# Conduction and shape factors (ht.conduction)¶

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.

$(hA)_{\text{cylinder}}=\frac{k}{\ln(D_o/D_i)} \cdot 2\pi L\\ R_{\text{cylinder}}=\frac{1}{(hA)_{\text{cylinder}}}= \frac{\ln(D_o/D_i)}{2\pi Lk}$
Parameters
Difloat

Inner diameter of the cylinder, [m]

Dofloat

Outer diameter of the cylinder, [m]

kfloat

Thermal conductivity of the cylinder, [W/m/K]

Lfloat

Length of the cylinder, [m]

Returns
Rfloat

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)
8.38432343682705e-05

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
Rfloat

Thermal resistance of a substance, (K/W) if A is 1 m^2, otherwise must be [m^2*K/W]

tfloat

Thickness of the substance used in the measurement of R, [m]

Afloat, optional

Area; normally 1, [m^2]

Returns
kfloat

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.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_valuefloat

R-value of a substance [m^2 K/(W*inch) or ft^2 deg F*h/(BTU*inch)]

SIbool, optional

Whether to use the SI conversion or not

Returns
kfloat

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.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
D1float

Diameter of inner pipe, [m]

D2float

Diameter of outer pipe, [m]

Zfloat

Distance from the middle of inner pipe to the center of the other, [m]

Lfloat, optional

Length of the pipe, [m]

Returns
Sfloat

Shape factor [m]

Notes

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

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

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

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
Dfloat

Diameter of the pipe, [m]

Lfloat

Length of the pipe, [m]

Returns
Sfloat

Shape factor [m]

Notes

L should be much larger than D.

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

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 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
D1float

Diameter of one pipe, [m]

D2float

Diameter of the other pipe, [m]

Wfloat

Distance from the middle of one pipe to the middle of the other, [m]

Lfloat, optional

Length of the pipe, [m]

Returns
Sfloat

Shape factor [m]

Notes

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

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

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_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
Dfloat

Diameter of the pipe, [m]

Zfloat

Distance from the middle of the pipe to the infinite plane, [m]

Lfloat, optional

Length of the pipe, [m]

Returns
Sfloat

Shape factor [m]

Notes

L should be much larger than D.

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

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_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
Dfloat

Diameter of the pipe, [m]

Zfloat

Distance from the middle of the pipe to either of the planes, [m]

Lfloat, optional

Length of the pipe, [m]

Returns
Sfloat

Shape factor [m]

Notes

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

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

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_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
Dfloat

Diameter of the sphere, [m]

Zfloat

Distance from the middle of the sphere to the infinite plane, [m]

Returns
Sfloat

Shape factor [m]

Notes

No restrictions on the use of this equation.

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

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.cylindrical_heat_transfer(Ti, To, hi, ho, Di, ts, ks)[source]

Calculation for the heat transfer through a cylindrical wall, as occurs in pipes and cylindrical vessels. This is the core method which calculates the temperatures of each layer - and allows an outer layer to iterate on temperature or duty to meet a fixed specification, or include things like temperature dependent thermal conductivities or radiation.

Parameters
Tifloat

Temperature of the inside of the cylinder, [K]

Tofloat

External temperature outside the cylinder, away from the cylinder wall, [K]

hifloat

Inside heat transfer coefficient, [W/m^2/K]

hofloat

Outside heat transfer coefficient, [W/m^2/K]

Difloat

Inside diameter of cylinder, [m]

tslist[float]

List of thicknesses of each layer of the cylinder, [m]

kslist[float]

List of thermal conductivities of each layer of the cylinder, [w/m/K]

Returns
resultsdict
• Q : Heat exchanged through the cylinder (per meter of length), [W/m]

• Rs : Thermal resistances of each of the layers, [m*K/W]

• Ts : Temperatures of the outside of each of the layers, [K]

• UAHeat transfer coefficient times area (on a per-meter of

cylinder) basis, [W/K/m]

• U_innerHeat transfer coefficient with respect to the inside

diameter, [W/K]

• U_outerHeat transfer coefficient with respect to the exterior

diameter, [W/K]

• qSpecific heat exchanged (per square meter) through the cylinder

(per meter of length), [W/m^3]

Examples

>>> from pprint import pprint
>>> pprint(cylindrical_heat_transfer(Ti=453.15, To=301.15, hi=1e12, ho=22.697193, Di=0.0779272, ts=[0.0054864, .05], ks=[56.045, 0.0598535265]))
{'Q': 73.12000884069367,
'Rs': [0.00022201030738405449, 1.189361782070256],
'Ts': [453.15, 453.1226455779877, 306.578530147744],
'UA': 0.48105268974140575,
'U_inner': 1.9649599487726137,
'U_outer': 0.8106078714663484,
'q': 123.21239646288495}

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
kfloat

Thermal conductivity of a substance [W/m/K]

tfloat

Thickness of the substance for a given value of R, [m]

Afloat, optional

Area; normally 1, [m^2]

Returns
Rfloat

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_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
kfloat

Thermal conductivity of a substance [W/m/K]

SIbool, optional

Whether to use the SI conversion or not

Returns
R_valuefloat

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

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.k_to_thermal_resistivity(k)[source]

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

$r = \frac{1}{k}$
Parameters
kfloat

Thermal conductivity of a substance [W/m/K]

Returns
rfloat

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

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
rfloat

Thermal resistivity of a substance [m*K/W]

Returns
kfloat

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

1

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

Examples

>>> thermal_resistivity_to_k(4)
0.25