Scaling relationships for fluidized beds

SCALING

RELATIONSHIPS LEON

Department

of Mechanical

Engineering,

independent

nondimensional

BEDS

R. GLICKSMAN

Massachusetts Institute of Technology, U.S.A. (Received

Abstrati-The

FOR FLUIDIZED

Augusr

parameters

Cambridge,

MA 02139,

1982) governing

the dynamics

of

fluklized

beds

are

governing fluid and particle behavior. The parameters include the Reynolds number, the Froude number, the ratio of solid to fluid densities, bed geometric ratios, particle size distribution and shape. At low Reynolds numbers, corresponding to a particle diameter of about 0.2 mm at ambient air conditions, fluid inertial effects can be neglected. At high Reynolds numbers, corresponding to a particle diameter of about 2 mm at ambient air conditions, viscous effects can be neglected. These limits change as the fluid pressure and temperature are varied. By proper adjustment of the length Kales, the particle density, and the bed sup&&al velocity it is possible to obtain exact similitudebetween geometrically similar beds fluidizedby dissimilar fluids. found

by

nondimensionalizing

the

differential

equations

1. INTRODUCTION

To properly design a fluidized bed combustor or reactor the fluid dynamics of the bed must be well understood. For example, in a bubbling bed the size and frequency of bubbles control particle mixing and gas exchange between the bubble phase and the dense phase.. The flow regime boundaries must he well known to allow confident design of the bed. Although the fluid mechanics is crucial to the proper design of large beds, there is a dearth of information currently available for large beds, especially for beds with large particles at elevated temperature and pressure. Because of the complexity of the phenomena, it is unreaJistic to expect a theoretical solution for the overall bed behavior based on iirst principles. There is a large body of experimental data taken on small beds operated at low velocities with small particles. There arc a few investigations of somewhat larger laboratory beds, up to about 1 m2 crosssection, run at ambient temperature and pressure, see e.g. Whitehead[l], Cranfield and Geldart[Z], Fitzgerald[3] and Staub et ul.[4]. It is not clear how results from larger laboratory beds can he applied to commerical beds. In many cases commercial beds operate at elevated temperature with fluid density and viscosity far from ambient conditions. In addition, wall effects in the test bed may still be much more predominant than they are in the full-scale beds. Detailed fluid dynamic testing can be done much more conveniently in beds at ambient temperature and pressure. However, there must be a technique to confidently relate these results to beds at the actual operating conditions. The purpose of this work is to systematically develop a set of scaling laws which can be used to properly design a bed at ambient conditions which is

an accurate model of a bed at elevated pressure and temperature. In addition, the scaling laws will yield a set of general independent dimensionless parameters which can be used to characterize flow regimes and fluid dynamic behavior of any fluid&d bed. PAW WORK

The literature is repleat with numerous parameters, dimensional and non-dimensional, which have been proposed to characterize the fluid dynamics of a fluid&d bed. For bubbling beds, the use of the velocity in excess of that for minimum fluidization, Cl0- U,+ has been used to characterized the bubble flow; this is based on an over-simplified form of the two phase hypothesis[S]. Correlations have been given based on the ratio of u,,/u,~ The Froude number based on the minimum velocity, (u,$/dpg), has been proposed as the parameter to characterize the boundary between particulate and aggregation fluidization[6] and the Archimedes number has been used to correlate a wide array of phenomena[7]. Geldart established a different set of parameters which he used to characterize the fluidizing characteristics of particles which he demonstrates can be. written in terms of dimensionless parameters[8]. Romero and Johanson[9] suggested four nondimensional groups to characterize the quality of fluidization, they are the Froude number and Reynolds number, both based on minimum fluidization velocity, the ratio of solid-to-fluid density, and the ratio of bed height at minimum Buidization to the hed diameter. Broadhurst and Beeker[lO] developed a list similar to that of Romero and Johanson except Broadhurst used the superficial velocity in place of the minimum fluidization velocity. This was developed from the pi theorem. These parameters were used by Broadhurst

