70000 2.5

Excel formula:

=CHEN_EDELSTEIN(0.08, 0.15, 0.018, 930, 7.5, 0.00025, 0.000013, 0.13, 3400, 230000, 0.028, 70000, 2.5)

Expected output:

5167.53

Example 4: Mid-range inputs

Inputs:

m x D rhol rhog mul mug kl Cpl Hvap sigma dPsat Te
0.12 0.3 0.024 740 10 0.00019 0.000009 0.095 2600 195000 0.019 85000 4.5

Excel formula:

=CHEN_EDELSTEIN(0.12, 0.3, 0.024, 740, 10, 0.00019, 0.000009, 0.095, 2600, 195000, 0.019, 85000, 4.5)

Expected output:

3944.98

Python Code

Show Code 
from ht.boiling_flow import Chen_Edelstein as ht_Chen_Edelstein

def Chen_Edelstein(m, x, D, rhol, rhog, mul, mug, kl, Cpl, Hvap, sigma, dPsat, Te):
    """
    Compute the Chen-Edelstein boiling heat transfer coefficient.

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

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

    Args:
        m (float): Mass flow rate (kg/s).
        x (float): Quality at the tube interval (dimensionless).
        D (float): Tube diameter (m).
        rhol (float): Liquid density (kg/m^3).
        rhog (float): Gas density (kg/m^3).
        mul (float): Liquid viscosity (Pa*s).
        mug (float): Gas viscosity (Pa*s).
        kl (float): Liquid thermal conductivity (W/m/K).
        Cpl (float): Liquid heat capacity (J/kg/K).
        Hvap (float): Heat of vaporization (J/kg).
        sigma (float): Surface tension (N/m).
        dPsat (float): Saturation pressure difference (Pa).
        Te (float): Excess wall temperature (K).

    Returns:
        float: Heat transfer coefficient (W/m^2/K), or an error message if invalid.
    """
    try:
        return ht_Chen_Edelstein(m=m, x=x, D=D, rhol=rhol, rhog=rhog, mul=mul, mug=mug,
            kl=kl, Cpl=Cpl, Hvap=Hvap, sigma=sigma, dPsat=dPsat, Te=Te)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Mass flow rate (kg/s).
Quality at the tube interval (dimensionless).
Tube diameter (m).
Liquid density (kg/m^3).
Gas density (kg/m^3).
Liquid viscosity (Pa*s).
Gas viscosity (Pa*s).
Liquid thermal conductivity (W/m/K).
Liquid heat capacity (J/kg/K).
Heat of vaporization (J/kg).
Surface tension (N/m).
Saturation pressure difference (Pa).
Excess wall temperature (K).

COOPER

This function computes a nucleate pool-boiling heat transfer coefficient using the Cooper correlation. The method requires either excess wall temperature or heat flux and incorporates reduced pressure, molecular weight, and a roughness term.

In common form, the relation scales as:

h \propto q^{0.67}\left(\frac{P}{P_c}\right)^{a}\left[-\log_{10}\left(\frac{P}{P_c}\right)\right]^{-0.55}MW^{-0.5}

where the exponent a depends on the roughness parameter. This is a standard empirical approach for nucleate boiling estimation.

Excel Usage

=COOPER(P, Pc, MW, Te, q, Rp)
  • P (float, required): Saturation pressure (Pa).
  • Pc (float, required): Critical pressure (Pa).
  • MW (float, required): Molecular weight (g/mol).
  • Te (float, optional, default: null): Excess wall temperature (K).
  • q (float, optional, default: null): Heat flux (W/m^2).
  • Rp (float, optional, default: 0.000001): Surface roughness parameter (m).

Returns (float): Heat transfer coefficient (W/m^2/K), or an error message if invalid.

Example 1: Example with excess temperature

Inputs:

P Pc MW Te
101325 22048321 18.02 4.3

Excel formula:

=COOPER(101325, 22048321, 18.02, 4.3)

Expected output:

1558.14

Example 2: Using heat flux input

Inputs:

P Pc MW q
101325 22048321 18.02 100000

Excel formula:

=COOPER(101325, 22048321, 18.02, 100000)

Expected output:

9530.96

Example 3: Rough surface with higher pressure

Inputs:

P Pc MW Te Rp
500000 4000000 44 10 0.000002

Excel formula:

=COOPER(500000, 4000000, 44, 10, 0.000002)

Expected output:

52990.3

Example 4: Alternate fluid with heat flux

Inputs:

P Pc MW q Rp
200000 5000000 30 50000 0.000005

Excel formula:

=COOPER(200000, 5000000, 30, 50000, 0.000005)

Expected output:

12524.6

Python Code

Show Code 
from ht.boiling_flow import Cooper as ht_Cooper

def Cooper(P, Pc, MW, Te=None, q=None, Rp=1e-06):
    """
    Compute the Cooper nucleate boiling heat transfer coefficient.

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

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

    Args:
        P (float): Saturation pressure (Pa).
        Pc (float): Critical pressure (Pa).
        MW (float): Molecular weight (g/mol).
        Te (float, optional): Excess wall temperature (K). Default is None.
        q (float, optional): Heat flux (W/m^2). Default is None.
        Rp (float, optional): Surface roughness parameter (m). Default is 1e-06.

    Returns:
        float: Heat transfer coefficient (W/m^2/K), or an error message if invalid.
    """
    try:
        if Te is None and q is None:
            return "Error: Te or q must be provided"
        return ht_Cooper(P=P, Pc=Pc, MW=MW, Te=Te, q=q, Rp=Rp)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Saturation pressure (Pa).
Critical pressure (Pa).
Molecular weight (g/mol).
Excess wall temperature (K).
Heat flux (W/m^2).
Surface roughness parameter (m).

FORSTER_ZUBER

This function calculates nucleate boiling heat transfer using the Forster-Zuber correlation. It supports either excess wall temperature or heat flux as the driving input and uses thermophysical properties of the liquid-vapor system.

A representative expression is:

h \propto \left(\frac{k_l^{0.79} c_{p,l}^{0.45} \rho_l^{0.49}}{\sigma^{0.5} \mu_l^{0.29} H_{vap}^{0.24} \rho_g^{0.24}}\right) \Delta T_e^{0.24} \Delta P_{sat}^{0.75}

The correlation is frequently used as a nucleate-boiling component in broader flow-boiling models.

