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Mass transfer correlations for rotating drum bioreactors
Journal of Biotechnology 97 (2002) 89 – 101 www.elsevier.com/locate/jbiotec
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 Zealand b Department of Chemical Engineering, Uni6ersity of Queensland, St Lucia, Qld 4072, Australia c Departamento de Bioquı´mica, 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 ms − 1 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%=2.32 × 10 − 3Peeff. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Mass transfer; Rotating drum; Solid state fermentation; Scale-up; Sherwood number
Nomenclature a a% c C cair CAIR cbran
area available for mass transfer per unit of void volume (m − 1) area available for mass transfer per unit of headspace (m − 1) dimensionless water concentration calculated from Eq. (13) water concentration in dry air (kg kg − 1) dimensionless water concentration in headspace water concentration in dry air of headspace (kg kg − 1) dimensionless concentration of water at the bran surface
* 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
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
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cdead CIN COUT cplug CSAT CV D d Ddrum Dh g h K k n N Peeff Qmix Re s Sc Sh t u0 ueff ug up Vdead 6g x z h% i lg k s ~ V V1 V2 x
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 kg − 1) water concentration in dry air exiting the drum (kg kg − 1) 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 kg − 1) constant of bed viscosity (dimensionless) axial diffusion coefficient from the Central Jet RTD (dimensionless) particle diameter (m) drum diameter (m) hydraulic diameter of drum (m) acceleration due to gravity (ms − 2) maximum height of the bed (m) constant in Eq. (3) defined in Eq. (4) mass transfer coefficient estimated using the Central Jet RTD (dimensionless) area factor used by Stuart (1996) (dimensionless) constant of bed porosity (dimensionless) effective Peclet number across the surface of the bran (dimensionless) degree of exchange between the different regions in Central Jet RTD (dimensionless) Reynolds number (dimensionless) thickness of the mobile layer at the bed surface (m) Schmidt number (dimensionless) Sherwood number (dimensionless) dimensionless time (number of residence times) velocity of the particles at the bottom of the moving layer (ms − 1) effective velocity of gas over the surface of the bed (ms − 1) superficial gas velocity through drum (ms − 1) average particle velocity in the moving layer (ms − 1) volume of the stagnant region in the Central Jet RTD (dimensionless) gas kinematic viscosity (m2 s − 1) length along the drum (dimensionless) rotational speed of the drum (s − 1) generic expression for mass transfer area available per unit volume (Eq. (1)) mass transfer coefficient used by Blumberg and Schlu¨ nder coefficient for fitted correlation of Blumberg and Schlu¨ nder diffusivity of water in air at the particle surface (m2 s − 1) dynamic angle of repose of bed (°) generic expression for mass transfer coefficient (Eq. (1)) generic expression for time term (Eq. (1)) generic concentration term (Eq. (1)) concentration term in Eq. (1) concentration term in Eq. (1) factor in Eq. (35)
1. Introduction Rotating drum bioreactors (RDBs) have potential to provide better heat and mass transfer
characteristics than solid state fermentation (SSF) bioreactors with static beds, while providing gentler agitation than bioreactors with internal stirrers. Furthermore, the absence of internal moving
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
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 motion 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 widespread 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 increase in scale evaporative cooling becomes more important 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 generalised models are comparatively recent (Riquelme and Navarro, 1986). There have been few attempts to quantify these effects for SSF. In the current work, a series of experiments was performed 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%(V1 − V2) (~
91
(1)
where V is a concentration of the component in question (e.g. mass water per unit mass dry air), V1 − V2 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 transport occurs 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 number of residence times). For simplicity it is commonly assumed in rotating drums that the mass transfer coefficient is constant over the length of the drum (Blumberg and Schlu¨ nder, 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 the headspace to estimate the concentration of the species being transported as a function of position. 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 controlled by water sprays (Barstow et al., 1988). This means that the driving force for evaporation depends only on the gas phase water concentration 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 diffusion 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, perfect mixing and the Central Jet residence time distribution (RTD), derived from earlier experiments in our laboratory (Hardin et al., 2001) with an empirical correlation from literature (Blumberg and Schlu¨ nder, 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 the drum. 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-
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.
age at 4 °C, was autoclaved at 121 °C for 1 h to simulate the treatment the substrate normally undergoes. 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 to each 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 periodically 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 particle cohesiveness and is difficult to predict. Flow regimes beyond tumbling were not observed in the rotational speed range of these experiments.
