3 Packed Beds: The Behaviour of Particles in Bulk

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Fig. 13.7 Flow through a packed bed



Fig. 13.8 Kozeny’s model of a packed bed



u =



Pd2

32LB μ



(13.38)



where the velocity u represents the interstitial velocity, that is, the velocity between the particles in

a packed bed. Kozeny, and later Carman, proposed that the interstitial and superficial velocities are

related by the bed voidage,

u =



u

ε



(13.39)



In other words, the cross-sectional area which determines the interstitial velocity for a given volumetric flow rate is proportional to the inter-particle voidage. Kozeny further suggested that the

equivalent pore space diameter dB is given by

dB =



ε

SB



(13.40)



where SB is the particle surface area per unit bed volume which comes into contact with the fluid

passing through the bed. In turn this quantity is related to the specific surface by the fraction of the

bed occupied by particles (1 − ε). Thus



13.3



Packed Beds: The Behaviour of Particles in Bulk



357



SB = S (1 − ε)



(13.41)



Substituting each of these assumptions into Eq. (13.38), and further assuming that the equivalent

pore space length is proportional to the bed depth, results in the Carman–Kozeny equation

u=



ε3 P



(13.42)



KS2 (1 − ε)2 μL



which can be used to predict fluid velocity or flow rate as a function of pressure drop for a bed of

incompressible particles, that is, where the bed voidage is constant. The dimensionless constant K is

known as Kozeny’s constant and has a value of approximately 5.0, although strictly it is a function of

both intra-particle porosity and particle shape.

A comparison can now be made between the Darcy and Kozeny relationships. Thus from Eqs.

(13.37) and (13.42) the permeability of the bed is

β=



ε3



(13.43)



KS2 (1 − ε)2



In other words permeability depends upon the geometry of the bed, that is, bed voidage and specific

surface, which in turn is a function of particle size. Permeability has dimensions of m2 and for fine

6

particles values in the range 10−10 to 10−12 m2 can be expected. Note that for spheres S = d . Bed

permeability is often expressed in terms of the specific resistance α where

α=



1

β



(13.44)



Thus α has dimensions of m−2 and values of the order of 1010 to 1012 m−2 for fine particles. This

definition is used in the analysis of filtration.

Example 13.13

Coffee particles, which may be assumed to be spheres 400 μm in diameter, are to be dried in a stream

of warm air. If the particle and bulk densities are 618 and 1030 kg m−3 , respectively, calculate the

permeability of the bed of coffee particles.

The inter-particle voidage is given by Eq. (13.18) and therefore

ε =1−



618

1030



or

ε = 0.40

6

For spheres S = d . Now assuming Kozeny’s constant to be equal to 5, the permeability of the bed

becomes

2



β=



(0.40)3 (400 × 10−6 )

180(1 − 0.40)2



m2



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or

β = 3.56 × 10−10 m2

For large particles (greater than about 600 or 700 μm) the Carman–Kozeny relationship is inadequate and predicts far too low a pressure drop. Therefore Ergun suggested a semi-empirical equation

for the pressure drop per unit bed depth, containing two terms. Ergun’s equation may be expressed as

150(1 − ε)2 μu 1.75 (1 − ε) ρf u2

P

+

=

L

ε 3 d2

ε3 d



(13.45)



The first term represents the pressure loss due to viscous drag (this is essentially the

Carman–Kozeny equation) whilst the second term represents kinetic energy losses, which are significant at higher velocities (kinetic energy being proportional to velocity squared). Equation (13.45)

is valid in the range 1 < Re < 2000 where the Reynolds number is defined by

Re =



uρ

S (1 − ε) μ



(13.46)



13.4 Fluidisation

13.4.1 Introduction

Fluidisation is a technique which enables solid particles to take on some of the properties of a fluid.

For example, fluidised solids will adopt the shape of the container in which they are held and can be

made to flow, under pressure, from an orifice or overflow a weir. Solids may be fluidised either by a

liquid or by a gas. These phenomena give rise to a series of characteristics (for example good mixing

and good heat transfer) which are exploited in a wide range of food processing operations such as

freezing, drying, mixing and granulation.