Excel Usage

=FORSTER_ZUBER(rhol, rhog, mul, kl, Cpl, Hvap, sigma, dPsat, Te, q)
  • rhol (float, required): Liquid density (kg/m^3).
  • rhog (float, required): Gas density (kg/m^3).
  • mul (float, required): Liquid viscosity (Pa*s).
  • kl (float, required): Liquid thermal conductivity (W/m/K).
  • Cpl (float, required): Liquid heat capacity (J/kg/K).
  • Hvap (float, required): Heat of vaporization (J/kg).
  • sigma (float, required): Surface tension (N/m).
  • dPsat (float, required): Saturation pressure difference (Pa).
  • Te (float, optional, default: null): Excess wall temperature (K).
  • q (float, optional, default: null): Heat flux (W/m^2).

Returns (float): Heat transfer coefficient (W/m^2/K), or an error message if invalid.

Example 1: Example with excess temperature

Inputs:

Te dPsat Cpl kl mul sigma Hvap rhol rhog
4.3 16885.8 4180 0.688 0.000275 0.0588 2250000 958 0.597

Excel formula:

=FORSTER_ZUBER(4.3, 16885.8, 4180, 0.688, 0.000275, 0.0588, 2250000, 958, 0.597)

Expected output:

3534.06

Example 2: Using heat flux input

Inputs:

q dPsat Cpl kl mul sigma Hvap rhol rhog
120000 15000 3500 0.12 0.0003 0.025 200000 900 10

Excel formula:

=FORSTER_ZUBER(120000, 15000, 3500, 0.12, 0.0003, 0.025, 200000, 900, 10)

Expected output:

1890

Example 3: Alternate fluid properties with Te

Inputs:

Te dPsat Cpl kl mul sigma Hvap rhol rhog
6 20000 3000 0.1 0.00022 0.02 180000 850 12

Excel formula:

=FORSTER_ZUBER(6, 20000, 3000, 0.1, 0.00022, 0.02, 180000, 850, 12)

Expected output:

1255.8

Example 4: Higher heat flux case

Inputs:

q dPsat Cpl kl mul sigma Hvap rhol rhog
200000 25000 3800 0.15 0.00028 0.03 210000 920 8

Excel formula:

=FORSTER_ZUBER(200000, 25000, 3800, 0.15, 0.00028, 0.03, 210000, 920, 8)

Expected output:

3325

Python Code

Show Code 
from ht.boiling_flow import Forster_Zuber as ht_Forster_Zuber

def Forster_Zuber(rhol, rhog, mul, kl, Cpl, Hvap, sigma, dPsat, Te=None, q=None):
    """
    Compute the Forster-Zuber nucleate boiling heat transfer coefficient.

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

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

    Args:
        rhol (float): Liquid density (kg/m^3).
        rhog (float): Gas density (kg/m^3).
        mul (float): Liquid viscosity (Pa*s).
        kl (float): Liquid thermal conductivity (W/m/K).
        Cpl (float): Liquid heat capacity (J/kg/K).
        Hvap (float): Heat of vaporization (J/kg).
        sigma (float): Surface tension (N/m).
        dPsat (float): Saturation pressure difference (Pa).
        Te (float, optional): Excess wall temperature (K). Default is None.
        q (float, optional): Heat flux (W/m^2). Default is None.

    Returns:
        float: Heat transfer coefficient (W/m^2/K), or an error message if invalid.
    """
    try:
        if Te is None and q is None:
            return "Error: Te or q must be provided"
        return ht_Forster_Zuber(rhol=rhol, rhog=rhog, mul=mul, kl=kl, Cpl=Cpl,
            Hvap=Hvap, sigma=sigma, dPsat=dPsat, Te=Te, q=q)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Liquid density (kg/m^3).
Gas density (kg/m^3).
Liquid viscosity (Pa*s).
Liquid thermal conductivity (W/m/K).
Liquid heat capacity (J/kg/K).
Heat of vaporization (J/kg).
Surface tension (N/m).
Saturation pressure difference (Pa).
Excess wall temperature (K).
Heat flux (W/m^2).

LAZAREK_BLACK

This function estimates flow-boiling heat transfer with the Lazarek-Black correlation for small channels. The method is commonly applied when either wall heat flux or excess wall temperature is known.

The correlation is commonly written as:

h_{tp} = 30\,Re_{lo}^{0.857} Bg^{0.714}\frac{k_l}{D}

where Re_{lo} is a liquid-only Reynolds number and Bg is a boiling group containing heat flux and latent heat effects.

Excel Usage

=LAZAREK_BLACK(m, D, mul, kl, Hvap, q, Te)
  • m (float, required): Mass flow rate (kg/s).
  • D (float, required): Channel diameter (m).
  • mul (float, required): Liquid viscosity (Pa*s).
  • kl (float, required): Liquid thermal conductivity (W/m/K).
  • Hvap (float, required): Heat of vaporization (J/kg).
  • q (float, optional, default: null): Heat flux (W/m^2).
  • Te (float, optional, default: null): Excess wall temperature (K).

Returns (float): Heat transfer coefficient (W/m^2/K), or an error message if invalid.

Example 1: Example with excess temperature

Inputs:

m D mul kl Hvap Te
10 0.3 0.001 0.6 2000000 100

Excel formula:

=LAZAREK_BLACK(10, 0.3, 0.001, 0.6, 2000000, 100)

Expected output:

9501.93

Example 2: Using heat flux input

Inputs:

m D mul kl Hvap q
5 0.02 0.0002 0.12 180000 80000

Excel formula:

=LAZAREK_BLACK(5, 0.02, 0.0002, 0.12, 180000, 80000)

Expected output:

20829.3

Example 3: Small channel with low temperature rise

Inputs:

m D mul kl Hvap Te
2 0.01 0.00015 0.1 200000 8

Excel formula:

=LAZAREK_BLACK(2, 0.01, 0.00015, 0.1, 200000, 8)

Expected output:

223929

Example 4: Higher heat flux at larger diameter

Inputs:

m D mul kl Hvap q
8 0.04 0.00025 0.15 220000 120000

Excel formula:

=LAZAREK_BLACK(8, 0.04, 0.00025, 0.15, 220000, 120000)

Expected output:

19773.5

Python Code

Show Code 
from ht.boiling_flow import Lazarek_Black as ht_Lazarek_Black

def Lazarek_Black(m, D, mul, kl, Hvap, q=None, Te=None):
    """
    Compute the Lazarek-Black boiling heat transfer coefficient.

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

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

    Args:
        m (float): Mass flow rate (kg/s).
        D (float): Channel diameter (m).
        mul (float): Liquid viscosity (Pa*s).
        kl (float): Liquid thermal conductivity (W/m/K).
        Hvap (float): Heat of vaporization (J/kg).
        q (float, optional): Heat flux (W/m^2). Default is None.
        Te (float, optional): Excess wall temperature (K). Default is None.