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tion of radial gradients. The thickness of the mobile layer depends on the material, fill depth and the rotation speed. Blumberg and Schlu¨ nder (1996) studied mass transfer in rotating drums and developed an implicit equation for the thickness (s) of this mobile layer: pz[Ddrum(h−s)− (h− s)2]− Ks 5/2 + 2pzs(Ddrum/2−h +s)=0
Fig. 2. Flow regimes in rotating drums showing direction of rotation (anti-clockwise) and the direction of motion for the substrate bed.
The maximum rotation rate used in the experiments 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. Conversely, particles deeper in the bed have little contact with the gas. Movement of particles within the bed ensures that, ultimately, the entire bed is exposed to the air and minimises the forma-
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 gsink 3 C Vd 2 ×
1/2
3 12 18 12 3 − N+ N 2 − N 3 + N 4 5 11 17 23 29
(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 viscosity (0.6 for most materials), g is the acceleration due to gravity (ms − 2), k is the dynamic angle of repose of the bed and d is the particle diameter in metres (Savage, 1979). Blumberg and Schlu¨ nder (1996) give the full derivation of these equations. Note that Eq. (4) is different from the equation given in Blumberg and Schlu¨ nder due to a typographical error in that paper. In the current work the dynamic angle of repose was measured as 379 2° through the Perspex end of the drum. Blumberg and Schlu¨ nder (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
Fig. 3. Detail of drum in the tumbling regime showing the direction of travel for the mobile layer as well as the nomenclature for equations for calculating the mobile layer thickness.
(3)
(5)
where up is the average flow velocity of the particles 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/2−h +s)
(6)
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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 paper to determine an estimate for the mass transfer coefficient based on realistic RTDs. Blumberg and Schlu¨ nder (1996) developed other correlations that are based on the Reynolds number of the gas flow through the drum and over the particles. Their correlations are described in Appendix A. The overall mass transfer depends on the effective velocity of the gas over the particles, which is simply the vector sum of the gas velocity and the particle velocity (Eq. (5)). If plug flow is assumed, the two are at right angles to one another for a horizontal drum so the effective velocity is as given in Eq. (7): ueff = u 2g +u 2p
(7)
In Blumberg and Schlu¨ nder (1996), an effective Peclet number, Peeff, was calculated on the basis of the plug flow assumption, regardless of the real flow pattern. This simplification is helpful for scale-up and allows comparison with literature results. The effective velocity is combined with the diffusion coefficient (lg in m2 s − 1) 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 (representing 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-mixed flow patterns. Plug flow assumes a front of gas moving through the drum with an axial concentration 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 concentration of water in the space above the bran. For the plug flow analysis the concentration of water in the air above the bran (CAIR) was estimated as the log mean of the inlet and outlet driving forces. For the well-mixed case, the outlet 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 surrounding 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 1 ( 2cplug (cplug − =D 2 (1− Vdead) (x (t (x +
Qmix (c − cplug) (1−Vdead) dead
(cdead Q = − mix (cdead − cplug) (t Vdead
(9) (10)
where c refers to the dimensionless water concentration (Eq. (12)), D refers to the axial dispersion in the Central Jet, Vdead is the relative volume of the stagnant region and Qmix is the
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
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 description of the parameters is found in Hardin et al. (2001). If it is assumed that the bran lies entirely within the 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 Qmix (cdead −cplug) +ka(cbran −cdead) =− (t Vdead (11) Peeff =
uPd lg
(12)
The concentrations in the above differential equations are dimensionless. The dimensional concentrations (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=
CSAT − CAIR CSAT −CIN
(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 concentration 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.
