# Convection to packed beds (ht.conv_packed_bed)¶

ht.conv_packed_bed.Nu_packed_bed_Gnielinski(dp, voidage, vs, rho, mu, Pr, fa=None)[source]

Calculates Nusselt number of a fluid passing over a bed of particles using a correlation shown in [R517523] and cited as from [R515523] and [R516523]. Likely the best available model as the author of [R515523] is the same as [R516523] and [R517523].

\begin{align}\begin{aligned}Nu = f_a Nu_{sphere}\\Nu_{sphere} = 2 + \sqrt{Nu_{m,lam}^2 + Nu_{m,turb}^2}\\Nu_{m,lam} = 0.664Re^{0.5} Pr^{1/3}\\Nu_{m,turb} = \frac{0.037Re^{0.8} Pr}{1 + 2.443Re^{-0.1}(Pr^{2/3} -1)}\\Re = \frac{\rho v_s d_p}{\mu \epsilon}\end{aligned}\end{align}
Parameters: dp : float Equivalent spherical particle diameter of packing [m] voidage : float Void fraction of bed packing [-] vs : float Superficial velocity of the fluid [m/s] rho : float Density of the fluid [kg/m^3] mu : float Viscosity of the fluid, [Pa*s] Pr : float Prandtl number of the fluid [] fa : float, optional Fator increasing heat transfer [] Nu : float Nusselt number for heat transfer to the packed bed [-]

Notes

fa is a factor relating how much more heat transfer happens than would normally, around one sphere. For spheres of the same size, $$f_a = 1 + 1.5(1-\epsilon)$$. For cylinders with l/d ratio of 0.24 < l/d < 1.2 use fa = 1.6. For cubes, use fa = 1.6 For Raschig rings, use fa = 2.1 For Berl saddles, use fa = 2.3. fa is calculated with the relationship for spheres if not provided.

Confirmed with experimental data for a range of $$1E-1 < Re <1,000$$ and $$0.4 < Pr < 1000$$ for spheres. Limits are smaller for other shapes.

References

 [R515523] (1, 2, 3) Gnielinski, V. (1981) “Equations for the calculation of heat and mass transfer during flow through stationary spherical packings at moderate and high Peclet numbers”. International Chemical Engineering 21 (3): 378-383
 [R516523] (1, 2, 3) Gnielinski, V. (1982) “Berechnung des Warmeund Stoffaustauschs in durchstomten ruhenden Schuttungen”. Verfahrenstechnik 16(1): 36-39
 [R517523] (1, 2, 3) Gnielinski, V. in G esellschaft, V. D. I., ed. VDI Heat Atlas. 2nd ed. 2010 edition. Berlin; New York: Springer, 2010.

Examples

>>> Nu_packed_bed_Gnielinski(dp=8E-4, voidage=0.4, vs=1, rho=1E3, mu=1E-3, Pr=0.7)
61.37823202546954

ht.conv_packed_bed.Nu_Wakao_Kagei(Re, Pr)[source]

Calculates Nusselt number of a fluid passing over a bed of particles using a correlation shown in [R518526] and also cited in the review of [R519526]. Relatively rough, as it has no dependence on voidage.

$Nu = 2 + 1.1Pr^{1/3}Re^{0.6}$
Parameters: Re : float Reynolds number with pebble diameter as characteristic dimension, [-] Pr : float Prandtl number of the fluid [] Nu : float Nusselt number for heat transfer to the packed bed [-]

Notes

Fit for Re from 3 to 3000; claimed reasonableness of fit to to 1E6.

References

 [R518526] (1, 2) Wakao, Noriaki, and Seiichirō Kagei. Heat and Mass Transfer in Packed Beds. Taylor & Francis, 1982.
 [R519526] (1, 2) Abdulmohsin, Rahman S., and Muthanna H. Al-Dahhan. “Characteristics of Convective Heat Transport in a Packed Pebble-Bed Reactor.” Nuclear Engineering and Design 284 (April 1, 2015): 143-52. doi:10.1016/j.nucengdes.2014.11.041.

Examples

>>> Nu_Wakao_Kagei(2000, 0.7)
95.40641328041248

ht.conv_packed_bed.Nu_Achenbach(Re, Pr, voidage)[source]

Calculates Nusselt number of a fluid passing over a bed of particles using a correlation shown in [R520528] and also cited in the review of [R521528].

$Nu = [(1.18Re^{0.58})^4 + (0.23\left(\frac{Re}{1-\epsilon} \right)^{0.75})^4]^{0.25}$
Parameters: Re : float Reynolds number with pebble diameter as characteristic dimension, [-] Pr : float Prandtl number of the fluid [] voidage : float Void fraction of bed packing [-] Nu : float Nusselt number for heat transfer to the packed bed [-]

Notes

Claimed value for Re/ε < 7.7E5 Developed with tests performed in a wind tunnel at conditions up to 30 bar.

References

 [R520528] (1, 2) Achenbach, E. “Heat and Flow Characteristics of Packed Beds.” Experimental Thermal and Fluid Science 10, no. 1 (January 1, 1995): 17-27. doi:10.1016/0894-1777(94)00077-L.
 [R521528] (1, 2) Abdulmohsin, Rahman S., and Muthanna H. Al-Dahhan. “Characteristics of Convective Heat Transport in a Packed Pebble-Bed Reactor.” Nuclear Engineering and Design 284 (April 1, 2015): 143-52. doi:10.1016/j.nucengdes.2014.11.041.

Examples

>>> Nu_Achenbach(2000, 0.7, 0.4)
117.70343608599121

ht.conv_packed_bed.Nu_KTA(Re, Pr, voidage)[source]

Calculates Nusselt number of a fluid passing over a bed of particles using a correlation shown in [R522530] and also cited in the review of [R523530].

$Nu = 1.27\frac{Pr^{1/3}}{\epsilon^{1.18}}Re^{0.36} + 0.033\frac{Pr^{0.5}}{\epsilon^{1.07}}Re^{0.86}$
Parameters: Re : float Reynolds number with pebble diameter as characteristic dimension, [-] Pr : float Prandtl number of the fluid [-] voidage : float Void fraction of bed packing [-] Nu : float Nusselt number for heat transfer to the packed bed [-]

Notes

100 < Re < 1E5; 0.36 < ε < 0.42; D/d > 20 with D as bed diameter, d as particle diameter; H > 4d with H as bed height.

References

 [R522530] (1, 2) Reactor Core Design of High-Temperature Gas-Cooled Reactors Part 2: Heat Transfer in Spherical Fuel Elements (June 1983). http://www.kta-gs.de/e/standards/3100/3102_2_engl_1983_06.pdf
 [R523530] (1, 2) Abdulmohsin, Rahman S., and Muthanna H. Al-Dahhan. “Characteristics of Convective Heat Transport in a Packed Pebble-Bed Reactor.” Nuclear Engineering and Design 284 (April 1, 2015): 143-52. doi:10.1016/j.nucengdes.2014.11.041.

Examples

>>> Nu_KTA(2000, 0.7, 0.4)
102.08516480718129