# External convection (ht.conv_external)¶

ht.conv_external.Nu_cylinder_Churchill_Bernstein(Re, Pr)[source]

Calculates Nusselt number for crossflow across a single tube at a specified Re and Pr, both evaluated at the film temperature. No other wall correction is necessary for this formulation. Method is shown without modification in [2] and many other texts.

$Nu_D = 0.3 + \frac{0.62 Re_D^{0.5} Pr^{1/3}}{[1 + (0.4/Pr)^{2/3} ]^{0.25}}\left[1 + \left(\frac{Re_D}{282000}\right)^{5/8}\right]^{0.8}$
Parameters
Refloat

Reynolds number with respect to cylinder diameter, [-]

Prfloat

Prandtl number at film temperature, [-]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Notes

May underestimate heat transfer in some cases, as it the formula is described in [1] as “appears to provide a lower bound for RePr > 0.4”. An alternate exponent for a smaller range is also presented in [1].

This method applies to both the laminar and turbulent regimes.

References

1(1,2)

Churchill, S. W., and M. Bernstein. “A Correlating Equation for Forced Convection From Gases and Liquids to a Circular Cylinder in Crossflow.” Journal of Heat Transfer 99, no. 2 (May 1, 1977): 300-306. doi:10.1115/1.3450685.

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

Example 7.3 in [2], matches.

>>> Nu_cylinder_Churchill_Bernstein(6071, 0.7)
40.63708594124974

ht.conv_external.Nu_cylinder_Fand(Re, Pr)[source]

Calculates Nusselt number for crossflow across a single tube at a specified Re and Pr, both evaluated at the film temperature. No other wall correction is necessary for this formulation. Also shown in [2].

$Nu = (0.35 + 0.34Re^{0.5} + 0.15Re^{0.58})Pr^{0.3}$
Parameters
Refloat

Reynolds number with respect to cylinder diameter, [-]

Prfloat

Prandtl number at film temperature, [-]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Notes

Developed with test results for water, and Re from 1E4 to 1E5, but also compared with other data in the literature. Claimed validity of Re from 1E-1 to 1E5.

This method applies to both the laminar and turbulent regimes.

References

1

Fand, R. M. “Heat Transfer by Forced Convection from a Cylinder to Water in Crossflow.” International Journal of Heat and Mass Transfer 8, no. 7 (July 1, 1965): 995-1010. doi:10.1016/0017-9310(65)90084-0.

2

Sanitjai, S., and R. J. Goldstein. “Forced Convection Heat Transfer from a Circular Cylinder in Crossflow to Air and Liquids.” International Journal of Heat and Mass Transfer 47, no. 22 (October 2004): 4795-4805. doi:10.1016/j.ijheatmasstransfer.2004.05.012.

Examples

>>> Nu_cylinder_Fand(6071, 0.7)
45.19984325481126


Calculates Nusselt number for crossflow across a single tube at a specified Re and Pr, both evaluated at the film temperature. No other wall correction is necessary for this formulation. Also shown in [2].

$Nu = (0.35 + 0.56 Re^{0.52})Pr^{0.3}$
Parameters
Refloat

Reynolds number with respect to cylinder diameter, [-]

Prfloat

Prandtl number at film temperature, [-]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Notes

Developed with very limited test results for water only.

This method applies to both the laminar and turbulent regimes.

References

1

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

2

Fand, R. M. “Heat Transfer by Forced Convection from a Cylinder to Water in Crossflow.” International Journal of Heat and Mass Transfer 8, no. 7 (July 1, 1965): 995-1010. doi:10.1016/0017-9310(65)90084-0.

Examples

>>> Nu_cylinder_McAdams(6071, 0.7)
46.98179235867934

ht.conv_external.Nu_cylinder_Perkins_Leppert_1962(Re, Pr, mu=None, muw=None)[source]

Calculates Nusselt number for crossflow across a single tube as shown in [1] at a specified Re and Pr, both evaluated at the free stream temperature. Recommends a viscosity exponent correction of 0.25, which is applied only if provided. Also shown in [2].

