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) if A is 1 m^2, otherwise must be [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]
Returns:  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
[1] 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]
Returns:  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
[1] 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]
Returns:  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 Rvalue, but is of the same dimensionality.
References
[1] 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]
Returns:  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 Rvalue, but is of the same dimensionality.
References
[1] 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 Rvalue, 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
Rvalue 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
Returns:  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
[1] 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 Rvalue of a substance given its thermal conductivity, Will return Rvalue in SI units unless SI is false. SI units are m^2 K/(W*inch); Imperial units of Rvalue 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
Returns:  R_value : float
Rvalue 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
[1] 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]
Returns:  R : float
Thermal resistance [K/W]
References
[1] 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) 8.38432343682705e05

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]
Returns:  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
[1] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010. [2] 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]
Returns:  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
[1] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010. [2] 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]
Returns:  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
[1] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010. [2] 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 diameter`D2`. Length L must be provided, but can be set to 1 to obtain a dimensionless shape factor used in some sources.
\[S = \frac{2\pi L}{\cosh^{1}\left(\frac{4w^2D_1^2D_2^2}{2D_1D_2}\right)}\]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]
Returns:  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
[1] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010. [2] 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]
Returns:  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
[1] Shape Factors for Heat Conduction Through Bodies with Isothermal or Convective Boundary Conditions, J. E. Sunderland, K. R. Johnson, ASHRAE Transactions, Vol. 70, 1964. [2] 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 diameter`D2`. 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]
Returns:  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
[1] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010. [2] 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