Correlaciones de Transferencia de Masa Para Hacer Girar Biorreactores de Tambor

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Journal of Biotechnology 97 (2002) 89101

Mass transfer correlations for rotating drum bioreactors

Matthew T. Hardin a,*, Tony Howes b, David A. Mitchell c

a Chemical and Materials Engineering Department, Uni6ersity of Auckland, Auckland, New Zealandb Department of Chemical Engineering, Uni6ersity of Queensland, St Lucia, Qld 4072, Australia

c Departamento de Bioqumica, Uni6ersidade Federal do Parana, Cx. P. 19046, 81531-990 Curitiba, Parana, Brazil

Received 6 December 2001; received in revised form 18 March 2002; accepted 27 March 2002

Abstract

Evaporative cooling is extremely important for large-scale operation of rotating drum bioreactors (RDBs). Outlet

water vapour concentrations were measured for a RDB containing wet wheat bran with the aim of determining the

mass transfer coefficient for evaporation from the bran bed to the headspace. Mass transfer was expressed as the mass

transfer coefficient times the area for transfer per unit volume of void space in the drum. Values of ka% were

determined under combinations of aeration superficial velocities ranging from 0.006 to 0.017 ms1 and rotation rates

ranging from 0 to 9 rpm. Mass transfer coefficients were evaluated using a variety of residence time distributions

(RTDs) for flow in the gas phase including plug flow and well-mixed and a Central Jet RTD based on RTD studies.

If plug flow is assumed, the degree of holdup at low effective Peclet ( Peeff) numbers gives an apparent under-estimate

of ka% compared with empirical correlations. Values of ka% calculated using the Central Jet RTD agree well with values

of ka% from literature correlations. There was a linear relationship between ka% and effective Peclet number:ka%=2.32103Peeff. 2002 Elsevier Science B.V. All rights reserved.

Keywords: Mass transfer; Rotating drum; Solid state fermentation; Scale-up; Sherwood number

Nomenclature

area available for mass transfer per unit of void volume (m1)a

area available for mass transfer per unit of headspace (m1)a %

dimensionless water concentration calculated from Eq. (13)cC water concentration in dry air (kg kg1)

dimensionless water concentration in headspacecairwater concentration in dry air of headspace (kg kg1)CAIR

cbran dimensionless concentration of water at the bran surface

www.elsevier.com/locate/jbiotec

* Correponding author.

E-mail address: [email protected] (M.T. Hardin).

0168-1656/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 1 6 5 6 ( 0 2 ) 0 0 0 5 9 - 7
mailto:[email protected]:[email protected]


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M.T. Hardin et al. /Journal of Biotechnology 97 (2002) 8910190

cdead dimensionless concentration of water in the dead zone of the drum in the Central Jet RTD

water concentration in dry air entering the drum (kg kg1)CINwater concentration in dry air exiting the drum (kg kg1)COUT

cplug dimensionless concentration of water in the Central Jet of the Central Jet RTD

saturation concentration of water in dry air at the bed temperature (kg kg1)CSATconstant of bed viscosity (dimensionless)CV

D axial diffusion coefficient from the Central Jet RTD (dimensionless)

particle diameter (m)ddrum diameter (m)Ddrum

Dh hydraulic diameter of drum (m)

acceleration due to gravity (ms2)g

h maximum height of the bed (m)

constant in Eq. (3) defined in Eq. (4)K

mass transfer coefficient estimated using the Central Jet RTD (dimensionless)k

area factor used by Stuart (1996) (dimensionless)n

constant of bed porosity (dimensionless)N

effective Peclet number across the surface of the bran (dimensionless)Peeffdegree of exchange between the different regions in Central Jet RTD (dimensionless)QmixReynolds number (dimensionless)Re

s thickness of the mobile layer at the bed surface (m)

Schmidt number (dimensionless)Sc

Sherwood number (dimensionless)Sh

t dimensionless time (number of residence times)

velocity of the particles at the bottom of the moving layer (ms1)u0ueff effective velocity of gas over the surface of the bed (ms

