# Free convection to immersed bodies (ht.conv_free_immersed)¶

Calculates Nusselt number for natural convection around a vertical or horizontal helical coil suspended in a fluid without forced convection.

For horizontal cases:

$Nu_D = 0.318 Ra_D^{0.293},\; 5 \times {10}^{3} < Ra < 1 \times {10}^5$

For vertical cases:

$Nu_D = 0.290 Ra_D^{0.293},\; 5 \times {10}^{3} < Ra < 1 \times {10}^5$
Parameters
Prfloat

Prandtl number calculated with the film temperature - wall and temperature very far from the coil average, [-]

Grfloat

Grashof number calculated with the film temperature - wall and temperature very far from the coil average, and using the outer diameter of the coil [-]

horizontalbool, optional

Whether the coil is horizontal or vertical, [-]

Returns
Nufloat

Nusselt number using the outer diameter of the coil and the film temperature, [-]

Notes

This correlation is also reviewed in [2].

References

1

Xin, R. C., and M. A. Ebadian. “Natural Convection Heat Transfer from Helicoidal Pipes.” Journal of Thermophysics and Heat Transfer 10, no. 2 (1996): 297-302.

2

Prabhanjan, Devanahalli G., Timothy J. Rennie, and G. S. Vijaya Raghavan. “Natural Convection Heat Transfer from Helical Coiled Tubes.” International Journal of Thermal Sciences 43, no. 4 (April 1, 2004): 359-65.

Examples

>>> Nu_coil_Xin_Ebadian(0.7, 2E4, horizontal=False)
4.755689726250451
>>> Nu_coil_Xin_Ebadian(0.7, 2E4, horizontal=True)
5.2148597687849785

ht.conv_free_immersed.Nu_free_horizontal_plate(Pr, Gr, buoyancy, L=None, W=None, Method=None)[source]

This function calculates the heat transfer coefficient for external free convection from a horizontal plate.

Requires at a minimum a fluid’s Prandtl number Pr, and the Grashof number Gr for the system fluid, temperatures, and geometry.

L and W are not used by any correlations presently, but are included for future support.

If no correlation’s name is provided as Method, the ‘VDI’ correlation is selected.

Parameters
Prfloat

Prandtl number with respect to fluid properties [-]

Grfloat

Grashof number with respect to fluid properties and plate - fluid temperature difference [-]

buoyancybool, optional

Whether or not the plate’s free convection is buoyancy assisted (hot plate) or not, [-]

Lfloat, optional

Length of horizontal plate, [m]

Wfloat, optional

Width of the horizontal plate, [m]

Returns
Nufloat

Nusselt number with respect to plate length, [-]

Other Parameters
Methodstring, optional

A string of the function name to use, as in the dictionary conv_free_horizontal_plate_methods

Examples

Turbulent example

>>> Nu_free_horizontal_plate(5.54, 3.21e8, buoyancy=True)
203.89681224927565

>>> Nu_free_horizontal_plate(5.54, 3.21e8, buoyancy=True, Method='McAdams')
181.73121274384457

ht.conv_free_immersed.Nu_free_horizontal_plate_methods(Pr, Gr, buoyancy, L=None, W=None, check_ranges=True)[source]

This function returns a list of methods for calculating heat transfer coefficient for external free convection from a verical plate.

Requires at a minimum a fluid’s Prandtl number Pr, and the Grashof number Gr for the system fluid, temperatures, and geometry.

L and W are not used by any correlations presently, but are included for future support.

Parameters
Prfloat

Prandtl number with respect to fluid properties [-]

Grfloat

Grashof number with respect to fluid properties and plate - fluid temperature difference [-]

buoyancybool, optional

Whether or not the plate’s free convection is buoyancy assisted (hot plate) or not, [-]

Lfloat, optional

Length of horizontal plate, [m]

Wfloat, optional

Width of the horizontal plate, [m]

check_rangesbool, optional

Whether or not to return only correlations suitable for the provided data, [-]

Returns
methodslist[str]

List of methods which can be used to calculate Nu with the given inputs, [-]

Examples

>>> Nu_free_horizontal_plate_methods(0.69, 2.63E9, True)

ht.conv_free_immersed.Nu_free_vertical_plate(Pr, Gr, buoyancy=None, H=None, W=None, Method=None)[source]

This function calculates the heat transfer coefficient for external free convection from a verical plate.