1373

L. R. GLKXC~MAN

1374

and Becker to determine the criteria for minimum fluidization. However they did not realize the potential for using the parameters for scaling beds with different fluid conditions. Recently, Fitzgerald and Scharf [l l] have mentioned the same parameters in conjunction with bed scaling. However a systematic development has not been presented, nor has the limiting assumptions inherent in the proposed parameters been set out. In the next section the general equations of motion for a fluidized bed will be nondimensionalized. This will yield the governing nondiiensional independent parameters and the general nondimensional dependent variables. The application of these parameters, their limitations, and the method of verification will be discussed in subsequent sections. In the companion paper to this one, an experimental verification of the scaling laws will be presented.

where Tis the unit vector in the vertical direction and the drag force between the fluid and the particles is represented by fi(rS - 6). Note that in general fi is not a constant, rather it must be found from a general expression for the drag. The equation of motion of the particles becomes, &(1--E)

[

1

+Sp,g(l-C)-/?((P-6)

;;+,_.grad,_

-0.

(4)

The boundary conditions must also be specified for the bed. The velocities of the particles and the fluid must be specified at all boundaries. For example, at the base of the bed with a porous plate distributor with uniform flow, I =iii/(l

--d)

(5)

u=o DERIVATION

OF SCALING

(6)

RELATIONSHIPS

When the equations governing a particular phenomena can be written, the most insightful way to derive the scaling relationships is to nondimensionalize the governing equations[l2]. Thus, the equations reveal useful information even though they cannot be solved in general. Governing equations The equation of motion and conservation of mass can be written for both the fluid and the particles. The governing equations are devetoped similar to those presented by Jackson[l3]. For simplicity the fluid will be assumed to be incompressible, although the scaling results do not depend on this condition. For the fluid the conservation of mass becomes, div (~6) = 0

(1)

where 6 is the void fraction, the volume fraction of the bed occupied by the fluid, and U is the gas velocity in vectorial notation. Note that c can, in general, vary throughout the bed. For the particles, the conservation of mass is,

where LQis the superficial fluid velocity and A is the voidage of the porous plate. At the walls, zi and 6 are zero and above the bed U = it+ and d = 0. Similarly, conditions for the pressure must be specified at the boundaries: e.g. p =pO at the distrihutor. The equations of motion and continuity are far too difficult to solve especially for aggregative fluidization where L varies throughout the bed. However, by nondimensionalizing the governing equations, the governing non-dimensional parameters can be found. The velocities will be nondimensionalized with respect to the superEcia1 gas velocity 6’ = C/uO;li’ = ii/uQ additional dimensionless

quantities

V’ = dpV, t’ =;

(7)

are: t.

(8)

I, The nondimensional tions are simply,

form of the continuity

equa-

(2)

div (&‘) = 0

8)

where B is the particle velocity. The equations of motion for the fluid and the particles require some care. In order to write explicit expressions for all of the forces, the inter-particle forces will be omitted. These forces include those due to particle-to-particle collisions as well as electrostatic forces. The consequences of this omission will be taken up in a later section of the paper. The equation of motion of the fluid becomes,

div (~a’) = 0

(10)

div [(1 - c)c?] = 0

To nondimensionalii the equations of motion each term will be multiplied by the term dp/psz+,2. For the particles the equation of motion becomes,

-~($_8’+-0.

p/t gi+J-gradri c

1

(11)

+Tp,gc+gradp+p(ii-8)=0 (3)

The nondimensional

equation

of motion

for the

1375

Scaling relationships for fluidized beds fluid becomes,

. V’)U’

1 (12) , The boundary conditions are written in terms of nondimensional length scales such as D/d,, L/d,, etc. At the distributor, U’=i/(l

-A);iT’=O

(131

at the side walls, a = fj’= 0.