Consider a bed of particles, say of a size similar to sand. When a fluid is passed upwards through

the particles, the bed remains packed at low fluid velocities; the particles do not move. However, if the

fluid velocity is increased sufficiently, a point will be reached at which the drag force on a particle will

be balanced by the net weight of the particle. The particles are suspended in the upward moving fluid

and move away from one another. This is the point of incipient fluidisation at, and beyond which, the

bed is said to be fluidised. The superficial fluid velocity in the bed at the point of incipient fluidisation

is called the minimum fluidising velocity umf . At velocities in excess of that required for minimum

fluidisation one of two phenomena will occur.

(i) The bed may continue to expand and the particles will space themselves uniformly. This is known

as particulate fluidisation and in general occurs when the fluidising medium is a liquid (Fig. 13.9).

(ii) Alternatively, the excess fluid may pass through the bed in the form of bubbles. This is called

aggregative fluidisation and usually occurs where the fluidising medium is a gas. This type of

behaviour gives rise to the analogy of a boiling liquid (Fig. 13.10).

A fluidised bed requires a distributor plate which supports the bed when it is not fluidised, prevents particles from passing through and promotes uniform fluidisation by distributing the fluidising

medium evenly. The nature of the distributor plate influences the number and size of bubbles formed

in aggregative fluidisation. Several types of plate are possible including porous or sintered ceramics

and metals, layers of wire mesh and drilled plates. The pressure drop across the plate should be high

to promote even gas distribution and is usually some fraction of bed pressure drop, often up to 50%.



13.4



Fluidisation



359



Fig. 13.9 Particulate fluidisation



Fig. 13.10 Aggregative fluidisation



Fluidised beds can be operated either in batch or in continuous mode. In continuous operation the

solids are fed via screw conveyors, weigh feeders or pneumatic conveying lines and are withdrawn

via standpipes or weirs.



13.4.2 Minimum Fluidising Velocity in Aggregative Fluidisation

The behaviour of a fluidised bed and its effectiveness as a mixer, drier or freezer depend crucially upon

the superficial gas velocity in the bed relative to the minimum fluidising velocity. It is essential that

the minimum fluidising velocity is known; it may be determined experimentally or may be calculated

from elementary fluid mechanics concepts. In addition much effort has been expended on producing

accurate semi-empirical relationships to predict umf .

The relationship between bed pressure drop and superficial fluidising velocity is shown in

Fig. 13.11. As the gas velocity increases, the pressure drop increases in the fixed bed, or packed bed,



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Fig. 13.11 Pressure drop as a function of superficial gas velocity



region and then levels out as the bed becomes fluidised. Ideally, the pressure drop then remains constant as the weight of the particles is supported by the fluid. However, if the velocity is then reduced

marked hysteresis is observed. This is because the bed voidage remains at the minimum fluidising

value whereas with increasing gas velocity considerable vibration of the particles takes place, the

voidage is lower, and the pressure drop correspondingly is slightly greater. In practice too there will

be a maximum pressure drop through which the curve passes because of particle interlocking. Also,

as the velocity is reduced, the transition between the fluidised and fixed curves is gradual rather than

sudden.

(a) Experimental measurement: The standardised procedure to measure minimum fluidising velocity is to fluidise the bed of particles vigorously for some minutes and then reduce gas velocity in

small increments, recording the bed pressure drop each time. This may be done with a simple water

manometer with one leg open to atmosphere and one leg connected to a narrow tube placed in the bed.

The data is then interpreted as in Fig. 13.12; umf corresponds to the intersection of the straight lines

representing the fixed and fluidised beds.

