Conduction

Overview

Thermal conduction describes heat flow through solids or stationary media due to a temperature gradient. In engineering practice, conduction models are central to insulation selection, pipe and vessel thermal design, and steady-state heat-loss estimation. This category focuses on the resistance-network form of Fourier’s law, where geometry and material properties are translated into directly usable design calculations. The tools are especially useful when moving between material datasheet formats (conductivity, resistivity, and R-value) and model-ready parameters.

Core concepts in this group include thermal conductivity k, thermal resistance R, thermal resistivity r, and shape factor S for 2D conduction approximations. For planar and cylindrical problems, heat rate is often written as Q=\Delta T/R_{\text{total}}; for shape-factor methods, the compact relation is Q = kS\Delta T. These formulations let analysts combine material layers and boundary films in the same circuit-like framework and quickly compare design alternatives.

Implementation: The functions are implemented with the Python ht library, specifically the ht.conduction module for conduction and shape-factor correlations. The library is designed for practical heat-transfer engineering, combining textbook equations with ready-to-use computational helpers.

The property-conversion tools map between common material metrics used across mechanical and building domains. K_TO_R, R_TO_K, K_TO_R_VALUE, and R_VALUE_TO_K convert between conductivity and thermal resistance forms (including insulation R-value conventions). K_TO_THERM_RESIST and THERM_RESIST_TO_K convert between conductivity and thermal resistivity, which is common in some references and material tables. The legacy aliases LEGACY_K_THERM_RES and LEGACY_THERM_RES_K preserve older naming while supporting the same workflows.

The cylindrical resistance and multilayer tools support pipe- and shell-style radial conduction calculations. R_CYLINDER provides the canonical cylindrical conduction resistance term, R_{\text{cyl}}=\frac{\ln(D_o/D_i)}{2\pi kL}, while CYL_HEAT_TRANSFER evaluates full multilayer cylinders with internal and external film coefficients, returning heat duty, layer resistances, and interface temperatures. This is useful for insulation thickness checks, process line heat-loss estimates, and wall-temperature constraints. LEGACY_CYL_HT is the backward-compatible wrapper for the same class of calculation.

The shape-factor estimators address geometries where exact full-field solutions are unnecessary but geometry still dominates heat flow. S_PIPE_ECC_TO_PIPE, S_PIPE_NORM_PLANE, S_PIPE_TO_PIPE, S_PIPE_TO_PLANE, S_PIPE_TWO_PLANES, and S_SPHERE_TO_PLANE provide compact S values for common buried or embedded configurations, then pair with Q=kS\Delta T. The matching legacy interfaces LEGACY_S_PIPE_ECC, LEGACY_S_PIPE_NORM, LEGACY_S_PIPE_PAIR, LEGACY_S_PIPE_PLANE, LEGACY_S_PIPE_PLNS, and LEGACY_S_SPH_PLANE maintain compatibility with existing models.

Two mathematical helper functions support closed-form conduction expressions used by these correlations: LOG for logarithms in radial resistance and shape-factor formulas, and ACOSH for inverse hyperbolic cosine terms that arise in eccentric and near-boundary geometry relations. Together, these helpers and engineering correlations provide a complete toolkit for steady conduction screening, design iteration, and standards-aligned reporting.

ACOSH

Computes the inverse hyperbolic cosine of the input value. This operation is the inverse of the hyperbolic cosine and is commonly used in analytical solutions of conduction and shape-factor expressions.

The function returns \operatorname{arcosh}(x), defined for x \ge 1:

y = \operatorname{arcosh}(x) \iff x = \cosh(y)

Excel Usage

=ACOSH(x)
  • x (float, required): Input value (-).

Returns (float): Inverse hyperbolic cosine of the input (-).

Example 1: Acosh of two

Inputs:

x
2

Excel formula:

=ACOSH(2)

Expected output:

1.31696

Example 2: Acosh of three

Inputs:

x
3

Excel formula:

=ACOSH(3)

Expected output:

1.76275

Example 3: Acosh of ten

Inputs:

x
10

Excel formula:

=ACOSH(10)

Expected output:

2.99322

Example 4: Acosh of one

Inputs:

x
1

Excel formula:

=ACOSH(1)

Expected output:

0

Python Code

Show Code 
from ht.conduction import acosh as ht_acosh

def acosh(x):
    """
    Compute the inverse hyperbolic cosine.

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

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

    Args:
        x (float): Input value (-).

    Returns:
        float: Inverse hyperbolic cosine of the input (-).
    """
    try:
        return ht_acosh(x)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Input value (-).

CYL_HEAT_TRANSFER

Computes conductive and convective heat transfer through a multilayer cylindrical wall using inside/outside film coefficients and per-layer material properties. It returns heat rate and intermediate resistance and temperature values for each layer.

The radial resistance model is built from cylindrical conduction and convection terms:

R_{\text{cond}} = \frac{\ln(D_o/D_i)}{2\pi kL},\quad R_{\text{conv}} = \frac{1}{hA},\quad Q = \frac{\Delta T}{\sum R}

Excel Usage

=CYL_HEAT_TRANSFER(Ti, To, hi, ho, Di, ts, ks)
  • Ti (float, required): Inside temperature (K).
  • To (float, required): Outside bulk temperature (K).
  • hi (float, required): Inside heat transfer coefficient (W/m^2/K).
  • ho (float, required): Outside heat transfer coefficient (W/m^2/K).
  • Di (float, required): Inside diameter (m).
  • ts (list[list], required): Layer thicknesses (m).
  • ks (list[list], required): Layer thermal conductivities (W/m/K).

Returns (list[list]): Table of heat transfer results and layer properties.

Example 1: Two-layer cylinder example

Inputs:

Ti To hi ho Di ts ks
453.15 301.15 1000000000000 22.697193 0.0779272 0.0054864 0.05 56.045 0.0598535265

Excel formula:

=CYL_HEAT_TRANSFER(453.15, 301.15, 1000000000000, 22.697193, 0.0779272, {0.0054864,0.05}, {56.045,0.0598535265})

Expected output:

Result
Q 73.12
UA 0.481053
U_inner 1.96496
U_outer 0.810608
q 123.212
Rs 0.00022201 1.18936
Ts 453.15 453.123 306.579
Example 2: Single layer cylinder

Inputs:

Ti To hi ho Di ts ks
400 300 200 50 0.1 0.02 15

Excel formula:

=CYL_HEAT_TRANSFER(400, 300, 200, 50, 0.1, {0.02}, {15})

Expected output:

Result
Q 1539.45
UA 15.3945
U_inner 49.0021
U_outer 35.0015
q 3500.15
Rs 0.0015702
Ts 400 394.504
Example 3: Moderate coefficients and two layers

Inputs:

Ti To hi ho Di ts ks
350 290 500 60 0.08 0.01 0.03 12 0.2

Excel formula:

=CYL_HEAT_TRANSFER(350, 290, 500, 60, 0.08, {0.01,0.03}, {12,0.2})

Expected output:

Result
Q 143.509
UA 2.39182
U_inner 9.51675
U_outer 4.75838
q 285.503
Rs 0.00148762 0.188001
Ts 350 349.575 295.9
Example 4: Higher outside transfer coefficient

Inputs:

Ti To hi ho Di ts ks
375 295 800 150 0.12 0.015 0.02 20 0.5

Excel formula:

=CYL_HEAT_TRANSFER(375, 295, 800, 150, 0.12, {0.015,0.02}, {20,0.5})

Expected output:

Result
Q 874.268
UA 10.9284
U_inner 28.9884
U_outer 18.3084
q 1464.67
Rs 0.00105993 0.0449139
Ts 375 373.448 307.663

Python Code

Show Code 
from ht.conduction import cylindrical_heat_transfer as ht_cylindrical_heat_transfer

def cyl_heat_transfer(Ti, To, hi, ho, Di, ts, ks):
    """
    Compute heat transfer through a multilayer cylindrical wall.

