# External convection (ht.conv_external)¶

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].

$Nu_{D}=CRe^{m}Pr^{n}\left(\frac{Pr}{Pr_s}\right)^{1/4}$
Parameters: Re : float Reynolds number with respect to cylinder diameter, [-] Pr : float Prandtl number at free stream temperature [-] Prw : float, optional Prandtl number at wall temperature, [-] Nu : float 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] (1, 2) 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, 3) 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_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: Re : float Reynolds number with respect to cylinder diameter, [-] Pr : float Prandtl number at film temperature, [-] Nu : float 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].

References

 [1] (1, 2, 3) 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, 3) 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_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: Re : float Reynolds number with respect to cylinder diameter, [-] Pr : float Prandtl number at film temperature, [-] Nu : float 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.

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_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: Re : float Reynolds number with respect to cylinder diameter, [-] Pr : float Prandtl number at film temperature, [-] Nu : float 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.

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] (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_Fand(6071, 0.7)
45.19984325481126

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: Re : float Reynolds number with respect to cylinder diameter, [-] Pr : float Prandtl number at free stream temperature, [-] mu : float, optional Viscosity of fluid at the free stream temperature [Pa*s] muw : float, optional Viscosity of fluid at the wall temperature [Pa*s] Nu : float 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.

References

 [1] (1, 2) 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] (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_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: Re : float Reynolds number with respect to cylinder diameter, [-] Pr : float Prandtl number at free stream temperature, [-] mu : float, optional Viscosity of fluid at the free stream temperature [Pa*s] muw : float, optional Viscosity of fluid at the wall temperature [Pa*s] Nu : float 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.

References

 [1] (1, 2) 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] (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_1962(6071, 0.7)
49.97164291175499

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: Re : float Reynolds number with respect to cylinder diameter, [-] Pr : float Prandtl number at free stream temperature, [-] mu : float, optional Viscosity of fluid at the free stream temperature [Pa*s] muw : float, optional Viscosity of fluid at the wall temperature [Pa*s] Nu : float 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%.

References

 [1] (1, 2) 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] (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_Whitaker(6071, 0.7)
45.94527461589126

ht.conv_external.Nu_cylinder_McAdams(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].

$Nu = (0.35 + 0.56 Re^{0.52})Pr^{0.3}$
Parameters: Re : float Reynolds number with respect to cylinder diameter, [-] Pr : float Prandtl number at film temperature, [-] Nu : float Nusselt number with respect to cylinder diameter, [-]

Notes

Developed with very limited test results for water only.

References

 [1] McAdams, William Henry. Heat Transmission. 3E. Malabar, Fla: Krieger Pub Co, 1985.
 [2] (1, 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