Internal convection (ht.conv_internal)

ht.conv_internal.laminar_T_const()[source]

Returns internal convection Nusselt number for laminar flows in pipe according to [R357431], [R358431] and [R359431]. Wall temperature is assumed constant. This is entirely theoretically derived and reproduced experimentally.

\[Nu = 3.66\]
Returns:

Nu : float

Nusselt number, [-]

Notes

This applies only for fully thermally and hydraulically developed and flows.

References

[R357431](1, 2) Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. New York: McGraw-Hill Education, 2007.
[R358431](1, 2) Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.
[R359431](1, 2) Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd ed. 2010 edition. Berlin ; New York: Springer, 2010.
ht.conv_internal.laminar_Q_const()[source]

Returns internal convection Nusselt number for laminar flows in pipe according to [R360434], [R361434], and [R362434]. Heat flux is assumed constant. This is entirely theoretically derived and reproduced experimentally.

\[Nu = 4.354\]
Returns:

Nu : float

Nusselt number, [-]

Notes

This applies only for fully thermally and hydraulically developed and flows. Many sources round to 4.36, but [R362434] does not.

References

[R360434](1, 2) Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. New York: McGraw-Hill Education, 2007.
[R361434](1, 2) Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.
[R362434](1, 2, 3) Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd ed. 2010 edition. Berlin ; New York: Springer, 2010.
ht.conv_internal.laminar_entry_thermal_Hausen(Re=None, Pr=None, L=None, Di=None)[source]

Calculates average internal convection Nusselt number for laminar flows in pipe during the thermal entry region according to [R363437] as shown in [R364437] and cited by [R365437].

\[Nu_D=3.66+\frac{0.0668\frac{D}{L}Re_{D}Pr}{1+0.04{(\frac{D}{L} Re_{D}Pr)}^{2/3}}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

L : float

Length of pipe [m]

Di : float

Diameter of pipe [m]

Returns:

Nu : float

Nusselt number, [-]

Notes

If Pr >> 1, (5 is a common requirement) this equation also applies to flows with developing velocity profile. As L gets larger, this equation becomes the constant-temperature Nusselt number.

References

[R363437](1, 2) Hausen, H. Darstellung des Warmeuberganges in Rohren durch verallgeminerte Potenzbeziehungen, Z. Ver deutsch. Ing Beih. Verfahrenstech., 4, 91-98, 1943
[R364437](1, 2) W. M. Kays. 1953. Numerical Solutions for Laminar Flow Heat Transfer in Circular Tubes.
[R365437](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

>>> laminar_entry_thermal_Hausen(Re=100000, Pr=1.1, L=5, Di=.5)
39.01352358988535
ht.conv_internal.laminar_entry_Seider_Tate(Re, Pr, L, Di, mu=None, mu_w=None)[source]

Calculates average internal convection Nusselt number for laminar flows in pipe during the thermal entry region as developed in [R366440], also shown in [R367440].

\[Nu_D=1.86\left(\frac{D}{L}Re_DPr\right)^{1/3}\left(\frac{\mu_b} {\mu_s}\right)^{0.14}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

L : float

Length of pipe [m]

Di : float

Diameter of pipe [m]

mu : float

Viscosity of fluid, [Pa*s]

mu_w : float

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

Returns:

Nu : float

Nusselt number, [-]

Notes

Reynolds number should be less than 10000. This should be calculated using pipe diameter. Prandlt number should be no less than air and no more than liquid metals; 0.7 < Pr < 16700 Viscosities should be the bulk and surface properties; they are optional. Outside the boundaries, this equation is provides very false results.

References

[R366440](1, 2) Sieder, E. N., and G. E. Tate. “Heat Transfer and Pressure Drop of Liquids in Tubes.” Industrial & Engineering Chemistry 28, no. 12 (December 1, 1936): 1429-35. doi:10.1021/ie50324a027.
[R367440](1, 2) Serth, R. W., Process Heat Transfer: Principles, Applications and Rules of Thumb. 2E. Amsterdam: Academic Press, 2014.

Examples

>>> laminar_entry_Seider_Tate(Re=100000, Pr=1.1, L=5, Di=.5)
41.366029684589265
ht.conv_internal.laminar_entry_Baehr_Stephan(Re=None, Pr=None, L=None, Di=None)[source]

Calculates average internal convection Nusselt number for laminar flows in pipe during the thermal and velocity entry region according to [R368442] as shown in [R369442].

\[ \begin{align}\begin{aligned}Nu_D=\frac{\frac{3.657}{\tanh[2.264 Gz_D^{-1/3}+1.7Gz_D^{-2/3}]} +0.0499Gz_D\tanh(Gz_D^{-1})}{\tanh(2.432Pr^{1/6}Gz_D^{-1/6})}\\Gz = \frac{D}{L}Re_D Pr\end{aligned}\end{align} \]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

L : float

Length of pipe [m]

Di : float

Diameter of pipe [m]

Returns:

Nu : float

Nusselt number, [-]

Notes

As L gets larger, this equation becomes the constant-temperature Nusselt number.