Requires at a minimum a fluid’s Prandtl number Pr, and the Grashof number Gr for the system fluid (which require T and P to obtain).

L and W are not used by any correlations presently, but are included for future support.

If no correlation’s name is provided as Method, the ‘Churchill’ correlation is selected.

Parameters
Prfloat

Prandtl number with respect to fluid properties [-]

Grfloat

Grashof number with respect to fluid properties and plate - fluid temperature difference [-]

buoyancybool, optional

Whether or not the plate’s free convection is buoyancy assisted (hot plate) or not, [-]

Hfloat, optional

Height of vertical plate, [m]

Wfloat, optional

Width of the vertical plate, [m]

Returns
Nufloat

Nusselt number with respect to plate height, [-]

Other Parameters
Methodstring, optional

A string of the function name to use; one of (‘Churchill’, ).

Examples

Turbulent example

>>> Nu_free_vertical_plate(0.69, 2.63E9, False)
147.16185223770603

ht.conv_free_immersed.Nu_free_vertical_plate_methods(Pr, Gr, H=None, W=None, check_ranges=True)[source]

This function returns a list of methods for calculating heat transfer coefficient for external free convection from a verical plate.

Requires at a minimum a fluid’s Prandtl number Pr, and the Grashof number Gr for the system fluid (which require T and P to obtain).

L and W are not used by any correlations presently, but are included for future support.

Parameters
Prfloat

Prandtl number with respect to fluid properties [-]

Grfloat

Grashof number with respect to fluid properties and plate - fluid temperature difference [-]

Hfloat, optional

Height of vertical plate, [m]

Wfloat, optional

Width of the vertical plate, [m]

check_rangesbool, optional

Whether or not to return only correlations suitable for the provided data, [-]

Returns
methodslist[str]

List of methods which can be used to calculate Nu with the given inputs, [-]

Examples

>>> Nu_free_vertical_plate_methods(0.69, 2.63E9)
['Churchill']

ht.conv_free_immersed.Nu_horizontal_cylinder(Pr, Gr, Method=None)[source]

This function handles choosing which horizontal cylinder free convection correlation is used. Generally this is used by a helper class, but can be used directly. Will automatically select the correlation to use if none is provided; returns None if insufficient information is provided.

Preferred functions are ‘Morgan’ when discontinuous results are acceptable and ‘Churchill-Chu’ otherwise.

Parameters
Prfloat

Prandtl number with respect to film temperature [-]

Grfloat

Grashof number with respect to cylinder diameter, [-]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Other Parameters
Methodstring, optional

A string of the function name to use, as in the dictionary horizontal_cylinder_correlations

Notes

All fluid properties should be evaluated at the film temperature, the average between the outer surface temperature of the solid, and the fluid temperature far away from the heat transfer interface - normally the same as the temperature before any cooling or heating occurs.

$T_f = (T_{\text{surface}} + T_\infty)/2$

Examples

>>> Nu_horizontal_cylinder(0.72, 1E7)
24.864192615468973

ht.conv_free_immersed.Nu_horizontal_cylinder_Churchill_Chu(Pr, Gr)[source]

Calculates Nusselt number for natural convection around a horizontal cylinder according to the Churchill-Chu [1] correlation, also presented in [2]. Cylinder must be isothermal; an alternate expression exists for constant heat flux.

$Nu_{D}=\left[0.60+\frac{0.387Ra_{D}^{1/6}} {[1+(0.559/Pr)^{9/16}]^{8/27}}\right]^2$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number with respect to cylinder diameter, [-]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Notes

Although transition from laminar to turbulent is discrete in reality, this equation provides a smooth transition in value from laminar to turbulent. Checked with the original source, which has its powers unsimplified but is equivalent.

[1] recommends 1E-5 as the lower limit for Ra, but no upper limit. [2] suggests an upper limit of 1E12.

References

1(1,2)

Churchill, Stuart W., and Humbert H. S. Chu. “Correlating Equations for Laminar and Turbulent Free Convection from a Horizontal Cylinder.” International Journal of Heat and Mass Transfer 18, no. 9 (September 1975): 1049-53. doi:10.1016/0017-9310(75)90222-7.

