# Conduction and shape factors (ht.conduction)¶

ht.conduction.R_to_k(R, t, A=1.0)[source]

Returns the thermal conductivity of a substance given its thickness and thermal resistance.

$k = \frac{t}{RA}$
Parameters: R : float Thermal resistance of a substance [K/W or m^2*K/W] t : float Thickness of the substance used in the measurement of R, [m] A : float, optional Area; normally 1, [m^2] k : float Thermal conductivity of a substance [W/m/K]

Notes

When solving problems of changing areas, this value may be calculated with an area other than 1 m^2. Values in tables reported as properties of materials are often divided by area already; the conversion holds if A is 1.

References

 [R181199] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> R_to_k(R=0.05, t=0.025)
0.5

ht.conduction.k_to_R(k, t, A=1.0)[source]

Returns the thermal resistance of a substance given its thickness and thermal conductivity.

$R = \frac{t}{kA}$
Parameters: k : float Thermal conductivity of a substance [W/m/K] t : float Thickness of the substance for a given value of R, [m] A : float, optional Area; normally 1, [m^2] R : float Thermal resistance of a substance [K/W]

Notes

When solving problems of changing areas, this value may be calculated with an area other than 1 m^2. Values in tables reported as properties of materials are often divided by area already; the conversion holds if A is 1.

References

 [R182200] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> k_to_R(k=0.5, t=0.025)
0.05

ht.conduction.k_to_thermal_resistivity(k)[source]

Returns the thermal resistivity of a substance given its thermal conductivity.

$r = \frac{1}{k}$
Parameters: k : float Thermal conductivity of a substance [W/m/K] r : float Thermal resistivity of a substance [m*K/W]

Notes

Do not confuse this with thermal resistance! Often not introduced in heat transfer textbooks to avoid further confusion. Used almost exclusively as a description of solids. Thermal resistivity has different units than R-value, but is of the same dimensionality.

References

 [R183201] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

>>> k_to_thermal_resistivity(0.25)
4.0

ht.conduction.thermal_resistivity_to_k(r)[source]

Returns the thermal resistivity of a substance given its thermal conductivity.

$k = \frac{1}{r}$
Parameters: r : float Thermal resistivity of a substance [m*K/W] k : float Thermal conductivity of a substance [W/m/K]

Notes

Do not confuse this with thermal resistance! Often not introduced in heat as a description of solids. Thermal resistivity has different units than R-value, but is of the same dimensionality.

References

 [R184202] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

>>> thermal_resistivity_to_k(4)
0.25

ht.conduction.R_value_to_k(R_value, SI=True)[source]

Returns the thermal conductivity of a substance given its R-value, which can be in either SI units of m^2 K/(W*inch) or the Imperial units of ft^2 deg F*h/(BTU*inch).

Parameters: R_value : float R-value of a substance [m^2 K/(W*inch) or ft^2 deg F*h/(BTU*inch)] SI : bool, optional Whether to use the SI conversion or not k : float Thermal conductivity of a substance [W/m/K]

Notes

If given input is SI, it is divided by 0.0254 (multiplied by 39.37) and then inversed. Otherwise, it is multiplied by 6.93347 and then inversed.

References

 [R185203] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

>>> R_value_to_k(0.12), R_value_to_k(0.71, SI=False)
(0.2116666666666667, 0.20313787163983468)

>>> R_value_to_k(1., SI=False)/R_value_to_k(1.)
5.678263341113488

ht.conduction.k_to_R_value(k, SI=True)[source]

Returns the R-value of a substance given its thermal conductivity, Will return R-value in SI units unless SI is false. SI units are m^2 K/(W*inch); Imperial units of R-value are ft^2 deg F*h/(BTU*inch).

Parameters: k : float Thermal conductivity of a substance [W/m/K] SI : bool, optional Whether to use the SI conversion or not R_value : float R-value of a substance [m^2 K/(W*inch) or ft^2 deg F*h/(BTU*inch)]

Notes

Provides the reverse conversion of R_value_to_k.

References

 [R186204] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

>>> k_to_R_value(R_value_to_k(0.12)), k_to_R_value(R_value_to_k(0.71, SI=False), SI=False)
(0.11999999999999998, 0.7099999999999999)

ht.conduction.R_cylinder(Di, Do, k, L)[source]

Returns the thermal resistance R of a cylinder of constant thermal conductivity k, of inner and outer diameter Di and Do, and with a length L.

$\begin{split}(hA)_{\text{cylinder}}=\frac{k}{\ln(D_o/D_i)} \cdot 2\pi L\\ R_{\text{cylinder}}=\frac{1}{(hA)_{\text{cylinder}}}= \frac{\ln(D_o/D_i)}{2\pi Lk}\end{split}$
Parameters: Di : float Inner diameter of the cylinder, [m] Do : float Outer diameter of the cylinder, [m] k : float Thermal conductivity of the cylinder, [W/m/K] L : float Length of the cylinder, [m] R : float Thermal resistance [K/W]

References

 [R187205] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> R_cylinder(0.9, 1., 20, 10)
8.38432343682705e-05

ht.conduction.S_isothermal_sphere_to_plane(D, Z)[source]

Returns the Shape factor S of a sphere of constant temperature and of outer diameter D which is Z distance from an infinite plane.