1)

superficial gas velocity through drum (ms1)ugaverage particle velocity in the moving layer (ms1)upvolume of the stagnant region in the Central Jet RTD (dimensionless)Vdeadgas kinematic viscosity (m2 s1)6g

x length along the drum (dimensionless)

rotational speed of the drum (s1)z

h % generic expression for mass transfer area available per unit volume (Eq. (1))

mass transfer coefficient used by Blumberg and Schlunderi

coefficient for fitted correlation of Blumberg and Schlunder

diffusivity of water in air at the particle surface (m2 s1)lgdynamic angle of repose of bed ()k

s generic expression for mass transfer coefficient (Eq. (1))

generic expression for time term (Eq. (1))~

V generic concentration term (Eq. (1))

concentration term in Eq. (1)V1V2 concentration term in Eq. (1)

factor in Eq. (35)x

1. Introduction

Rotating drum bioreactors (RDBs) have poten-

tial to provide better heat and mass transfer

characteristics than solid state fermentation (SSF)

bioreactors with static beds, while providing gen-

tler agitation than bioreactors with internal stir-

rers. Furthermore, the absence of internal moving


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M.T. Hardin et al. /Journal of Biotechnology 97 (2002) 89101 91

parts for mixing makes design, construction and

operation simpler, and the low pressure-drop

across the bioreactor greatly reduces the operating

costs associated with the aeration system. The

gentle agitation associated with the tumbling mo-

tion of the substrate bed minimises damage to the

substrate particles, and makes RDBs ideal for

those micro-organisms which can tolerate some

shear damage, but which are affected deleteriously

by more vigorous mixing.

A major hurdle that currently prevents wide-

spread use of SSF is the difficulty of regulating

the temperature in large-scale fermentations. In

an RDB, heat exchange, and hence removal of

metabolic heat, is limited to convective cooling

(from the surface of the drum to the surroundings

or from the substrate to the headspace air) and

evaporative cooling. As the fermentations increasein scale evaporative cooling becomes more impor-

tant because the ratio of heat produced to surface

area for convection declines. One of the main

advantages of the mixed bed in an RDB is that it

is practical to add water to the substrate during

the fermentation, for example, by spraying a fine

mist onto the surface of the moving bed (Barstow

et al., 1988). This means that evaporative cooling

can be used as the major mechanism of heat

removal without the danger of restricting growth

due to drying out of the substrate.

Since mass transfer is directly proportional to

the surface area available for particle/gas contact,

it follows that any activity that improves this

surface area will improve mass transfer. Both

increasing the rotation speed and adding lifters

will improve particle/gas contact. Evaporative

mass transfer in rotating drums has been studied

from the point of view of drying (Friedman and

Marshall, 1949) but attempts to develop gener-

alised models are comparatively recent (Riquelmeand Navarro, 1986). There have been few at-

tempts to quantify these effects for SSF. In the

current work, a series of experiments was per-

formed to quantify the mass transfer as a function

of fill depth, rotation speed and air inlet.

Mass transfer is typically characterised by an

equation of the form:

#V

(~=sh %(V1V2) (1)

where V is a concentration of the component in

question (e.g. mass water per unit mass dry air),

V1V2 is the driving force for mass transfer

(both V1 and V2 are concentration terms), s is the

mass transfer coefficient (length per unit time),which characterises the rate at which transportoccurs in response to the driving force, h % is the

area available for mass transfer per unit volume

being transferred into (area per unit volume) and

~ is an expression of time. All of these terms may

be expressed in a variety of units or even in

dimensionless terms (e.g. time expressed as num-

ber of residence times).

For simplicity it is commonly assumed in rotat-

ing drums that the mass transfer coefficient is

constant over the length of the drum (Blumbergand Schlunder, 1996; Jauhari et al., 1998) and

therefore, the challenge lies in establishing the

driving force as a function of axial position within

the drum. In order to estimate the driving force as

a function of position, it is necessary to employ a

model that describes the flow pattern in theheadspace to estimate the concentration of the

species being transported as a function of posi-

tion. Once a model for the concentration profile

down the length of the drum is established, the

mass transfer coefficient can be found easily.For the purposes of this analysis, it assumed

that the moisture content of the substrate is con-

trolled by water sprays (Barstow et al., 1988).

This means that the driving force for evaporation

depends only on the gas phase water concentra-

tion as the substrate surface is saturated with

water. This implies that film resistance from the

liquid to the gas phase limits mass transfer hence,

the system acts as if it is in the constant-rate

drying period. In practice, many SSFs are carried

out in the falling rate period. This means that

evaporative transfer is limited by internal diffu-

sion of water within the substrate and hence, is

greatly affected by substrate composition and

temperature.

