**7. Nomenclature**

### **7.1 General**



### **7.2 Greek symbols**

106 Mass Transfer in Chemical Engineering Processes

methods to facilitate and represent complex geometries and reduce computational time and

Although CFD techniques are widely used, the modeller should bear in mind many of the pitfalls that characterize them. Some of these pitfalls are related to but not limited to the choice of the meshing technique; the numerical formulation; the physical correlations; the coding of meaningful and case specific UDFs; the choice from a spectrum of low and high order schemes for the formulation of the governing equations; and last but not least, the

In addition, due to the complex nature of the processes occurring in the drying systems, extensive simulations must be carried out to demonstrate that the solution is time- and gridindependent, and that the numerical schemes used have high level of accuracy by validating them with either experimental data or parametric and sensitivity analysis. This is particularly crucial in the approximation of the convective terms, as low order schemes are stable but diffusive, whereas high order schemes are more accurate but harder to converge.

convergence problems.

**7. Nomenclature** 

A Surface area [m2]

*ds* Particle diameter [m]

**7.1 General** 

*k*

*Dv*

*Fvm*

choice of iterative and solution dependent parameters.

*b* Coefficient in turbulence model [dimensionless]

C Turbulence coefficient = 0.09 [dimensionless] *cp* Specific heat capacity of the gas phase [J/kg K]

*cvm* Virtual mass coefficient = 0.5 [dimensionless] *Cg* Vapor concentration in the gas phase [kmol/m3] *Cp,s* Vapor concentration at the particle surface [kmol/m3]

*Ds,Dt,sg* Turbulent quantities for the dispersed phase

*h* Heat transfer coefficient [W/m2K] *Hpq* Interphase enthalpy [J/kg] *Hq* Enthalpy of the *q* phase [J/kg] *k* Turbulence kinetic energy [m2/s2]

C1,C2,C3 Turbulence coefficients [=1.42, 1.68, 1.2, respectively]

*CD* Drag coefficient, defined different ways [dimensionless]

Diffusion Coefficient of water vapor in air [m2/s]

*ess* Particle-particle restitution coefficient [dimensionless] *ew* Particle-wall restitution coefficient [dimensionless] Virtual mass force per unit volume [N/m3] *Gk,g* Production of turbulence kinetic energy *go* Radial distribution function [dimensionless]

*g* Gravitational acceleration constant [m/s2] ; The gas phase

*Kgs* Interphase momentum exchange coefficient [kg/m3s]

*kcond* Thermal conductivity of gas phase [W/m K] *kc* Convective mass transfer coefficient [m/s]

Diffusion coefficient for granular energy

*KErgun* Fluid-particle interaction coefficient of the Ergun equation [kg/m3s]

*c* Particle fluctuation velocity [m/s]


Numerical Simulation of Pneumatic and Cyclonic Dryers Using Computational Fluid Dynamics 109

[2] Mujumdar, A.S.; Wu, Z. Thermal drying technologies — Cost effective innovation aided by mathematical modeling approach. Drying Technology 2008, *26*, 146 - 154. [3] Jamaleddine, T.J.; Ray, M.B. Application of computational fluid dynamics for simulation of drying processes: A review. Drying Technology 2010, *28* (2), 120 - 154. [4] Massah, H.; Oshinowo, L. Advanced gas-solid multiphase flow models offer significant

[5] Enwald, H.; Peirano, E.; Almstedt, A.E. Eulerian two-phase flow theory applied to fluidization. International Journal of Multiphase Flow 1996, *22* (suppl.), 21 - 66. [6] Wen, C.Y.; Yu, Y.H. Mechanics of fluidization. Chemical Engineering Progress

[7] Ergun, S. Fluid flow through packed columns. Chemical Engineering Progress 1952, *48*,

[8] Gidaspow, D.; Bezburuah, R.; Ding, J. Hydrodynamics of circulating fluidized beds,

Foundation Conference on Fluidization, Gold Coast, Australia 1992, 75 - 82. [9] Chapman, S.; Cowling, T.G. *The mathematical theory of non-uniform gases*. 3rd ed.,

[10] Jenkins, J.T.; Savage, S.B. A theory for the rapid flow of identical, smooth, nearly

[11] Ding, J.; Gidaspow, D. A bubbling fluidization model using kinetic theory of granular

[12] Gidaspow, D. *Multiphase Flow and Fluidization*. Academic Press, Inc., New York, 1994. [13] Tsuji, Y.; Kawagushi, T.; Tanaka, T. Discrete particle simulation of two-dimensional

[14] Hoomans, B.P.B.; Kuipers, J.A.M.; Briels, W.J.; Van Swaaij, W.P.M. Discrete particle

[16] Gallagher, R.H. *Finite Element Analysis: Fundamentals*. Prentice-Hall: Englewood Cliffs,

[17] Skuratovsky, I.; Levy, A.; Borde, I. Two-fluid two-dimensional model for pneumatic

[18] Lun, C.K.K.; Savage, S.B.; Jeffrey, D.J.; Chepurnity, N. Kinetic theories for granular

[19] Gidaspow, D.; Huilin, L. Equation of State and Radial Distribution Function of FCC

[20] Baeyens, J.; Gauwbergen, D. van; Vinckier, I. Pneumatic drying: the use of large-scale experimental data in a design procedure. Powder Technology 1995, *83*, 139 – 148.

[22] Fick, A. *Ueber Diffusion*. Poggendorff's Annals of Physics 1855, *94*, 59 - 86. [23] FLUENT 6.3 User's Guide. Fluent Incorporated, Lebanon, NH, 2006. [24] Hinze, J. O. *Turbulence*. McGraw-Hill Publishing Co., New York, 1975.

flow: Inelastic particles in couette flow and slightly inelastic particles in a general

hard-sphere approach. Chemical Engineering Science 1996, 51, 99–118. [15] Strang, G.; Fix, G. *An Analysis of the Finite Element Method*. Prentice-Hall: Englewood

simulation of bubble and slug formation in a two-dimensional gas-fluidised bed: A

Symposium Series 1996, *62*, 100 – 111.

flow. AIChE J. 1990, *36*(4), 523 - 538.

Cambridge University Press: Cambridge, U.K., 1970.

fluidized bed. Powder Technology 1993, *77*, 79 - 87.

drying. Drying Technology 2003, *21*(9), 1649 – 1672.

flow field, J. Fluid Mechanics 1984, 140, 223 - 256.

Particles in a CFB. AIChE J. 1998, 279.

[21] De Brandt, IEC Proc. Des. Dev. 1974, *13*, 396.

elastic, spherical particles. J. Fluid Mech. 1983, *130*, 187 - 202.

89 - 94.

Cliffs, NJ, 1973.

NJ, 1975.

process improvements. Journal Articles by Fluent Software Users 2000, *JA112*, 1 - 6.

kinetic theory approach. Fluidization VII Proceedings of the 7th Engineering

