Yang Fluidization, Solids Handling, and Processing

506 Fluidization, Solids Handling, and Processing

For G/S particle systems, enhancement in convective heat transfer is achieved at the expense of increased pressure drop in moving the gas at higher velocities. A measure of the relative benefit of enhanced heat transfer to added expenditure for fluid movement can be approximated by an effectiveness factor, E, defined as the ratio of the heat transfer coefficient to some kind of a pressure drop factor. For G/S systems in which particles are buoyed by the flowing gas stream, this pressure drop factor is expressed by the Archimedes number Ar, and E can be written

E = Nu Pr-1/3/Ar

By definition

and for fixed bed operation, it is well known that j f. Therefore,

E = 1/Re

showing that E drops in inverse proportion to increased flow. This is shown schematically by the downwardly directed curve on the left-hand side of Fig. 9. For liquid/solid (L/S) fluidization, experimental mass transfer data indicated that the transfer factor Sh Sc -1/3 remained essentially constant as liquid velocity varied all the way from incipient fluidization to free fall. For G/S fluidization, however, as soon as the particles start to fluidize, gas bypassing through the formation of bubbles induces a sudden drop in the transfer factor. Recovery in the efficiency of G/S contacting starts somewhere in the vicinity of the transition to turbulent fluidization, and continues into the regime of pneumatic transport or free fall, where the transfer factor could even exceed that for the single particles due to the turbulence caused by the proximity of neighboring particles.

These considerations of contact efficiency and pressure drop in relation to bubbles in fluidization points to an area of endeavor where bubbles are absent.

Figure 9. Effectiveness factor for particle-fluid systems. (Kwauk and Tai, 1964.)

3.2Species of Bubbleless G/S Contacting

The following four species of bubbleless G/S contacting will be dealt with in this chapter:

(i)Dilute-phase fluidization:

-countercurrent G/S flow with cogravity fall of particles

-cocurrent, or transport, irrespective of direction of flow

(ii)Fast fluidization:

cocurrent, countergravity transport, with continual replenishment of solids at bottom by recycle from top

(iii)Shallow fluid bed:

the region immediately above a distributor where bubbles have not yet taken shape

(iv)Fluidization with no net fluid flow:

-periodic fluidization through jigging

-levitation by fluid oscillation

508 Fluidization, Solids Handling, and Processing

4.0DILUTE RAINING FLUIDIZATION

One method of improving G/S contacting consists of showering solids in dilute suspension from the top into an upflowing gas stream. Experiments verified that gas/solid heat transfer coefficient for such a system is essentially the same as for the discrete particles, and that pressure drop for gas flow is extremely low.

4.1Raining Particles Heat Exchanger

Figure 10 shows the differential heat exchange between a gas and a solids stream flowing countercurrently and cocurrently. The efficiency of the heat transfer equipment is to represented by the number of heat transfer stages

where τ = GCp /SCs is the relative flowing heat capacity between the gas and the solids. Integration gives

Thus a single stage, NH = 1, corresponds to a solids temperature rise (or drop) equal to the average gas-solids temperature difference, as shown in the lower left-hand side of Fig. 10. On this basis, it is convenient to define equipment efficiency in terms of a unitary heat transfer stage:

–Unitary heat transfer time, θ/NH

–Unitary heat transfer distance, z/NH

–Unitary heat transfer pressure drop, P/NH

For good equipment performance, it is, therefore, desirable to look for a low height z/NH and low pressure drop P/NH..

510 Fluidization, Solids Handling, and Processing

Comparison with the integrated form for NH given yields the following expressions for G/S heat exchange

1− e − NH (1−τ )

countercurrent η = 1−τe− NH (1−τ )

1 −e− NH (1−τ )

cocurrent η =

1+τ

The particles, while exchanging heat with the flowing gas stream, are in accelerative motion in accordance with the equation

where, for uniform spherical particles having a diameter of dp

Integration of the above equation can be represented by three dimensionless numbers

in which Re0 is the Reynolds number for gas flowing at velocity uo, Res is the slip velocity between the particles and the fluid, and m is the exponent with which pressure drop varies with velocity: P um.

These three dimensionless numbers all involve what can be called the acceleration integral

While the particle is experiencing the accelerating motion as described above, heat is being transferred between it and the surrounding gas stream also in an unsteady state:

(π d 3p / 6)ρ s C s dTs = (π d 2p )h(Tg -Ts )dθ

Integration of this equation can be expressed in terms of the dimensionless numbers used already

θ1

N H = ò0 Nu dθ1

512 Fluidization, Solids Handling, and Processing

where

K = (6/Pr) (Cp/MCs)

and the Nusselt number Nu can be correlated to the Reynolds number by the Kramers (1946; also known as the Ranz-Marshall) relation

Nu = a + b Req

Thus the number of heat transfer stages can be expressed in terms of the acceleration integral

NH = (K/τ) [a òF (0) + b òF (q)]

For G/S heat exchange, altogether eight cases may be differentiated, according to whether the operation is countercurrent or cocurrent, whether the solids are being heated or cooled, and whether the value of τ is less or greater than unity.

The three unitary heat transfer parameters descriptive of equipment efficiency can now be redefined in terms of the above dimensionless integrals:

Unitary heat transfer time:θ / NH = ( d 2p ρs /μ) (θ1 / NH)

Unitary heat transfer distance: z / NH = (dp ρs /ρf) (Z1 / NH)

Unitary heat transfer pressure drop: P / NH = z (Φ / NH)

A method has also been developed for treating polydisperse particles (Kwauk, 1964c).

4.2Experimental Verification

Experiments for verifying the efficiency of heat transfer in the dilute phase were carried out in the equipment shown in Fig. 11 (Kwauk and Tai, 1964). It consisted of two vertical heat transfer columns, i.d. = 300 mm for