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

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

    Args:
        Ti (float): Inside temperature (K).
        To (float): Outside bulk temperature (K).
        hi (float): Inside heat transfer coefficient (W/m^2/K).
        ho (float): Outside heat transfer coefficient (W/m^2/K).
        Di (float): Inside diameter (m).
        ts (list[list]): Layer thicknesses (m).
        ks (list[list]): Layer thermal conductivities (W/m/K).

    Returns:
        list[list]: Table of heat transfer results and layer properties.
    """
    try:
        def to2d(x):
            return [[x]] if not isinstance(x, list) else x

        def flatten_numeric(x, name):
            data = to2d(x)
            if not isinstance(data, list) or not all(isinstance(row, list) for row in data):
                return None, f"Error: {name} must be a 2D list"
            flat = []
            for row in data:
                for val in row:
                    try:
                        flat.append(float(val))
                    except (TypeError, ValueError):
                        return None, f"Error: {name} must contain only numbers"
            if not flat:
                return None, f"Error: {name} must contain at least one value"
            return flat, None

        ts_list, err = flatten_numeric(ts, "ts")
        if err:
            return err
        ks_list, err = flatten_numeric(ks, "ks")
        if err:
            return err
        if len(ts_list) != len(ks_list):
            return "Error: ts and ks must have the same length"

        result = ht_cylindrical_heat_transfer(
            Ti=Ti,
            To=To,
            hi=hi,
            ho=ho,
            Di=Di,
            ts=ts_list,
            ks=ks_list,
        )

        rows = [
            ["Q", result.get("Q")],
            ["UA", result.get("UA")],
            ["U_inner", result.get("U_inner")],
            ["U_outer", result.get("U_outer")],
            ["q", result.get("q")],
            ["Rs"] + list(result.get("Rs", [])),
            ["Ts"] + list(result.get("Ts", [])),
        ]
        max_len = max(len(row) for row in rows)
        return [row + [""] * (max_len - len(row)) for row in rows]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Inside temperature (K).
Outside bulk temperature (K).
Inside heat transfer coefficient (W/m^2/K).
Outside heat transfer coefficient (W/m^2/K).
Inside diameter (m).
Layer thicknesses (m).
Layer thermal conductivities (W/m/K).

K_TO_R

Converts thermal conductivity into thermal resistance for a slab of specified thickness and area. This is useful for moving between tabulated material conductivity and resistance-based design calculations.

The conversion is:

R = \frac{t}{kA}

Excel Usage

=K_TO_R(k, t, A)
  • k (float, required): Thermal conductivity (W/m/K).
  • t (float, required): Thickness (m).
  • A (float, optional, default: 1): Area (m^2).

Returns (float): Thermal resistance (K/W).

Example 1: Default area conversion

Inputs:

k t
0.5 0.025

Excel formula:

=K_TO_R(0.5, 0.025)

Expected output:

0.05

Example 2: Larger area conversion

Inputs:

k t A
1.2 0.05 2

Excel formula:

=K_TO_R(1.2, 0.05, 2)

Expected output:

0.0208333

Example 3: Thin layer with higher conductivity

Inputs:

k t
2 0.01

Excel formula:

=K_TO_R(2, 0.01)

Expected output:

0.005

Example 4: Thick layer with lower conductivity

Inputs:

k t
0.2 0.1

Excel formula:

=K_TO_R(0.2, 0.1)

Expected output:

0.5

Python Code

Show Code 
from ht.conduction import k_to_R as ht_k_to_R

def k_to_R(k, t, A=1):
    """
    Compute thermal resistance from thermal conductivity.

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

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

    Args:
        k (float): Thermal conductivity (W/m/K).
        t (float): Thickness (m).
        A (float, optional): Area (m^2). Default is 1.

    Returns:
        float: Thermal resistance (K/W).
    """
    try:
        return ht_k_to_R(k=k, t=t, A=A)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Thermal conductivity (W/m/K).
Thickness (m).
Area (m^2).

K_TO_R_VALUE

Converts thermal conductivity to insulation R-value per inch in either SI or Imperial conventions. This enables direct comparison with building-material specifications.

Conceptually, the conversion is the inverse of conductivity with a unit-system scaling factor:

R_{\text{value}} \propto \frac{1}{k}

Excel Usage

=K_TO_R_VALUE(k, SI)
  • k (float, required): Thermal conductivity (W/m/K).
  • SI (bool, optional, default: true): Whether to return the SI R-value (-).

Returns (float): R-value (m^2K/(Winch) or ft^2degFh/(BTU*inch)).

Example 1: SI R-value example

Inputs:

k
0.2

Excel formula:

=K_TO_R_VALUE(0.2)

Expected output:

0.127

Example 2: Imperial R-value example

Inputs:

k SI
0.2 false

Excel formula:

=K_TO_R_VALUE(0.2, FALSE)

Expected output:

0.721139

Example 3: Higher conductivity example

Inputs:

k
1.5

Excel formula:

=K_TO_R_VALUE(1.5)

Expected output:

0.0169333

Example 4: Low conductivity in imperial

Inputs:

k SI
0.05 false

Excel formula:

=K_TO_R_VALUE(0.05, FALSE)

Expected output:

2.88456

Python Code

Show Code 
from ht.conduction import k_to_R_value as ht_k_to_R_value

def k_to_R_value(k, SI=True):
    """
    Convert thermal conductivity to R-value.

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

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

    Args:
        k (float): Thermal conductivity (W/m/K).
        SI (bool, optional): Whether to return the SI R-value (-). Default is True.

    Returns:
        float: R-value (m^2*K/(W*inch) or ft^2*degF*h/(BTU*inch)).
    """
    try:
        return ht_k_to_R_value(k=k, SI=SI)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Thermal conductivity (W/m/K).
Whether to return the SI R-value (-).

K_TO_THERM_RESIST

Converts thermal conductivity to thermal resistivity, which is the reciprocal material property often used for solids in conduction references.

The relation is:

r = \frac{1}{k}

Excel Usage

=K_TO_THERM_RESIST(k)
  • k (float, required): Thermal conductivity (W/m/K).

