Paper No. : 04 Paper Title: Unit Operations in Food Processing Module- 18: Circulation of fluids through porous bed 18.1 Introduction A typical packed bed is a cylindrical column that is filled with a suitable packing material. The packing material in the bed may be sphere, cylinders or various kinds of irregular size particles. Several unit operations such as absorption, adsorption, distillation and extraction are carried out in packed columns. These packing enhance the surface area available for transfer operations. The packed bed configuration also facilitates the intimate mixing of fluids with mismatched densities, largely due to increased surface area for contact. As a fluid passes through a packed bed it experiences pressure loss due to factors such as friction. The essential factors determining the energy loss, i.e. pressure drop, in packed beds are: 1. Rate of fluid flow 2. Viscosity and density of the fluid 3. Closeness and orientation of packing 4. Size shape and surface of the particles The first two variables concern the fluid, while the last two the solids. In the discussion to follow it is assumed that the packing is everywhere uniform and that little or no channelling occurs. The ratio of the diameter of the tower to the packing diameter should be a minimum of 8:1 to 10:1 for wall effects to be small. In the theoretical approach used, the packed column is regarded as a bundle of crooked tubes of varying cross-sectional area. 18.2 Geometric relations for particles in packed beds Certain geometric relations for particles in packed beds are used in the derivations for flow. 18.2.1 Void fraction The void fraction ε in a packed bed is the fraction of the flow cross-sectional area occupied by the gas phase and represented as the ratio between volumes of voids in the bed to total volume of the bed. (1) 18.2.2 Specific surface The specific surface of particle (a v ) in m -1 is defined as the ratio between the surface area of a particle in m 2 and the volume of a particle in m 3.
(2) (3) (4) D P is the diameter in m. For a packed bed of non-spherical particles, the effective particle diameter D P is defined as (5) Since (1-ε) is the volume fraction of particles in the bed,... (6) Where, a is the ratio of total surface area in the bed to total volume of bed (void volume plus particle volume) in m -1. 18.2.3 Average interstitial velocity in the packed bed The average interstitial velocity in the bed is v (m/s) and it is related to the superficial velocity v based on the cross section of the empty container by 18.2.4 Hydraulic radius (7) The hydraulic radius wetted perimeter. is the ratio between the cross sectional area available for flow and (8) Hence (9) Since the equivalent diameter D for a channel is D= 4, 18.2.5 Reynolds number for a packed bed The Reynolds number for a packed bed can be presented as following. (10)
For packed beds Ergun defined the Reynolds number as above but by eliminating 4/6 term. Hence Reynolds number for packed bed can be stated as below. (11) As, G = v ρ. (12) 18.3 Laminar flow through a packed bed The pressure drop for laminar fluid flow through a packed bed of spheres with diameter D may be calculated using the Hagen-Poiseuille equation 18.3.1 Hagen-Poiseuille equation For laminar flow, the Hagen-Poiseuille equation states that Where L = length in m, D=inside diameter in m, Putting D= 4 and the expression for can be presented as follows. (13) (14) 18.3.2 Blake- Kozeny equation for laminar flow Experimental data shows that the constant should be 150. For void fraction less than 0.5, effective particle diameter D P andn Rep <10. (15) 18.3.3 Pressure drop for Turbulent flow in packed bed For turbulent flow pressure drop is expressed as (16) Where f is the fanning friction factor. Putting D= 4 = and the expression for can be presented as follows. (17)
18.4.1 Burke-Plummer equation for turbulent flow in packed bed For highly turbulent flow the friction factor should approach a constant value. Also, it is assumed that all packed beds should have the same roughness. Experimental data indicated that 3f=1.75. Hence, the final equation for turbulent flow for N Rep > 1000, which is called the Burke- Plummer equation, becomes (18) 18.5 General equation for flow, intermediate, and high Reynolds numbers Ergun proposed the following general equation for low, intermediate, and high Reynolds numbers which is a combination of Blake-Kozeny equation for laminar flow and Burke- Plummer equation for turbulent flow.. (19) Rewriting Eqn. 19 in terms of dimensionless groups, (20) 18.6 Shape factors Many particles in packed beds are often irregular in shape. The equivalent diameter of particle is defined as the diameter of a sphere having the same volume as this particle. The sphericity. Shape factor of a particle is the ratio of the surface area of this sphere having the same volume as the particle to the actual surface area of the particle. For a sphere, the surface area S p =πd p 2 and the volume is V p =πd p 3 /6. Hence, for any particle, = πd p 2 / S p (21) Where, S p is the actual surface area of the particle D p is the particle diameter (equivalent diameter) of the sphere having the same volume as the particle. (22) (23)