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( 2cplug 1 (cplug 0= D − 2 (x (1− Vdead) (x +
Qmix (c − cplug) (1−Vdead) dead
(14)
Qmix (cdead − cplug)+ ka(cbran − cdead) Vdead
(15)
0= −
Combining Eq. (14) and Eq. (15) to eliminate cdead gives: 1 (cplug ( 2cplug − 2 (1− Vdead) (x (x  ÁQmixcplug + Kc à bran Qmix à Vdead + à − cplugà (1−Vdead)à Qmix à +K Å Ä Vdead
0= D
(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
x= 1,
cplug =
CSAT − COUT CSAT − CIN
(18) (19)
Eq. (17) has the solution: cplug = Aexp(R1x)+ Bexp(R2x)
(20)
where A and B are constants and R1 and R2 are the roots of the equation y Qmix + (1− Vdead) (1−Vdead) ÁQmixy + kac  ÃV bran à dead à −yà = 0 à Qmix + ka Ã Ä V Å dead Dy 2 −
(21)
The second boundary condition is used to eliminate 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 R2 B 0, 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 represented as: cplug x = 1 = exp(R2)
(23)
Rearranging R2 allows the lumped mass transfer coefficient ka% to be calculated from: f NTU = Qmix 1−f
(24)
It can be shown that the NTU will be ln(c) for plug flow and (1/c − 1) for the well-mixed case (Levenspiel, 1999), where c is the water concentration as defined in Eqs. (5)–(13). To calculate ka% using the Blumberg and Schlu¨ nder (1996) correlation, the mass transfer coefficients derived from the Blumberg and Schlu¨ nder (1996) correlations were multiplied by the flat area of the bed surface and divided by the void space of the drum.
where f is given by the expression: f=
1 −Vdead 4QmixD −
1 −2DlnC* 1− Vdead
1 1−Vdead
n
3. Results
2
2
(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.
Outlet humidities were measured for the drum operated under different conditions of aeration rate, rotation rate and fractional filling. The results 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 velocity 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.
Air rate (l min−1)
155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 92.6 118 137 155 174
Rotation speed (rpm)
0.9 1.8 3.6 5.4 7.2 9 0.9 1.8 3.6 5.4 7.2 9 0.9 1.8 3.6 5.4 7.2 9 5.4 5.4 5.4 5.4 5.4
30 30 30 30 30 30 22.5 22.5 22.5 22.5 22.5 22.5 15 15 15 15 15 15 30 30 30 30 30
% Filling
Table 1 Comparison of ka% derived from different methods
21.9 32.9 49.1 61.7 72.5 82.3 19.8 29.5 43.7 54.9 63.9 72.2 17.3 25.6 37.5 46.5 54.1 60.6 61.3 61.8 61.8 61.8 61.9
Peeff
0.0649 0.0940 0.1041 0.1124 0.1180 0.1248 0.0731 0.1016 0.1264 0.1314 0.1419 0.1569 0.0630 0.0722 0.0864 0.1011 0.1095 0.1225 0.1203 0.1562 0.1617 0.1707 0.1852
Well-mixed
Estimated ka%
0.0369 0.0460 0.0487 0.0507 0.0520 0.0536 0.0505 0.0616 0.0694 0.0708 0.0736 0.0777 0.0468 0.0511 0.0570 0.0625 0.0656 0.0698 0.0492 0.0635 0.0694 0.0760 0.0837
Plug flow
0.0472 0.0504 0.1323 0.1373 0.1252 0.2024 0.0538 0.0880 0.1337 0.1400 0.1742 0.1141 0.0449 0.0564 0.0746 0.0923 0.1164 0.1179 0.1423 0.1286 0.1746 0.1799 0.1692
Central Jet (Hardin et al., 2001) 0.0545 0.0822 0.1241 0.1584 0.1885 0.2156 0.0714 0.0861 0.1125 0.1351 0.1556 0.1741 0.0570 0.0666 0.0838 0.0985 0.1114 0.1232 0.1587 0.1587 0.1583 0.1584 0.1583
Blumberg and Schlu¨ nder (1996) correlation
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101 97
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M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
Estimating a ka% using either the simple wellmixed 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 Schlu¨ nder (1996). Blumberg and Schlu¨ nder (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 approach of Blumberg and Schlu¨ nder (1996). Blumberg and Schlu¨ nder validated their approach against a wide range of experimental data, and therefore, the numbers obtained using their approach should be reasonably close to the real values. Interestingly, the results of Blumberg and Schlu¨ nder (1996) were for smaller scales (maximum 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 Schlu¨ nder (1996). These deviations from plug flow tend to increase 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 the approach of Blumberg and Schlu¨ nder (1996). The method of Blumberg and Schlu¨ nder 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 drying studies.