References

[R368442](1, 2) Baehr, Hans Dieter, and Karl Stephan. Heat and Mass Transfer. Springer, 2013.
[R369442](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

>>> laminar_entry_Baehr_Stephan(Re=100000, Pr=1.1, L=5, Di=.5)
72.65402046550976
ht.conv_internal.turbulent_Dittus_Boelter(Re, Pr, heating=True, revised=True)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R370444], and [R371444], a reprint of [R372444].

\[Nu = m*Re_D^{4/5}Pr^n\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

heating : bool

Indicates if the process is heating or cooling, optional

revised : bool

Indicates if revised coefficients should be used or not

Returns:

Nu : float

Nusselt number, [-]

Notes

The revised coefficient is m = 0.023. The original form of Dittus-Boelter has a linear coefficient of 0.0243 for heating and 0.0265 for cooling. These are sometimes rounded to 0.024 and 0.026 respectively. The default, heating, provides n = 0.4. Cooling makes n 0.3.

0.6 ≤ Pr ≤ 160 Re_{D} ≥ 10000 L/D ≥ 10

References

[R370444](1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R371444](1, 2) Dittus, F. W., and L. M. K. Boelter. “Heat Transfer in Automobile Radiators of the Tubular Type.” International Communications in Heat and Mass Transfer 12, no. 1 (January 1985): 3-22. doi:10.1016/0735-1933(85)90003-X
[R372444](1, 2) Dittus, F. W., and L. M. K. Boelter, University of California Publications in Engineering, Vol. 2, No. 13, pp. 443-461, October 17, 1930.

Examples

>>> turbulent_Dittus_Boelter(Re=1E5, Pr=1.2)
247.40036409449127
>>> turbulent_Dittus_Boelter(Re=1E5, Pr=1.2, heating=False)
242.9305927410295
ht.conv_internal.turbulent_Sieder_Tate(Re, Pr, mu=None, mu_w=None)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R373447] and supposedly [R374447].

\[Nu = 0.027Re^{4/5}Pr^{1/3}\left(\frac{\mu}{\mu_s}\right)^{0.14}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

mu : float

Viscosity of fluid, [Pa*s]

mu_w : float

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

Returns:

Nu : float

Nusselt number, [-]

Notes

A linear coefficient of 0.023 is often listed with this equation. The source of the discrepancy is not known. The equation is not present in the original paper, but is nevertheless the source usually cited for it.

References

[R373447](1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R374447](1, 2) Sieder, E. N., and G. E. Tate. “Heat Transfer and Pressure Drop of Liquids in Tubes.” Industrial & Engineering Chemistry 28, no. 12 (December 1, 1936): 1429-35. doi:10.1021/ie50324a027.

Examples

>>> turbulent_Sieder_Tate(Re=1E5, Pr=1.2)
286.9178136793052
>>> turbulent_Sieder_Tate(Re=1E5, Pr=1.2, mu=0.01, mu_w=0.067)
219.84016455766044
ht.conv_internal.turbulent_entry_Hausen(Re, Pr, Di, x)[source]

Calculates internal convection Nusselt number for the entry region of a turbulent flow in pipe according to [R376449] as in [R375449].

\[Nu = 0.037(Re^{0.75} - 180)Pr^{0.42}[1+(x/D)^{-2/3}]\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

Di : float

Inside diameter of pipe, [m]

x : float

Length inside of pipe for calculation, [m]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R375449] is 0.7 < Pr ≤ 3 and 10^4 ≤ Re ≤ 5*10^6.

References

[R375449](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R376449](1, 2) H. Hausen, “Neue Gleichungen fÜr die Wärmeübertragung bei freier oder erzwungener Stromung,”Allg. Warmetchn., (9): 75-79, 1959.

Examples

>>> turbulent_entry_Hausen(Re=1E5, Pr=1.2, Di=0.154, x=0.05)
677.7228275901755
ht.conv_internal.turbulent_Colburn(Re, Pr)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R378451] as in [R377451].

\[Nu = 0.023Re^{0.8}Pr^{1/3}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R377451] is 0.5 < Pr < 3 and 10^4 < Re < 10^5.