Returns (float): Thermal resistivity (m*K/W).

Example 1: Example resistivity

Inputs:

k
0.25

Excel formula:

=K_TO_THERM_RESIST(0.25)

Expected output:

4

Example 2: Higher conductivity

Inputs:

k
2

Excel formula:

=K_TO_THERM_RESIST(2)

Expected output:

0.5

Example 3: Lower conductivity

Inputs:

k
0.1

Excel formula:

=K_TO_THERM_RESIST(0.1)

Expected output:

10

Example 4: Moderate conductivity

Inputs:

k
0.5

Excel formula:

=K_TO_THERM_RESIST(0.5)

Expected output:

2

Python Code

Show Code 
from ht.conduction import k_to_thermal_resistivity as ht_k_to_thermal_resistivity

def k_to_therm_resist(k):
    """
    Convert thermal conductivity to thermal resistivity.

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

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

    Args:
        k (float): Thermal conductivity (W/m/K).

    Returns:
        float: Thermal resistivity (m*K/W).
    """
    try:
        return ht_k_to_thermal_resistivity(k=k)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Thermal conductivity (W/m/K).

LEGACY_CYL_HT

Provides a backward-compatible wrapper for multilayer cylindrical wall heat-transfer calculations. It evaluates conduction through each layer with inside/outside convection and returns a table of thermal results.

The governing resistance-network form is:

Q = \frac{T_i - T_o}{R_{i,\text{conv}} + \sum R_{\text{layer}} + R_{o,\text{conv}}}

Excel Usage

=LEGACY_CYL_HT(Ti, To, hi, ho, Di, ts, ks)
  • Ti (float, required): Inside temperature (K).
  • To (float, required): Outside bulk temperature (K).
  • hi (float, required): Inside heat transfer coefficient (W/m^2/K).
  • ho (float, required): Outside heat transfer coefficient (W/m^2/K).
  • Di (float, required): Inside diameter (m).
  • ts (list[list], required): Layer thicknesses (m).
  • ks (list[list], required): Layer thermal conductivities (W/m/K).

Returns (list[list]): Table of heat transfer results and layer properties.

Example 1: Two-layer cylinder example

Inputs:

Ti To hi ho Di ts ks
453.15 301.15 1000000000000 22.697193 0.0779272 0.0054864 0.05 56.045 0.0598535265

Excel formula:

=LEGACY_CYL_HT(453.15, 301.15, 1000000000000, 22.697193, 0.0779272, {0.0054864,0.05}, {56.045,0.0598535265})

Expected output:

Result
Q 73.12
UA 0.481053
U_inner 1.96496
U_outer 0.810608
q 123.212
Rs 0.00022201 1.18936
Ts 453.15 453.123 306.579
Example 2: Single layer cylinder

Inputs:

Ti To hi ho Di ts ks
400 300 200 50 0.1 0.02 15

Excel formula:

=LEGACY_CYL_HT(400, 300, 200, 50, 0.1, {0.02}, {15})

Expected output:

Result
Q 1539.45
UA 15.3945
U_inner 49.0021
U_outer 35.0015
q 3500.15
Rs 0.0015702
Ts 400 394.504
Example 3: Moderate coefficients and two layers

Inputs:

Ti To hi ho Di ts ks
350 290 500 60 0.08 0.01 0.03 12 0.2

Excel formula:

=LEGACY_CYL_HT(350, 290, 500, 60, 0.08, {0.01,0.03}, {12,0.2})

Expected output:

Result
Q 143.509
UA 2.39182
U_inner 9.51675
U_outer 4.75838
q 285.503
Rs 0.00148762 0.188001
Ts 350 349.575 295.9
Example 4: Higher outside transfer coefficient

Inputs:

Ti To hi ho Di ts ks
375 295 800 150 0.12 0.015 0.02 20 0.5

Excel formula:

=LEGACY_CYL_HT(375, 295, 800, 150, 0.12, {0.015,0.02}, {20,0.5})

Expected output:

Result
Q 874.268
UA 10.9284
U_inner 28.9884
U_outer 18.3084
q 1464.67
Rs 0.00105993 0.0449139
Ts 375 373.448 307.663

Python Code

Show Code 
from ht.conduction import cylindrical_heat_transfer as ht_cylindrical_heat_transfer

def legacy_cyl_ht(Ti, To, hi, ho, Di, ts, ks):
    """
    Deprecated alias for cyl_heat_transfer.

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

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

    Args:
        Ti (float): Inside temperature (K).
        To (float): Outside bulk temperature (K).
        hi (float): Inside heat transfer coefficient (W/m^2/K).
        ho (float): Outside heat transfer coefficient (W/m^2/K).
        Di (float): Inside diameter (m).
        ts (list[list]): Layer thicknesses (m).
        ks (list[list]): Layer thermal conductivities (W/m/K).

    Returns:
        list[list]: Table of heat transfer results and layer properties.
    """
    try:
        def to2d(x):
            return [[x]] if not isinstance(x, list) else x

        def flatten_numeric(x, name):
            data = to2d(x)
            if not isinstance(data, list) or not all(isinstance(row, list) for row in data):
                return None, f"Error: {name} must be a 2D list"
            flat = []
            for row in data:
                for val in row:
                    try:
                        flat.append(float(val))
                    except (TypeError, ValueError):
                        return None, f"Error: {name} must contain only numbers"
            if not flat:
                return None, f"Error: {name} must contain at least one value"
            return flat, None

        ts_list, err = flatten_numeric(ts, "ts")
        if err:
            return err
        ks_list, err = flatten_numeric(ks, "ks")
        if err:
            return err
        if len(ts_list) != len(ks_list):
            return "Error: ts and ks must have the same length"

        result = ht_cylindrical_heat_transfer(
            Ti=Ti,
            To=To,
            hi=hi,
            ho=ho,
            Di=Di,
            ts=ts_list,
            ks=ks_list,
        )

        rows = [
            ["Q", result.get("Q")],
            ["UA", result.get("UA")],
            ["U_inner", result.get("U_inner")],
            ["U_outer", result.get("U_outer")],
            ["q", result.get("q")],
            ["Rs"] + list(result.get("Rs", [])),
            ["Ts"] + list(result.get("Ts", [])),
        ]
        max_len = max(len(row) for row in rows)
        return [row + [""] * (max_len - len(row)) for row in rows]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Inside temperature (K).
Outside bulk temperature (K).
Inside heat transfer coefficient (W/m^2/K).
Outside heat transfer coefficient (W/m^2/K).
Inside diameter (m).
Layer thicknesses (m).
Layer thermal conductivities (W/m/K).

LEGACY_K_THERM_RES

Backward-compatible wrapper that computes thermal resistivity from thermal conductivity. It preserves legacy naming while returning the same reciprocal material property.

The conversion is:

r = \frac{1}{k}

Excel Usage

=LEGACY_K_THERM_RES(k)
  • k (float, required): Thermal conductivity (W/m/K).