This paper describes an investigation of the

effect on the mass transfer coefficient, expressed

as the lumped parameter ka%, of varying the drum

rotation rate and the velocity of the gas, for a


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drum operating in the slumping and tumbling

regimes during a constant-rate drying period. It

compares three gas flow patterns; plug flow, per-

fect mixing and the Central Jet residence time

distribution (RTD), derived from earlier experi-

ments in our laboratory (Hardin et al., 2001) with

an empirical correlation from literature (Blum-

berg and Schlunder, 1996).

2. Materials and methods

2.1. Experimental procedure

The reactor was a stainless steel rotating drum

(Fig. 1), with inside dimensions of a radius of 280

mm and a length of 800 mm. A variable speed

drive capable of between 0 and 9 rpm turned thedrum.

For simplicity, no organisms were grown. Since

the rate of metabolic heat and water production

varies with phase of growth of the organism and

substrate composition, the use of organisms

would have added complicating factors, making it

difficult to compare different runs solely on the

basis of drying. Damp Defiance type 2032 wheat

bran (47% water wet weight, 2 mm diameter) was

used to imitate one type of substrate typically

used in these processes (Stuart, 1996; Marsh et al.,2000). The bran and water were mixed in the

drum at full speed (9 rpm) for approximately 1 h.

Many granules of bran formed during this time,

and were broken up by hand before the bran was

used for experimental purposes. The bran was

removed from the drum and, after overnight stor-

age at 4 C, was autoclaved at 121 C for 1 h to

simulate the treatment the substrate normally un-

dergoes. The bran was cooled to approximately

ambient temperature before recording of results

began.

The humidity of the inlet air was measured at

the inlet point of the drum while the bran was

cooling. The inlet humidity was checked prior toeach series of trials. While there was variation in

inlet humidity between series of trials, within each

series the inlet air humidity was constant. The

outlet humidity was measured on-line using an

Electro-tech Model 5612B humidity probe and the

relative humidity logged. The relative humidities

were used to find water concentrations in the gas

streams by use of the Antoine equation.

2.2. Analysis of experimental results

The actual surface area available for the mass

transfer is a function of particle size and exposure

of individual particles to the gas stream. Since this

is extremely difficult to calculate, in this analysis

the surface area per unit volume of headspace, is

combined with the mass transfer coefficient k to

give a ka% term. This is very similar to the way

that oxygen transfer is handled in submerged

liquid culture.

The effect of rotation rate on mass transfer

between the bed and headspace will be related to

the flow regimes that correspond to the various

rotation rates (Fig. 2). As the speed of rotation of

the drum is increased, the regime of flow of the

particles changes. At very low speeds the particles

exhibit slumping flow where the solid bed periodi-

cally slips back down the wall. At higher speeds

the material on the top of the bed tumbles back

down the face of the bed in a continuous cascade.

At still higher speeds material is thrown into the

air before landing again (cataracting). Beyond a

certain critical speed there is sufficient centrifugal

force to hold the bed against the wall. The exact

transition between these regimes is a function of

rotation speed particle size distribution and parti-

cle cohesiveness and is difficult to predict. Flow

regimes beyond tumbling were not observed in the

rotational speed range of these experiments.

Fig. 1. Experimental apparatus. The drum is 800 mm long

560 mm diameter. The humidity probes measure the moisture

content of the air as close as possible (within 50 mm) of the

drum inlet and outlet.


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Fig. 2. Flow regimes in rotating drums showing direction of

rotation (anti-clockwise) and the direction of motion for the

substrate bed.

tion of radial gradients. The thickness of the

mobile layer depends on the material, fill depth

and the rotation speed.

Blumberg and Schlunder (1996) studied mass

transfer in rotating drums and developed an im-

plicit equation for the thickness (s) of this mobile

layer:

pz [Ddrum(hs)(hs)2]Ks5/2

+2pzs(Ddrum/2h+s)=0 (3)

where z is the rotation rate in revolutions per s,

Ddrum is the drum diameter, h is maximum bed

height (Fig. 3) and K is a constant that depends

on the material. This equation can be solved fairly

simply by numerical methods. K is a function of

the porosity of the bed and the dynamic angle of

repose of the material (k). K can be calculated as

follows:

K=2

3gsink

CVd21/2

3

5

12

11N+

18

17N2

12

23N3+

3

29N4

(4)

where N is a dimensionless constant factor of

porosity which is equal to 0.8 for most materials,

CV is a dimensionless constant factor of bed vis-

cosity (0.6 for most materials), g is the accelera-

tion due to gravity (ms

2

), k is the dynamic angleof repose of the bed and d is the particle diameter

in metres (Savage, 1979). Blumberg and Schlunder

(1996) give the full derivation of these equations.