The similarity, at small drum scales, between the values of the Central Jet ka% and ka% estimated using the Blumberg and Schlu¨ nder (1996) correlation 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-limiting step is diffusion of water from the saturated surface of the particle. The materials used by Blumberg and Schlu¨ nder 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 the Central Jet flow RTD (solid circles) and also using the Blumberg and Schlu¨ nder 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% = 02.32× 10 − 3Peeff. 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 Schlu¨ nder (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).
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
Peclet numbers with a reasonable correlation (R 2 =0.75).
4. Discussion
4.1. Comparison of methods of estimation The Blumberg and Schlu¨ nder (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 assumptions are, firstly, that the area for mass transfer is simply the flat area of the bed surface, and secondly, 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 whole surface area available for mass transfer. The correlations 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. Stuart’s 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 between 1 and 5. Note that this assumption (a very thin layer) was explicit in the work of Blumberg and Schlu¨ nder (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 Central Jet RTD is there is currently a lack of information on how to predict Central Jet parameters for different drum geometries and sizes. This implies that each different reactor must be character-
99
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 conditions. Characterisation of the RTD using the Central Jet RTD allows a more accurate estimate of ka% when back mixing becomes more prevalent. Also, only one calculation is needed in our approach whereas, Blumberg and Schlu¨ nder require the worker to calculate four Sherwood numbers and combine these with a Reynolds-numberderived 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 processes in RDBs was developed by Stuart (1996). This model dealt with variations in rotation speed by a factor n, which varied the surface area available 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 operated 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). Sherwood numbers increased by a factor of approximately 4 over the range of rotation speeds from zero to 9 rpm. From a design perspective the correlations developed in this paper (and shown in Fig. 5) give a mass transfer coefficient for a given Peclet number. The effective Peclet number takes into account the fill depth, the gas flow rate and the
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M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
rotational speed. This figure allows the worker to estimate the ka% that corresponds to their calculated 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 RTD parameters. At the moment the correlations for Vdead, Qmix and D are valid only for the drum as described in Hardin et al. (2001). Different correlations 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 correlation. If these are found, scale-up on the basis of mass transfer phenomena within RDBs will be simplified.
5. Conclusion
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-section of the drum. These values were combined using: ig,eff = 1.8ig,P + (1− )ig,tube
(28)
where is evaluated from the fitted equation: u d = 0.0015 eff 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 = 2+ Sh 2g,lam + Sh 2g,turb lg
where Shg,lam = 0.664 Red(Scg)1/3
The Central Jet RTD correlation offers a simple yet accurate estimate of Sherwood numbers in RDBs for Peclet numbers up to 85. This will allow easy design of RDBs by improving estimates of evaporative cooling. Implicit in this correlation 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 constant-rate drying period.
(30)
Shg,turb = Red =
(31)
0.037Re 0.8 d Scg − 0.1 1+ 2.443Re d (Sc 2/3 g − 1)
(32)
ugd 6g
(33)
Mass transfer for forced convection in tubes: Shg,tube,lam =
ig,tubeDh lg
= 3.663 + 0.73 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 Schlu¨nder correlation The effective mass transfer coefficient, ig,eff estimated using the Blumberg and Schlu¨ nder correlation is calculated by combining the mass transfer
+
+ (1.615(ReDScgDh/L)1/3 − 0.7)3
2 1+ 22Scg
Shg,tube,turb =
n
1/6
( ReDScgDh/L)3
1/3
(34)
ig,tubeDh (x/8)(ReD − 1000)Scg = lg 1+ 12.7 x/8(Sc 2/3 g − 1)
n 1+
Dh L
2/3
(35)
where: x = (1.82logReD − 1.64) − 2 and
(36)
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
ReD =
ueffDh 6g
(37)
References Barstow, L.M., Dale, B.E., Tengerdy, R.P., 1988. Evaporative temperature and moisture control in solid substrate fermentation. Biotechnol. Tech. 2, 237 –242. Blumberg, W., Schlu¨ nder, E.-U., 1996. Transversale Schuettgu¨ tbewegung und konvektiver Stoffu¨ bergang in Drehrohen. Teil 1: Ohne Hubschaufeln. Chem. Eng. Proc. 35, 395 – 404. Friedman, S.J., Marshall, W.R., 1949. Studies in rotary drying, part II —heat and mass transfer. Chem. Eng. Prog. 45, 573 – 588. Hardin, M.T., Howes, T., Mitchell, D.A., 2001. Residence time distributions of gas flowing through rotating drum bioreactors. Biotechnol. Bioeng. 74, 145 –153.
