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 [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].
\[ \begin{align}\begin{aligned}Nu = f_a Nu_{sphere}\\Nu_{sphere} = 2 + \sqrt{Nu_{m,lam}^2 + Nu_{m,turb}^2}\\Nu_{m,lam} = 0.664Re^{0.5} Pr^{1/3}\\Nu_{m,turb} = \frac{0.037Re^{0.8} Pr}{1 + 2.443Re^{0.1}(Pr^{2/3} 1)}\\Re = \frac{\rho v_s d_p}{\mu \epsilon}\end{aligned}\end{align} \]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 []
Returns:  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 \(1E1 < Re <1,000\) and \(0.4 < Pr < 1000\) for spheres. Limits are smaller for other shapes.
References
[1] (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): 378383 [2] (1, 2, 3) Gnielinski, V. (1982) “Berechnung des Warmeund Stoffaustauschs in durchstomten ruhenden Schuttungen”. Verfahrenstechnik 16(1): 3639 [3] (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=8E4, voidage=0.4, vs=1, rho=1E3, mu=1E3, 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 [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:  Re : float
Reynolds number with pebble diameter as characteristic dimension, []
 Pr : float
Prandtl number of the fluid []
Returns:  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
[1] (1, 2) Wakao, Noriaki, and Seiichirō Kagei. Heat and Mass Transfer in Packed Beds. Taylor & Francis, 1982. [2] (1, 2) Abdulmohsin, Rahman S., and Muthanna H. AlDahhan. “Characteristics of Convective Heat Transport in a Packed PebbleBed Reactor.” Nuclear Engineering and Design 284 (April 1, 2015): 14352. 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 [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:  Re : float
Reynolds number with pebble diameter as characteristic dimension, []
 Pr : float
Prandtl number of the fluid []
 voidage : float
Void fraction of bed packing []
Returns:  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
[1] (1, 2) Achenbach, E. “Heat and Flow Characteristics of Packed Beds.” Experimental Thermal and Fluid Science 10, no. 1 (January 1, 1995): 1727. doi:10.1016/08941777(94)00077L. [2] (1, 2) Abdulmohsin, Rahman S., and Muthanna H. AlDahhan. “Characteristics of Convective Heat Transport in a Packed PebbleBed Reactor.” Nuclear Engineering and Design 284 (April 1, 2015): 14352. 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:  Re : float
Reynolds number with pebble diameter as characteristic dimension, []
 Pr : float
Prandtl number of the fluid []
 voidage : float
Void fraction of bed packing []
Returns:  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
[1] (1, 2) Reactor Core Design of HighTemperature GasCooled Reactors Part 2: Heat Transfer in Spherical Fuel Elements (June 1983). http://www.ktags.de/e/standards/3100/3102_2_engl_1983_06.pdf [2] (1, 2) Abdulmohsin, Rahman S., and Muthanna H. AlDahhan. “Characteristics of Convective Heat Transport in a Packed PebbleBed Reactor.” Nuclear Engineering and Design 284 (April 1, 2015): 14352. doi:10.1016/j.nucengdes.2014.11.041. Examples
>>> Nu_KTA(2000, 0.7, 0.4) 102.08516480718129