(14)

The boundary conditions for the pressure level can be nondimensionalized by the term P,Jp~‘. From eqns (9)-(12) the controlling non-dimensional parameters can bc identified as,

and the ratios of other geometrical bed dimensions. If the particles are not all the same size, a nondimensional size distribution should also be included. The 6rst non-dimensional parameter represents the ratio of particle drag to inertia forces, and the second parameter represents the ratio of the gravity force acting on the particle to the particle inertia force. The term p0/psu2 can be ignored when the fluid velocity is small compared to sonic velocity or the absolute pressure does not change enough to influence the thermodynamic properties of the fluid; the term will be omitted in this paper. The coefficient B is not a quantity set independently, rather /I must be related to the other bed properties. There are two extremes, if the particles are closely spaced, approaching that of a packed bed, the Ergun expression written in terms of the interstitial velocities is appropriate,

+ 1.75?(1

63 This can be rearranged

-c)p,Iii-I7~ p. 4A

(16)

P

+ 1.75(1

P,Wkdp

where $, is the particle

which is of the same order of magnitude as the ratio of inertia to the viscous term in the Ergun equation, eqn (17). Thus, when the ratio of inertia to viscous terms is negligible in the Ergun equation, the fluid inertia is negligible compared to the particle drag term. This approximation takes the upper limit of the gradent of fluid velocity as u/d,. This may only be exceeded in the boundary layer around the particle; if fluid inertia is not important in the drag on the particle as given by Ergun or equivalent equation, it is not important in the boundary layer of the particle. When the Reynolds number is 400 or greater the viscous term is less than one-tenth of the inertia term and (Bd,/p,q,)is only a function of p,/p,, Eand 4,; the gas viscosity is unimportant. Between the two extremes in Reynolds numbers the drag is a function of the terms including both the gas density and viscosity. In the other extreme, when the bed voidage is very large the drag should be of a form similar to the drag on single particles. For the case of spheres, ad, _ 3G~/Ifi--b(l--e) 4 PJ

P&l

L3

In eqn (12) the ratio of the inertia term (the bracketed term) to the last term is,

P.%

to read,

Bd,ds ~- _ 15O41 -6Y

dimensional parameter including p in eqn (15) can be replaced by the Reynolds number, p&,dJp. The first term on the r.h.s. of eqn (17) represents the viscous contribution to the drag and the second term represents the fluid inertia. The viscous term is ten times the inertia term when the Reynolds number, based on the fluid density, @,u,,d,~,/p) is 4 or lower, (assuming 1 - .Sis approximately one-half). For this case the inertia term can be omitted and @?dp/Ppo)is only a function of (p.t+,d+Jp), t and 4,; the density of the gas is unimportant. Note that E is not an independent parameter, it is a dependent variable determined by the bed conditions. The density ratio also appears in the nondimensionalized momentum equation for the fluid, eqn (12). When the viscous term is ten times the inertia term in the Ergun equation, eqns (16) and (17), then

--)pfIlT--ld2

(,,)

E’P, sphericity.

Thus the non-

f%

(201 . ,

In this instance there is also a viscous and a fluid inertia dominated regime, but they occur at Reynolds numbers below about 3 and above 1000 respectively. The nondimensional drag is a funciton of the same dimensionless parameters found when using the Ergun equation. For bubbling beds where the dense phase containing the particles is near the minimum

1376

L. R. GLICKSMAN

Combining the first two terms to produce a term without the supefiical velocity. The governing parameters can be. rewritten as

fluidization state., the drag is more properly represented by the Ergun equation or an equation of similar form. The viscous limit

3,

Looking at the viscous limit determined from the Ergun equation,

P

uo %

i, z, 4,, particle size distribution, bed P P geometry.

The first term is the Froude number, Fr, and the second is a modified form of the Archimedes number, Ar’. In terms of these dimensionless parameters, the more usual parameters used in fluid&d beds become,

if u, is some multiple of the minimum fluidization velocity, u,,~, then, Re=4JVU+!P --

y,

(22) and

u, -=%

and at the viscous limit, with Ed,= 0.5 and 4, = 1,

(24)

Table 1 tabulates the particle diameter at the viscous limit when the particle density is 2.4 g/cm3. Note that at one atmosphere beds with particles of one quarter millimeter or less are within the viscous limit. This may explain the applicability of bubble models such as Davidson and Harrison’s which assume D’Arcy’s Law holds, i.e. only viscous drag forces in the dense phase. At elevated pressures, beds with much smaller particles have inertial as well as viscous forces which are important. For the viscous limit, the governing parameters become, -&, s

u, - Mti’ 0.07 (go)“2

(27)