Returns (float): Thermal resistivity (m*K/W).

Example 1: Example resistivity

Inputs:

k
0.25

Excel formula:

=LEGACY_K_THERM_RES(0.25)

Expected output:

4

Example 2: Higher conductivity

Inputs:

k
2

Excel formula:

=LEGACY_K_THERM_RES(2)

Expected output:

0.5

Example 3: Lower conductivity

Inputs:

k
0.1

Excel formula:

=LEGACY_K_THERM_RES(0.1)

Expected output:

10

Example 4: Moderate conductivity

Inputs:

k
0.5

Excel formula:

=LEGACY_K_THERM_RES(0.5)

Expected output:

2

Python Code

Show Code 
from ht.conduction import k_to_thermal_resistivity as ht_k_to_thermal_resistivity

def legacy_k_therm_res(k):
    """
    Deprecated alias for k_to_therm_resist.

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

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

    Args:
        k (float): Thermal conductivity (W/m/K).

    Returns:
        float: Thermal resistivity (m*K/W).
    """
    try:
        return ht_k_to_thermal_resistivity(k=k)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Thermal conductivity (W/m/K).

LEGACY_S_PIPE_ECC

Backward-compatible wrapper for the eccentric-pipe conduction shape factor. It preserves legacy naming while returning the same geometric factor used in simplified steady conduction analysis.

The heat-rate relation is:

Q = Sk(T_1 - T_2)

Excel Usage

=LEGACY_S_PIPE_ECC(D_one, D_two, Z, L)
  • D_one (float, required): Diameter of inner pipe (m).
  • D_two (float, required): Diameter of outer pipe (m).
  • Z (float, required): Offset between pipe centers (m).
  • L (float, optional, default: 1): Pipe length (m).

Returns (float): Shape factor (m).

Example 1: Example eccentric pipes

Inputs:

D_one D_two Z L
0.1 0.4 0.05 10

Excel formula:

=LEGACY_S_PIPE_ECC(0.1, 0.4, 0.05, 10)

Expected output:

47.7098

Example 2: Unit length eccentric pipes

Inputs:

D_one D_two Z
0.2 0.6 0.08

Excel formula:

=LEGACY_S_PIPE_ECC(0.2, 0.6, 0.08)

Expected output:

6.19483

Example 3: Larger offset with longer length

Inputs:

D_one D_two Z L
0.15 0.5 0.12 5

Excel formula:

=LEGACY_S_PIPE_ECC(0.15, 0.5, 0.12, 5)

Expected output:

34.9225

Example 4: Small diameter pipes

Inputs:

D_one D_two Z L
0.05 0.2 0.03 2

Excel formula:

=LEGACY_S_PIPE_ECC(0.05, 0.2, 0.03, 2)

Expected output:

9.78228

Python Code

Show Code 
from ht.conduction import S_isothermal_pipe_eccentric_to_isothermal_pipe as ht_S_isothermal_pipe_eccentric_to_isothermal_pipe

def legacy_S_pipe_ecc(D_one, D_two, Z, L=1):
    """
    Deprecated alias for S_pipe_ecc_to_pipe.

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

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

    Args:
        D_one (float): Diameter of inner pipe (m).
        D_two (float): Diameter of outer pipe (m).
        Z (float): Offset between pipe centers (m).
        L (float, optional): Pipe length (m). Default is 1.

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_pipe_eccentric_to_isothermal_pipe(
            D1=D_one,
            D2=D_two,
            Z=Z,
            L=L,
        )
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Diameter of inner pipe (m).
Diameter of outer pipe (m).
Offset between pipe centers (m).
Pipe length (m).

LEGACY_S_PIPE_NORM

Backward-compatible wrapper for the shape factor of a long isothermal pipe normal to a plane. It keeps legacy naming and returns the same geometric conduction factor.

The shape-factor heat-rate form is:

Q = Sk(T_1 - T_2)

Excel Usage

=LEGACY_S_PIPE_NORM(D, L)
  • D (float, required): Pipe diameter (m).
  • L (float, required): Pipe length (m).

Returns (float): Shape factor (m).

Example 1: Example normal pipe

Inputs:

D L
1 100

Excel formula:

=LEGACY_S_PIPE_NORM(1, 100)

Expected output:

104.869

Example 2: Shorter pipe length

Inputs:

D L
0.5 20

Excel formula:

=LEGACY_S_PIPE_NORM(0.5, 20)

Expected output:

24.7605

Example 3: Slender pipe geometry

Inputs:

D L
0.1 10

Excel formula:

=LEGACY_S_PIPE_NORM(0.1, 10)

Expected output:

10.4869

Example 4: Moderate geometry values

Inputs:

D L
0.8 40

Excel formula:

=LEGACY_S_PIPE_NORM(0.8, 40)

Expected output:

47.4353

Python Code

Show Code 
from ht.conduction import S_isothermal_pipe_normal_to_plane as ht_S_isothermal_pipe_normal_to_plane

def legacy_S_pipe_norm(D, L):
    """
    Deprecated alias for S_pipe_norm_plane.

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

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

    Args:
        D (float): Pipe diameter (m).
        L (float): Pipe length (m).

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_pipe_normal_to_plane(D=D, L=L)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Pipe diameter (m).
Pipe length (m).

LEGACY_S_PIPE_PAIR

Backward-compatible wrapper for the two-pipe conduction shape factor. It preserves older naming while returning the same geometry-dependent factor.

This factor is used in:

Q = Sk(T_1 - T_2)

Excel Usage

=LEGACY_S_PIPE_PAIR(D_one, D_two, W, L)
  • D_one (float, required): Diameter of one pipe (m).
  • D_two (float, required): Diameter of the other pipe (m).
  • W (float, required): Center-to-center distance (m).
  • L (float, optional, default: 1): Pipe length (m).

Returns (float): Shape factor (m).

Example 1: Example pipe pair

Inputs:

D_one D_two W L
0.1 0.2 1 1

Excel formula:

=LEGACY_S_PIPE_PAIR(0.1, 0.2, 1, 1)

Expected output:

1.18871

Example 2: Wider spacing and longer length

Inputs:

D_one D_two W L
0.15 0.25 1.5 2

Excel formula:

=LEGACY_S_PIPE_PAIR(0.15, 0.25, 1.5, 2)

Expected output:

2.29685

Example 3: Short length spacing

Inputs:

D_one D_two W L
0.08 0.12 0.6 0.8

Excel formula:

=LEGACY_S_PIPE_PAIR(0.08, 0.12, 0.6, 0.8)

Expected output:

1.00611

Example 4: Moderate geometry values

Inputs:

D_one D_two W L
0.2 0.3 1.2 1.5

Excel formula:

=LEGACY_S_PIPE_PAIR(0.2, 0.3, 1.2, 1.5)

Expected output:

2.0753

Python Code

Show Code 
from ht.conduction import S_isothermal_pipe_to_isothermal_pipe as ht_S_isothermal_pipe_to_isothermal_pipe

def legacy_S_pipe_pair(D_one, D_two, W, L=1):
    """
    Deprecated alias for S_pipe_to_pipe.