$Nu = \left[0.30Re^{0.5} + 0.10Re^{0.67}\right]Pr^{0.4} \left(\frac{\mu}{\mu_w}\right)^{0.25}$
Parameters
Refloat

Reynolds number with respect to cylinder diameter, [-]

Prfloat

Prandtl number at free stream temperature, [-]

mufloat, optional

Viscosity of fluid at the free stream temperature [Pa*s]

muwfloat, optional

Viscosity of fluid at the wall temperature [Pa*s]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Notes

Considered results with Re from 40 to 1E5, Pr from 1 to 300; and viscosity ratios of 0.25 to 4.

This method applies to both the laminar and turbulent regimes.

References

1

Perkins, Jr., H. C., and G. Leppert. “Forced Convection Heat Transfer From a Uniformly Heated Cylinder.” Journal of Heat Transfer 84, no. 3 (August 1, 1962): 257-261. doi:10.1115/1.3684359.

2

Sanitjai, S., and R. J. Goldstein. “Forced Convection Heat Transfer from a Circular Cylinder in Crossflow to Air and Liquids.” International Journal of Heat and Mass Transfer 47, no. 22 (October 2004): 4795-4805. doi:10.1016/j.ijheatmasstransfer.2004.05.012.

Examples

>>> Nu_cylinder_Perkins_Leppert_1962(6071, 0.7)
49.97164291175499

ht.conv_external.Nu_cylinder_Perkins_Leppert_1964(Re, Pr, mu=None, muw=None)[source]

Calculates Nusselt number for crossflow across a single tube as shown in [1] at a specified Re and Pr, both evaluated at the free stream temperature. Recommends a viscosity exponent correction of 0.25, which is applied only if provided. Also shown in [2].

$Nu = \left[0.31Re^{0.5} + 0.11Re^{0.67}\right]Pr^{0.4} \left(\frac{\mu}{\mu_w}\right)^{0.25}$
Parameters
Refloat

Reynolds number with respect to cylinder diameter, [-]

Prfloat

Prandtl number at free stream temperature, [-]

mufloat, optional

Viscosity of fluid at the free stream temperature [Pa*s]

muwfloat, optional

Viscosity of fluid at the wall temperature [Pa*s]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Notes

Considers new data since Nu_cylinder_Perkins_Leppert_1962, Re from 2E3 to 1.2E5, Pr from 1 to 7, and surface to bulk temperature differences of 11 to 66.

This method applies to both the laminar and turbulent regimes.

References

1

Perkins Jr., H. C., and G. Leppert. “Local Heat-Transfer Coefficients on a Uniformly Heated Cylinder.” International Journal of Heat and Mass Transfer 7, no. 2 (February 1964): 143-158. doi:10.1016/0017-9310(64)90079-1.

2

Sanitjai, S., and R. J. Goldstein. “Forced Convection Heat Transfer from a Circular Cylinder in Crossflow to Air and Liquids.” International Journal of Heat and Mass Transfer 47, no. 22 (October 2004): 4795-4805. doi:10.1016/j.ijheatmasstransfer.2004.05.012.

Examples

>>> Nu_cylinder_Perkins_Leppert_1964(6071, 0.7)
53.61767038619986

ht.conv_external.Nu_cylinder_Sanitjai_Goldstein(Re, Pr)[source]

Calculates Nusselt number for crossflow across a single tube at a specified Re and Pr, both evaluated at the film temperature. No other wall correction is necessary for this formulation. Method is the most recent implemented here and believed to be more accurate than other formulations available.

$Nu = 0.446Re^{0.5} Pr^{0.35} + 0.528\left[(6.5\exp(Re/5000))^{-5} + (0.031Re^{0.8})^{-5}\right]^{-1/5}Pr^{0.42}$
Parameters
Refloat

Reynolds number with respect to cylinder diameter, [-]

Prfloat

Prandtl number at film temperature, [-]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Notes

Developed with test results for water, mixtures of ethylene glycol and water, and air (Pr = 0.7 to 176). Re range from 2E3 to 9E4. Also presents results for local heat transfer coefficients.

This method applies to both the laminar and turbulent regimes.

References

1

Sanitjai, S., and R. J. Goldstein. “Forced Convection Heat Transfer from a Circular Cylinder in Crossflow to Air and Liquids.” International Journal of Heat and Mass Transfer 47, no. 22 (October 2004): 4795-4805. doi:10.1016/j.ijheatmasstransfer.2004.05.012.