References

[R377451](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R378451](1, 2) Colburn, Allan P. “A Method of Correlating Forced Convection Heat-Transfer Data and a Comparison with Fluid Friction.” International Journal of Heat and Mass Transfer 7, no. 12 (December 1964): 1359-84. doi:10.1016/0017-9310(64)90125-5.

Examples

>>> turbulent_Colburn(Re=1E5, Pr=1.2)
244.41147091200068
ht.conv_internal.turbulent_Drexel_McAdams(Re, Pr)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R380453] as in [R379453].

\[Nu = 0.021Re^{0.8}Pr^{0.4}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R379453] is Pr ≤ 0.7 and 10^4 ≤ Re ≤ 5*10^6.

References

[R379453](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R380453](1, 2) Drexel, Rober E., and William H. Mcadams. “Heat-Transfer Coefficients for Air Flowing in Round Tubes, in Rectangular Ducts, and around Finned Cylinders,” February 1, 1945. http://ntrs.nasa.gov/search.jsp?R=19930090924.

Examples

>>> turbulent_Drexel_McAdams(Re=1E5, Pr=0.6)
171.19055301724387
ht.conv_internal.turbulent_von_Karman(Re, Pr, fd)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R382455] as in [R381455].

\[Nu = \frac{(f/8)Re Pr}{1 + 5(f/8)^{0.5}\left[Pr-1+\ln\left(\frac{5Pr+1} {6}\right)\right]}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R381455] is 0.5 ≤ Pr ≤ 3 and 10^4 ≤ Re ≤ 10^5.

References

[R381455](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R382455](1, 2) T. von Karman, “The Analogy Between Fluid Friction and Heat Transfer,” Trans. ASME, (61):705-710,1939.

Examples

>>> turbulent_von_Karman(Re=1E5, Pr=1.2, fd=0.0185)
255.7243541243272
ht.conv_internal.turbulent_Prandtl(Re, Pr, fd)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R384457] as in [R383457].

\[Nu = \frac{(f/8)RePr}{1 + 8.7(f/8)^{0.5}(Pr-1)}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R383457] 0.5 ≤ Pr ≤ 5 and 10^4 ≤ Re ≤ 5*10^6

References

[R383457](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R384457](1, 2) L. Prandt, Fuhrrer durch die Stomungslehre, Vieweg, Braunschweig, p. 359, 1944.

Examples

>>> turbulent_Prandtl(Re=1E5, Pr=1.2, fd=0.0185)
256.073339689557
ht.conv_internal.turbulent_Friend_Metzner(Re, Pr, fd)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R386459] as in [R385459].

\[Nu = \frac{(f/8)RePr}{1.2 + 11.8(f/8)^{0.5}(Pr-1)Pr^{-1/3}}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R385459] 50 < Pr ≤ 600 and 5*10^4 ≤ Re ≤ 5*10^6. The extreme limits on range should be considered!

References

[R385459](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R386459](1, 2) Friend, W. L., and A. B. Metzner. “Turbulent Heat Transfer inside Tubes and the Analogy among Heat, Mass, and Momentum Transfer.” AIChE Journal 4, no. 4 (December 1, 1958): 393-402. doi:10.1002/aic.690040404.

Examples

>>> turbulent_Friend_Metzner(Re=1E5, Pr=100., fd=0.0185)
1738.3356262055322
ht.conv_internal.turbulent_Petukhov_Kirillov_Popov(Re=None, Pr=None, fd=None)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R388461] and [R389461] as in [R387461].

\[\begin{split}Nu = \frac{(f/8)RePr}{C+12.7(f/8)^{1/2}(Pr^{2/3}-1)}\\ C = 1.07 + 900/Re - [0.63/(1+10Pr)]\end{split}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R387461] is 0.5 < Pr ≤ 10^6 and 4000 ≤ Re ≤ 5*10^6

References

[R387461](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R388461](1, 2) B. S. Petukhov, and V. V. Kirillov, “The Problem of Heat Exchange in the Turbulent Flow of Liquids in Tubes,” (Russian) Teploenergetika, (4): 63-68, 1958
[R389461](1, 2) B. S. Petukhov and V. N. Popov, “Theoretical Calculation of Heat Exchange in Turbulent Flow in Tubes of an Incompressible Fluidwith Variable Physical Properties,” High Temp., (111): 69-83, 1963.