    Returns:
        float: Heat transfer coefficient (W/m^2/K), or an error message if invalid.
    """
    try:
        if Te is None and q is None:
            return "Error: Te or q must be provided"
        return ht_Lazarek_Black(m=m, D=D, mul=mul, kl=kl, Hvap=Hvap, q=q, Te=Te)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Mass flow rate (kg/s).
Channel diameter (m).
Liquid viscosity (Pa*s).
Liquid thermal conductivity (W/m/K).
Heat of vaporization (J/kg).
Heat flux (W/m^2).
Excess wall temperature (K).

LI_WU

This function estimates saturated flow-boiling heat transfer using the Li-Wu correlation, developed for mini and microchannel applications. It requires either heat flux or excess wall temperature.

A common representation is:

h_{tp} = 334\,Bg^{0.3}(Bo\,Re_l^{0.36})^{0.4}\frac{k_l}{D}

where Re_l is based on the liquid-phase flow, and Bg and Bo capture boiling intensity effects.

Excel Usage

=LI_WU(m, x, D, rhol, rhog, mul, kl, Hvap, sigma, q, Te)
  • m (float, required): Mass flow rate (kg/s).
  • x (float, required): Quality at the tube interval (dimensionless).
  • D (float, required): Tube diameter (m).
  • rhol (float, required): Liquid density (kg/m^3).
  • rhog (float, required): Gas density (kg/m^3).
  • mul (float, required): Liquid viscosity (Pa*s).
  • kl (float, required): Liquid thermal conductivity (W/m/K).
  • Hvap (float, required): Heat of vaporization (J/kg).
  • sigma (float, required): Surface tension (N/m).
  • q (float, optional, default: null): Heat flux (W/m^2).
  • Te (float, optional, default: null): Excess wall temperature (K).

Returns (float): Heat transfer coefficient (W/m^2/K), or an error message if invalid.

Example 1: Example with heat flux

Inputs:

m x D rhol rhog kl mul sigma Hvap q
1 0.2 0.3 567 18.09 0.086 0.000156 0.02 900000 100000

Excel formula:

=LI_WU(1, 0.2, 0.3, 567, 18.09, 0.086, 0.000156, 0.02, 900000, 100000)

Expected output:

5345.41

Example 2: Using excess wall temperature

Inputs:

m x D rhol rhog kl mul sigma Hvap Te
0.8 0.3 0.02 850 10 0.12 0.0002 0.025 180000 6

Excel formula:

=LI_WU(0.8, 0.3, 0.02, 850, 10, 0.12, 0.0002, 0.025, 180000, 6)

Expected output:

4267.15

Example 3: Small diameter with moderate quality

Inputs:

m x D rhol rhog kl mul sigma Hvap q
0.5 0.4 0.01 900 12 0.1 0.00018 0.03 200000 60000

Excel formula:

=LI_WU(0.5, 0.4, 0.01, 900, 12, 0.1, 0.00018, 0.03, 200000, 60000)

Expected output:

3783.53

Example 4: Mid-range properties with Te

Inputs:

m x D rhol rhog kl mul sigma Hvap Te
1.2 0.25 0.05 700 15 0.09 0.00022 0.018 160000 5

Excel formula:

=LI_WU(1.2, 0.25, 0.05, 700, 15, 0.09, 0.00022, 0.018, 160000, 5)

Expected output:

3753.9

Python Code

Show Code 
from ht.boiling_flow import Li_Wu as ht_Li_Wu

def Li_Wu(m, x, D, rhol, rhog, mul, kl, Hvap, sigma, q=None, Te=None):
    """
    Compute the Li-Wu boiling heat transfer coefficient.

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

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

    Args:
        m (float): Mass flow rate (kg/s).
        x (float): Quality at the tube interval (dimensionless).
        D (float): Tube diameter (m).
        rhol (float): Liquid density (kg/m^3).
        rhog (float): Gas density (kg/m^3).
        mul (float): Liquid viscosity (Pa*s).
        kl (float): Liquid thermal conductivity (W/m/K).
        Hvap (float): Heat of vaporization (J/kg).
        sigma (float): Surface tension (N/m).
        q (float, optional): Heat flux (W/m^2). Default is None.
        Te (float, optional): Excess wall temperature (K). Default is None.

    Returns:
        float: Heat transfer coefficient (W/m^2/K), or an error message if invalid.
    """
    try:
        if Te is None and q is None:
            return "Error: Te or q must be provided"
        return ht_Li_Wu(m=m, x=x, D=D, rhol=rhol, rhog=rhog, mul=mul, kl=kl,
            Hvap=Hvap, sigma=sigma, q=q, Te=Te)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Mass flow rate (kg/s).
Quality at the tube interval (dimensionless).
Tube diameter (m).
Liquid density (kg/m^3).
Gas density (kg/m^3).
Liquid viscosity (Pa*s).
Liquid thermal conductivity (W/m/K).
Heat of vaporization (J/kg).
Surface tension (N/m).
Heat flux (W/m^2).
Excess wall temperature (K).

LIU_WINTERTON

This function computes the Liu-Winterton flow-boiling heat transfer coefficient for saturated boiling in tubes. The method combines a convective liquid term and a nucleate-boiling term into a single composite prediction.

The combined form is:

h_{tp} = \sqrt{(F h_l)^2 + (S h_{nb})^2}

where F and S are empirical factors and h_{nb} is typically linked to Cooper-type nucleate boiling behavior.

Excel Usage

=LIU_WINTERTON(m, x, D, rhol, rhog, mul, kl, Cpl, MW, P, Pc, Te)
  • m (float, required): Mass flow rate (kg/s).
  • x (float, required): Quality at the tube interval (dimensionless).
  • D (float, required): Tube diameter (m).
  • rhol (float, required): Liquid density (kg/m^3).
  • rhog (float, required): Gas density (kg/m^3).
  • mul (float, required): Liquid viscosity (Pa*s).
  • kl (float, required): Liquid thermal conductivity (W/m/K).
  • Cpl (float, required): Liquid heat capacity (J/kg/K).
  • MW (float, required): Molecular weight (g/mol).
  • P (float, required): Pressure (Pa).
  • Pc (float, required): Critical pressure (Pa).
  • Te (float, required): Excess wall temperature (K).

Returns (float): Heat transfer coefficient (W/m^2/K), or an error message if invalid.