Examples

>>> Nu_cylinder_Sanitjai_Goldstein(6071, 0.7)
40.38327083519522

ht.conv_external.Nu_cylinder_Whitaker(Re, Pr, mu=None, muw=None)[source]

Calculates Nusselt number for crossflow across a single tube as shown in [1] at a specified Re and Pr, both evaluated at the free stream temperature. Recommends a viscosity exponent correction of 0.25, which is applied only if provided. Also shown in [2].

$Nu_D = (0.4 Re_D^{0.5} + 0.06Re_D^{2/3})Pr^{0.4} \left(\frac{\mu}{\mu_w}\right)^{0.25}$
Parameters
Refloat

Reynolds number with respect to cylinder diameter, [-]

Prfloat

Prandtl number at free stream temperature, [-]

mufloat, optional

Viscosity of fluid at the free stream temperature [Pa*s]

muwfloat, optional

Viscosity of fluid at the wall temperature [Pa*s]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Notes

Developed considering data from 1 to 1E5 Re, 0.67 to 300 Pr, and range of viscosity ratios from 0.25 to 5.2. Found experimental data to generally agree with it within 25%.

This method applies to both the laminar and turbulent regimes.

References

1

Whitaker, Stephen. “Forced Convection Heat Transfer Correlations for Flow in Pipes, Past Flat Plates, Single Cylinders, Single Spheres, and for Flow in Packed Beds and Tube Bundles.” AIChE Journal 18, no. 2 (March 1, 1972): 361-371. doi:10.1002/aic.690180219.

2

Sanitjai, S., and R. J. Goldstein. “Forced Convection Heat Transfer from a Circular Cylinder in Crossflow to Air and Liquids.” International Journal of Heat and Mass Transfer 47, no. 22 (October 2004): 4795-4805. doi:10.1016/j.ijheatmasstransfer.2004.05.012.

Examples

>>> Nu_cylinder_Whitaker(6071, 0.7)
45.94527461589126

ht.conv_external.Nu_cylinder_Zukauskas(Re, Pr, Prw=None)[source]

Calculates Nusselt number for crossflow across a single tube at a specified Re. Method from [1], also shown without modification in [2]. This method applies to both the laminar and turbulent regimes.

$Nu_{D}=CRe^{m}Pr^{n}\left(\frac{Pr}{Pr_s}\right)^{1/4}$
Parameters
Refloat

Reynolds number with respect to cylinder diameter, [-]

Prfloat

Prandtl number at free stream temperature [-]

Prwfloat, optional

Prandtl number at wall temperature, [-]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Notes

If Prandtl number at wall are not provided, the Prandtl number correction is not used and left to an outside function.

n is 0.37 if Pr <= 10; otherwise n is 0.36.

C and m are from the following table. If Re is outside of the ranges shown, the nearest range is used blindly.

Re

C

m

1-40

0.75

0.4

40-1E3

0.51

0.5

1E3-2E5

0.26

0.6

2E5-1E6

0.076

0.7

References

1

Zukauskas, A. Heat transfer from tubes in crossflow. In T.F. Irvine, Jr. and J. P. Hartnett, editors, Advances in Heat Transfer, volume 8, pages 93-160. Academic Press, Inc., New York, 1972.

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

Example 7.3 in [2], matches.

>>> Nu_cylinder_Zukauskas(7992, 0.707, 0.69)
50.523612661934386

ht.conv_external.Nu_external_cylinder(Re, Pr, Prw=None, mu=None, muw=None, Method=None)[source]

Calculates Nusselt number for crossflow across a single tube at a specified Re and Pr according to the specified method. Optional parameters are Prw, mu, and muw. This function has eight methods available. The ‘Sanitjai-Goldstein’ method is the default.

The front of the cyliner is normally always in a laminar regime; whereas the back is turbulent. The proportions change with Re; all correlations take this into account. For this heat transfer case, there is no separation between laminar and turbulent methods.