2(1,2,3)

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

Examples

From [2], Example 9.2, matches:

>>> Nu_horizontal_cylinder_Churchill_Chu(0.69, 2.63E9)
139.13493970073597

ht.conv_free_immersed.Nu_horizontal_cylinder_Kuehn_Goldstein(Pr, Gr)[source]

Calculates Nusselt number for natural convection around a horizontal cylinder according to the Kuehn-Goldstein [1] correlation, also shown in [2]. Cylinder must be isothermal.

$\frac{2}{Nu_D} = \ln\left[1 + \frac{2}{\left[\left\{0.518Ra_D^{0.25} \left[1 + \left(\frac{0.559}{Pr}\right)^{3/5}\right]^{-5/12} \right\}^{15} + (0.1Ra_D^{1/3})^{15}\right]^{1/15}}\right]$
Parameters
Prfloat

Prandtl number with respect to film temperature [-]

Grfloat

Grashof number with respect to cylinder diameter, [-]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Notes

[1] suggests this expression is valid for all cases except low-Pr fluids. [2] suggests no restrictions.

References

1(1,2)

Kuehn, T. H., and R. J. Goldstein. “Correlating Equations for Natural Convection Heat Transfer between Horizontal Circular Cylinders.” International Journal of Heat and Mass Transfer 19, no. 10 (October 1976): 1127-34. doi:10.1016/0017-9310(76)90145-9

2(1,2)

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_horizontal_cylinder_Kuehn_Goldstein(0.69, 2.63E9)
122.99323525628186

ht.conv_free_immersed.Nu_horizontal_cylinder_Morgan(Pr, Gr)[source]

Calculates Nusselt number for natural convection around a horizontal cylinder according to the Morgan [1] correlations, a product of a very large review of the literature. Sufficiently common as to be shown in [2]. Cylinder must be isothermal.

$Nu_D = C Ra_D^n$

Gr min

Gr max

C

n

10E-10

10E-2

0.675

0.058

10E-2

10E2

1.02

0.148

10E2

10E4

0.850

0.188

10E4

10E7

0.480

0.250

10E7

10E12

0.125

0.333

Parameters
Prfloat

Prandtl number with respect to film temperature [-]

Grfloat

Grashof number with respect to cylinder diameter, [-]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Notes

Most comprehensive review with a new proposed equation to date. Discontinuous among the jumps in range. Blindly runs outside if upper and lower limits without warning.

References

1

Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975.

2

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_horizontal_cylinder_Morgan(0.69, 2.63E9)
151.3881997228419

ht.conv_free_immersed.Nu_horizontal_cylinder_methods(Pr, Gr, check_ranges=True)[source]

This function returns a list of correlation names for free convetion to a horizontal cylinder.

Preferred functions are ‘Morgan’ when discontinuous results are acceptable and ‘Churchill-Chu’ otherwise.

Parameters
Prfloat

Prandtl number with respect to film temperature [-]

Grfloat

Grashof number with respect to cylinder diameter, [-]

check_rangesbool, optional

Whether or not to return only correlations suitable for the provided data, [-]

Returns
methodslist[str]

List of methods which can be used to calculate Nu with the given inputs

Examples

>>> Nu_horizontal_cylinder_methods(0.72, 1E7)[0]
'Morgan'


Calculates the Nusselt number for natural convection above a horizontal plate according to the McAdams [1] correlations. The plate must be isothermal. Four different equations are used, two each for laminar and turbulent; the two sets of correlations are required because if the plate is hot, buoyancy lifts the fluid off the plate and enhances free convection whereas if the plate is cold, the cold fluid above it settles on it and decreases the free convection.

Parameters
Prfloat

Prandtl number with respect to fluid properties [-]

Grfloat

Grashof number with respect to fluid properties and plate - fluid temperature difference [-]

buoyancybool, optional

Whether or not the plate’s free convection is buoyancy assisted (hot plate) or not, [-]

Returns
Nufloat

Nusselt number with respect to length, [-]

References

1

McAdams, William Henry. Heat Transmission. 3E. Malabar, Fla: Krieger Pub Co, 1985.

