Free convection to immersed bodies (ht.conv_free_immersed)

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:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

Returns:
Nu : float

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.

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](1, 2) 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, 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_vertical_plate_Churchill(0.69, 2.63E9)
147.16185223770603
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:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

Returns:
Nu : float

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](1, 2) 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_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].

\[ \begin{align}\begin{aligned}Nu_H = 0.67 Ra_H^{0.25},\; 10^{7} < Ra < 10^{9}\\Nu_H = 0.0782 Ra_H^{0.357}, \; 10^{9} < Ra < 10^{11}\end{aligned}\end{align} \]
Parameters:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

turbulent : bool 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:
Nu : float

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](1, 2) Griffiths, Ezer, A. H. Davis, and Great Britain. The Transmission of Heat by Radiation and Convection. London: H. M. Stationery off., 1922.
[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_Griffiths_Davis_Morgan(.7, 2E10)
327.6230596100138
ht.conv_free_immersed.Nu_vertical_cylinder_Jakob_Linke_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].

\[ \begin{align}\begin{aligned}Nu_H = 0.555 Ra_H^{0.25},\; 10^{4} < Ra < 10^{8}\\Nu_H = 0.129 Ra_H^{1/3},\; 10^{8} < Ra < 10^{12}\end{aligned}\end{align} \]
Parameters:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

turbulent : bool 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:
Nu : float

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](1, 2) Jakob, M., and Linke, W., Warmeubergang beim Verdampfen von Flussigkeiten an senkrechten und waagerechten Flaschen, Phys. Z., vol. 36, pp. 267-280, 1935.
[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, 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](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_Jakob_Linke_Morgan(.7, 2E10)
310.90835207860454
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].

\[ \begin{align}\begin{aligned}Nu_H = 1.07 Ra_H^{0.28},\; 2\times 10^{6} < Ra < 2\times 10^{8}\\Nu_H = 0.152 Ra_H^{0.38},\; 2\times 10^{8} < Ra < 2\times 10^{11}\end{aligned}\end{align} \]
Parameters:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

turbulent : bool 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:
Nu : float

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, 3) 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](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, 3, 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.
[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_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].

\[ \begin{align}\begin{aligned}Nu_H = 0.48 Ra_H^{0.25},\; 10^{9} < Ra\\Nu_H = 51.5 + 0.0000726 Ra_H^{0.63},\; 10^{9} < Ra < 1.69 \times 10^{10}\\Nu_H = 0.148 Ra_H^{1/3} - 127.6 ,\; 1.69 \times 10^{10} < Ra\end{aligned}\end{align} \]
Parameters:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

turbulent : bool 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:
Nu : float

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](1, 2) 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](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, 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](1, 2, 3, 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_Eigenson_Morgan(0.7, 2E10)
230.55946525499715
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].

\[ \begin{align}\begin{aligned}Nu_H = 0.726 Ra_H^{0.25},\; 2\times 10^{8} < Ra < 4\times 10^{10}\\Nu_H = 0.0674 (Gr_H Pr^{1.29})^{1/3},\; 4\times 10^{10} < Ra < 9\times 10^{11}\end{aligned}\end{align} \]
Parameters:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

turbulent : bool 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:
Nu : float

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](1, 2) 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, 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.
[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](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_Touloukian_Morgan(.7, 2E10)
249.72879961097854
ht.conv_free_immersed.Nu_vertical_cylinder_McAdams_Weiss_Saunders(Pr, Gr, turbulent=None)[source]

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].

\[ \begin{align}\begin{aligned}Nu_H = 0.59 Ra_H^{0.25},\; 10^{4} < Ra < 10^{9}\\Nu_H = 0.13 Ra_H^{1/3.},\; 10^{9} < Ra < 10^{12}\end{aligned}\end{align} \]
Parameters:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

turbulent : bool 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
——-
Nu : float

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](1, 2) 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](1, 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](1, 2) McAdams, William Henry. Heat Transmission. 3E. Malabar, Fla: Krieger Pub Co, 1985.
[4](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.
[5](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.
[6](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_McAdams_Weiss_Saunders(.7, 2E10)
313.31849434277973
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].

\[ \begin{align}\begin{aligned}Nu_H = 0.555 Ra_H^{0.25},\; 10^{5} < Ra < 10^{9}\\Nu_H = 0.021 Ra_H^{0.4},\; 10^{9} < Ra < 10^{12}\end{aligned}\end{align} \]
Parameters:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

turbulent : bool 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:
Nu : float

Nusselt number, [-]

Notes

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

References

[1](1, 2) 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](1, 2) Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat Transfer. Cengage, 2010.
[3](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.
[4](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.
[5](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_Kreith_Eckert(.7, 2E10)
240.25393473033196
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:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