Parameters
Refloat

Reynolds number of fluid with respect to cylinder diameter, [-]

Prfloat

Prandtl number at either the free stream or wall temperature depending on the method, [-]

Prwfloat, optional

Prandtl number at wall temperature, [-]

mufloat, optional

Viscosity of fluid at the free stream temperature [Pa*s]

muwfloat, optional

Viscosity of fluid at the wall temperature [Pa*s]

Returns
Nufloat

Nusselt number with respect to cylinder diameter, [-]

Other Parameters
Methodstr, optional

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

Notes

A comparison of the methods for various Prandtl and Reynolds number ranges is plotted below.

Examples

>>> Nu_external_cylinder(6071, 0.7)
40.38327083519522

ht.conv_external.Nu_external_cylinder_methods(Re, Pr, Prw=None, mu=None, muw=None, check_ranges=True)[source]

This function returns a list of correlation names for forced convection over an external cylinder.

The preferred method ‘Sanitjai-Goldstein’.

Parameters
Refloat

Reynolds number of fluid with respect to cylinder diameter, [-]

Prfloat

Prandtl number at either the free stream or wall temperature depending on the method, [-]

Prwfloat, optional

Prandtl number at wall temperature, [-]

mufloat, optional

Viscosity of fluid at the free stream temperature [Pa*s]

muwfloat, optional

Viscosity of fluid at the wall temperature [Pa*s]

check_rangesbool, optional

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

Returns
methods

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

Examples

>>> Nu_external_cylinder_methods(0.72, 1E7)[0]
'Sanitjai-Goldstein'

ht.conv_external.Nu_external_horizontal_plate(Re, Pr, L=None, x=None, Method=None, laminar_method='Baehr', turbulent_method='Schlichting', Re_transition=500000.0)[source]

This function calculates the heat transfer coefficient for external forced convection along a horizontal plate.

Requires at a minimum a flow’s Reynolds and Prandtl numbers Re and Pr. L and x are not used by any correlations presently, but are included for future support.

If no correlation’s name is provided as Method, the most accurate applicable correlation is selected.

Parameters
Refloat

Reynolds number with respect to bulk properties and plate length, [-]

Prfloat

Prandtl number with respect to bulk properties, [-]

Lfloat, optional

Length of horizontal plate, [m]

xfloat, optional

Length of horizontal plate for specific calculation distance, [m]

Returns
Nufloat

Nusselt number with respect to plate length, [-]

Other Parameters
Methodstr, optional

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

laminar_methodstr, optional

The prefered method for laminar flow, [-]

turbulent_methodstr, optional

The prefered method for turbulent flow, [-]

Re_transitionfloat, optional

The transition Reynolds number for laminar changing to turbulent flow, [-]

Examples

Turbulent example

>>> Nu_external_horizontal_plate(Re=1E7, Pr=.7)
11496.952599969829

ht.conv_external.Nu_external_horizontal_plate_methods(Re, Pr, L=None, x=None, check_ranges=True)[source]

Returns a list of correlation names for calculating Nusselt number for forced convection across a horizontal plate, supporting both laminar and turbulent regimes.

Parameters
Refloat

Reynolds number with respect to bulk properties and plate length, [-]

Prfloat

Prandtl number with respect to bulk properties, [-]

Lfloat, optional

Length of horizontal plate, [m]

xfloat, optional

Length of horizontal plate for specific calculation distance, [m]

check_rangesbool, optional

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

Returns
methods

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

Examples

>>> Nu_external_horizontal_plate_methods(Re=1e7, Pr=.7)[0]
'Schlichting'

ht.conv_external.Nu_horizontal_plate_laminar_Baehr(Re, Pr)[source]

Calculates Nusselt number for laminar flow across an isothermal flat plate at a specified Re and Pr, both evaluated at the bulk temperature. No other wall correction is necessary for this formulation. Four different equations are used for different Prandtl number ranges.