Examples

>>> Nu_horizontal_plate_McAdams(5.54, 3.21e8, buoyancy=True)
181.73121274384457
>>> Nu_horizontal_plate_McAdams(5.54, 3.21e8, buoyancy=False)
55.44564799362829

>>> Nu_horizontal_plate_McAdams(.01, 3.21e8, buoyancy=True)
22.857041558492334
>>> Nu_horizontal_plate_McAdams(.01, 3.21e8, buoyancy=False)
11.428520779246167

ht.conv_free_immersed.Nu_horizontal_plate_Rohsenow(Pr, Gr, buoyancy=True)[source]

Calculates the Nusselt number for natural convection above a horizontal plate according to the Rohsenow, Hartnett, and Cho (1998) [1] correlations. The plate must be isothermal. Three different equations are used, one each for laminar and turbulent for the heat transfer happening at upper surface case and one for the case of heat transfer happening at the lower surface.

The lower surface correlation is recommened for the laminar flow regime. The two different sets of correlations are required because if the plate is hot, buoyancy lifts the fluid off the plate and enhances free convection whereas if the plate is cold, the cold fluid above it settles on it and decreases the free convection.

Parameters
Prfloat

Prandtl number with respect to fluid properties [-]

Grfloat

Grashof number with respect to fluid properties and plate - fluid temperature difference [-]

buoyancybool, optional

Whether or not the plate’s free convection is buoyancy assisted (hot plate) or not, [-]

Returns
Nufloat

Nusselt number with respect to length, [-]

Notes

The characteristic length suggested for use is as follows, with a and b being the length and width of the plate.

$L = \frac{ab}{2(a+b)}$

References

1

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

Examples

>>> Nu_horizontal_plate_Rohsenow(5.54, 3.21e8, buoyancy=True)
175.91054716322836
>>> Nu_horizontal_plate_Rohsenow(5.54, 3.21e8, buoyancy=False)
35.95799244863986

ht.conv_free_immersed.Nu_horizontal_plate_VDI(Pr, Gr, buoyancy=True)[source]

Calculates the Nusselt number for natural convection above a horizontal plate according to the VDI [1] correlations. The plate must be isothermal. Three different equations are used, one each for laminar and turbulent for the heat transfer happening at upper surface case and one for the case of heat transfer happening at the lower surface. The lower surface correlation is recommened for the laminar flow regime. The two different sets of correlations are required because if the plate is hot, buoyancy lifts the fluid off the plate and enhances free convection whereas if the plate is cold, the cold fluid above it settles on it and decreases the free convection.

Parameters
Prfloat

Prandtl number with respect to fluid properties [-]

Grfloat

Grashof number with respect to fluid properties and plate - fluid temperature difference [-]

buoyancybool, optional

Whether or not the plate’s free convection is buoyancy assisted (hot plate) or not, [-]

Returns
Nufloat

Nusselt number with respect to length, [-]

Notes

The characteristic length suggested for use is as follows, with a and b being the length and width of the plate.

$L = \frac{ab}{2(a+b)}$

The buoyancy enhanced cases are from [2]; the other is said to be from [3], although the equations there not quite the same and do not include the Prandtl number correction.

References

1

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

2

Stewartson, Keith. “On the Free Convection from a Horizontal Plate.” Zeitschrift Für Angewandte Mathematik Und Physik ZAMP 9, no. 3 (September 1, 1958): 276-82. https://doi.org/10.1007/BF02033031.

3

Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1987.

Examples

>>> Nu_horizontal_plate_VDI(5.54, 3.21e8, buoyancy=True)
203.89681224927565
>>> Nu_horizontal_plate_VDI(5.54, 3.21e8, buoyancy=False)
39.16864971535617

ht.conv_free_immersed.Nu_sphere_Churchill(Pr, Gr)[source]

Calculates Nusselt number for natural convection around a sphere according to the Churchill [1] correlation. Sphere must be isothermal.

$Nu_D=2+\frac{0.589Ra_D^{1/4}} {\left[1+(0.469/Pr)^{9/16}\right]^{4/9}} \cdot\left\{1 + \frac{7.44\times 10^{-8}Ra} {[1+(0.469/Pr)^{9/16}]^{16/9}}\right\}^{1/12}$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number [-]

Returns
Nufloat

Nusselt number, [-]

Notes

Although transition from laminar to turbulent is discrete in reality, this equation provides a smooth transition in value from laminar to turbulent. Checked with the original source.