Note that Eq. (4) is different from the equation

given in Blumberg and Schlunder due to a typo-

graphical error in that paper. In the current work

the dynamic angle of repose was measured as

3792 through the Perspex end of the drum.

Blumberg and Schlunder (1996) go on to show

that the average velocity of the particles in the

moving layer can then be calculated from:up=Ks

3/2+u0 (5)

where up is the average flow velocity of the parti-

cles in the moving layer and u0 is the velocity of

the particles in the moving layer in contact with

the non-moving bed and is given by:

u0=2pz(Ddrum/2h+s) (6)

The maximum rotation rate used in the experi-

ments was 9 rpm. This did not allow material to

go beyond the cascading flow regime (Fig. 2).

When material is in the cascading regime there is

a mobile top layer that slides down the face of the

bed (Fig. 3). It is assumed that mass transfer

between the headspace and bed occurs only in this

layer as individual particles in this layer are in

rapid motion with respect to the gas phase. Con-

versely, particles deeper in the bed have little

contact with the gas. Movement of particleswithin the bed ensures that, ultimately, the entire

bed is exposed to the air and minimises the forma-

Fig. 3. Detail of drum in the tumbling regime showing the

direction of travel for the mobile layer as well as the nomen-

clature for equations for calculating the mobile layer thickness.


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The effect of superficial air velocity on mass

transfer was also investigated. The air flow rate

is expressed in terms of the superficial velocity

(volumetric gas flow rate divided by unfilled

cross-sectional area). The superficial velocity has

an effect on the RTD of the air in the drum.

The Central Jet RTD of Hardin et al. (2001),

which is summarised below, is used in this paperto determine an estimate for the mass transfer

coefficient based on realistic RTDs. Blumberg

and Schlunder (1996) developed other correla-

tions that are based on the Reynolds number of

the gas flow through the drum and over the

particles. Their correlations are described in Ap-

pendix A.

The overall mass transfer depends on the ef-

fective velocity of the gas over the particles,

which is simply the vector sum of the gas veloc-

ity and the particle velocity (Eq. (5)). If plugflow is assumed, the two are at right angles to

one another for a horizontal drum so the effec-

tive velocity is as given in Eq. (7):

ueff=ug2+up2 (7)

In Blumberg and Schlunder (1996), an effec-

tive Peclet number, Peeff, was calculated on the

basis of the plug flow assumption, regardless of

the real flow pattern. This simplification is help-

ful for scale-up and allows comparison with lit-erature results. The effective velocity is

combined with the diffusion coefficient (lg in

m2 s1) and the particle diameter (d, in m) into

an effective Peclet number, Peeff (Eq. (8)):

Peeff=ueffd

lg(8)

The diffusivities used in this work are derived

from correlations in McCabe et al. (1985).

The Peclet number is a dimensionless group

giving the ratio of convective flow to diffusive

flow. The two extremes for Pe are (represent-

ing a case where diffusion is insignificant) and 0

(representing a case where convective transfer is

insignificant). Peeff is used as a characteristic for

our correlations, as it is a simple description of

the gas flow over the particles. It also allows

simple comparison between the different meth-

ods of estimating ka%. Note, however, that the

gas RTD is far from plug flow and the Peeff is

not a physical quantity within the context of the

Central Jet RTD.

The driving force for the mass transfer is the

most difficult part to estimate accurately because

it depends on the RTD. Two extreme idealised

flow patterns are the plug flow and well-mixedflow patterns. Plug flow assumes a front of gas

moving through the drum with an axial concen-

tration gradient along the drum with no axial

mixing of the gas. The well-mixed distribution

assumes that the gas is uniformly well-mixed

within the drum and the exit concentration of a

substance is equal to the internal concentration

which is the same at every point in the drum.

The flow pattern of the gas affects the concen-

tration of water in the space above the bran.

For the plug flow analysis the concentration ofwater in the air above the bran (CAIR) was esti-

mated as the log mean of the inlet and outlet

driving forces. For the well-mixed case, the out-

let humidity was used to calculate the outlet

moisture content of the air and this value was

used as CAIR. Neither of these RTDs is strictly

true. RTD studies of the system (Hardin et al.,

2001) led to a three-parameter RTD involving a

Central Jet region with axial dispersion, a sur-

rounding stagnant region and a rate of exchange

between the two regions. This Central Jet RTD

accounts for back mixing within the drum, and

is also used to analyse the mass transfer results

obtained in the current work.