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

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

    Args:
        D_one (float): Diameter of one pipe (m).
        D_two (float): Diameter of the other pipe (m).
        W (float): Center-to-center distance (m).
        L (float, optional): Pipe length (m). Default is 1.

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_pipe_to_isothermal_pipe(
            D1=D_one,
            D2=D_two,
            W=W,
            L=L,
        )
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Diameter of one pipe (m).
Diameter of the other pipe (m).
Center-to-center distance (m).
Pipe length (m).

LEGACY_S_PIPE_PLANE

Backward-compatible wrapper for the pipe-to-plane conduction shape factor. It keeps legacy naming while producing the same geometric multiplier for steady conduction calculations.

The model relation is:

Q = Sk(T_1 - T_2)

Excel Usage

=LEGACY_S_PIPE_PLANE(D, Z, L)
  • D (float, required): Pipe diameter (m).
  • Z (float, required): Distance to plane (m).
  • L (float, optional, default: 1): Pipe length (m).

Returns (float): Shape factor (m).

Example 1: Example pipe to plane

Inputs:

D Z L
1 100 3

Excel formula:

=LEGACY_S_PIPE_PLANE(1, 100, 3)

Expected output:

3.14607

Example 2: Unit length near plane

Inputs:

D Z
0.4 5

Excel formula:

=LEGACY_S_PIPE_PLANE(0.4, 5)

Expected output:

1.60629

Example 3: Moderate spacing and length

Inputs:

D Z L
0.6 8 2

Excel formula:

=LEGACY_S_PIPE_PLANE(0.6, 8, 2)

Expected output:

3.16039

Example 4: Small pipe near plane

Inputs:

D Z L
0.2 2 1

Excel formula:

=LEGACY_S_PIPE_PLANE(0.2, 2, 1)

Expected output:

1.70357

Python Code

Show Code 
from ht.conduction import S_isothermal_pipe_to_plane as ht_S_isothermal_pipe_to_plane

def legacy_S_pipe_plane(D, Z, L=1):
    """
    Deprecated alias for S_pipe_to_plane.

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

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

    Args:
        D (float): Pipe diameter (m).
        Z (float): Distance to plane (m).
        L (float, optional): Pipe length (m). Default is 1.

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_pipe_to_plane(D=D, Z=Z, L=L)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Pipe diameter (m).
Distance to plane (m).
Pipe length (m).

LEGACY_S_PIPE_PLNS

Backward-compatible wrapper for the shape factor of a pipe between two planes. It preserves older naming while returning the same geometry factor for conduction estimates.

The heat-transfer relation is:

Q = Sk(T_1 - T_2)

Excel Usage

=LEGACY_S_PIPE_PLNS(D, Z, L)
  • D (float, required): Pipe diameter (m).
  • Z (float, required): Distance to each plane (m).
  • L (float, optional, default: 1): Pipe length (m).

Returns (float): Shape factor (m).

Example 1: Example pipe between planes

Inputs:

D Z L
0.1 5 1

Excel formula:

=LEGACY_S_PIPE_PLNS(0.1, 5, 1)

Expected output:

1.29637

Example 2: Unit length with smaller spacing

Inputs:

D Z
0.2 3

Excel formula:

=LEGACY_S_PIPE_PLNS(0.2, 3)

Expected output:

1.72484

Example 3: Larger pipe diameter

Inputs:

D Z L
0.5 10 2

Excel formula:

=LEGACY_S_PIPE_PLNS(0.5, 10, 2)

Expected output:

3.19719

Example 4: Moderate geometry values

Inputs:

D Z L
0.3 6 1.5

Excel formula:

=LEGACY_S_PIPE_PLNS(0.3, 6, 1.5)

Expected output:

2.39789

Python Code

Show Code 
from ht.conduction import S_isothermal_pipe_to_two_planes as ht_S_isothermal_pipe_to_two_planes

def legacy_S_pipe_plns(D, Z, L=1):
    """
    Deprecated alias for S_pipe_two_planes.

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

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

    Args:
        D (float): Pipe diameter (m).
        Z (float): Distance to each plane (m).
        L (float, optional): Pipe length (m). Default is 1.

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_pipe_to_two_planes(D=D, Z=Z, L=L)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Pipe diameter (m).
Distance to each plane (m).
Pipe length (m).

LEGACY_S_SPH_PLANE

Backward-compatible wrapper for the sphere-to-plane conduction shape factor. It preserves legacy naming while returning the same geometric factor for steady-state conduction approximations.

The shape-factor form is:

Q = Sk(T_1 - T_2)

Excel Usage

=LEGACY_S_SPH_PLANE(D, Z)
  • D (float, required): Sphere diameter (m).
  • Z (float, required): Distance to plane (m).

Returns (float): Shape factor (m).

Example 1: Example sphere to plane

Inputs:

D Z
1 100

Excel formula:

=LEGACY_S_SPH_PLANE(1, 100)

Expected output:

6.29893

Example 2: Closer plane distance

Inputs:

D Z
0.4 4

Excel formula:

=LEGACY_S_SPH_PLANE(0.4, 4)

Expected output:

2.57772

Example 3: Small sphere near plane

Inputs:

D Z
0.2 2

Excel formula:

=LEGACY_S_SPH_PLANE(0.2, 2)

Expected output:

1.28886

Example 4: Moderate geometry values

Inputs:

D Z
0.6 8

Excel formula:

=LEGACY_S_SPH_PLANE(0.6, 8)

Expected output:

3.84195

Python Code

Show Code 
from ht.conduction import S_isothermal_sphere_to_plane as ht_S_isothermal_sphere_to_plane

def legacy_S_sph_plane(D, Z):
    """
    Deprecated alias for S_sphere_to_plane.

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

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

    Args:
        D (float): Sphere diameter (m).
        Z (float): Distance to plane (m).

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_sphere_to_plane(D=D, Z=Z)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Sphere diameter (m).
Distance to plane (m).

LEGACY_THERM_RES_K

Backward-compatible wrapper that computes thermal conductivity from thermal resistivity. It keeps legacy naming while using the same reciprocal conversion.

The conversion is:

k = \frac{1}{r}

Excel Usage

=LEGACY_THERM_RES_K(r)
  • r (float, required): Thermal resistivity (m*K/W).

Returns (float): Thermal conductivity (W/m/K).

Example 1: Example conductivity

Inputs:

r
4

Excel formula:

=LEGACY_THERM_RES_K(4)

Expected output:

0.25

Example 2: Lower resistivity

Inputs:

r
2

Excel formula:

=LEGACY_THERM_RES_K(2)

Expected output:

0.5

Example 3: Higher resistivity

Inputs:

r
10

Excel formula:

=LEGACY_THERM_RES_K(10)

Expected output:

0.1

Example 4: Moderate resistivity

Inputs:

r
1.5

Excel formula:

=LEGACY_THERM_RES_K(1.5)

Expected output:

0.666667

Python Code

Show Code 
from ht.conduction import thermal_resistivity_to_k as ht_thermal_resistivity_to_k

def legacy_therm_res_k(r):
    """
    Deprecated alias for therm_resist_to_k.