Good for Ra < 1E13. Limit of Nu is 2 at low Grashof numbers.

References

1

Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1987.

Examples

>>> Nu_sphere_Churchill(.7, 1E7)
25.670869440317578

ht.conv_free_immersed.Nu_vertical_cylinder(Pr, Gr, L=None, D=None, Method=None)[source]

This function handles choosing which vertical cylinder free convection correlation is used. Generally this is used by a helper class, but can be used directly. Will automatically select the correlation to use if none is provided; returns None if insufficient information is provided.

Preferred functions are ‘Popiel & Churchill’ for fully defined geometries, and ‘McAdams, Weiss & Saunders’ otherwise.

Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number with respect to cylinder height [-]

Lfloat, optional

Length of vertical cylinder, [m]

Dfloat, optional

Diameter of cylinder, [m]

Returns
Nufloat

Nusselt number, [-]

Other Parameters
Methodstring, optional

A string of the function name to use, as in the dictionary vertical_cylinder_correlations

Examples

>>> Nu_vertical_cylinder(0.72, 1E7)
30.562236756513943

ht.conv_free_immersed.Nu_vertical_cylinder_Al_Arabi_Khamis(Pr, Gr, L, D, turbulent=None)[source]

Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to [1], also as presented in [2] and [3].

$Nu_H = 2.9Ra_H^{0.25}/Gr_D^{1/12},\; 9.88 \times 10^7 \le Ra_H \le 2.7\times10^{9}$
$Nu_H = 0.47 Ra_H^{0.333}/Gr_D^{1/12},\; 2.7 \times 10^9 \le Ra_H \le 2.95\times10^{10}$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number with respect to cylinder height [-]

Lfloat

Length of vertical cylinder, [m]

Dfloat

Diameter of cylinder, [m]

turbulentbool or None, optional

Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection, [-]

Returns
Nufloat

Nusselt number, [-]

Notes

For air. Local Nusselt number results also given in [1]. D from 12.75 to 51 mm; H from 300 to 2000 mm. Temperature kept constant by steam condensing.

If outside of range, no warning is given. Applies for range of:

$1.08 \times 10^4 \le Gr_D \le 6.9 \times 10^5$

References

1(1,2)

Al-Arabi, M., and M. Khamis. “Natural Convection Heat Transfer from Inclined Cylinders.” International Journal of Heat and Mass Transfer 25, no. 1 (January 1982): 3-15. doi:10.1016/0017-9310(82)90229-0.

2

Popiel, Czeslaw O. “Free Convection Heat Transfer from Vertical Slender Cylinders: A Review.” Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557.

3

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_vertical_cylinder_Al_Arabi_Khamis(.71, 2E10, 10, 1)
280.39793209114765

ht.conv_free_immersed.Nu_vertical_cylinder_Carne_Morgan(Pr, Gr, turbulent=None)[source]

Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1] correlated by [2], as presented in [3] and [4].

$Nu_H = 1.07 Ra_H^{0.28},\; 2\times 10^{6} < Ra < 2\times 10^{8}$
$Nu_H = 0.152 Ra_H^{0.38},\; 2\times 10^{8} < Ra < 2\times 10^{11}$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number [-]

turbulentbool or None, optional

Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection

Returns
Nufloat

Nusselt number, [-]

Notes

Cylinder of diameters 0.475 cm to 7.62 cm, L/D from 8 to 127. Isothermal boundary condition was assumed, but not verified. Transition between ranges is not smooth. If outside of range, no warning is given. The higher range of [1] is not shown in [3], and the formula for the first is actually for the second in [3].

References

1(1,2)

J. B. Carne. “LIX. Heat Loss by Natural Convection from Vertical Cylinders.” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 24, no. 162 (October 1, 1937): 634-53. doi:10.1080/14786443708565140.

2

Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975.

3(1,2,3)

Popiel, Czeslaw O. “Free Convection Heat Transfer from Vertical Slender Cylinders: A Review.” Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557.

4

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_vertical_cylinder_Carne_Morgan(.7, 2E8)
204.31470629065677

ht.conv_free_immersed.Nu_vertical_cylinder_Eigenson_Morgan(Pr, Gr, turbulent=None)[source]

Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1] correlated by [2], presented in [3] and in more detail in [4].