(b) Carman–Kozeny equation: The Carman–Kozeny expression for minimum fluidising velocity

is derived by substituting for the pressure drop in Eq. (13.42). The pressure drop across a fluidised

bed of bed height h can be obtained by treating the fluidised solids as a fluid with a density equal to

the difference between the solid and fluid densities. Thus the pressure drop is given by Eq. (2.13) but

includes the term (1 − ε) to take account of the fraction of the bed volume occupied by solids. Hence

at minimum fluidising velocity

Pmf = (ρS − ρf ) (1 − εmf ) ghmf



(13.47)



and therefore

umf =



εmf 3 (ρS − ρf )g

KS2 (1 − εmf )μ



(13.48)



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Fluidisation



361



Fig. 13.12 Measurement of minimum fluidising velocity: aggregative fluidisation



6

Now for K = 5, and for spherical particles where the specific surface is equal to d , the minimum

fluidising velocity becomes



umf =



εmf 3 (ρS − ρf )gd 2

180 (1 − εmf ) μ



(13.49)



At minimum fluidisation the drag force acting on a particle due to the flow of fluidising gas over the

particle is balanced by the net weight of the particle. The former is a function of surface area and the

latter is proportional to particle volume. Consequently the surface-volume mean diameter is the most

appropriate particle size to use in expressions for minimum fluidising velocity. Note that Eq. (13.49)

suggests that umf is proportional to the difference in density between particle and fluid, proportional

to the square of particle diameter and inversely proportional to fluid viscosity.



Example 13.14

A food powder is to be dried in a 0.5 m diameter fluidised bed using air at 50◦ C. It is found that

minimum fluidising conditions are obtained when the bed pressure drop is 6000 Pa for a bed height

of 0.50 m. Using the Carman–Kozeny relationship, determine the minimum fluidising velocity if the

surface-volume mean particle diameter is 180 μm and the particle density is 2300 kg m−3 .

For air at 50◦ C the density is 1.1 kg m−3 and the viscosity is 1.98 × 10−5 Pa s. The voidage at

minimum fluidising velocity is obtained from Eq. (13.47) and hence

6000 = (2300 − 1.1) (1 − εmf )9.81 × 0.50

from which the voidage is

εmf = 0.468



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Thus the Carman–Kozeny equation gives the minimum fluidising velocity as

2



umf =



(0.468)3 (2300 − 1.1)(180 × 10−6 ) 9.81

180(1 − 0.468)1.98 × 10−5



m s−1



or

umf = 0.0395 m s−1

(c) Ergun equation: The Carman–Kozeny equation works well for fine particles. However, for large

particles, for example peas in a fluidised bed freezer, the minimum fluidising velocity is high and the

kinetic energy losses are significant. In these circumstances the Carman–Kozeny expression vastly

over estimates umf and the Ergun equation must be used.

Writing Ergun’s equation for minimum fluidising conditions gives

P

hmf



=



150(1 − εmf )2 μumf

1.75(1 − εmf )ρf umf 2

+

εmf 3 d 2

εmf 3 d



(13.50)



and substituting for pressure drop from Eq. (13.47) gives

(ρS − ρf )g =



150(1 − εmf )μumf

1.75ρf umf 2

+

3 d2

εmf

εmf 3 d



(13.51)



This expression is unwieldy but can be simplified considerably. Multiplying through by

results in

1.75ρf2 d 2 umf 2

150 (1 − εmf ) umf dρf

ρf (ρS − ρf ) d3 g

+

=

μ2

εmf 3 μ

εmf 3 μ2



ρf d 3

μ2



(13.52)



which can be put into the form

Ga =



1.75Re2

150 (1 − εmf )

mf

Remf +

3

3

εmf

εmf



(13.53)



This is a quadratic equation where the Galileo number Ga is defined by

Ga =



ρf (ρS − ρf ) d3 g

μ2



(13.54)



and the Reynolds number at minimum fluidisation by

Remf =



ρf umf d

μ



(13.55)



Example 13.15

A novel method for germinating tomato seeds includes a relatively rapid drying stage in a gas–solid

fluidised bed. The seeds have a flat, irregular disc-like shape but may be assumed to have a diameter

of 2 mm. The solid density of the seeds is 1600 kg m−3 . Determine the minimum fluidising velocity

of the seeds in air at 300 K if the pressure drop across a 0.2 m deep bed at minimum fluidisation is

1568 Pa.



13.4



Fluidisation



363



At 300 K the density and viscosity of air are 1.177 kg m−3 and 1.846 × 10−5 Pa s, respectively.