As, (24) For a sphere, =1.0. For a cylinder where the diameter= length, is the calculated to be 0.874 and for a cube, is calculated volume and surface as 0.806. Typical values for manycrushed materials are between 0.6 and 0.7. For convenience for the cylinder and the cube, the nominal diameter is sometimes used (instead of the equivalent diameter) which then gives a shape factor of 1.0. 18.7 Mixtures of particles For mixtures of particles of various sizes mean specific surface a vm can be defined as following. (25) Where x i is the volume fraction. As, Hence, (26) The effective mean diameter for the mixture (D pm ) can be expressed as (27) 18.8 Flow in Fluidized Beds 18.8.1 Minimum velocity and porosity for fluidization 18.8.1.1 Minimum fluidization velocity When a fluid flows upward through a packed bed of particles at low velocities, the particles remain stationary. As the fluid velocity is increased, the pressure drop increases according to the Ergun equation (19). Upon further increases in velocity, conditions finally occur where the force of the pressure drop times the cross-sectional area just equals the gravitational force on the mass of particles. Then the particles just begin to move, and this is the onset of fluidization or minimum fluidization. The fluid velocity at which fluidization begins is the minimum fluidization velocity v mf in m/s based on the empty cross section of the tower (superficial velocity). Let ε mf = The porosity of the bed when true fluidization initiates or the minimum porosity for fluidization. L mf = height of the bed in m.
18.8.1.2 Experimental process The bed with particles is subjected to rising gas stream. As the gas velocity is increased the pressure drop is also increased until the onset of minimum fluidization. Then as the velocity is further increased, the pressure drop decreases very slightly and then remains practically unchanged as the bed continues to expand or increase in porosity with increase in velocity. The bed resembles a boiling liquid. As the bed expands with increase in velocity, the bed continues to retain its top horizontal surface. Eventually, as the velocity is increased much further, entrainment of particles from the actual fluidized bed becomes appreciable. For a bed having a uniform cross-sectional area A, the relation between bed height L and porosity ε is expressed as, Where, (28) L 1 is height of the bed with porosity L 2 is height of the bed with porosity. 18.8.1.3 Pressure drop and minimum fluidization velocity (29) As a first approximate, the pressure drop at the start of fluidization can be determined as follows the force obtained from the pressure drop times the cross- sectional area must be equal the gravitational force exerted by the mass of the particles minus the buoyant force of the displaced fluid. Force obtained from the pressure drop = Pressure drop at the start of fluidization cross- sectional area = gravitational force exerted by the mass of the particles- buoyant force of the displaced fluid Hence, (30) (31) 18.8.1.4 Pressure drop for irregular shaped particles For irregular shaped particles in bed the particle size and shape factor are considered. Parameter Sphere Irregular shaped particles Effective mean diameter D p
Pressure drop in a packed bed for irregular shaped particles becomes (32) Where L =L, bed length in m. 18.8.1.5 Minimum fluid velocity v mf for packed beds The minimum fluid velocity v mf at which fluidization begins can be obtained by small extrapolation of the equation obtained for packed beds.(eqn.32) and by the following substitutions. Parameter Fluidized bed Packed bed Fluid velocity v v mf Porosity ε ε mf Bed length L L mf Reynolds number (33) (34) Eq.(33) becomes (35) Case1: When N Re,mf < 20 (small particles) (36) Case2: When N Re,mf > 1000 (large particles) (37) 18.8.1.6 Expansion of fluidized beds For the case of small particles and where N Re,mf variation of porosity or bed height L as follows. = D p v ρ/µ< 20, we can estimate the (38)
(Or) (39) Solving for (Or) (40) In the above Eqn.(40) all terms except ε are constant for the particular system.. v ε depends upon 18.8.1.7 Applicability Equation (40) can be used with liquids to estimate ε with ε<0.80. However, because of clumping and other factors, errors can occur when used for gases. References 1 Transport Processes and Unit Operations (3 rd Edition), C. J. Geankoplis, Prentice Hall Inc. Publ., 1993 2 Transport Phenomena in Food Process Engineering (3 rd Edition), A. K. Datta, Himalaya Publ., 2001 3 Flow through Packed Beds and Fluidized Beds, Subramanian, R. Shankar. Clarkson University, http://www. clarkson. edu/subramanian/ch301/notes/packfluidbed. pdf (2004).. ***