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

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

    Args:
        r (float): Thermal resistivity (m*K/W).

    Returns:
        float: Thermal conductivity (W/m/K).
    """
    try:
        return ht_thermal_resistivity_to_k(r=r)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Thermal resistivity (m*K/W).

LOG

Computes the logarithm of a positive value with either the natural base or a user-specified base. This helper appears in closed-form heat-transfer expressions involving logarithmic geometry terms.

The operation is:

y = \log_b(x) = \frac{\ln(x)}{\ln(b)}

Excel Usage

=LOG(x, base)
  • x (float, required): Input value (-).
  • base (float, optional, default: 2.718281828): Logarithm base (-).

Returns (float): Logarithm of the input (-).

Example 1: Natural log of one

Inputs:

x
1

Excel formula:

=LOG(1)

Expected output:

0

Example 2: Log base ten of one thousand

Inputs:

x base
1000 10

Excel formula:

=LOG(1000, 10)

Expected output:

3

Example 3: Log base two of eight

Inputs:

x base
8 2

Excel formula:

=LOG(8, 2)

Expected output:

3

Example 4: Natural log of ten

Inputs:

x
10

Excel formula:

=LOG(10)

Expected output:

2.30259

Python Code

Show Code 
from ht.conduction import log as ht_log

def log(x, base=2.718281828):
    """
    Compute the logarithm of a value with optional base.

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

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

    Args:
        x (float): Input value (-).
        base (float, optional): Logarithm base (-). Default is 2.718281828.

    Returns:
        float: Logarithm of the input (-).
    """
    try:
        return ht_log(x, base)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Input value (-).
Logarithm base (-).

R_CYLINDER

Computes conductive thermal resistance for a cylindrical wall from inner and outer diameters, conductivity, and length. This is the standard radial conduction form for pipes and cylindrical layers.

The cylinder resistance is:

R = \frac{\ln(D_o/D_i)}{2\pi Lk}

Excel Usage

=R_CYLINDER(Di, Do, k, L)
  • Di (float, required): Inner diameter (m).
  • Do (float, required): Outer diameter (m).
  • k (float, required): Thermal conductivity (W/m/K).
  • L (float, required): Length (m).

Returns (float): Thermal resistance of the cylinder (K/W).

Example 1: Example cylinder resistance

Inputs:

Di Do k L
0.9 1 20 10

Excel formula:

=R_CYLINDER(0.9, 1, 20, 10)

Expected output:

0.0000838432

Example 2: Thin wall with high conductivity

Inputs:

Di Do k L
0.95 1 45 5

Excel formula:

=R_CYLINDER(0.95, 1, 45, 5)

Expected output:

0.0000362826

Example 3: Short cylinder length

Inputs:

Di Do k L
0.2 0.4 15 0.5

Excel formula:

=R_CYLINDER(0.2, 0.4, 15, 0.5)

Expected output:

0.014709

Example 4: Moderate dimensions and conductivity

Inputs:

Di Do k L
0.5 0.8 12 2

Excel formula:

=R_CYLINDER(0.5, 0.8, 12, 2)

Expected output:

0.00311681

Python Code

Show Code 
from ht.conduction import R_cylinder as ht_R_cylinder

def R_cylinder(Di, Do, k, L):
    """
    Compute thermal resistance of a cylindrical wall.

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

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

    Args:
        Di (float): Inner diameter (m).
        Do (float): Outer diameter (m).
        k (float): Thermal conductivity (W/m/K).
        L (float): Length (m).

    Returns:
        float: Thermal resistance of the cylinder (K/W).
    """
    try:
        return ht_R_cylinder(Di=Di, Do=Do, k=k, L=L)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Inner diameter (m).
Outer diameter (m).
Thermal conductivity (W/m/K).
Length (m).

R_TO_K

Converts measured or specified thermal resistance into thermal conductivity using thickness and area. This is the inverse operation of resistance from conductivity for planar layers.

The conversion is:

k = \frac{t}{RA}

Excel Usage

=R_TO_K(R, t, A)
  • R (float, required): Thermal resistance (K/W or m^2*K/W).
  • t (float, required): Thickness (m).
  • A (float, optional, default: 1): Area (m^2).

Returns (float): Thermal conductivity (W/m/K).

Example 1: Default area conversion

Inputs:

R t
0.05 0.025

Excel formula:

=R_TO_K(0.05, 0.025)

Expected output:

0.5

Example 2: Larger area conversion

Inputs:

R t A
0.08 0.04 2

Excel formula:

=R_TO_K(0.08, 0.04, 2)

Expected output:

0.25

Example 3: Thin layer with small R

Inputs:

R t
0.01 0.005

Excel formula:

=R_TO_K(0.01, 0.005)

Expected output:

0.5

Example 4: Thick layer with higher R

Inputs:

R t
0.2 0.1

Excel formula:

=R_TO_K(0.2, 0.1)

Expected output:

0.5

Python Code

Show Code 
from ht.conduction import R_to_k as ht_R_to_k

def R_to_k(R, t, A=1):
    """
    Compute thermal conductivity from thermal resistance.

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

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

    Args:
        R (float): Thermal resistance (K/W or m^2*K/W).
        t (float): Thickness (m).
        A (float, optional): Area (m^2). Default is 1.

    Returns:
        float: Thermal conductivity (W/m/K).
    """
    try:
        return ht_R_to_k(R=R, t=t, A=A)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Thermal resistance (K/W or m^2*K/W).
Thickness (m).
Area (m^2).

R_VALUE_TO_K

Converts insulation R-value per inch into thermal conductivity, supporting both SI and Imperial R-value definitions. It is useful when translating building-material specs into engineering heat-transfer models.

The relationship is inverse with unit-dependent scaling:

k \propto \frac{1}{R_{\text{value}}}

Excel Usage

=R_VALUE_TO_K(R_value, SI)
  • R_value (float, required): R-value (m^2K/(Winch) or ft^2degFh/(BTU*inch)).
  • SI (bool, optional, default: true): Whether the R-value is in SI units (-).

Returns (float): Thermal conductivity (W/m/K).

Example 1: SI R-value example

Inputs:

R_value
0.12

Excel formula:

=R_VALUE_TO_K(0.12)

Expected output:

0.211667

Example 2: Imperial R-value example

Inputs:

R_value SI
0.71 false

Excel formula:

=R_VALUE_TO_K(0.71, FALSE)

Expected output:

0.203138

Example 3: Larger SI R-value

Inputs:

R_value
0.25

Excel formula:

=R_VALUE_TO_K(0.25)

Expected output:

0.1016

Example 4: Imperial reference value

Inputs:

R_value SI
1 false

Excel formula:

=R_VALUE_TO_K(1, FALSE)

Expected output:

0.144228

Python Code

Show Code 
from ht.conduction import R_value_to_k as ht_R_value_to_k

def R_value_to_k(R_value, SI=True):
    """
    Convert R-value to thermal conductivity.