$S = \frac{2\pi D}{1 - \frac{D}{4Z}}$
Parameters: D : float Diameter of the sphere, [m] Z : float Distance from the middle of the sphere to the infinite plane, [m] S : float Shape factor [m]

Notes

No restrictions on the use of this equation.

$\begin{split}Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk}\end{split}$

References

 [R188206] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010.
 [R189206] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> S_isothermal_sphere_to_plane(1, 100)
6.298932638776527

ht.conduction.S_isothermal_pipe_to_plane(D, Z, L=1)[source]

Returns the Shape factor S of a pipe of constant outer temperature and of outer diameter D which is Z distance from an infinite plane. Length L must be provided, but can be set to 1 to obtain a dimensionless shape factor used in some sources.

$S = \frac{2\pi L}{\cosh^{-1}(2z/D)}$
Parameters: D : float Diameter of the pipe, [m] Z : float Distance from the middle of the pipe to the infinite plane, [m] L : float, optional Length of the pipe, [m] S : float Shape factor [m]

Notes

L should be much larger than D.

$\begin{split}Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk}\end{split}$

References

 [R190208] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010.
 [R191208] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> S_isothermal_pipe_to_plane(1, 100, 3)
3.146071454894645

ht.conduction.S_isothermal_pipe_normal_to_plane(D, L)[source]

Returns the Shape factor S of a pipe of constant outer temperature and of outer diameter D which extends into an infinite medium below an an infinite plane.

$S = \frac{2\pi L}{\ln(4L/D)}$
Parameters: D : float Diameter of the pipe, [m] L : float Length of the pipe, [m] S : float Shape factor [m]

Notes

L should be much larger than D.

$\begin{split}Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk}\end{split}$

References

 [R192210] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010.
 [R193210] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> S_isothermal_pipe_normal_to_plane(1, 100)
104.86893910124888

ht.conduction.S_isothermal_pipe_to_isothermal_pipe(D1, D2, W, L=1.0)[source]

Returns the Shape factor S of a pipe of constant outer temperature and of outer diameter D1 which is w distance from another infinite pipe of outer diameterD2. Length L must be provided, but can be set to 1 to obtain a dimensionless shape factor used in some sources.

$S = \frac{2\pi L}{\cosh^{-1}\left(\frac{4w^2-D_1^2-D_2^2}{2D_1D_2}\right)}$
Parameters: D1 : float Diameter of one pipe, [m] D2 : float Diameter of the other pipe, [m] W : float Distance from the middle of one pipe to the middle of the other, [m] L : float, optional Length of the pipe, [m] S : float Shape factor [m]

Notes

L should be much larger than both diameters. L should be larger than W.

$\begin{split}Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk}\end{split}$

References

 [R194212] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010.
 [R195212] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> S_isothermal_pipe_to_isothermal_pipe(.1, .2, 1, 1)
1.188711034982268

ht.conduction.S_isothermal_pipe_to_two_planes(D, Z, L=1.0)[source]

Returns the Shape factor S of a pipe of constant outer temperature and of outer diameter D which is Z distance from two infinite isothermal planes of equal temperatures, parallel to each other and enclosing the pipe. Length L must be provided, but can be set to 1 to obtain a dimensionless shape factor used in some sources.

$S = \frac{2\pi L}{\ln\frac{8z}{\pi D}}$
Parameters: D : float Diameter of the pipe, [m] Z : float Distance from the middle of the pipe to either of the planes, [m] L : float, optional Length of the pipe, [m] S : float Shape factor [m]

Notes

L should be much larger than both diameters. L should be larger than W.

$\begin{split}Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk}\end{split}$

References

 [R196214] Shape Factors for Heat Conduction Through Bodies with Isothermal or Convective Boundary Conditions, J. E. Sunderland, K. R. Johnson, ASHRAE Transactions, Vol. 70, 1964.
 [R197214] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> S_isothermal_pipe_to_two_planes(.1, 5, 1)
1.2963749299921428

ht.conduction.S_isothermal_pipe_eccentric_to_isothermal_pipe(D1, D2, Z, L=1.0)[source]

Returns the Shape factor S of a pipe of constant outer temperature and of outer diameter D1 which is Z distance from the center of another pipe of outer diameterD2. Length L must be provided, but can be set to 1 to obtain a dimensionless shape factor used in some sources.

$S = \frac{2\pi L}{\cosh^{-1} \left(\frac{D_2^2 + D_1^2 - 4Z^2}{2D_1D_2}\right)}$
Parameters: D1 : float Diameter of inner pipe, [m] D2 : float Diameter of outer pipe, [m] Z : float Distance from the middle of inner pipe to the center of the other, [m] L : float, optional Length of the pipe, [m] S : float Shape factor [m]

Notes

L should be much larger than both diameters. D2 should be larger than D1.

$\begin{split}Q = Sk(T_1 - T_2) \\ R_{\text{shape}}=\frac{1}{Sk}\end{split}$

References

 [R198216] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010.
 [R199216] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> S_isothermal_pipe_eccentric_to_isothermal_pipe(.1, .4, .05, 10)
47.709841915608976