Examples

>>> turbulent_Petukhov_Kirillov_Popov(Re=1E5, Pr=1.2, fd=0.0185)
250.11935088905105
ht.conv_internal.turbulent_Webb(Re, Pr, fd)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R391464] as in [R390464].

\[Nu = \frac{(f/8)RePr}{1.07 + 9(f/8)^{0.5}(Pr-1)Pr^{1/4}}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R390464] is 0.5 < Pr ≤ 100 and 10^4 ≤ Re ≤ 5*10^6

References

[R390464](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R391464](1, 2) Webb, Dr R. L. “A Critical Evaluation of Analytical Solutions and Reynolds Analogy Equations for Turbulent Heat and Mass Transfer in Smooth Tubes.” Wärme - Und Stoffübertragung 4, no. 4 (December 1, 1971): 197–204. doi:10.1007/BF01002474.

Examples

>>> turbulent_Webb(Re=1E5, Pr=1.2, fd=0.0185)
239.10130376815872
ht.conv_internal.turbulent_Sandall(Re, Pr, fd)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R393466] as in [R392466].

\[\begin{split}Nu = \frac{(f/8)RePr}{12.48Pr^{2/3} - 7.853Pr^{1/3} + 3.613\ln Pr + 5.8 + C}\\ C = 2.78\ln((f/8)^{0.5} Re/45)\end{split}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R392466] is 0.5< Pr ≤ 2000 and 10^4 ≤ Re ≤ 5*10^6.

References

[R392466](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R393466](1, 2) Sandall, O. C., O. T. Hanna, and P. R. Mazet. “A New Theoretical Formula for Turbulent Heat and Mass Transfer with Gases or Liquids in Tube Flow.” The Canadian Journal of Chemical Engineering 58, no. 4 (August 1, 1980): 443–47. doi:10.1002/cjce.5450580404.

Examples

>>> turbulent_Sandall(Re=1E5, Pr=1.2, fd=0.0185)
229.0514352970239
ht.conv_internal.turbulent_Gnielinski(Re, Pr, fd)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R395468] as in [R394468]. This is the most recent general equation, and is strongly recommended.

\[Nu = \frac{(f/8)(Re-1000)Pr}{1+12.7(f/8)^{1/2}(Pr^{2/3}-1)}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R394468] is 0.5 < Pr ≤ 2000 and 2300 ≤ Re ≤ 5*10^6.

References

[R394468](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R395468](1, 2) Gnielinski, V. (1976). New Equation for Heat and Mass Transfer in Turbulent Pipe and Channel Flow, International Chemical Engineering, Vol. 16, pp. 359–368.

Examples

>>> turbulent_Gnielinski(Re=1E5, Pr=1.2, fd=0.0185)
254.62682749359632
ht.conv_internal.turbulent_Gnielinski_smooth_1(Re, Pr)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R397470] as in [R396470]. This is a simplified case assuming smooth pipe.

\[Nu = 0.0214(Re^{0.8}-100)Pr^{0.4}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R396470] is 0.5 < Pr ≤ 1.5 and 10^4 ≤ Re ≤ 5*10^6.

References

[R396470](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R397470](1, 2) Gnielinski, V. (1976). New Equation for Heat and Mass Transfer in Turbulent Pipe and Channel Flow, International Chemical Engineering, Vol. 16, pp. 359–368.

Examples

>>> turbulent_Gnielinski_smooth_1(Re=1E5, Pr=1.2)
227.88800494373442
ht.conv_internal.turbulent_Gnielinski_smooth_2(Re, Pr)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R399472] as in [R398472]. This is a simplified case assuming smooth pipe.

\[Nu = 0.012(Re^{0.87}-280)Pr^{0.4}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Range according to [R398472] is 1.5 < Pr ≤ 500 and 3*10^3 ≤ Re ≤ 10^6.

References

[R398472](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R399472](1, 2) Gnielinski, V. (1976). New Equation for Heat and Mass Transfer in Turbulent Pipe and Channel Flow, International Chemical Engineering, Vol. 16, pp. 359–368.

Examples

>>> turbulent_Gnielinski_smooth_2(Re=1E5, Pr=7.)
577.7692524513449
ht.conv_internal.turbulent_Churchill_Zajic(Re, Pr, fd)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R401474] as developed in [R400474]. Has yet to obtain popularity.

\[ \begin{align}\begin{aligned}Nu = \left\{\left(\frac{Pr_T}{Pr}\right)\frac{1}{Nu_{di}} + \left[1-\left(\frac{Pr_T}{Pr}\right)^{2/3}\right]\frac{1}{Nu_{D\infty}} \right\}^{-1}\\Nu_{di} = \frac{Re(f/8)}{1 + 145(8/f)^{-5/4}}\\Nu_{D\infty} = 0.07343Re\left(\frac{Pr}{Pr_T}\right)^{1/3}(f/8)^{0.5}\\Pr_T = 0.85 + \frac{0.015}{Pr}\end{aligned}\end{align} \]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

No restrictions on range. This is equation is developed with more theoretical work than others.