Example 1: Example from reference

Inputs:

m x D rhol rhog kl mul Cpl P Pc MW Te
1 0.4 0.3 567 18.09 0.086 0.000156 2300 1000000 22000000 44.02 7

Excel formula:

=LIU_WINTERTON(1, 0.4, 0.3, 567, 18.09, 0.086, 0.000156, 2300, 1000000, 22000000, 44.02, 7)

Expected output:

4747.75

Example 2: Lower quality with higher pressure

Inputs:

m x D rhol rhog kl mul Cpl P Pc MW Te
0.6 0.2 0.02 820 10 0.12 0.0002 3000 1500000 4000000 30 5

Excel formula:

=LIU_WINTERTON(0.6, 0.2, 0.02, 820, 10, 0.12, 0.0002, 3000, 1500000, 4000000, 30, 5)

Expected output:

60702.1

Example 3: Higher quality in small tube

Inputs:

m x D rhol rhog kl mul Cpl P Pc MW Te
0.8 0.6 0.015 900 12 0.1 0.00018 2800 800000 3000000 18 6

Excel formula:

=LIU_WINTERTON(0.8, 0.6, 0.015, 900, 12, 0.1, 0.00018, 2800, 800000, 3000000, 18, 6)

Expected output:

107814

Example 4: Mid-range properties

Inputs:

m x D rhol rhog kl mul Cpl P Pc MW Te
1.2 0.35 0.05 700 15 0.09 0.00022 2600 1200000 6000000 40 8

Excel formula:

=LIU_WINTERTON(1.2, 0.35, 0.05, 700, 15, 0.09, 0.00022, 2600, 1200000, 6000000, 40, 8)

Expected output:

34338.6

Python Code

Show Code 
from ht.boiling_flow import Liu_Winterton as ht_Liu_Winterton

def Liu_Winterton(m, x, D, rhol, rhog, mul, kl, Cpl, MW, P, Pc, Te):
    """
    Compute the Liu-Winterton boiling heat transfer coefficient.

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

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

    Args:
        m (float): Mass flow rate (kg/s).
        x (float): Quality at the tube interval (dimensionless).
        D (float): Tube diameter (m).
        rhol (float): Liquid density (kg/m^3).
        rhog (float): Gas density (kg/m^3).
        mul (float): Liquid viscosity (Pa*s).
        kl (float): Liquid thermal conductivity (W/m/K).
        Cpl (float): Liquid heat capacity (J/kg/K).
        MW (float): Molecular weight (g/mol).
        P (float): Pressure (Pa).
        Pc (float): Critical pressure (Pa).
        Te (float): Excess wall temperature (K).

    Returns:
        float: Heat transfer coefficient (W/m^2/K), or an error message if invalid.
    """
    try:
        return ht_Liu_Winterton(m=m, x=x, D=D, rhol=rhol, rhog=rhog, mul=mul,
            kl=kl, Cpl=Cpl, MW=MW, P=P, Pc=Pc, Te=Te)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Mass flow rate (kg/s).
Quality at the tube interval (dimensionless).
Tube diameter (m).
Liquid density (kg/m^3).
Gas density (kg/m^3).
Liquid viscosity (Pa*s).
Liquid thermal conductivity (W/m/K).
Liquid heat capacity (J/kg/K).
Molecular weight (g/mol).
Pressure (Pa).
Critical pressure (Pa).
Excess wall temperature (K).

LOCKHART_XTT

This function calculates the Lockhart-Martinelli two-phase parameter X_{tt}, an empirical ratio used in flow-boiling and two-phase pressure-drop modeling. It supports default exponents or deriving selected exponents from an input n value.

The standard form is:

X_{tt} = \left(\frac{1-x}{x}\right)^{0.9}\left(\frac{\rho_g}{\rho_l}\right)^{0.5}\left(\frac{\mu_l}{\mu_g}\right)^{0.1}

This parameter is dimensionless and is often used as an intermediate term in two-phase correlations.

Excel Usage

=LOCKHART_XTT(x, rhol, rhog, mul, mug, pow_x, pow_rho, pow_mu, n)
  • x (float, required): Quality at the tube interval (dimensionless).
  • rhol (float, required): Liquid density (kg/m^3).
  • rhog (float, required): Gas density (kg/m^3).
  • mul (float, required): Liquid viscosity (Pa*s).
  • mug (float, required): Gas viscosity (Pa*s).
  • pow_x (float, optional, default: 0.9): Power for phase ratio (dimensionless).
  • pow_rho (float, optional, default: 0.5): Power for density ratio (dimensionless).
  • pow_mu (float, optional, default: 0.1): Power for viscosity ratio (dimensionless).
  • n (float, optional, default: null): Exponent parameter to derive powers (dimensionless).

Returns (float): Lockhart-Martinelli parameter (dimensionless), or an error message if invalid.

Example 1: Example with default powers

Inputs:

x rhol rhog mul mug
0.4 800 2.5 0.001 0.00001

Excel formula:

=LOCKHART_XTT(0.4, 800, 2.5, 0.001, 0.00001)

Expected output:

0.127617

Example 2: Using n to derive powers

Inputs:

x rhol rhog mul mug n
0.3 900 8 0.0002 0.00002 0.2

Excel formula:

=LOCKHART_XTT(0.3, 900, 8, 0.0002, 0.00002, 0.2)

Expected output:

0.25445

Example 3: Custom power values

Inputs:

x rhol rhog mul mug pow_x pow_rho pow_mu
0.6 700 20 0.0003 0.000015 0.8 0.6 0.2

Excel formula:

=LOCKHART_XTT(0.6, 700, 20, 0.0003, 0.000015, 0.8, 0.6, 0.2)

Expected output:

0.155917

Example 4: Higher quality with default powers

Inputs:

x rhol rhog mul mug
0.7 950 12 0.00025 0.00002

Excel formula:

=LOCKHART_XTT(0.7, 950, 12, 0.00025, 0.00002)

Expected output:

0.0674902

Python Code

Show Code 
from ht.boiling_flow import Lockhart_Martinelli_Xtt as ht_Lockhart_Martinelli_Xtt

def Lockhart_Xtt(x, rhol, rhog, mul, mug, pow_x=0.9, pow_rho=0.5, pow_mu=0.1, n=None):
    """
    Compute the Lockhart-Martinelli Xtt two-phase flow parameter.

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

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

    Args:
        x (float): Quality at the tube interval (dimensionless).
        rhol (float): Liquid density (kg/m^3).
        rhog (float): Gas density (kg/m^3).
        mul (float): Liquid viscosity (Pa*s).
        mug (float): Gas viscosity (Pa*s).
        pow_x (float, optional): Power for phase ratio (dimensionless). Default is 0.9.
        pow_rho (float, optional): Power for density ratio (dimensionless). Default is 0.5.
        pow_mu (float, optional): Power for viscosity ratio (dimensionless). Default is 0.1.
        n (float, optional): Exponent parameter to derive powers (dimensionless). Default is None.