, -%f_Q&y

(28)

1- e

= 0.07(8-;a

Inertial hit At high particle

(Air)

(mm)

Particles of this diameter or smaller are in the viscous limit

dp) Inertial

15

0.2

2.6

1

8OU

0.6

7.3

10

15 800

P*

=

Uo/umf

=

0.09

1.3

0.3

3.4

2.4 3

o/cd

BOW

(mn)

Particles of this diameter or larger are in the inertial limit

1

10

the viscous the gas are In this limit, bed dynam-

The minimum size of the particle diameter at the

dJViscous

T(*C)

(29)

sd, Pf &, $, $, particle size distribution, bed T p’ S p p geometry

Table 1. Limits for viscous dominated flow and inertial dominated Gas Conditions

(D,dp)l’z.

Reynolds numbers drag forces between the particle and negligible compared to the inertia forces. the fluid viscosity is unimportant to the ics. The governing parameters become,

4, $, t$,, particle size distribution, bed PPP geometry.

P (ati)

(26)

where U, is the terminal velocity of the particle. Other flow regime relationships such as Stewart’s slugging criterion can be rewritten in terms of the dimensionless parameters;

Combining eqns (22) and (23) the limiting value of the particle diameter becomes.

gd -$,

1650 J&z

1377

Scaling relationships for fluidized beds

.Ol

IO

1

0.1

PARTICLE

DIAMETER

dp

100

(mm)

Fig. 1. Regimes where inertia forces and viscous forces can be neglected in a bed fluidizad by air at ambient temperatures. inertial limit can be found in a development similar to that for the viscous limit. It is found that dPm has the same form ad dPm.._ and the former is about ten times larger than tb.e latter for the same particle density and gas properties, see Table 1.t At one atmosphere d,,,, is approximately 2.6mm and 7.3mm for bed temperatures of 15°C and SOO”C, respectively. For example, the dynamics of an atmospheric bed at 800°C with particles of 1.5 mm diameter is primarily controlled by viscous forces, while the dynamics of a bed with the same sized particles run at room temperature is primarily controlled by inertial forces. Thus, modelling of a heated bed by use of a room temperature bed with particles of the same size and density leads to results which may not be even qualitatively similar. Figure 1 illustrates the influence of pressure and y/u_, on the viscous and inertial limits at ambient temperature. Iniermediaie region Between the upper and lower Reynolds number limits, both the viscous and inertial forces are important to the fluid dynamics. In this regime no simplification can be made to the number of governing parameters, both the gas density and viscosity are important. The nondimensional parameters can be written as, pN,dp’g 2,

P

q2 P/ L D ga’ ;;’ ;r’ a’ q3,, particle size distirbution, P

s P P

bed geometry.

(30)

To attain completely similar behavior between a hot tFor

the inertial

limit., a Re value of

determine the upper limit.

1000 was used to

bed and a model at atmospheric temperature the value of each non-dimensional parameter must be the same for the two beds. If the hot bed is in the viscous or inertial dominated region the requirements are less stringent since there are fewer controlling nondimensional parameters. In the most general case all the parameters must be considered. Dependent variables When all the independent nondimensional ters are set, then the dependent variables are fixed. TrneThe dependent variables include the fluid and particle velocities throughout the bed, the pressure distribution and the voidage distribution in the bed. From the voidage distribution it follows that the bubble size and distribution must also be a dependent variable. If two beds are designed and operated to have identical values of all of the independent nondimensional parameters, then the dependent nondimensional variables of the two beds must also be identical at every location within the beds. Kline[l2]. The variables include the ratio of the bubble diameter to bed length or particle size and the ratio of the particle velocity to the superficial bed velocity. Note that the time (and frequency) scales will also be related between the two beds when properly nondimensionalized, see eqn (8). The governing equations of motion can be written treating the particles individually without invoking the assumption of a continuum for the mass of particles in the bed. Although it is unrealistic to expect a solution for the resulting set of equations describing a collection of individual particels making up a tluidized bed, the results of a dimensional analysis is instructive. At this level of detail, variation in voidage or voidage gradients occuring at bubble