The equation for the common Prandtl number range is also recommended in [2] and [3].

if $\text{Pr} < 0.005$:

$\text{Nu}_L = 1.128\text{Re}^{0.5}\text{Pr}^{0.5}$

if $0.005 < \text{Pr} < 0.05$:

$\text{Nu}_L = 1.0\text{Re}^{0.5}\text{Pr}^{0.5}$

if $0.6 < \text{Pr} < 10$:

$\text{Nu}_L = 0.664\text{Re}^{0.5}\text{Pr}^{1/3}$

if $\text{Pr} > 10$:

$\text{Nu}_L = 0.678\text{Re}^{0.5}\text{Pr}^{1/3}$
Parameters
Refloat

Reynolds number with respect to plate length and bulk fluid properties, [-]

Prfloat

Prandtl number at bulk temperature, [-]

Returns
Nufloat

Nusselt number with respect to plate length and bulk temperature, [-]

Notes

Does not take into account the impact of free convection, which can increase the convection substantially.

References

1

Baehr, Hans Dieter, and Karl Stephan. Heat and Mass Transfer. Springer, 2013.

2

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

3

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

Examples

>>> Nu_horizontal_plate_laminar_Baehr(1e5, 0.7)
186.4378528752262

ht.conv_external.Nu_horizontal_plate_laminar_Churchill_Ozoe(Re, Pr)[source]

Calculates Nusselt number for laminar flow across an isothermal flat plate at a specified Re and Pr, both evaluated at the bulk temperature. No other wall correction is necessary for this formulation. A single equation covers all Prandtl number ranges.

$Nu_L = \frac{0.6774Re_L^{1/2}Pr^{1/3}}{[1+(0.0468/Pr)^{2/3}]^{1/4}}$
Parameters
Refloat

Reynolds number with respect to plate length and bulk fluid properties, [-]

Prfloat

Prandtl number at bulk temperature, [-]

Returns
Nufloat

Nusselt number with respect to plate length and bulk temperature, [-]

Notes

Does not take into account the impact of free convection, which can increase the convection substantially.

References

1

Churchill, Stuart W., and Hiroyuki Ozoe. “Correlations for Laminar Forced Convection in Flow Over an Isothermal Flat Plate and in Developing and Fully Developed Flow in an Isothermal Tube.” Journal of Heat Transfer 95, no. 3 (August 1, 1973): 416 https://doi.org/10.1115/1.3450078.

2

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

Examples

>>> Nu_horizontal_plate_laminar_Churchill_Ozoe(1e5, 0.7)
183.08600782591418

ht.conv_external.Nu_horizontal_plate_turbulent_Kreith(Re, Pr)[source]

Calculates Nusselt number for turbulent flow across an isothermal flat plate at a specified Re and Pr, both evaluated at the bulk temperature. The formulation of Kreith is used.

$\text{Nu}_L = 0.036\text{Re}_L^{0.8} \text{Pr}^{1/3}$
Parameters
Refloat

Reynolds number with respect to plate length and bulk fluid properties, [-]

Prfloat

Prandtl number at bulk temperature, [-]

Returns
Nufloat

Nusselt number with respect to plate length and bulk temperature, [-]

Notes

Does not take into account the impact of free convection, which can increase the convection substantially. Applies for turbulent flow only.

References

1

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

Examples

>>> Nu_horizontal_plate_turbulent_Kreith(1.03e6, 0.71)
2074.8740070411122

ht.conv_external.Nu_horizontal_plate_turbulent_Schlichting(Re, Pr)[source]

Calculates Nusselt number for turbulent flow across an isothermal flat plate at a specified Re and Pr, both evaluated at the bulk temperature. The formulation of Schlichting is used, which adds a surface friction term to a formulation from Petukhov and Popov.

$\text{Nu}_L = \frac{0.037\text{Re}_L^{0.8} \text{Pr}} {1 + 2.443\text{Re}_L^{-0.1}(\text{Pr}^{2/3} - 1)}$
Parameters
Refloat

Reynolds number with respect to plate length and bulk fluid properties, [-]

Prfloat

Prandtl number at bulk temperature, [-]

Returns
Nufloat

Nusselt number with respect to plate length and bulk temperature, [-]

Notes

Does not take into account the impact of free convection, which can increase the convection substantially.

References

1

Schlichting, H., and Klaus Gersten. Grenzschicht-Theorie. 9th ed. Berlin Heidelberg: Springer-Verlag, 1997. http://www.springer.com/de/book/9783662075548.

2

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

Examples

>>> Nu_horizontal_plate_turbulent_Schlichting(1e5, 0.7)
309.620048541267