    Returns:
        float: Lockhart-Martinelli parameter (dimensionless), or an error message if invalid.
    """
    try:
        return ht_Lockhart_Martinelli_Xtt(x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug,
            pow_x=pow_x, pow_rho=pow_rho, pow_mu=pow_mu, n=n)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Quality at the tube interval (dimensionless).
Liquid density (kg/m^3).
Gas density (kg/m^3).
Liquid viscosity (Pa*s).
Gas viscosity (Pa*s).
Power for phase ratio (dimensionless).
Power for density ratio (dimensionless).
Power for viscosity ratio (dimensionless).
Exponent parameter to derive powers (dimensionless).

SUN_MISHIMA

This function estimates saturated flow-boiling heat transfer using the Sun-Mishima correlation, which is often applied to mini-channel conditions. The model requires either heat flux or excess wall temperature.

A representative relation is:

h_{tp} = \frac{6\,Re_{lo}^{1.05}Bg^{0.54}}{We_l^{0.191}(\rho_l/\rho_g)^{0.142}}\frac{k_l}{D}

It combines liquid-only Reynolds effects with boiling and Weber-number scaling for empirical prediction.

Excel Usage

=SUN_MISHIMA(m, D, rhol, rhog, mul, kl, Hvap, sigma, q, Te)
  • m (float, required): Mass flow rate (kg/s).
  • D (float, required): Tube diameter (m).
  • rhol (float, required): Liquid density (kg/m^3).
  • rhog (float, required): Gas density (kg/m^3).
  • mul (float, required): Liquid viscosity (Pa*s).
  • kl (float, required): Liquid thermal conductivity (W/m/K).
  • Hvap (float, required): Heat of vaporization (J/kg).
  • sigma (float, required): Surface tension (N/m).
  • q (float, optional, default: null): Heat flux (W/m^2).
  • Te (float, optional, default: null): Excess wall temperature (K).

Returns (float): Heat transfer coefficient (W/m^2/K), or an error message if invalid.

Example 1: Example with excess temperature

Inputs:

m D rhol rhog kl mul sigma Hvap Te
1 0.3 567 18.09 0.086 0.000156 0.02 900000 10

Excel formula:

=SUN_MISHIMA(1, 0.3, 567, 18.09, 0.086, 0.000156, 0.02, 900000, 10)

Expected output:

507.671

Example 2: Using heat flux input

Inputs:

m D rhol rhog kl mul sigma Hvap q
0.8 0.02 900 12 0.12 0.0002 0.025 180000 70000

Excel formula:

=SUN_MISHIMA(0.8, 0.02, 900, 12, 0.12, 0.0002, 0.025, 180000, 70000)

Expected output:

15391.5

Example 3: Small diameter with higher heat flux

Inputs:

m D rhol rhog kl mul sigma Hvap q
0.6 0.01 850 9 0.11 0.00018 0.03 200000 90000

Excel formula:

=SUN_MISHIMA(0.6, 0.01, 850, 9, 0.11, 0.00018, 0.03, 200000, 90000)

Expected output:

21451

Example 4: Mid-range properties with Te

Inputs:

m D rhol rhog kl mul sigma Hvap Te
1.2 0.04 700 15 0.09 0.00022 0.018 160000 6

Excel formula:

=SUN_MISHIMA(1.2, 0.04, 700, 15, 0.09, 0.00022, 0.018, 160000, 6)

Expected output:

5898.6

Python Code

Show Code 
from ht.boiling_flow import Sun_Mishima as ht_Sun_Mishima

def Sun_Mishima(m, D, rhol, rhog, mul, kl, Hvap, sigma, q=None, Te=None):
    """
    Compute the Sun-Mishima boiling heat transfer coefficient.

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

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

    Args:
        m (float): Mass flow rate (kg/s).
        D (float): Tube diameter (m).
        rhol (float): Liquid density (kg/m^3).
        rhog (float): Gas density (kg/m^3).
        mul (float): Liquid viscosity (Pa*s).
        kl (float): Liquid thermal conductivity (W/m/K).
        Hvap (float): Heat of vaporization (J/kg).
        sigma (float): Surface tension (N/m).
        q (float, optional): Heat flux (W/m^2). Default is None.
        Te (float, optional): Excess wall temperature (K). Default is None.

    Returns:
        float: Heat transfer coefficient (W/m^2/K), or an error message if invalid.
    """
    try:
        if Te is None and q is None:
            return "Error: Te or q must be provided"
        return ht_Sun_Mishima(m=m, D=D, rhol=rhol, rhog=rhog, mul=mul, kl=kl,
            Hvap=Hvap, sigma=sigma, q=q, Te=Te)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Mass flow rate (kg/s).
Tube diameter (m).
Liquid density (kg/m^3).
Gas density (kg/m^3).
Liquid viscosity (Pa*s).
Liquid thermal conductivity (W/m/K).
Heat of vaporization (J/kg).
Surface tension (N/m).
Heat flux (W/m^2).
Excess wall temperature (K).

THOME

This function computes flow-boiling heat transfer using the Thome microchannel model, a detailed slug/film-based approach for saturated evaporation in small channels. It accepts either heat flux directly or excess wall temperature for an implicit solution.

The model forms a weighted average of regime-specific contributions:

h(z) = \frac{t_l}{\tau}h_l(z) + \frac{t_{film}}{\tau}h_{film}(z) + \frac{t_{dry}}{\tau}h_g(z)

where time fractions and local film behavior are estimated from flow properties, boiling intensity, and fluid thermophysical data.

Excel Usage

=THOME(m, x, D, rhol, rhog, mul, mug, kl, kg, Cpl, Cpg, Hvap, sigma, Psat, Pc, q, Te)
  • m (float, required): Mass flow rate (kg/s).
  • x (float, required): Quality at the tube interval (dimensionless).
  • D (float, required): Tube diameter (m).
  • rhol (float, required): Liquid density (kg/m^3).
  • rhog (float, required): Gas density (kg/m^3).
  • mul (float, required): Liquid viscosity (Pa*s).
  • mug (float, required): Gas viscosity (Pa*s).
  • kl (float, required): Liquid thermal conductivity (W/m/K).
  • kg (float, required): Gas thermal conductivity (W/m/K).
  • Cpl (float, required): Liquid heat capacity (J/kg/K).
  • Cpg (float, required): Gas heat capacity (J/kg/K).
  • Hvap (float, required): Heat of vaporization (J/kg).
  • sigma (float, required): Surface tension (N/m).
  • Psat (float, required): Saturation pressure (Pa).
  • Pc (float, required): Critical pressure (Pa).
  • q (float, optional, default: null): Heat flux (W/m^2).
  • Te (float, optional, default: null): Excess wall temperature (K).

