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