The voidage at minimum fluidisation is obtained from Eq. (13.47), thus

1568 = (1 − εmf )(1600 − 1.177)9.81 × 0.20

from which

εmf = 0.50

The Galileo number can be calculated from Eq. (13.54) to give

3



Ga =



1.177(1600 − 1.177)9.81(2 × 10−3 )

2



(1.846 × 10−5 )



or

Ga = 4.333 × 105

Substituting Ga and ε mf into the Ergun equation yields the quadratic expression

4.333 × 105 = 600Remf + 14Re2

mf

from which the positive root is Remf = 155.8. Clearly the negative root has no physical significance

and can be ignored. Thus

155.8 =



1.177umf 2 × 10−3

1.846 × 10−5



and the minimum fluidising velocity is then

umf = 1.22 m s−1



Example 13.16

Compare the Carman–Kozeny and Ergun relationships for determining the minimum fluidising velocity in air of the following particles: (a) surface-volume mean diameter = 600 μm, particle density =

2400 kg m−3 ; (b) surface-volume mean diameter = 9 mm, particle density = 1200 kg m−3 . In each

case assume that the bed voidage at minimum fluidisation is 0.45 and that the density and viscosity of

air are 1.1 kg m−3 and 2 × 10−5 Pa s, respectively.

For the 600 μm diameter particles the Carman–Kozeny equation gives the minimum fluidising

velocity as

2



umf =



(0.45)3 (2400 − 1.1)(600 × 10−6 ) 9.81

180(1 − 0.45)2 × 10



−5



or

umf = 0.389 m s−1



m s−1



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The Galileo number is

3



Ga =



1.1(2400 − 1.1)9.81(600 × 10−6 )

(2 × 10−5 )



2



Ga = 1.398 × 104

which on substitution into the Ergun equation gives

1.398 × 104 = 905.4Remf + 19.20Re2

mf

The positive root is Remf = 12.25 and therefore

12.25 =



1.1umf 600 × 10−6

2 × 10−5



from which

umf = 0.371 m s−1

In other words there is good agreement between the Carman–Kozeny and Ergun models for the

600 μm particles. However, for the 9 mm diameter particles the Carman–Kozeny equation gives

2



umf =



(0.45)3 (1200 − 1.1)(9 × 10−3 ) 9.81

180(1 − 0.45)2 × 10−5



m s−1



or

umf = 43.8 m s−1

This is an unrealistically high velocity. In contrast, using the Ergun model, Ga = 2.358 × 107 and

2.358 × 107 = 905.4Remf + 19.20Re2

mf

This gives a Reynolds number of Remf = 1084.9 and therefore

1084.9 =



1.1umf 9 × 10−3

2 × 10−5



from which

umf = 2.192 m s−1

Example 13.16 shows that for large particles the kinetic energy losses far outweigh the losses due

to viscous drag. For large food particles, such as peas and other vegetable pieces in a fluidised bed

freezer or drier, the Carman–Kozeny equation is wholly inadequate to predict minimum fluidising

velocity and the Ergun equation must be used. However, this is a little cumbersome and it is possible

to ignore the first term in Eq. (13.53) which then reduces to

Re2 =

mf



3

εmf Ga

1.75



(13.56)



13.4



Fluidisation



365



For the 9 mm diameter particles of Example 13.16, Eq. (13.56) gives the minimum fluidising

velocity as 2.238 m s−1 which is a good approximation to that predicted by the full Ergun equation.

(d) Semi-empirical correlations: Probably the most useful and accurate of the many semi-empirical

equations available is that due to Leva:

d1.82 (ρS − ρf )0.94

umf = 0.0079

(13.57)

μ0.88

This equation allows the prediction of minimum fluidising velocity from a knowledge of the mean

particle diameter, the particle density, the density of fluidising medium and the viscosity of fluidising

medium (SI units).



13.4.3 Gas-Solid Fluidised Bed Behaviour

(a) Influence of gas velocity: As the superficial gas velocity is increased beyond the minimum fluidising velocity a greater proportion of gas passes through the bed in the form of bubbles, the bubbles

grow larger, particle movement is more rapid and there is a greater degree of ‘turbulence’. Fluidising

gas velocity is the single most important variable affecting the behaviour of a bed of given particles

and it is expressed usually as either

(i) multiples of umf , for example uu = 3 implies that the gas velocity is three times that required for

mf

minimum fluidisation, or as

(ii) excess gas velocity, u − umf for example u − umf = 1.2 m s−1 implies that the gas velocity is 1.2

m s−1 greater than that required for minimum fluidisation.