$Nu_H = 0.48 Ra_H^{0.25},\; 10^{9} < Ra$
$Nu_H = 51.5 + 0.0000726 Ra_H^{0.63},\; 10^{9} < Ra < 1.69 \times 10^{10}$
$Nu_H = 0.148 Ra_H^{1/3} - 127.6 ,\; 1.69 \times 10^{10} < Ra$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number [-]

turbulentbool or None, optional

Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection

Returns
Nufloat

Nusselt number, [-]

Notes

Author presents results as appropriate for both flat plates and cylinders. Height of 2.5 m with diameters of 2.4, 7.55, 15, 35, and 50 mm. Another experiment of diameter 58 mm and length of 6.5 m was considered. Cylinder of diameters 0.475 cm to 7.62 cm, L/D from 8 to 127.Transition between ranges is not smooth. If outside of range, no warning is given. Formulas are presented similarly in [3] and [4], but only [4] shows the transition formula.

References

1

Eigenson L (1940). Les lois gouvernant la transmission de la chaleur aux gaz biatomiques par les parois des cylindres verticaux dans le cas de convection naturelle. Dokl Akad Nauk SSSR 26:440-444

2

Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975.

3(1,2)

Popiel, Czeslaw O. “Free Convection Heat Transfer from Vertical Slender Cylinders: A Review.” Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557.

4(1,2,3)

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_vertical_cylinder_Eigenson_Morgan(0.7, 2E10)
230.55946525499715

ht.conv_free_immersed.Nu_vertical_cylinder_Griffiths_Davis_Morgan(Pr, Gr, turbulent=None)[source]

Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1] correlated by [2], as presented in [3] and [4].

$Nu_H = 0.67 Ra_H^{0.25},\; 10^{7} < Ra < 10^{9}$
$Nu_H = 0.0782 Ra_H^{0.357}, \; 10^{9} < Ra < 10^{11}$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number [-]

turbulentbool or None, optional

Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection

Returns
Nufloat

Nusselt number, [-]

Notes

Cylinder of diameter 17.43 cm, length from 4.65 to 263.5 cm. Air as fluid. Transition between ranges is not smooth. If outside of range, no warning is given.

References

1

Griffiths, Ezer, A. H. Davis, and Great Britain. The Transmission of Heat by Radiation and Convection. London: H. M. Stationery off., 1922.

2

Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975.

3

Popiel, Czeslaw O. “Free Convection Heat Transfer from Vertical Slender Cylinders: A Review.” Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557.

4

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_vertical_cylinder_Griffiths_Davis_Morgan(.7, 2E10)
327.6230596100138

ht.conv_free_immersed.Nu_vertical_cylinder_Hanesian_Kalish_Morgan(Pr, Gr)[source]

Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1] correlated by [2], also as presented in [3] and [4].

$Nu_H = 0.48 Ra_H^{0.23},\; 10^{6} < Ra < 10^{8}$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number [-]

Returns
Nufloat

Nusselt number, [-]

Notes

For air and fluoro-carbons. If outside of range, no warning is given. Laminar range only!

References

1

Hanesian, D. and Kalish, R. “Heat Transfer by Natural Convection with Fluorocarbon Gases.” IEEE Transactions on Parts, Materials and Packaging 6, no. 4 (December 1970): 147-148. doi:10.1109/TPMP.1970.1136270.

2

Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975.

3

Popiel, Czeslaw O. “Free Convection Heat Transfer from Vertical Slender Cylinders: A Review.” Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557.

4

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_vertical_cylinder_Hanesian_Kalish_Morgan(.7, 1E7)
18.014150492696604


Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1] correlated by [2], as presented in [3] and [4].

$Nu_H = 0.555 Ra_H^{0.25},\; 10^{4} < Ra < 10^{8}$
$Nu_H = 0.129 Ra_H^{1/3},\; 10^{8} < Ra < 10^{12}$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number [-]

turbulentbool or None, optional

Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection

Returns
Nufloat

Nusselt number, [-]

Notes

Cylinder of diameter 3.5 cm, length from L/D = 4.3. Air as fluid. Transition between ranges is not smooth. If outside of range, no warning is given. Results are presented rounded in [4], and the second range is not shown in [3].