The Central Jet RTD can be described by two

partial differential equations:

(cplug

(t=D

(2cplug

(x2

1

(1Vdead)

(cplug

(x

+Qmix

(1Vdead)

(cdeadcplug) (9)

(cdead

(t=

Qmix

Vdead(cdeadcplug) (10)

where c refers to the dimensionless water con-

centration (Eq. (12)), D refers to the axial dis-

persion in the Central Jet, Vdead is the relative

volume of the stagnant region and Qmix is the


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relative rate of exchange between the stagnant

(dead) and jet (plug) regions. The parameters D,

Vdead and Qmix are strictly empirical and depend of

the superficial gas velocity and possibly other

factors including drum geometry. A fuller descrip-

tion of the parameters is found in Hardin et al.

(2001).

If it is assumed that the bran lies entirely withinthe dead region then Eq. (10) can be modified to

describe the transfer of water from the bran (Eq.

(11)). Also the effective Peclet number should be

expressed only in terms of the surface particle

velocity, as this is the only motion of particles

relative to the gas in the dead region. Hence, the

effective Peclet number used in this paper is as given

in Eq. (12).

(cdead

(t=

Qmix

Vdead(cdeadcplug)+ka(cbrancdead)

(11)

Peeff=uPd

lg(12)

The concentrations in the above differential

equations are dimensionless. The dimensional con-

centrations (C) are normalised with the saturation

water concentration at the temperature of the bran

(CSAT) (i.e. assuming that the water activity of the

bran is 1) to give the variable c, where:

c=CSATCAIR

CSATCIN(13)

This implies cbran=0 at the surface of the bran

and cplug=1 at the entry to the drum. The actual

water concentrations, CAIR and CIN, are calculated

from the measured relative humidity. The concen-

tration of water at the surface of the bran particle

is assumed to be equal to CSAT. This must be true

if the steady state analysis is to hold, that is, the

data are collected during the constant-rate drying

period. Tests on the water activity of the bran after

the trials showed this to be so. The saturation

humidity for the temperature is found from the

Antoine equation and by assuming that the air

behaves as an ideal gas.

During the constant-rate drying period, the drum

is at steady state therefore, the left hand sides of Eq.

(9) and Eq. (11) are equal to zero i.e.

0=D(2cplug

(x2

1

(1Vdead)

(cplug

(x

+Qmix

(1Vdead)(cdeadcplug) (14)

0=Qmix

Vdead(cdeadcplug)+ka(cbrancdead) (15)

Combining Eq. (14) and Eq. (15) to eliminate

cdead gives:

0=D(2cplug

(x2

1

(1Vdead)

(cplug

(x

+Qmix

(1Vdead)

Qmixcplug

Vdead+Kcbran

Qmix

Vdead+K

cplug

(16)

which is a second order differential equation of the

form:

0=Ay+By %+f(y) (17)

with the boundary conditions:

x=0, cplug=1 (18)

x=1, cplug=CSATCOUT

CSATCIN(19)

Eq. (17) has the solution:

cplug=Aexp(R1x)+Bexp(R2x) (20)

where A and Bare constants and R1 and R2 are the

roots of the equation

Dy2y

(1Vdead)+

Qmix

(1Vdead)

Qmixy

Vdead+kacbran

Qmix

Vdead

+ka

y

=0 (21)

The second boundary condition is used to elim-

inate either A or B, based on the magnitude of the

roots, with the first boundary condition used to set

the retained constant equal to 1. Arranging the

roots so that the magnitude of R1\0 and R2B0,

the final solution is:

cplug=exp(R2x) (22)


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Fig. 4. Conceptual model of water transport from bran bed to stagnant region to central jet region and thence to the exit of the

drum. Note that the bran bed is assumed to be in contact only with the stagnant region.

The outlet concentration is required and is rep-

resented as:

cplugx=1=exp(R2) (23)Rearranging R2 allows the lumped mass trans-

fer coefficient ka% to be calculated from:

NTU=Qmixf

1f(24)

where f is given by the expression:

f=1Vdead

4QmixD

11Vdead

2DlnC*2

1

1Vdead

2n(25)

and NTU represents the number of transfer units.

ka=NTU

Qmix(26)

where ka is in terms of transfer area per unit of

volume of the dead space in m3. Thence

ka%=kaVdead (27)

Conceptually, the movement of water vapour

from the bran to the exit air can be imagined as

shown in Fig. 4.