As an approximation, it may be assumed that all the gas over and above that required for minimum

fluidisation flows up through the bed in the form of bubbles. This is the assumption of the ‘two-phase

theory’; in other words the bed consists of a bubble phase and a dilute or lean phase of fluidised solids.

If the total volumetric flow of gas is Q then, according to the two-phase theory,

Q = Qmf + QB



(13.58)



where Qmf is the volumetric flow of gas at minimum fluidisation and QB is the volumetric bubble

flow rate. In practice, proportionately more gas flows interstitially (i.e. between the particles) as the

velocity is increased than at umf . In addition, there is a limited interchange of gas between the bubble

phase and the dilute phase. As the gas velocity is increased further the very smallest particles are

likely to be carried out of the bed in the exhaust stream. This is because at any realistic fluidising gas

velocity the terminal falling velocity of the very smallest particles will be exceeded. The loss of bed

material in this way is known as elutriation and will increase as uu increases. Further increases in gas

mf

velocity result in greater elutriation and a more dilute concentration of the solids remaining in the bed.

Eventually all the particles will be transported in the gas stream at the onset of pneumatic conveying.

(b) Geldart’s classification: Geldart suggested classifying fluidised particles into four groups. This

classification is shown diagrammatically in Fig. 13.13 in the form of a plot of the density difference

between particle and fluid against mean particle size. Group A particles are typically between 20 and

100 μm in diameter with a particle density less than 1400 kg m−3 . These particles exhibit considerable bed expansion as the fluidising velocity increases and collapses only slowly as the velocity is

decreased. In other words they tend to retain the fluidising gas. On the other hand the larger and denser

group B particles (40–500 μm, particle density in the range 1400–4000 kg m−3 ) form freely bubbling

fluidised beds at the minimum fluidising velocity. This is the classical fluidised bed behaviour; group

B particles can be defined by



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Fig. 13.13 Geldart’s classification of fluidised particles (Source: Adapted from Geldart, 1973)



(ρS − ρf )1.17 ≥ 9.06 × 105



(13.59)



Groups A and B are the most frequently encountered classes of particles, giving stable fluidisation.

Group C, with particle diameters below 30 μm, consists of cohesive powders which display a

tendency to agglomerate and are very difficult to fluidise. This group is characteristic of a number of

food materials such as flour or very fine spray-dried particles. Large particles with a mean diameter

greater than 600 μm and a density above 4000 kg m−3 are classed as Group D. Such particles display

very little bed expansion and generally give unstable fluidisation; channelling of the gas is prevalent.

Food examples include seeds and vegetables pieces. Group D particles can be defined by

(ρS − ρf )d2 ≥ 109



(13.60)



13.4.4 Bubbles and Particle Mixing

The size of bubbles at the bottom of the bed is determined by the nature of the distributor and they

grow as they rise through the bed. Small bubbles are overtaken by larger ones and coalescence takes

place at the base of the larger bubble. Consequently, the number of bubbles at any bed cross-section

decreases with bed height. Small bubbles are approximately spherical but with a slight indentation

at the base. However, the size of the indentation increases with bubble diameter and the bubbles

take on the characteristic shape shown in Fig. 13.14. As gas velocity increases the nature of the

bubbles changes. Especially in deep beds, the bubbles grow to the size of the bed container and

push plugs of material up the bed as they rise. The particles then stream past the slugs at the bed

walls on their downward path. This is known as slugging and is to be avoided in food processing

applications.

Particle mixing in a fluidised bed is brought about solely by the movement of bubbles. As a bubble

rises in the bed (Fig. 13.15) it gathers a wake of particles and then draws up a spout of particles behind

it. The wake grows as the bubble rises and a proportion of it may be shed before the bubble reaches

the bed surface. Growth and shedding of the wake may be repeated several times in the life of a single