Returns (float): Heat transfer coefficient (W/m^2/K), or an error message if invalid.

Example 1: Example with heat flux

Inputs:

m x D rhol rhog kl kg mul mug Cpl Cpg sigma Hvap Psat Pc q
1 0.4 0.3 567 18.09 0.086 0.2 0.000156 0.00001 2300 1400 0.02 900000 100000 22000000 100000

Excel formula:

=THOME(1, 0.4, 0.3, 567, 18.09, 0.086, 0.2, 0.000156, 0.00001, 2300, 1400, 0.02, 900000, 100000, 22000000, 100000)

Expected output:

1633.01

Example 2: Using excess wall temperature

Inputs:

m x D rhol rhog kl kg mul mug Cpl Cpg sigma Hvap Psat Pc Te
0.8 0.3 0.02 850 12 0.12 0.18 0.0002 0.000012 3000 1500 0.025 180000 900000 4000000 6

Excel formula:

=THOME(0.8, 0.3, 0.02, 850, 12, 0.12, 0.18, 0.0002, 0.000012, 3000, 1500, 0.025, 180000, 900000, 4000000, 6)

Expected output:

96633.1

Example 3: Small diameter with higher heat flux

Inputs:

m x D rhol rhog kl kg mul mug Cpl Cpg sigma Hvap Psat Pc q
0.6 0.5 0.01 900 9 0.1 0.16 0.00018 0.000011 2800 1400 0.03 200000 800000 3000000 70000

Excel formula:

=THOME(0.6, 0.5, 0.01, 900, 9, 0.1, 0.16, 0.00018, 0.000011, 2800, 1400, 0.03, 200000, 800000, 3000000, 70000)

Expected output:

20435

Example 4: Mid-range properties with Te

Inputs:

m x D rhol rhog kl kg mul mug Cpl Cpg sigma Hvap Psat Pc Te
1.2 0.35 0.04 700 15 0.09 0.19 0.00022 0.000013 2600 1450 0.018 160000 1200000 6000000 7

Excel formula:

=THOME(1.2, 0.35, 0.04, 700, 15, 0.09, 0.19, 0.00022, 0.000013, 2600, 1450, 0.018, 160000, 1200000, 6000000, 7)

Expected output:

28555.1

Python Code

Show Code 
from ht.boiling_flow import Thome as ht_Thome

def Thome(m, x, D, rhol, rhog, mul, mug, kl, kg, Cpl, Cpg, Hvap, sigma, Psat, Pc, q=None, Te=None):
    """
    Compute the Thome microchannel boiling heat transfer coefficient.

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

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

    Args:
        m (float): Mass flow rate (kg/s).
        x (float): Quality at the tube interval (dimensionless).
        D (float): Tube diameter (m).
        rhol (float): Liquid density (kg/m^3).
        rhog (float): Gas density (kg/m^3).
        mul (float): Liquid viscosity (Pa*s).
        mug (float): Gas viscosity (Pa*s).
        kl (float): Liquid thermal conductivity (W/m/K).
        kg (float): Gas thermal conductivity (W/m/K).
        Cpl (float): Liquid heat capacity (J/kg/K).
        Cpg (float): Gas heat capacity (J/kg/K).
        Hvap (float): Heat of vaporization (J/kg).
        sigma (float): Surface tension (N/m).
        Psat (float): Saturation pressure (Pa).
        Pc (float): Critical pressure (Pa).
        q (float, optional): Heat flux (W/m^2). Default is None.
        Te (float, optional): Excess wall temperature (K). Default is None.

    Returns:
        float: Heat transfer coefficient (W/m^2/K), or an error message if invalid.
    """
    try:
        if Te is None and q is None:
            return "Error: Te or q must be provided"
        return ht_Thome(m=m, x=x, D=D, rhol=rhol, rhog=rhog, mul=mul, mug=mug,
            kl=kl, kg=kg, Cpl=Cpl, Cpg=Cpg, Hvap=Hvap, sigma=sigma, Psat=Psat,
            Pc=Pc, q=q, Te=Te)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Mass flow rate (kg/s).
Quality at the tube interval (dimensionless).
Tube diameter (m).
Liquid density (kg/m^3).
Gas density (kg/m^3).
Liquid viscosity (Pa*s).
Gas viscosity (Pa*s).
Liquid thermal conductivity (W/m/K).
Gas thermal conductivity (W/m/K).
Liquid heat capacity (J/kg/K).
Gas heat capacity (J/kg/K).
Heat of vaporization (J/kg).
Surface tension (N/m).
Saturation pressure (Pa).
Critical pressure (Pa).
Heat flux (W/m^2).
Excess wall temperature (K).

TO_SOLVE_Q_THOME

This function evaluates the residual equation used to solve for heat flux in the Thome flow-boiling model when excess wall temperature is prescribed. It is primarily an internal numerical helper for implicit root-finding workflows.

Conceptually, it evaluates:

R(q) = T_{e,model}(q,\ldots) - T_{e,target}

A root at R(q)=0 indicates the heat flux that is consistent with the specified operating state and target excess wall temperature.

Excel Usage

=TO_SOLVE_Q_THOME(q, m, x, D, rhol, rhog, kl, kg, mul, mug, Cpl, Cpg, sigma, Hvap, Psat, Pc, Te)
  • q (float, required): Heat flux (W/m^2).
  • m (float, required): Mass flow rate (kg/s).
  • x (float, required): Quality at the tube interval (dimensionless).
  • D (float, required): Tube diameter (m).
  • rhol (float, required): Liquid density (kg/m^3).
  • rhog (float, required): Gas density (kg/m^3).
  • kl (float, required): Liquid thermal conductivity (W/m/K).
  • kg (float, required): Gas thermal conductivity (W/m/K).
  • mul (float, required): Liquid viscosity (Pa*s).
  • mug (float, required): Gas viscosity (Pa*s).
  • Cpl (float, required): Liquid heat capacity (J/kg/K).
  • Cpg (float, required): Gas heat capacity (J/kg/K).
  • sigma (float, required): Surface tension (N/m).
  • Hvap (float, required): Heat of vaporization (J/kg).
  • Psat (float, required): Saturation pressure (Pa).
  • Pc (float, required): Critical pressure (Pa).
  • Te (float, required): Excess wall temperature (K).

Returns (float): Heat flux residual value (W/m^2), or an error message if invalid.