It can be shown that the NTU will be ln(c) for

plug flow and (1/c1) for the well-mixed case

(Levenspiel, 1999), where c is the water concentra-

tion as defined in Eqs. (5)(13). To calculate ka%

using the Blumberg and Schlunder (1996) correla-

tion, the mass transfer coefficients derived from

the Blumberg and Schlunder (1996) correlations

were multiplied by the flat area of the bed surface

and divided by the void space of the drum.

3. Results

Outlet humidities were measured for the drum

operated under different conditions of aeration

rate, rotation rate and fractional filling. The re-

sults were analysed using four different methods

of estimating the apparent ka% number for each

operating condition. The results of this analysis

are in Table 1. The variation in fill depth changes

the moving layer thickness and the average veloc-

ity of the moving layer. This change is captured in

the effective Peclet number (Peeff), however,

changes in gas flow rate do not substantially

change the Peeff for a given rotation speed. The

calculated Peeff is more sensitive to changes in the

rotation rate, which affect the average velocity of

the particles in the mobile layer.


9/13


Table1

Comparisonofka%derived

fromdifferentmethods

Peeff

Estimatedka%

%Filling

Rotationspeed

Airr

ate

(lmin

1)

(rpm)

Plugflow

Ce

ntralJet

BlumbergandSchlunder(1996)

Well-mixed

correlation

(H

ardinetal.,

2001)

0.0

472

0.0

545

30

21.9

0.0

649

155

0.9

0.0

369

0.0

940

0.0

460

0.0

504

0.0

822

30

1.8

32.9

155

3.6

0.1

323

155

0.1

241

30

49.1

0.1

041

0.0

487

0.1

373

0.1

584

0.0

507

61.7

30

0.1

124

5.4

155

0.0

520

155

0.1

252

0.1

885

30

72.5

0.1

180

7.2

0.1

248

0.0

536

0.2

024

0.2

156

9

155

30

82.3

0.0

505

155

0.0

538

0.0

714

22.5

19.8

0.0

731

0.9

0.0

880

0.0

861

0.0

616

1.8

155

0.1

016

29.5

22.5

0.0

694

155

0.1

337

0.1

125

22.5

43.7

0.1

264

3.6

0.1

400

5.4

0.1

351

155

22.5

54.9

0.1

314

0.0

708

0.1

742

0.1

556

0.0

736

0.1

419

63.9

7.2

155

22.5

0.0

777

155

0.1

141

0.1

741

22.5

72.2

0.1

569

9

0.0

468

155

0.0

449

0.0

570

15

17.3

0.0

630

0.9

0.0

564

0.0

666

0.0

511

0.0

722

1.8

25.6

15

155

0.0

864

0.0

570

0.0

746

0.0

838

3.6

155

15

37.5

0.0

923

0.0

985

0.0

625

5.4

155

0.1

011

46.5

15

0.0

656

155

0.1

164

0.1

114

15

54.1

0.1

095

7.2

0.1

225

0.0

698

0.1

179

0.1

232

155

9

15

60.6

0.1

423

0.1

587

0.0

492

61.3

30

0.1

203

5.4

92.6

0.0

635

118

0.1

286

0.1

587

30

61.8

0.1

562

5.4

0.1

617

0.0

694

0.1

746

0.1

583

5.4

137

30

61.8

0.1

707

0.0

760

0.1

799

0.1

584

155

5.4

30

61.8

0.1

692

0.1

583

30

0.0

837

5.4

174

0.1

852

61.9


10/13


Estimating a ka% using either the simple well-

mixed RTD or the plug flow RTD yields values of

ka% quite different from those estimated using the

Central Jet RTD and also quite different from the

values of ka% derived from the correlations of

Blumberg and Schlunder (1996). Blumberg and

Schlunder (1996) validated their correlation with a

large number of different literature values. The

values of ka% estimated using the more complex

flow patterns of Hardin et al. (2001) seem to be

more realistic than those estimated using the plug

flow and well-mixed assumptions, based on their

similarity to the numbers obtained using the ap-

proach of Blumberg and Schlunder (1996). Blum-

berg and Schlunder validated their approach

against a wide range of experimental data, and

therefore, the numbers obtained using their ap-

proach should be reasonably close to the realvalues.

