Convection to packed beds (ht.conv_packed_bed)¶

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 [1] and also cited in the review of [2].

$Nu = [(1.18Re^{0.58})^4 + (0.23\left(\frac{Re}{1-\epsilon} \right)^{0.75})^4]^{0.25}$
Parameters
Refloat

Reynolds number with pebble diameter as characteristic dimension, [-]

Prfloat

Prandtl number of the fluid []

voidagefloat

Void fraction of bed packing [-]

Returns
Nufloat

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

1

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.

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 [1] and also cited in the review of [2].

$Nu = 1.27\frac{Pr^{1/3}}{\epsilon^{1.18}}Re^{0.36} + 0.033\frac{Pr^{0.5}}{\epsilon^{1.07}}Re^{0.86}$
Parameters
Refloat

Reynolds number with pebble diameter as characteristic dimension, [-]

Prfloat

Prandtl number of the fluid [-]

voidagefloat

Void fraction of bed packing [-]

Returns
Nufloat

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

1

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

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

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 [1] and also cited in the review of [2]. Relatively rough, as it has no dependence on voidage.

$Nu = 2 + 1.1Pr^{1/3}Re^{0.6}$
Parameters
Refloat

Reynolds number with pebble diameter as characteristic dimension, [-]

Prfloat

Prandtl number of the fluid []

Returns
Nufloat

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

1

Wakao, Noriaki, and Seiichirō Kagei. Heat and Mass Transfer in Packed Beds. Taylor & Francis, 1982.

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_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 [3] and cited as from [1] and [2]. Likely the best available model as the author of [1] is the same as [2] and [3].

$Nu = f_a Nu_{sphere}$
$Nu_{sphere} = 2 + \sqrt{Nu_{m,lam}^2 + Nu_{m,turb}^2}$
$Nu_{m,lam} = 0.664Re^{0.5} Pr^{1/3}$
$Nu_{m,turb} = \frac{0.037Re^{0.8} Pr}{1 + 2.443Re^{-0.1}(Pr^{2/3} -1)}$
$Re = \frac{\rho v_s d_p}{\mu \epsilon}$
Parameters
dpfloat

Equivalent spherical particle diameter of packing [m]

voidagefloat

Void fraction of bed packing [-]

vsfloat

Superficial velocity of the fluid [m/s]

rhofloat

Density of the fluid [kg/m^3]

mufloat

Viscosity of the fluid, [Pa*s]

Prfloat

Prandtl number of the fluid []

fafloat, optional

Fator increasing heat transfer []

Returns
Nufloat

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

1(1,2)

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

2(1,2)

Gnielinski, V. (1982) “Berechnung des Warmeund Stoffaustauschs in durchstomten ruhenden Schuttungen”. Verfahrenstechnik 16(1): 36-39

3(1,2)

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