Returns:
Nu : float

Nusselt number, [-]

Notes

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

References

[1](1, 2) 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](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_Hanesian_Kalish_Morgan(.7, 1E7)
18.014150492696604
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].

\[ \begin{align}\begin{aligned}Nu_H = 2.9Ra_H^{0.25}/Gr_D^{1/12},\; 9.88 \times 10^7 \le Ra_H \le 2.7\times10^{9}\\Nu_H = 0.47 Ra_H^{0.333}/Gr_D^{1/12},\; 2.7 \times 10^9 \le Ra_H \le 2.95\times10^{10}\end{aligned}\end{align} \]
Parameters:
Pr : float

Prandtl number [-]

Gr : float

Grashof number with respect to cylinder height [-]

L : float

Length of vertical cylinder, [m]

D : float

Diameter of cylinder, [m]

turbulent : bool 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:
Nu : float

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, 3) 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](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.
[3](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_Al_Arabi_Khamis(.71, 2E10, 10, 1)
280.39793209114765
ht.conv_free_immersed.Nu_vertical_cylinder_Popiel_Churchill(Pr, Gr, L, D, Nu_vertical_plate_correlation=<function Nu_vertical_plate_Churchill>)[source]

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

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

Prandtl number [-]

Gr : float

Grashof number with respect to cylinder height [-]

L : float

Length of vertical cylinder, [m]

D : float

Diameter of cylinder, [m]

Nu_vertical_plate_correlation : function, optional

Correlation for vertical plate heat transfer

Returns:
Nu : float

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](1, 2) 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](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.
[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)
228.8979005514989
ht.conv_free_immersed.Nu_vertical_cylinder(Pr, Gr, L=None, D=None, Method=None, AvailableMethods=False)[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:
Pr : float

Prandtl number [-]

Gr : float

Grashof number with respect to cylinder height [-]

L : float, optional

Length of vertical cylinder, [m]

D : float, optional

Diameter of cylinder, [m]

Returns:
Nu : float

Nusselt number, [-]

methods : list, only returned if AvailableMethods == True

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

Other Parameters:
 
Method : string, optional

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

AvailableMethods : bool, optional

If True, function will consider which methods which can be used to calculate Nu with the given inputs

Examples

>>> Nu_vertical_cylinder(0.72, 1E7)
30.562236756513943
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:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

Returns:
Nu : float

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, 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, 3) 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, 4) 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:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

Returns:
Nu : float

Nusselt number, [-]

Notes

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

References

[1](1, 2, 3) 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, 3) 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:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

Returns:
Nu : float

Nusselt number, [-]

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](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.
[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_Morgan(0.69, 2.63E9)
151.3881997228419
ht.conv_free_immersed.Nu_horizontal_cylinder(Pr, Gr, Method=None, AvailableMethods=False)[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.

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

Parameters:
Pr : float

Prandtl number [-]

Gr : float

Grashof number [-]

Returns:
Nu : float

Nusselt number, [-]

methods : list, only returned if AvailableMethods == True

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

Other Parameters:
 
Method : string, optional

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

AvailableMethods : bool, optional

If True, function will consider which methods which can be used to calculate Nu with the given inputs

Examples

>>> Nu_horizontal_cylinder(0.72, 1E7)
24.864192615468973
ht.conv_free_immersed.Nu_vertical_helical_coil_Ali(Pr, Gr)[source]

Calculates Nusselt number for natural convection around a vertical helical coil inside a tank or other vessel according to the Ali [1] correlation.

\[Nu_L = 0.555Gr_L^{0.301} Pr^{0.314}\]
Parameters:
Pr : float

Prandtl number of the fluid surrounding the coil with properties evaluated at bulk conditions or as described in the notes [-]

Gr : float

Prandtl number of the fluid surrounding the coil with properties evaluated at bulk conditions or as described in the notes (for the two temperatures, use the average coil fluid temperature and the temperature of the fluid outside the coil) [-]

Returns:
Nu : float

Nusselt number with respect to the total length of the helical coil (and bulk thermal conductivity), [-]

Notes

In [1], the temperature at which the fluid surrounding the coil’s properties were evaluated at was calculated in an unusual fashion. The average temperature of the fluid inside the coil \((T_{in} + T_{out})/2\) is averaged with the fluid outside the coil’s temperature.

The correlation is valid for Prandtl numbers between 4.4 and 345, and tank diameter/coil outer diameter ratios between 10 and 30.

References

[1](1, 2, 3) Ali, Mohamed E. “Natural Convection Heat Transfer from Vertical Helical Coils in Oil.” Heat Transfer Engineering 27, no. 3 (April 1, 2006): 79-85.

Examples

>>> Nu_vertical_helical_coil_Ali(4.4, 1E11)
1808.5774997297106