References

[R400474](1, 2) Churchill, Stuart W., and Stefan C. Zajic. “Prediction of Fully Developed Turbulent Convection with Minimal Explicit Empiricism.” AIChE Journal 48, no. 5 (May 1, 2002): 927–40. doi:10.1002/aic.690480503.
[R401474](1, 2) Plawsky, Joel L. Transport Phenomena Fundamentals, Third Edition. CRC Press, 2014.

Examples

>>> turbulent_Churchill_Zajic(Re=1E5, Pr=1.2, fd=0.0185)
260.5564907817961
ht.conv_internal.turbulent_ESDU(Re, Pr)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to the ESDU as shown in [R402476].

\[Nu = 0.0225Re^{0.795}Pr^{0.495}\exp(-0.0225\ln(Pr)^2)\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

4000 < Re < 1E6, 0.3 < Pr < 3000 and L/D > 60. This equation has not been checked. It was developed by a commercial group. This function is a small part of a much larger series of expressions accounting for many factors.

References

[R402476](1, 2) Hewitt, G. L. Shires, T. Reg Bott G. F., George L. Shires, and T. R. Bott. Process Heat Transfer. 1E. Boca Raton: CRC Press, 1994.

Examples

>>> turbulent_ESDU(Re=1E5, Pr=1.2)
232.3017143430645
ht.conv_internal.turbulent_Martinelli(Re, Pr, fd)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R404477] as shown in [R403477].

\[Nu = \frac{RePr(f/8)^{0.5}}{5[Pr + \ln(1+5Pr) + 0.5\ln(Re(f/8)^{0.5}/60)]}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

No range is given for this equation. Liquid metals are probably its only applicability.

References

[R403477](1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R404477](1, 2) Martinelli, R. C. (1947). “Heat transfer to molten metals”. Trans. ASME, 69, 947-959.

Examples

>>> turbulent_Martinelli(Re=1E5, Pr=100., fd=0.0185)
887.1710686396347
ht.conv_internal.turbulent_Nunner(Re, Pr, fd, fd_smooth)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R406479] as shown in [R405479].

\[Nu = \frac{RePr(f/8)}{1 + 1.5Re^{-1/8}Pr^{-1/6}[Pr(f/f_s)-1]}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

fd_smooth : float

Darcy friction factor of a smooth pipe [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

Valid for Pr ≅ 0.7; bad results for Pr > 1.

References

[R405479](1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R406479](1, 2) W. Nunner, “Warmeiibergang und Druckabfall in Rauhen Rohren,” VDI-Forschungsheft 445, ser. B,(22): 5-39, 1956

Examples

>>> turbulent_Nunner(Re=1E5, Pr=0.7, fd=0.0185, fd_smooth=0.005)
101.15841010919947
ht.conv_internal.turbulent_Dipprey_Sabersky(Re, Pr, fd, eD)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R408481] as shown in [R407481].

\[Nu = \frac{RePr(f/8)}{1 + (f/8)^{0.5}[5.19Re_\epsilon^{0.2} Pr^{0.44} - 8.48]}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

eD : float

Relative roughness, [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

According to [R407481], the limits are: 1.2 ≤ Pr ≤ 5.94 and 1.4*10^4 ≤ Re ≤ 5E5 and 0.0024 ≤ eD ≤ 0.049.

References

[R407481](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R408481](1, 2) Dipprey, D. F., and R. H. Sabersky. “Heat and Momentum Transfer in Smooth and Rough Tubes at Various Prandtl Numbers.” International Journal of Heat and Mass Transfer 6, no. 5 (May 1963): 329–53. doi:10.1016/0017-9310(63)90097-8

Examples

>>> turbulent_Dipprey_Sabersky(Re=1E5, Pr=1.2, fd=0.0185, eD=1E-3)
288.33365198566656
ht.conv_internal.turbulent_Gowen_Smith(Re, Pr, fd)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R410483] as shown in [R409483].

\[Nu = \frac{Re Pr (f/8)^{0.5}} {4.5 + [0.155(Re(f/8)^{0.5})^{0.54} + (8/f)^{0.5}]Pr^{0.5}}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

0.7 ≤ Pr ≤ 14.3 and 10^4 ≤ Re ≤ 5E4 and 0.0021 ≤ eD ≤ 0.095

References

[R409483](1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R410483](1, 2) Gowen, R. A., and J. W. Smith. “Turbulent Heat Transfer from Smooth and Rough Surfaces.” International Journal of Heat and Mass Transfer 11, no. 11 (November 1968): 1657–74. doi:10.1016/0017-9310(68)90046-X.