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

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

    Args:
        R_value (float): R-value (m^2*K/(W*inch) or ft^2*degF*h/(BTU*inch)).
        SI (bool, optional): Whether the R-value is in SI units (-). Default is True.

    Returns:
        float: Thermal conductivity (W/m/K).
    """
    try:
        return ht_R_value_to_k(R_value=R_value, SI=SI)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

R-value (m^2*K/(W*inch) or ft^2*degF*h/(BTU*inch)).
Whether the R-value is in SI units (-).

S_PIPE_ECC_TO_PIPE

Computes the conduction shape factor between two eccentric isothermal cylindrical surfaces. The result is used with conductivity and temperature difference to estimate heat rate in 2D conduction approximations.

The shape-factor heat-rate model is:

Q = Sk(T_1 - T_2)

Excel Usage

=S_PIPE_ECC_TO_PIPE(D_one, D_two, Z, L)
  • D_one (float, required): Diameter of inner pipe (m).
  • D_two (float, required): Diameter of outer pipe (m).
  • Z (float, required): Offset between pipe centers (m).
  • L (float, optional, default: 1): Pipe length (m).

Returns (float): Shape factor (m).

Example 1: Example eccentric pipes

Inputs:

D_one D_two Z L
0.1 0.4 0.05 10

Excel formula:

=S_PIPE_ECC_TO_PIPE(0.1, 0.4, 0.05, 10)

Expected output:

47.7098

Example 2: Unit length eccentric pipes

Inputs:

D_one D_two Z
0.2 0.6 0.08

Excel formula:

=S_PIPE_ECC_TO_PIPE(0.2, 0.6, 0.08)

Expected output:

6.19483

Example 3: Larger offset with longer length

Inputs:

D_one D_two Z L
0.15 0.5 0.12 5

Excel formula:

=S_PIPE_ECC_TO_PIPE(0.15, 0.5, 0.12, 5)

Expected output:

34.9225

Example 4: Small diameter pipes

Inputs:

D_one D_two Z L
0.05 0.2 0.03 2

Excel formula:

=S_PIPE_ECC_TO_PIPE(0.05, 0.2, 0.03, 2)

Expected output:

9.78228

Python Code

Show Code 
from ht.conduction import S_isothermal_pipe_eccentric_to_isothermal_pipe as ht_S_isothermal_pipe_eccentric_to_isothermal_pipe

def S_pipe_ecc_to_pipe(D_one, D_two, Z, L=1):
    """
    Compute the shape factor for eccentric isothermal pipes.

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

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

    Args:
        D_one (float): Diameter of inner pipe (m).
        D_two (float): Diameter of outer pipe (m).
        Z (float): Offset between pipe centers (m).
        L (float, optional): Pipe length (m). Default is 1.

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_pipe_eccentric_to_isothermal_pipe(
            D1=D_one,
            D2=D_two,
            Z=Z,
            L=L,
        )
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Diameter of inner pipe (m).
Diameter of outer pipe (m).
Offset between pipe centers (m).
Pipe length (m).

S_PIPE_NORM_PLANE

Computes the conduction shape factor for a long isothermal pipe normal to an isothermal plane. This factor can be combined with conductivity and temperature difference for approximate heat-flow calculations.

The associated heat-rate expression is:

Q = Sk(T_1 - T_2)

Excel Usage

=S_PIPE_NORM_PLANE(D, L)
  • D (float, required): Pipe diameter (m).
  • L (float, required): Pipe length (m).

Returns (float): Shape factor (m).

Example 1: Example normal pipe

Inputs:

D L
1 100

Excel formula:

=S_PIPE_NORM_PLANE(1, 100)

Expected output:

104.869

Example 2: Shorter pipe length

Inputs:

D L
0.5 20

Excel formula:

=S_PIPE_NORM_PLANE(0.5, 20)

Expected output:

24.7605

Example 3: Slender pipe geometry

Inputs:

D L
0.1 10

Excel formula:

=S_PIPE_NORM_PLANE(0.1, 10)

Expected output:

10.4869

Example 4: Moderate geometry values

Inputs:

D L
0.8 40

Excel formula:

=S_PIPE_NORM_PLANE(0.8, 40)

Expected output:

47.4353

Python Code

Show Code 
from ht.conduction import S_isothermal_pipe_normal_to_plane as ht_S_isothermal_pipe_normal_to_plane

def S_pipe_norm_plane(D, L):
    """
    Compute the shape factor for a pipe normal to a plane.

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

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

    Args:
        D (float): Pipe diameter (m).
        L (float): Pipe length (m).

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_pipe_normal_to_plane(D=D, L=L)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Pipe diameter (m).
Pipe length (m).

S_PIPE_TO_PIPE

Computes the conduction shape factor between two isothermal parallel pipes. The factor summarizes geometric influence on heat transfer for simplified steady-state conduction models.

It is used in:

Q = Sk(T_1 - T_2)

Excel Usage

=S_PIPE_TO_PIPE(D_one, D_two, W, L)
  • D_one (float, required): Diameter of one pipe (m).
  • D_two (float, required): Diameter of the other pipe (m).
  • W (float, required): Center-to-center distance (m).
  • L (float, optional, default: 1): Pipe length (m).

Returns (float): Shape factor (m).

Example 1: Example pipe pair

Inputs:

D_one D_two W L
0.1 0.2 1 1

Excel formula:

=S_PIPE_TO_PIPE(0.1, 0.2, 1, 1)

Expected output:

1.18871

Example 2: Wider spacing and longer length

Inputs:

D_one D_two W L
0.15 0.25 1.5 2

Excel formula:

=S_PIPE_TO_PIPE(0.15, 0.25, 1.5, 2)

Expected output:

2.29685

Example 3: Short length spacing

Inputs:

D_one D_two W L
0.08 0.12 0.6 0.8

Excel formula:

=S_PIPE_TO_PIPE(0.08, 0.12, 0.6, 0.8)

Expected output:

1.00611

Example 4: Moderate geometry values

Inputs:

D_one D_two W L
0.2 0.3 1.2 1.5

Excel formula:

=S_PIPE_TO_PIPE(0.2, 0.3, 1.2, 1.5)

Expected output:

2.0753

Python Code

Show Code 
from ht.conduction import S_isothermal_pipe_to_isothermal_pipe as ht_S_isothermal_pipe_to_isothermal_pipe

def S_pipe_to_pipe(D_one, D_two, W, L=1):
    """
    Compute the shape factor for two isothermal pipes.

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

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

    Args:
        D_one (float): Diameter of one pipe (m).
        D_two (float): Diameter of the other pipe (m).
        W (float): Center-to-center distance (m).
        L (float, optional): Pipe length (m). Default is 1.

