Packed Bed

6 Fluid Flow Through a Packed Bed of Particles

6.1

PRESS PR ESSURE URE DRO DROP–F P–FLOW LOW RELA RELATIO TIONSH NSHIP  IP 

6.1. 6. 1.11 La Lami mina narr Fl Flow ow In the nineteenth century Darcy (1856) observed that the flow of water through a packed bed of sand was governed by the relationship:

 pressure   liquid  gradient

/

velocity

or

ðÀÁpÞ / U   H 

ð6:1Þ

where U  is th thee su supe perfi rfici cial al flu fluid id ve velo loci city ty th thro roug ugh h th thee be bed d an and d ðÀÁpÞ is the fric fr icti tion onal al pr pres essu sure re dr drop op ac acro ross ss a be bed d de dept pth h H . (Su (Super perfici ficial al vel velocit ocity y ¼ fluid  A.) volumetric flow rate/cross-sectional area of bed, Q= A .) The flow of a fluid through a packed bed of solid particles may be analysed in termss of the fluid flow thro term through ugh tubes. The star starting ting point is the Hagen–Poiseui Hagen–Poiseuille lle equation for laminar flow through a tube: ðÀÁpÞ 32mU  ¼  H  D2

ð6:2Þ

where D is the tube diameter and m is the fluid viscosity. Consider the packed bed to be equivalent to many tubes of equivalent diameter De following tortuous paths of equivalent length H e and carrying fluid with a velocity U i . Then, from Equation (6.2), ðÀÁpÞ mU i ¼ K 1 2  H e De

 Introduction to Particle Technology - 2nd Edition Martin Rhodes # 2008 John Wiley & Sons Ltd.

ð6:3Þ

154

FLUID FLOW THROUGH A PACKED BED OF PARTICLES

U i is the actual velocity of fluid through the interstices of the packed bed and is related to superficial fluid velocity by: U i ¼ U =e

ð6:4Þ

where e is the voidage or void fraction of the packed bed. (Refer to Section 8.1.4 for discussion on actual and superficial velocities.) Although the paths of the tubes are tortuous, we can assume that their actual length is proportional to the bed depth, that is,  H e ¼ K 2 H 

ð6:5Þ

The tube equivalent diameter is defined as 4 Â flow area wetted perimeter where flow area ¼ e A, where A is the cross-sectional area of the vessel holding the bed; wetted perimeter ¼ SB A, where SB is the particle surface area per unit volume of the bed. That this is so may be demonstrated by comparison with pipe flow: Total particle surface area in the bed ¼ SB AH . For a pipe, wetted perimeter ¼

wetted surface pDL ¼ length L

SB AH  ¼ SB A. and so for the packed bed, wetted perimeter ¼  H  Now if  Sv is the surface area per unit volume of particles, then Sv ð1 À eÞ ¼ SB

ð6:6Þ

since

 surface of particles  volume of particles surface of particles volume of particles

Â

volume of bed

¼

volume of bed

and so equivalent diameter; De ¼

4e A 4e ¼ SB Sv ð1 À eÞ

ð6:7Þ

Substituting Equations (6.4), (6.5) and (6.7) in (6.3): ðÀÁpÞ ð1 À eÞ2 2 ¼ K 3 mUSv  H  e3

ð6:8Þ

where K 3 ¼ K 1 K 2 . Equation (6.8) is known as the Carman–Kozeny equation [after the work of Carman and Kozeny (Carman, 1937; Kozeny, 1927, 1933)] describing

155

PRESSURE DROP–FLOW RELATIONSHIP

laminar flow through randomly packed particles. The constant K 3 depends on particle shape and surface properties and has been found by experiment to have a value of about 5. Taking K 3 ¼ 5, for laminar flow through a randomly packed  bed of monosized spheres of diameter x (for which S ¼ 6=x) the Carman–Kozeny equation becomes: 2 ðÀÁpÞ mU ð1 À eÞ ¼ 180 2  H  x e3

ð6:9Þ

This is the most common form in which the Carman–Kozeny equation is quoted.

6.1.2 Turbulent Flow For turbulent flow through a randomly packed bed of monosized spheres of  diameter x the equivalent equation is: 2 ðÀÁpÞ rf U  ð1 À eÞ ¼ 1:75 e3  H  x

ð6:10Þ

6.1.3 General Equation for Turbulent and Laminar Flow Based on extensive experimental data covering a wide range of size and shape of  particles, Ergun (1952) suggested the following general equation for any flow conditions: 2 2 ðÀÁpÞ mU ð1 À eÞ rf U  ð1 À eÞ ¼ 150 2 þ 1:75 e3 e3  H  x x laminar turbulent component component



 



ð6:11Þ

This is known as the Ergun equation for flow through a randomly packed bed of  spherical particles of diameter x. Ergun’s equation additively combines the laminar and turbulent components of the pressure gradient. Under laminar conditions, the first term dominates and the equation reduces to the Carman– Kozeny equation [Equation (6.9)], but with the constant 150 rather than 180. (The difference in the values of the constants is probably due to differences in shape and packing of the particles.) In laminar flow the pressure gradient increases linearly with superficial fluid velocity and independent of fluid density. Under turbulent flow conditions, the second term dominates; the pressure gradient increases as the square of superficial fluid velocity and is independent of fluid viscosity. In terms of the Reynolds number defined in Equation (6.12), fully laminar condition exist for Reà less than about 10 and fully turbulent flow exists at Reynolds numbers greater than around 2000. Reà ¼

xU rf  mð1 À eÞ

ð6:12Þ

156

FLUID FLOW THROUGH A PACKED BED OF PARTICLES

In practice, the Ergun equation is often used to predict packed bed pressure gradient over the entire range of flow conditions. For simplicity, this practice is followed in the Worked Examples and Exercises in this chapter. Ergun also expressed flow through a packed bed in terms of a friction factor defined in Equation (6.13): 3 ðÀÁpÞ x e Friction factor; f  ¼  H  rf U 2 ð1 À eÞ Ã

ð6:13Þ

(Compare the form of this friction factor with the familiar Fanning friction factor for flow through pipes.) Equation (6.11) then becomes  f Ã ¼

150 þ 1:75 ReÃ

ð6:14Þ

with  f Ã ¼

150 for Reà < 10 and f Ã ¼ 1:75 for Reà > 2000 à Re

(see Figure 6.1).

6.1.4 Non-spherical Particles The Ergun and Carman–Kozeny equations also accommodate non-spherical particles if  x is replaced by xsv the diameter of a sphere having the same surface to volume ratio as the non-spherical particles in question. Use of  xsv gives the correct value of specific surface S (surface area of particles per unit volume of  particles). The relevance of this will be apparent if Equation (6.8) is recalled.

Figure 6.1 Friction factor versus Reynolds number plot for fluid flows through a packed  bed of spheres