Examples

>>> turbulent_Gowen_Smith(Re=1E5, Pr=1.2, fd=0.0185)
131.72530453824106
ht.conv_internal.turbulent_Kawase_Ulbrecht(Re, Pr, fd)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R412485] as shown in [R411485].

\[Nu = 0.0523RePr^{0.5}(f/4)^{0.5}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

No limits are provided.

References

[R411485](1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R412485](1, 2) Kawase, Yoshinori, and Jaromir J. Ulbrecht. “Turbulent Heat and Mass Transfer in Dilute Polymer Solutions.” Chemical Engineering Science 37, no. 7 (1982): 1039–46. doi:10.1016/0009-2509(82)80134-6.

Examples

>>> turbulent_Kawase_Ulbrecht(Re=1E5, Pr=1.2, fd=0.0185)
389.6262247333975
ht.conv_internal.turbulent_Kawase_De(Re, Pr, fd)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R414487] as shown in [R413487].

\[Nu = 0.0471 RePr^{0.5}(f/4)^{0.5}(1.11 + 0.44Pr^{-1/3} - 0.7Pr^{-1/6})\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

5.1 ≤ Pr ≤ 390 and 5000 ≤ Re ≤ 5E5 and 0.0024 ≤ eD ≤ 0.165.

References

[R413487](1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R414487](1, 2) Kawase, Yoshinori, and Addie De. “Turbulent Heat and Mass Transfer in Newtonian and Dilute Polymer Solutions Flowing through Rough Pipes.” International Journal of Heat and Mass Transfer 27, no. 1 (January 1984): 140–42. doi:10.1016/0017-9310(84)90246-1.

Examples

>>> turbulent_Kawase_De(Re=1E5, Pr=1.2, fd=0.0185)
296.5019733271324
ht.conv_internal.turbulent_Bhatti_Shah(Re, Pr, fd, eD)[source]

Calculates internal convection Nusselt number for turbulent flows in pipe according to [R416489] as shown in [R415489]. The most widely used rough pipe turbulent correlation.

\[Nu_D = \frac{(f/8)Re_DPr}{1+\sqrt{f/8}(4.5Re_{\epsilon}^{0.2}Pr^{0.5}-8.48)}\]
Parameters:

Re : float

Reynolds number, [-]

Pr : float

Prandtl number, [-]

fd : float

Darcy friction factor [-]

eD : float

Relative roughness, [-]

Returns:

Nu : float

Nusselt number, [-]

Notes

According to [R415489], the limits are: 0.5 ≤ Pr ≤ 10 0.002 ≤ ε/D ≤ 0.05 10,000 ≤ Re_{D} Another correlation is listed in this equation, with a wider variety of validity.

References

[R415489](1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.
[R416489](1, 2) M. S. Bhatti and R. K. Shah. Turbulent and transition flow convective heat transfer in ducts. In S. Kakaç, R. K. Shah, and W. Aung, editors, Handbook of Single-Phase Convective Heat Transfer, chapter 4. Wiley-Interscience, New York, 1987.

Examples

>>> turbulent_Bhatti_Shah(Re=1E5, Pr=1.2, fd=0.0185, eD=1E-3)
302.7037617414273
ht.conv_internal.Nu_conv_internal(Re, Pr, fd=None, eD=None, Di=None, x=None, fd_smooth=None, AvailableMethods=False, Method=None)[source]

This function handles choosing which internal flow heat transfer correlation to use, depending on the provided information. Generally this is used by a helper class, but can be used directly. Will automatically select the correlation to use if none is provided

ht.conv_internal.Morimoto_Hotta(Re, Pr, Dh, Rm)[source]

Calculates Nusselt number for flow inside a spiral heat exchanger of spiral mean diameter Rm and hydraulic diameter Dh according to [R417491], also as shown in [R418491] and [R419491].

\[ \begin{align}\begin{aligned}Nu = 0.0239\left(1 + 5.54\frac{D_h}{R_m}\right)Re^{0.806}Pr^{0.268}\\D_h = \frac{2HS}{H+S}\\R_m = \frac{R_{min} + R_{max}}{2}\end{aligned}\end{align} \]
Parameters:

Re : float

Reynolds number with bulk properties, [-]

Pr : float

Prandtl number with bulk properties [-]

Dh : float

Average hydraulic diameter, [m]

Rm : float

Average spiral radius, [m]

Returns:

Nu : float

Nusselt number with respect to Dh, [-]

Notes

[R417491] is in Japanese.