L. R. GLXXSMAN

1378

boundaries and elsewhere can be dealt with directly. Interparticle forces due to mechanical collisions can now be allowed. For the fluid between the particles, the nondimensional equation of continuity becomes, divti’=O

(31)

while for a Newtonian fluid, the equation of motion, the Navier-Stokes equation becomes

where the latter is the Navier-Stokes. The equation of motion for the ith particle with diameter di becomes, (33) Fi is a function of the fluid density and viscosity, the relative velocity of the particle and the fluid surrounding it, the location of the particle and the fluid surrounding particles and possibly the velocity of the particle relative to the velocity of the surrounding particles. The motion of the surrounding particles is given by the respective solution of eqn (33) for each particle. Thus the motion of all particles must be solved simultaneously. This task is far too formidable, but the pi theorem will permit the identification of the independent dimensionless parameters. The independent parameters controlling F, are the fluid density and viscosity, the superficial bed velocity, the shapes and diameters of the individual particles (as they influence the solution of the motion of surrounding particles, which are dependent variables), and the bed dimensions. Using the pi theorem the complete list of indepcnTable

2. Scaling

factors

dent nondimensional parameters is the same as that given in eqn (30). Since these parameters govern the behavior of the bed viewed as discrete particles, properly scaled beds should exhibit similar behavior in regions of large voidage gradients, i.e. near bubbles or void boundaries.

for proper modeling

SCALING

BETWEEN

DIFFERENT

F’LulD

BEDS WITH PROPERTIXYS

As pointed out previously, to design an accurate scale model of a given bed all of the independent nondimensional parameters must be identical. Consider the case where the bed in question is operated at an elevated temperature (e.g. 800”) and one atmospheric pressure with air. The scale model is to he operated with air at ambient pressure and temperature. The fluid density and viscosity will be significantly different between these two gas conditions, e.g. the gas density of the cold bed is 3.5 times as large as the density of the hot bed. Then to maintain a constant ratio of particle-to-Ruid density, the density of the solid particles in the cold bed must be 3.5 as large as the density in the hot bed. With the solid density of the model determined, the Archimedes number is used to determine the particle diameter of the model and the Froude number is used to determine the superficial velocity. Table 2 presents the proper scaling between the hot bed and the ambient model. Note similitude requires the two beds be geometrically similar in construction with identical normalized particle size distributions and sphericity. The length scales of the ambient temperature model are one-quarter of the heated bed. Thus, a modest size ambient bed can simulate a rather large hot bed at one atmosphere pressure. In the case where both inertial and viscous forces are important and the bed and its scale model have the same fluid properties, the model and the bed must have identical physical properties. However, in the viscous or inertial dominated limits, a scale model

of hot bed performance

(hot and cold beds at one atmosphere

pressure)

Hot

Bed Variable (800°C)

Superficial Velocity

uaoo

Particle Diameter

d

p 800

Particle

Scaled

Cold Bed (15'C)