Interestingly, the results of Blumberg and

Schlunder (1996) were for smaller scales (maxi-

mum diameter was 0.3 m) and, in general, higher

length to diameter ratios than the experiments

performed here. The effect of the difference in

geometry on the air RTD probably caused the

slight discrepancies between the estimates using

the Central Jet RTD and the estimates found

using the correlation of Blumberg and Schlunder

(1996). These deviations from plug flow tend toincrease as drums increase in diameter to length

ratio. Scaling up on the basis of a plug flow RTD

is ill-advised as the RTD deviates from plug flow

as scale increases with geometric similarity being

maintained (Mecklenburgh and Hartland, 1975).

The values of ka% estimated from the Central Jet

assumption are probably the most reliable of the

four estimations. They are based on a gas flow

RTD which was experimentally determined for

the drum (Hardin et al., 2001), and not on a plug

flow RTD, which was a key assumption of theapproach of Blumberg and Schlunder (1996). The

method of Blumberg and Schlunder for estimating

the ka% differs from the simple plug flow analysis.

The difference lies in the use of the correlations

described in Appendix A. These modify the plug

flow ka% based on correlations derived from dry-

ing studies.

The similarity, at small drum scales, between

the values of the Central Jet ka% and ka% estimated

using the Blumberg and Schlunder (1996) correla-

tion shows that the convective mass transfer is

independent of the composition of the particle

itself. This is probably to be expected in the

constant-rate drying period where the rate-limit-

ing step is diffusion of water from the saturatedsurface of the particle. The materials used by

Blumberg and Schlunder were mainly uniformly

sized, spherical, inorganic particles. The values of

ka% evaluated in the present trials were performed

using damp wheat bran, which is non-uniform in

size, non-spherical and organic. This similarity in

ka% suggests workers in SSF can access data from

the drying literature, even when it was obtained

with very different particle types, provided they

are operating in the constant-rate drying regime.

Fig. 5 shows ka% estimated on the basis of theCentral Jet flow RTD (solid circles) and also

using the Blumberg and Schlunder correlation

(hollow circles) as a function of effective Peclet

number. The line in the figure is the regression

line for estimates using the Central Jet RTD,

which was forced through the origin. The overall

correlation for estimates of ka% using the Central

Jet RTD is ka%=02.32103Peeff. As can be

seen from Fig. 5 it applies for a wide range of

Fig. 5. ka% vs. effective Peclet number. The estimates using the

Blumberg and Schlunder (1996) method are shown as hollow

circles and the estimates using the Central Jet RTD as solid

circles. The line is the line of best fit for the ka% estimated using

the Central Jet RTD (forced through the origin).


11/13


Peclet numbers with a reasonable correlation

(R2=0.75).

4. Discussion

4.1. Comparison of methods of estimation

The Blumberg and Schlunder (1996) method of

estimating ka% differs from the technique using the

Central Jet RTD developed in this chapter in

three main ways. The two major differing assump-

tions are, firstly, that the area for mass transfer is

simply the flat area of the bed surface, and sec-

ondly, that the gas flow regime through the drum

is plug flow. The third is the use of correlations

from the drying literature (Appendix A).

Calculation of the ka% using the Central Jet

RTD produces a term that includes the wholesurface area available for mass transfer. The cor-

relations from the Central Jet RTD correlation

therefore, include the effect on mass transfer of

increasing mobile layer thickness with increasing

rotational speed. The mobile layer means that the

headspace/bed interface is not something sharp,

but actually has a finite thickness, of the order of

3 cm. This contrasts with the assumption in the

work of Stuart (1996) of a very thin static air

layer of 3 mm thickness where mass transfer was

taking place. Stuarts thin layer was convoluted

by increasing the rotation speed thus, increasing

its surface area, with the fold increase in area

being described by a factor n which varied be-

tween 1 and 5. Note that this assumption (a very

thin layer) was explicit in the work of Blumberg

and Schlunder (1996). The values of ka% estimated

in the current work incorporate the area of mass

transfer so, do not make any assumption as to the

thickness of the layer in which mass transfer

occurs.

The Central Jet RTD is also, by its very nature,

a more realistic assessment of the gas flow regime

and hence, driving force down a given drum axis.

One disadvantage of predicting ka% from the Cen-

tral Jet RTD is there is currently a lack of infor-

mation on how to predict Central Jet parameters

for different drum geometries and sizes. This im-

plies that each different reactor must be character-

ised in terms of its gas flow RTD before any

attempt is made to estimate ka%. Further work is

required to develop correlations for geometry and

scale for Central Jet parameters. However, the use

of an empirical RTD better characterises the gas

flow and accommodates a wide variety of condi-

tions. Characterisation of the RTD using the Cen-

tral Jet RTD allows a more accurate estimate ofka% when back mixing becomes more prevalent.