References

[R417491](1, 2, 3) Morimoto, Eiji, and Kazuyuki Hotta. “Study of Geometric Structure and Heat Transfer Characteristics of Spiral Plate Heat Exchanger.” Transactions of the Japan Society of Mechanical Engineers Series B 52, no. 474 (1986): 926-33. doi:10.1299/kikaib.52.926.
[R418491](1, 2) Bidabadi, M. and Sadaghiani, A. and Azad, A. “Spiral heat exchanger optimization using genetic algorithm.” Transaction on Mechanical Engineering, International Journal of Science and Technology, vol. 20, no. 5 (2013): 1445-1454. http://www.scientiairanica.com/en/ManuscriptDetail?mid=47.
[R419491](1, 2) Turgut, Oğuz Emrah, and Mustafa Turhan Çoban. “Thermal Design of Spiral Heat Exchangers and Heat Pipes through Global Best Algorithm.” Heat and Mass Transfer, July 7, 2016, 1-18. doi:10.1007/s00231-016-1861-y.

Examples

>>> Morimoto_Hotta(1E5, 5.7, .05, .5)
634.4879473869859
ht.conv_internal.helical_turbulent_Nu_Mori_Nakayama(Re, Pr, Di, Dc)[source]

Calculates Nusselt number for a fluid flowing inside a curved pipe such as a helical coil under turbulent conditions, using the method of Mori and Nakayama [R420494], also shown in [R421494] and [R422494].

For \(Pr < 1\):

\[Nu = \frac{Pr}{26.2(Pr^{2/3}-0.074)}Re^{0.8}\left(\frac{D_i}{D_c} \right)^{0.1}\left[1 + \frac{0.098}{\left[Re\left(\frac{D_i}{D_c} \right)^2\right]^{0.2}}\right]\]

For \(Pr \ge 1\):

\[Nu = \frac{Pr^{0.4}}{41}Re^{5/6}\left(\frac{D_i}{D_c}\right)^{1/12} \left[1 + \frac{0.061}{\left[Re\left(\frac{D_i}{D_c}\right)^{2.5} \right]^{1/6}}\right]\]
Parameters:

Re : float

Reynolds number with D=Di, [-]

Pr : float

Prandtl number with bulk properties [-]

Di : float

Inner diameter of the coil, [m]

Dc : float

Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m]

Returns:

Nu : float

Nusselt number with respect to Di, [-]

Notes

At very low curvatures, the predicted heat transfer coefficient grows unbounded.

Applicable for \(Re\left(\frac{D_i}{D_c}\right)^2 > 0.1\)

References

[R420494](1, 2) Mori, Yasuo, and Wataru Nakayama. “Study on Forced Convective Heat Transfer in Curved Pipes.” International Journal of Heat and Mass Transfer 10, no. 5 (May 1, 1967): 681-95. doi:10.1016/0017-9310(67)90113-5.
[R421494](1, 2) El-Genk, Mohamed S., and Timothy M. Schriener. “A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes.” Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693.
[R422494](1, 2) Hardik, B. K., P. K. Baburajan, and S. V. Prabhu. “Local Heat Transfer Coefficient in Helical Coils with Single Phase Flow.” International Journal of Heat and Mass Transfer 89 (October 2015): 522-38. doi:10.1016/j.ijheatmasstransfer.2015.05.069.

Examples

>>> helical_turbulent_Nu_Mori_Nakayama(2E5, 0.7, 0.01, .2)
496.2522480663327
ht.conv_internal.helical_turbulent_Nu_Schmidt(Re, Pr, Di, Dc)[source]

Calculates Nusselt number for a fluid flowing inside a curved pipe such as a helical coil under turbulent conditions, using the method of Schmidt [R423497], also shown in [R424497], [R425497], and [R426497].

For \(Re_{crit} < Re < 2.2\times 10 ^4\):

\[Nu = 0.023\left[1 + 14.8\left(1 + \frac{D_i}{D_c}\right)\left( \frac{D_i}{D_c}\right)^{1/3}\right]Re^{0.8-0.22\left(\frac{D_i}{D_c} \right)^{0.1}}Pr^{1/3}\]

For \(2.2\times 10^4 < Re < 1.5\times 10^5\):

\[Nu = 0.023\left[1 + 3.6\left(1 - \frac{D_i}{D_c}\right)\left(\frac{D_i} {D_c}\right)^{0.8}\right]Re^{0.8}Pr^{1/3}\]
Parameters:

Re : float

Reynolds number with D=Di, [-]

Pr : float

Prandtl number with bulk properties [-]

Di : float

Inner diameter of the coil, [m]

Dc : float

Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m]

Returns:

Nu : float

Nusselt number with respect to Di, [-]

Notes

For very low curvatures, reasonable results are returned by both cases of Reynolds numbers.