Variable

o.5

‘a00

o.25

dP

800

3.5

Density

Ps

P5

800

Bed Dimension

L

800

0.25

L

8oo

Red Diameter

n

800

0.25

D

*oD

Time

t

800

0.5

t 800

Frequency

f

800

2

f

800

800

Sealing relationships for fluid&d beds be constructed operating with the same gas properties as the bed in question but different values of particle diameter, supetl%ial velocity and particle density. In this case the scaling criteria requires identical values of a reduced number of independent nondimensional parameters. Limitations of the scaling laws The scaling laws described here are only as valid as the governing set of equations. As mentioned at the outset, inter-particle forces other than mechanical forces due to collisions were omitted in the momentum equations. Thus, similarity will not exist, nor are the list of independent parameters complete, when such inter-particle forces are important. Fine particles may have significant electrostatic forces. In recent work in our laboratory, the addition of an anti-static spray to 0.7 mm glass particles in a Plexiglas bed operated with very low humidity air caused the minimum fluidization velocity to change by 33O,[ lo]. Electrostatic forces should become less important as the particle size and density is increased. An additional cause of concern is the simulation of a hot fluidized bed operation when the entering fluid stream is not at the same temperature as the bed. In the region of the distributor the incoming fluid experiences a rapid temperature change as well as a rapid change in its viscosity and density. Exact simulation of the distributor zone cannot be done in an ambient temperature bed fluidized by ambient temperature fluid. CONCLUSIONS Nondimensionalizing the governing equations of motion for fluids and solids in a fluidized bed will yield the governing independent nondimensional parameters. When the Reynolds number, based on the particle diameter and fluid properties is small, fluid inertial effects are unimportant and the number on non-dimensional parameters can be reduced by omission of the fluid density. Similarly, at high Reynolds number, the flnid viscosity is unimportant. In the general casz it is possible to obtain an exact match between geometrically similar beds operating with different fluid properties by properly adjusting the length scale, the particle density and the superficial velocity. The two beds will have identical values of the dependent variables such as bubble size, frequency and fluid velocity when these variables are expressed in dimensionless form. The governing parameters were derived neglecting some inter-particle forces. Electrostatic forces should be important for fine particle beds. Acknowledgements-This work was sponsored by the Tennessee _ Valley Authority, this support is gratefully aeknowl-

1379 NOTATION

can

particle diameter bed diameter f bubble frequency g acceleration due to gravity L bed height or bed dimension P pressure 1 time ti flnid velocity umf minimum fluidization velocity u, superficial velocity particle velocity coefficient of drag force, see eqn (3) ; A distributor plate voidage t bed voidage P fluid viscosity Pf fluid density P. particle density spheric&y 9, Ar’ modified Archimedes number @:dp’g/pz) Fr Froude number (gdp2/u,,*) Reynolds number @,d,h/p) (“;: dimensionless variable

4 D

REFERENCES

[t] Whitehead A. B. and Young A. D., Proc. Int. Symp. Fluidization, Eindhoven. Netherlands University Press 1967. [Z] Cranfield R. R. and Getdart D., Chem. Engng Sci. 1974 29 935. [3] Jovanovic G. N., Catipovic N. M., Fitzgerald T. J. and Levenspiel O., Fluidization (Edited by Grace J. R. and Matsen 1. M.). Plenum Press, New York 1980. [4] Staub F. W.. Two-phase f?ow and heat transfer in puidized beds. Final Report, EPRI CS-1456 1980. [S] Valenzuela J. A. and Glicksman L. R., Gas flow distribution in a bubbling fluidized bed. AIChE Symp. on Fundamentals of Huidization and Fluid-Particle Systems, Paper 62b. New Orleans 1981. [6] Wilhelm R. H. and Kwauk M., Chem. Engng Prog. 1948 44 201. [7l Zabrodsky S. S. Hydrodynamics and heat Tramfer in Fluidized Beds. The MIT Press, Cambridge 1966. [8] Baeyens J. and Geldart D. Proc. Int. Symp. on Fluidizotion and its Applications. Toulouse 1973. [9] Romero J. B. and Johanson L. N., Chem. Engng Prop. Symp. Ser. 1962 58(38), 28. [lo] Broadhurst T. E. and Becker H. A., Proc. Int. Symp. St&. Chimie Indtut. Toulouse, 1973. [l l] Fitzgerald T., Personal communication. [12] Kline S. J., Similitude and Approximation Theory. McGraw-Hill, New York 1965. [13] Jackson R., FIuidization (Edited by Davidson J. F. and Harrison, D.). Academic Press, New York 1971. [14] Valenzuela J., Ph.D. Thesis, MIT Department of Mechanical Engineering, 1982. [ 151 Potter 0.. Fluidization (Edited by Davidson J. F. and Harrison D.). Academic Press, New York 1971. 1161 Rowe P. N. and Partridge B. A., Trans. Inst. Chem. .^__ _^ -___