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_pipe_to_isothermal_pipe(
            D1=D_one,
            D2=D_two,
            W=W,
            L=L,
        )
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Diameter of one pipe (m).
Diameter of the other pipe (m).
Center-to-center distance (m).
Pipe length (m).

S_PIPE_TO_PLANE

Computes the conduction shape factor for an isothermal pipe near an isothermal plane. This provides a compact geometric term for estimating steady heat flow in surrounding solids.

The factor is used with:

Q = Sk(T_1 - T_2)

Excel Usage

=S_PIPE_TO_PLANE(D, Z, L)
  • D (float, required): Pipe diameter (m).
  • Z (float, required): Distance to plane (m).
  • L (float, optional, default: 1): Pipe length (m).

Returns (float): Shape factor (m).

Example 1: Example pipe to plane

Inputs:

D Z L
1 100 3

Excel formula:

=S_PIPE_TO_PLANE(1, 100, 3)

Expected output:

3.14607

Example 2: Unit length near plane

Inputs:

D Z
0.4 5

Excel formula:

=S_PIPE_TO_PLANE(0.4, 5)

Expected output:

1.60629

Example 3: Moderate spacing and length

Inputs:

D Z L
0.6 8 2

Excel formula:

=S_PIPE_TO_PLANE(0.6, 8, 2)

Expected output:

3.16039

Example 4: Small pipe near plane

Inputs:

D Z L
0.2 2 1

Excel formula:

=S_PIPE_TO_PLANE(0.2, 2, 1)

Expected output:

1.70357

Python Code

Show Code 
from ht.conduction import S_isothermal_pipe_to_plane as ht_S_isothermal_pipe_to_plane

def S_pipe_to_plane(D, Z, L=1):
    """
    Compute the shape factor for a pipe near a plane.

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

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

    Args:
        D (float): Pipe diameter (m).
        Z (float): Distance to plane (m).
        L (float, optional): Pipe length (m). Default is 1.

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_pipe_to_plane(D=D, Z=Z, L=L)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Pipe diameter (m).
Distance to plane (m).
Pipe length (m).

S_PIPE_TWO_PLANES

Computes the conduction shape factor for an isothermal pipe centered between two parallel isothermal planes at equal spacing. This captures geometry in compact conduction calculations.

The shape-factor model is:

Q = Sk(T_1 - T_2)

Excel Usage

=S_PIPE_TWO_PLANES(D, Z, L)
  • D (float, required): Pipe diameter (m).
  • Z (float, required): Distance to each plane (m).
  • L (float, optional, default: 1): Pipe length (m).

Returns (float): Shape factor (m).

Example 1: Example pipe between planes

Inputs:

D Z L
0.1 5 1

Excel formula:

=S_PIPE_TWO_PLANES(0.1, 5, 1)

Expected output:

1.29637

Example 2: Unit length with smaller spacing

Inputs:

D Z
0.2 3

Excel formula:

=S_PIPE_TWO_PLANES(0.2, 3)

Expected output:

1.72484

Example 3: Larger pipe diameter

Inputs:

D Z L
0.5 10 2

Excel formula:

=S_PIPE_TWO_PLANES(0.5, 10, 2)

Expected output:

3.19719

Example 4: Moderate geometry values

Inputs:

D Z L
0.3 6 1.5

Excel formula:

=S_PIPE_TWO_PLANES(0.3, 6, 1.5)

Expected output:

2.39789

Python Code

Show Code 
from ht.conduction import S_isothermal_pipe_to_two_planes as ht_S_isothermal_pipe_to_two_planes

def S_pipe_two_planes(D, Z, L=1):
    """
    Compute the shape factor for a pipe between two planes.

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

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

    Args:
        D (float): Pipe diameter (m).
        Z (float): Distance to each plane (m).
        L (float, optional): Pipe length (m). Default is 1.

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_pipe_to_two_planes(D=D, Z=Z, L=L)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Pipe diameter (m).
Distance to each plane (m).
Pipe length (m).

S_SPHERE_TO_PLANE

Computes the conduction shape factor for an isothermal sphere near an isothermal plane. The result is used in simplified conduction models for buried or embedded spherical geometries.

Heat transfer can then be estimated with:

Q = Sk(T_1 - T_2)

Excel Usage

=S_SPHERE_TO_PLANE(D, Z)
  • D (float, required): Sphere diameter (m).
  • Z (float, required): Distance to plane (m).

Returns (float): Shape factor (m).

Example 1: Example sphere to plane

Inputs:

D Z
1 100

Excel formula:

=S_SPHERE_TO_PLANE(1, 100)

Expected output:

6.29893

Example 2: Closer plane distance

Inputs:

D Z
0.4 4

Excel formula:

=S_SPHERE_TO_PLANE(0.4, 4)

Expected output:

2.57772

Example 3: Small sphere near plane

Inputs:

D Z
0.2 2

Excel formula:

=S_SPHERE_TO_PLANE(0.2, 2)

Expected output:

1.28886

Example 4: Moderate geometry values

Inputs:

D Z
0.6 8

Excel formula:

=S_SPHERE_TO_PLANE(0.6, 8)

Expected output:

3.84195

Python Code

Show Code 
from ht.conduction import S_isothermal_sphere_to_plane as ht_S_isothermal_sphere_to_plane

def S_sphere_to_plane(D, Z):
    """
    Compute the shape factor for a sphere near a plane.

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

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

    Args:
        D (float): Sphere diameter (m).
        Z (float): Distance to plane (m).

    Returns:
        float: Shape factor (m).
    """
    try:
        return ht_S_isothermal_sphere_to_plane(D=D, Z=Z)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Sphere diameter (m).
Distance to plane (m).

THERM_RESIST_TO_K

Converts thermal resistivity to thermal conductivity by taking the reciprocal. This is used when resistivity is available in material data but conductivity is required for model inputs.

The relation is:

k = \frac{1}{r}

Excel Usage

=THERM_RESIST_TO_K(r)
  • r (float, required): Thermal resistivity (m*K/W).

Returns (float): Thermal conductivity (W/m/K).

Example 1: Example conductivity

Inputs:

r
4

Excel formula:

=THERM_RESIST_TO_K(4)

Expected output:

0.25

Example 2: Lower resistivity

Inputs:

r
2

Excel formula:

=THERM_RESIST_TO_K(2)

Expected output:

0.5

Example 3: Higher resistivity

Inputs:

r
10

Excel formula:

=THERM_RESIST_TO_K(10)

Expected output:

0.1

Example 4: Moderate resistivity

Inputs:

r
1.5

Excel formula:

=THERM_RESIST_TO_K(1.5)

Expected output:

0.666667

Python Code

Show Code 
from ht.conduction import thermal_resistivity_to_k as ht_thermal_resistivity_to_k

def therm_resist_to_k(r):
    """
    Convert thermal resistivity to thermal conductivity.

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

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

    Args:
        r (float): Thermal resistivity (m*K/W).

    Returns:
        float: Thermal conductivity (W/m/K).
    """
    try:
        return ht_thermal_resistivity_to_k(r=r)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Thermal resistivity (m*K/W).