Example 1: Baseline residual calculation

Inputs:

q m x D rhol rhog kl kg mul mug Cpl Cpg sigma Hvap Psat Pc Te
100000 1 0.4 0.3 567 18.09 0.086 0.2 0.000156 0.00001 2300 1400 0.02 900000 100000 22000000 7

Excel formula:

=TO_SOLVE_Q_THOME(100000, 1, 0.4, 0.3, 567, 18.09, 0.086, 0.2, 0.000156, 0.00001, 2300, 1400, 0.02, 900000, 100000, 22000000, 7)

Expected output:

54.2367

Example 2: Lower quality with smaller diameter

Inputs:

q m x D rhol rhog kl kg mul mug Cpl Cpg sigma Hvap Psat Pc Te
70000 0.8 0.25 0.02 850 12 0.12 0.18 0.0002 0.000012 3000 1500 0.025 180000 900000 4000000 6

Excel formula:

=TO_SOLVE_Q_THOME(70000, 0.8, 0.25, 0.02, 850, 12, 0.12, 0.18, 0.0002, 0.000012, 3000, 1500, 0.025, 180000, 900000, 4000000, 6)

Expected output:

-3.1047

Example 3: Higher quality with higher heat flux

Inputs:

q m x D rhol rhog kl kg mul mug Cpl Cpg sigma Hvap Psat Pc Te
120000 0.6 0.6 0.01 900 9 0.1 0.16 0.00018 0.000011 2800 1400 0.03 200000 800000 3000000 8

Excel formula:

=TO_SOLVE_Q_THOME(120000, 0.6, 0.6, 0.01, 900, 9, 0.1, 0.16, 0.00018, 0.000011, 2800, 1400, 0.03, 200000, 800000, 3000000, 8)

Expected output:

-3.31727

Example 4: Mid-range properties

Inputs:

q m x D rhol rhog kl kg mul mug Cpl Cpg sigma Hvap Psat Pc Te
90000 1.2 0.35 0.04 700 15 0.09 0.19 0.00022 0.000013 2600 1450 0.018 160000 1200000 6000000 7

Excel formula:

=TO_SOLVE_Q_THOME(90000, 1.2, 0.35, 0.04, 700, 15, 0.09, 0.19, 0.00022, 0.000013, 2600, 1450, 0.018, 160000, 1200000, 6000000, 7)

Expected output:

-1.49611

Python Code

Show Code 
from ht.boiling_flow import to_solve_q_Thome as ht_to_solve_q_Thome

def to_solve_q_Thome(q, m, x, D, rhol, rhog, kl, kg, mul, mug, Cpl, Cpg, sigma, Hvap, Psat, Pc, Te):
    """
    Compute the Thome heat flux residual for a specified wall temperature.

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

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

    Args:
        q (float): Heat flux (W/m^2).
        m (float): Mass flow rate (kg/s).
        x (float): Quality at the tube interval (dimensionless).
        D (float): Tube diameter (m).
        rhol (float): Liquid density (kg/m^3).
        rhog (float): Gas density (kg/m^3).
        kl (float): Liquid thermal conductivity (W/m/K).
        kg (float): Gas thermal conductivity (W/m/K).
        mul (float): Liquid viscosity (Pa*s).
        mug (float): Gas viscosity (Pa*s).
        Cpl (float): Liquid heat capacity (J/kg/K).
        Cpg (float): Gas heat capacity (J/kg/K).
        sigma (float): Surface tension (N/m).
        Hvap (float): Heat of vaporization (J/kg).
        Psat (float): Saturation pressure (Pa).
        Pc (float): Critical pressure (Pa).
        Te (float): Excess wall temperature (K).

    Returns:
        float: Heat flux residual value (W/m^2), or an error message if invalid.
    """
    try:
        return ht_to_solve_q_Thome(q=q, m=m, x=x, D=D, rhol=rhol, rhog=rhog, kl=kl,
            kg=kg, mul=mul, mug=mug, Cpl=Cpl, Cpg=Cpg, sigma=sigma, Hvap=Hvap,
            Psat=Psat, Pc=Pc, Te=Te)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Heat flux (W/m^2).
Mass flow rate (kg/s).
Quality at the tube interval (dimensionless).
Tube diameter (m).
Liquid density (kg/m^3).
Gas density (kg/m^3).
Liquid thermal conductivity (W/m/K).
Gas thermal conductivity (W/m/K).
Liquid viscosity (Pa*s).
Gas viscosity (Pa*s).
Liquid heat capacity (J/kg/K).
Gas heat capacity (J/kg/K).
Surface tension (N/m).
Heat of vaporization (J/kg).
Saturation pressure (Pa).
Critical pressure (Pa).
Excess wall temperature (K).

TURBULENT_DITTUS

This function calculates turbulent internal-flow Nusselt number using the Dittus-Boelter correlation. It supports heating or cooling mode and either revised or original coefficient conventions.

The relation has the form:

Nu = m\,Re^{0.8}Pr^n

where m and n depend on mode and coefficient convention. The result is a dimensionless convective heat-transfer indicator.

Excel Usage

=TURBULENT_DITTUS(Re, Pr, heating, revised)
  • Re (float, required): Reynolds number (dimensionless).
  • Pr (float, required): Prandtl number (dimensionless).
  • heating (bool, optional, default: true): Whether the process is heating.
  • revised (bool, optional, default: true): Whether to use revised coefficients.

Returns (float): Nusselt number (dimensionless), or an error message if invalid.

Example 1: Example with heating

Inputs:

Re Pr
100000 1.2

Excel formula:

=TURBULENT_DITTUS(100000, 1.2)

Expected output:

247.4

Example 2: Example with cooling

Inputs:

Re Pr heating
100000 1.2 false

Excel formula:

=TURBULENT_DITTUS(100000, 1.2, FALSE)

Expected output:

242.931

Example 3: Using unrevised coefficients

Inputs:

Re Pr revised
50000 0.8 false

Excel formula:

=TURBULENT_DITTUS(50000, 0.8, FALSE)

Expected output:

127.649

Example 4: Cooling with unrevised coefficients

Inputs:

Re Pr heating revised
200000 1.5 false false

Excel formula:

=TURBULENT_DITTUS(200000, 1.5, FALSE, FALSE)

Expected output:

521.071

Python Code

Show Code 
from ht.boiling_flow import turbulent_Dittus_Boelter as ht_turbulent_Dittus_Boelter

def turbulent_Dittus(Re, Pr, heating=True, revised=True):
    """
    Compute the Dittus-Boelter turbulent Nusselt number.

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

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

    Args:
        Re (float): Reynolds number (dimensionless).
        Pr (float): Prandtl number (dimensionless).
        heating (bool, optional): Whether the process is heating. Default is True.
        revised (bool, optional): Whether to use revised coefficients. Default is True.