References

1

Jakob, M., and Linke, W., Warmeubergang beim Verdampfen von Flussigkeiten an senkrechten und waagerechten Flaschen, Phys. Z., vol. 36, pp. 267-280, 1935.

2

Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975.

3(1,2)

Popiel, Czeslaw O. “Free Convection Heat Transfer from Vertical Slender Cylinders: A Review.” Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557.

4(1,2)

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_vertical_cylinder_Jakob_Linke_Morgan(.7, 2E10)
310.90835207860454

ht.conv_free_immersed.Nu_vertical_cylinder_Kreith_Eckert(Pr, Gr, turbulent=None)[source]

Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1] correlated by [2], also as presented in [3], [4], and [5].

$Nu_H = 0.555 Ra_H^{0.25},\; 10^{5} < Ra < 10^{9}$
$Nu_H = 0.021 Ra_H^{0.4},\; 10^{9} < Ra < 10^{12}$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number [-]

turbulentbool or None, optional

Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection

Returns
Nufloat

Nusselt number, [-]

Notes

Transition between ranges is not smooth. If outside of range, no warning is given.

References

1

Eckert, E. R. G., Thomas W. Jackson, and United States. Analysis of Turbulent Free-Convection Boundary Layer on Flat Plate. National Advisory Committee for Aeronautics, no. 2207. Washington, D.C.: National Advisoty Committee for Aeronautics, 1950.

2

Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010.

3

Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975.

4

Popiel, Czeslaw O. “Free Convection Heat Transfer from Vertical Slender Cylinders: A Review.” Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557.

5

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_vertical_cylinder_Kreith_Eckert(.7, 2E10)
240.25393473033196


Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1] and [2] correlated by [3], as presented in [4], [5], and [6].

$Nu_H = 0.59 Ra_H^{0.25},\; 10^{4} < Ra < 10^{9}$
$Nu_H = 0.13 Ra_H^{1/3.},\; 10^{9} < Ra < 10^{12}$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number [-]

turbulentbool or None, optional

Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection

Returns
——-
Nufloat

Nusselt number, [-]

Notes

Transition between ranges is not smooth. If outside of range, no warning is given. For ranges under 10^4, a graph is provided, not included here.

References

1

Weise, Rudolf. “Warmeubergang durch freie Konvektion an quadratischen Platten.” Forschung auf dem Gebiet des Ingenieurwesens A 6, no. 6 (November 1935): 281-92. doi:10.1007/BF02592565.

2

Saunders, O. A. “The Effect of Pressure Upon Natural Convection in Air.” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 157, no. 891 (November 2, 1936): 278-91. doi:10.1098/rspa.1936.0194.

3

McAdams, William Henry. Heat Transmission. 3E. Malabar, Fla: Krieger Pub Co, 1985.

4

Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975.

5

Popiel, Czeslaw O. “Free Convection Heat Transfer from Vertical Slender Cylinders: A Review.” Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557.

6

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_vertical_cylinder_McAdams_Weiss_Saunders(.7, 2E10)
313.31849434277973

ht.conv_free_immersed.Nu_vertical_cylinder_Popiel_Churchill(Pr, Gr, L, D)[source]

Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to [1], also presented in [2].

$\frac{Nu}{Nu_{L,fp}} = 1 + B\left[32^{0.5}Gr_L^{-0.25}\frac{L}{D}\right]^C$
$B = 0.0571322 + 0.20305 Pr^{-0.43}$
$C = 0.9165 - 0.0043Pr^{0.5} + 0.01333\ln Pr + 0.0004809/Pr$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number with respect to cylinder height [-]

Lfloat

Length of vertical cylinder, [m]

Dfloat

Diameter of cylinder, [m]

Returns
Nufloat

Nusselt number, [-]

Notes

For 0.01 < Pr < 100. Requires a vertical flat plate correlation. Both [2], [3] present a power of 2 instead of 0.5 on the 32 in the equation, but the original has the correct form.

References

1

Popiel, C. O., J. Wojtkowiak, and K. Bober. “Laminar Free Convective Heat Transfer from Isothermal Vertical Slender Cylinder.” Experimental Thermal and Fluid Science 32, no. 2 (November 2007): 607-613. doi:10.1016/j.expthermflusci.2007.07.003.

2

Popiel, Czeslaw O. “Free Convection Heat Transfer from Vertical Slender Cylinders: A Review.” Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557.