References

[R423497](1, 2) Schmidt, Eckehard F. “Wärmeübergang Und Druckverlust in Rohrschlangen.” Chemie Ingenieur Technik 39, no. 13 (July 10, 1967): 781-89. doi:10.1002/cite.330391302.
[R424497](1, 2) El-Genk, Mohamed S., and Timothy M. Schriener. “A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes.” Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693.
[R425497](1, 2) Hardik, B. K., P. K. Baburajan, and S. V. Prabhu. “Local Heat Transfer Coefficient in Helical Coils with Single Phase Flow.” International Journal of Heat and Mass Transfer 89 (October 2015): 522-38. doi:10.1016/j.ijheatmasstransfer.2015.05.069.
[R426497](1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

Examples

>>> helical_turbulent_Nu_Schmidt(2E5, 0.7, 0.01, .2)
466.2569996832083
ht.conv_internal.helical_turbulent_Nu_Xin_Ebadian(Re, Pr, Di, Dc)[source]

Calculates Nusselt number for a fluid flowing inside a curved pipe such as a helical coil under turbulent conditions, using the method of Xin and Ebadian [R427501], also shown in [R428501] and [R429501].

For \(Re_{crit} < Re < 1\times 10^5\):

\[Nu = 0.00619Re^{0.92} Pr^{0.4}\left[1 + 3.455\left(\frac{D_i}{D_c} \right)\right]\]
Parameters:

Re : float

Reynolds number with D=Di, [-]

Pr : float

Prandtl number with bulk properties [-]

Di : float

Inner diameter of the coil, [m]

Dc : float

Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m]

Returns:

Nu : float

Nusselt number with respect to Di, [-]

Notes

For very low curvatures, reasonable results are returned.

The correlation was developed with data in the range of \(0.7 < Pr < 5; 0.0267 < \frac{D_i}{D_c} < 0.0884\).

References

[R427501](1, 2) Xin, R. C., and M. A. Ebadian. “The Effects of Prandtl Numbers on Local and Average Convective Heat Transfer Characteristics in Helical Pipes.” Journal of Heat Transfer 119, no. 3 (August 1, 1997): 467-73. doi:10.1115/1.2824120.
[R428501](1, 2) El-Genk, Mohamed S., and Timothy M. Schriener. “A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes.” Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693.
[R429501](1, 2) Hardik, B. K., P. K. Baburajan, and S. V. Prabhu. “Local Heat Transfer Coefficient in Helical Coils with Single Phase Flow.” International Journal of Heat and Mass Transfer 89 (October 2015): 522-38. doi:10.1016/j.ijheatmasstransfer.2015.05.069.

Examples

>>> helical_turbulent_Nu_Xin_Ebadian(2E5, 0.7, 0.01, .2)
474.11413424344755
ht.conv_internal.Nu_laminar_rectangular_Shan_London(a_r)[source]

Calculates internal convection Nusselt number for laminar flows in a rectangular pipe of varying aspect ratio, as developed in [R430504].

This model is derived assuming a constant wall heat flux from all sides. This is entirely theoretically derived and reproduced experimentally.

\[Nu_{lam} = 8.235\left(1 - 2.0421\alpha + 3.0853\alpha^2 - 2.4765\alpha^3 + 1.0578\alpha^4 - 0.1861\alpha^5\right)\]
Parameters:

a_r : float

The aspect ratio of the channel, from 0 to 1 [-]

Returns:

Nu : float

Nusselt number of flow in a rectangular channel, [-]

Notes

At an aspect ratio of 1 (square channel), the Nusselt number converges to 3.610224. The authors of [R430504] also published [R431504], which tabulates in their table 42 some less precise results that are used to check this function.

References

[R430504](1, 2, 3) Shah, R. K, and Alexander Louis London. Supplement 1: Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data. New York: Academic Press, 1978.
[R431504](1, 2) Shah, Ramesh K., and A. L. London. “Laminar Flow Forced Convection Heat Transfer and Flow Friction in Straight and Curved Ducts - A Summary of Analytical Solutions.” STANFORD UNIV CA DEPT OF MECHANICAL ENGINEERING, STANFORD UNIV CA DEPT OF MECHANICAL ENGINEERING, November 1971. http://www.dtic.mil/docs/citations/AD0736260.

Examples

>>> Nu_laminar_rectangular_Shan_London(.7)
3.751762675455