    Returns:
        float: Nusselt number (dimensionless), or an error message if invalid.
    """
    try:
        return ht_turbulent_Dittus_Boelter(Re=Re, Pr=Pr, heating=heating, revised=revised)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number (dimensionless).
Prandtl number (dimensionless).
Whether the process is heating.
Whether to use revised coefficients.

TURBULENT_GNIEL

This function computes turbulent internal-flow Nusselt number with the Gnielinski correlation, which links heat transfer to Reynolds number, Prandtl number, and Darcy friction factor.

The equation is:

Nu = \frac{(f/8)(Re-1000)Pr}{1+12.7(f/8)^{1/2}(Pr^{2/3}-1)}

This correlation is broadly used for turbulent pipe and channel convection over a wide operating range.

Excel Usage

=TURBULENT_GNIEL(Re, Pr, fd)
  • Re (float, required): Reynolds number (dimensionless).
  • Pr (float, required): Prandtl number (dimensionless).
  • fd (float, required): Darcy friction factor (dimensionless).

Returns (float): Nusselt number (dimensionless), or an error message if invalid.

Example 1: Example with standard inputs

Inputs:

Re Pr fd
100000 1.2 0.0185

Excel formula:

=TURBULENT_GNIEL(100000, 1.2, 0.0185)

Expected output:

254.627

Example 2: Higher Reynolds number

Inputs:

Re Pr fd
200000 1.5 0.02

Excel formula:

=TURBULENT_GNIEL(200000, 1.5, 0.02)

Expected output:

623.389

Example 3: Lower Reynolds number

Inputs:

Re Pr fd
50000 0.9 0.025

Excel formula:

=TURBULENT_GNIEL(50000, 0.9, 0.025)

Expected output:

144.785

Example 4: Higher Prandtl number

Inputs:

Re Pr fd
150000 5 0.017

Excel formula:

=TURBULENT_GNIEL(150000, 5, 0.017)

Expected output:

744.51

Python Code

Show Code 
from ht.boiling_flow import turbulent_Gnielinski as ht_turbulent_Gnielinski

def turbulent_Gniel(Re, Pr, fd):
    """
    Compute the Gnielinski turbulent Nusselt number.

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

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

    Args:
        Re (float): Reynolds number (dimensionless).
        Pr (float): Prandtl number (dimensionless).
        fd (float): Darcy friction factor (dimensionless).

    Returns:
        float: Nusselt number (dimensionless), or an error message if invalid.
    """
    try:
        return ht_turbulent_Gnielinski(Re=Re, Pr=Pr, fd=fd)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Reynolds number (dimensionless).
Prandtl number (dimensionless).
Darcy friction factor (dimensionless).

YUN_HEO_KIM

This function estimates saturated flow-boiling heat transfer with the Yun-Heo-Kim correlation, developed for microchannel evaporation. It accepts either heat flux or excess wall temperature as the driving input.

The correlation can be expressed as:

h_{tp} = 136876\,(Bg\,We_l)^{0.1993}Re_l^{-0.1626}

where Re_l and We_l are liquid-based dimensionless groups and Bg captures boiling intensity.

Excel Usage

=YUN_HEO_KIM(m, x, D, rhol, mul, Hvap, sigma, q, Te)
  • m (float, required): Mass flow rate (kg/s).
  • x (float, required): Quality at the tube interval (dimensionless).
  • D (float, required): Tube diameter (m).
  • rhol (float, required): Liquid density (kg/m^3).
  • mul (float, required): Liquid viscosity (Pa*s).
  • Hvap (float, required): Heat of vaporization (J/kg).
  • sigma (float, required): Surface tension (N/m).
  • q (float, optional, default: null): Heat flux (W/m^2).
  • Te (float, optional, default: null): Excess wall temperature (K).

Returns (float): Heat transfer coefficient (W/m^2/K), or an error message if invalid.

Example 1: Example with heat flux

Inputs:

m x D rhol mul sigma Hvap q
1 0.4 0.3 567 0.000156 0.02 900000 10000

Excel formula:

=YUN_HEO_KIM(1, 0.4, 0.3, 567, 0.000156, 0.02, 900000, 10000)

Expected output:

9479.31

Example 2: Using excess wall temperature

Inputs:

m x D rhol mul sigma Hvap Te
0.8 0.3 0.02 850 0.0002 0.025 180000 6

Excel formula:

=YUN_HEO_KIM(0.8, 0.3, 0.02, 850, 0.0002, 0.025, 180000, 6)

Expected output:

21308.7

Example 3: Small diameter with higher heat flux

Inputs:

m x D rhol mul sigma Hvap q
0.6 0.5 0.01 900 0.00018 0.03 200000 15000

Excel formula:

=YUN_HEO_KIM(0.6, 0.5, 0.01, 900, 0.00018, 0.03, 200000, 15000)

Expected output:

13677.9

Example 4: Mid-range properties with Te

Inputs:

m x D rhol mul sigma Hvap Te
1.2 0.35 0.04 700 0.00022 0.018 160000 7

Excel formula:

=YUN_HEO_KIM(1.2, 0.35, 0.04, 700, 0.00022, 0.018, 160000, 7)

Expected output:

26525.6

Python Code

Show Code 
from ht.boiling_flow import Yun_Heo_Kim as ht_Yun_Heo_Kim

def Yun_Heo_Kim(m, x, D, rhol, mul, Hvap, sigma, q=None, Te=None):
    """
    Compute the Yun-Heo-Kim boiling heat transfer coefficient.

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

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

    Args:
        m (float): Mass flow rate (kg/s).
        x (float): Quality at the tube interval (dimensionless).
        D (float): Tube diameter (m).
        rhol (float): Liquid density (kg/m^3).
        mul (float): Liquid viscosity (Pa*s).
        Hvap (float): Heat of vaporization (J/kg).
        sigma (float): Surface tension (N/m).
        q (float, optional): Heat flux (W/m^2). Default is None.
        Te (float, optional): Excess wall temperature (K). Default is None.

    Returns:
        float: Heat transfer coefficient (W/m^2/K), or an error message if invalid.
    """
    try:
        if Te is None and q is None:
            return "Error: Te or q must be provided"
        return ht_Yun_Heo_Kim(m=m, x=x, D=D, rhol=rhol, mul=mul, Hvap=Hvap,
            sigma=sigma, q=q, Te=Te)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Mass flow rate (kg/s).
Quality at the tube interval (dimensionless).
Tube diameter (m).
Liquid density (kg/m^3).
Liquid viscosity (Pa*s).
Heat of vaporization (J/kg).
Surface tension (N/m).
Heat flux (W/m^2).
Excess wall temperature (K).