Also, only one calculation is needed in our ap-

proach whereas, Blumberg and Schlunder require

the worker to calculate four Sherwood numbers

and combine these with a Reynolds-number-

derived factor to produce a predicted Sherwood

number. In our approach, after the RTD in a

drum has been characterised the calculation of

mass transfer coefficients for different conditions

is trivial.

4.2. Implications for design and operation of

RDBs

To date the most complete model of SSF pro-

cesses in RDBs was developed by Stuart (1996).

This model dealt with variations in rotation speed

by a factor n, which varied the surface area avail-

able for mass transfer from the flat area of the top

of the bed (n=1) to a factor of five times this

area for cataracting flow. However, the values

assumed for this factor n were simply educated

guesses and where not based on any theoretical or

experimental evidence. Note that, in the present

work, in which a 56-cm diameter drum was oper-

ated at 9 rpm, cataracting flow of the particles

was not observed. Nine rpm represents 16% of the

critical speed, and as expected, at this speed a

tumbling motion was observed. The correlations

predicted from the Central Jet RTD demonstrate

a much greater effect of rotation speed on mass

transfer than that assumed by Stuart (1996). Sher-

wood numbers increased by a factor of approxi-

mately 4 over the range of rotation speeds from

zero to 9 rpm.

From a design perspective the correlations de-

veloped in this paper (and shown in Fig. 5) give a

mass transfer coefficient for a given Peclet num-

ber. The effective Peclet number takes into ac-

count the fill depth, the gas flow rate and the


12/13


rotational speed. This figure allows the worker to

estimate the ka% that corresponds to their calcu-

lated Peeff, and therefore, to calculate the mass

transfer coefficient in their system. This greatly

simplifies design and calculation of mass transfer.

Further work is required to understand the effect

of scale on the Central Jet flow pattern, and

therefore, on the values of the Central Jet RTDparameters. At the moment the correlations for

Vdead, Qmix and D are valid only for the drum as

described in Hardin et al. (2001). Different corre-

lations based on scale and geometry, involving

more RTD trials using different drums, are

needed to improve the usefulness of the method

and to test the ka%-effective Peclet number correla-

tion. If these are found, scale-up on the basis of

mass transfer phenomena within RDBs will be

simplified.

5. Conclusion

The Central Jet RTD correlation offers a sim-

ple yet accurate estimate of Sherwood numbers in

RDBs for Peclet numbers up to 85. This will

allow easy design of RDBs by improving esti-

mates of evaporative cooling. Implicit in this cor-

relation are the assumptions that the observed

effects are independent of temperature and there

are no changes in volume due to mass transfer.

Also, this correlation is only tested for the con-

stant-rate drying period.

Acknowledgements

David Mitchell thanks the Brazilian National

Council of Research (Conselho Nacional de

Pesquisa, CNPq) for a research scholarship and

auxiliary funding.

Appendix A. Brief description of Blumberg and

Schlunder correlation

The effective mass transfer coefficient, ig,eff esti-

mated using the Blumberg and Schlunder correla-

tion is calculated by combining the mass transfer

coefficient around a single sphere, ig,p, and

through a tube in the entrance region, ig,tube,

using the hydraulic diameter of the free cross-sec-

tion of the drum. These values were combined

using:

ig,eff=1.8ig,P+(1)ig,tube (28)

where is evaluated from the fitted equation:

=0.0015ueffd

6g(29)

where 6g is the gas kinematic viscosity. The two

mass transfer coefficients from Eq. (28) are

derived from the following correlations.

Mass transfer around a single sphere:

Shg,P=ig,Pd

lg=2+

Shg,lam

2 +Shg,turb2 (30)

where

Shg,lam=0.664Red(Scg)1/3 (31)

Shg,turb=0.037Red

0.8Scg

1+2.443Red0.1(Scg

2/31)(32)

Red=ugd

6g(33)

Mass transfer for forced convection in tubes:

Shg,tube,lam=ig,tubeDh

lg

=3.663+0.73

+(1.615(ReDScgDh/L)1/30.7)3

+ 2

1+22Scg

1/6(ReDScgDh/L)3

n1/3(34)

Shg,tube,turb=ig,tubeDh

lg=

(x/8)(ReD1000)Scg

1+12.7x/8(Scg2/31)1+DhL2/3n (35)

where:

x= (1.82logReD1.64)2 (36)

and


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ReD=ueffDh

6g(37)

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