3

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_vertical_cylinder_Popiel_Churchill(0.7, 1E10, 2.5, 1)
228.89790055149896

ht.conv_free_immersed.Nu_vertical_cylinder_Touloukian_Morgan(Pr, Gr, turbulent=None)[source]

Calculates Nusselt number for natural convection around a vertical isothermal cylinder according to the results of [1] correlated by [2], as presented in [3] and [4].

$Nu_H = 0.726 Ra_H^{0.25},\; 2\times 10^{8} < Ra < 4\times 10^{10}$
$Nu_H = 0.0674 (Gr_H Pr^{1.29})^{1/3},\; 4\times 10^{10} < Ra < 9\times 10^{11}$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number [-]

turbulentbool or None, optional

Whether or not to force the correlation to return the turbulent result; will return the laminar regime if False; leave as None for automatic selection

Returns
Nufloat

Nusselt number, [-]

Notes

Cylinder of diameters 2.75 inch, with heights of 6, 18, and 36.25 inch. Temperature was controlled via multiple separately controlled heating sections. Fluids were water and ethylene-glycol. Transition between ranges is not smooth. If outside of range, no warning is given. [2], [3], and [4] are in complete agreement about this formulation.

References

1

Touloukian, Y. S, George A Hawkins, and Max Jakob. Heat Transfer by Free Convection from Heated Vertical Surfaces to Liquids. Trans. ASME 70, 13-18 (1948).

2(1,2)

Morgan, V.T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and J.P. Hartnett, V 11, 199-264, 1975.

3(1,2)

Popiel, Czeslaw O. “Free Convection Heat Transfer from Vertical Slender Cylinders: A Review.” Heat Transfer Engineering 29, no. 6 (June 1, 2008): 521-36. doi:10.1080/01457630801891557.

4(1,2)

Boetcher, Sandra K. S. “Natural Convection Heat Transfer From Vertical Cylinders.” In Natural Convection from Circular Cylinders, 23-42. Springer, 2014.

Examples

>>> Nu_vertical_cylinder_Touloukian_Morgan(.7, 2E10)
249.72879961097854

ht.conv_free_immersed.Nu_vertical_cylinder_methods(Pr, Gr, L=None, D=None, check_ranges=True)[source]

This function returns a list of correlation names for free convetion to a vertical cylinder.

The functions returned are ‘Popiel & Churchill’ for fully defined geometries, and ‘McAdams, Weiss & Saunders’ otherwise.

Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number with respect to cylinder height [-]

Lfloat, optional

Length of vertical cylinder, [m]

Dfloat, optional

Diameter of cylinder, [m]

check_rangesbool, optional

Whether or not to return only correlations suitable for the provided data, [-]

Returns
methodslist[str]

List of methods which can be used to calculate Nu with the given inputs

Examples

>>> Nu_vertical_cylinder_methods(0.72, 1E7)[0]
'McAdams, Weiss & Saunders'

ht.conv_free_immersed.Nu_vertical_plate_Churchill(Pr, Gr)[source]

Calculates Nusselt number for natural convection around a vertical plate according to the Churchill-Chu [1] correlation, also presented in [2]. Plate must be isothermal; an alternate expression exists for constant heat flux.

$Nu_{L}=\left[0.825+\frac{0.387Ra_{L}^{1/6}} {[1+(0.492/Pr)^{9/16}]^{8/27}}\right]^2$
Parameters
Prfloat

Prandtl number [-]

Grfloat

Grashof number [-]

Returns
Nufloat

Nusselt number with respect to height, [-]

Notes

Although transition from laminar to turbulent is discrete in reality, this equation provides a smooth transition in value from laminar to turbulent. Checked with the original source.

Can be applied to vertical cylinders as well, subject to the criteria below:

$\frac{D}{L}\ge \frac{35}{Gr_L^{1/4}}$

References

1

Churchill, Stuart W., and Humbert H. S. Chu. “Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate.” International Journal of Heat and Mass Transfer 18, no. 11 (November 1, 1975): 1323-29. doi:10.1016/0017-9310(75)90243-4.

2(1,2)

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

Examples

From [2], Example 9.2, matches:

>>> Nu_vertical_plate_Churchill(0.69, 2.63E9)
147.16185223770603