101
Jauhari, R., Gray, M.R., Masliyah, J.H., 1998. Gas – solid mass transfer in a rotating drum. Can. J. Chem. Eng. 76, 224 – 232. Levenspiel, O., 1999. Chemical Reaction Engineering. Wiley, New York. McCabe, W.L., Smith, J.C., Harriott, P., 1985. Unit Operations of Chemical Engineering. McGraw-Hill, Sydney. Marsh, A.J., Stuart, D.M., Mitchell, D.A., Howes, T., 2000. Characterizing mixing in a rotating drum bioreactor for solid-state fermentation. Biotechnol. Lett. 22, 473 – 477. Mecklenburgh, J.C., Hartland, S., 1975. The Theory of Backmixing. Wiley. Riquelme, G.D., Navarro, G., 1986. Analysis and modelling of rotary dryer —drying of copper concentrate in drying of Solids— recent international developments. In: Mujumdar, A.S. (Ed.), Wiley Eastern, New Delhi. Savage, S.B., 1979. Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92, 53 – 96. Stuart, D.M., 1996. Solid state fermentation in rotating drum bioreactors. PhD Thesis, University of Queensland.
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 Zealand b Department of Chemical Engineering, Uni6ersity of Queensland, St Lucia, Qld 4072, Australia c Departamento de Bioquı´mica, 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 ms − 1 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%=2.32 × 10 − 3Peeff. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Mass transfer; Rotating drum; Solid state fermentation; Scale-up; Sherwood number
Nomenclature a a% c C cair CAIR cbran
area available for mass transfer per unit of void volume (m − 1) area available for mass transfer per unit of headspace (m − 1) dimensionless water concentration calculated from Eq. (13) water concentration in dry air (kg kg − 1) dimensionless water concentration in headspace water concentration in dry air of headspace (kg kg − 1) dimensionless concentration of water at the bran surface
* 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
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
90
cdead CIN COUT cplug CSAT CV D d Ddrum Dh g h K k n N Peeff Qmix Re s Sc Sh t u0 ueff ug up Vdead 6g x z h% i lg k s ~ V V1 V2 x
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 kg − 1) water concentration in dry air exiting the drum (kg kg − 1) 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 kg − 1) constant of bed viscosity (dimensionless) axial diffusion coefficient from the Central Jet RTD (dimensionless) particle diameter (m) drum diameter (m) hydraulic diameter of drum (m) acceleration due to gravity (ms − 2) maximum height of the bed (m) constant in Eq. (3) defined in Eq. (4) mass transfer coefficient estimated using the Central Jet RTD (dimensionless) area factor used by Stuart (1996) (dimensionless) constant of bed porosity (dimensionless) effective Peclet number across the surface of the bran (dimensionless) degree of exchange between the different regions in Central Jet RTD (dimensionless) Reynolds number (dimensionless) thickness of the mobile layer at the bed surface (m) Schmidt number (dimensionless) Sherwood number (dimensionless) dimensionless time (number of residence times) velocity of the particles at the bottom of the moving layer (ms − 1) effective velocity of gas over the surface of the bed (ms − 1) superficial gas velocity through drum (ms − 1) average particle velocity in the moving layer (ms − 1) volume of the stagnant region in the Central Jet RTD (dimensionless) gas kinematic viscosity (m2 s − 1) length along the drum (dimensionless) rotational speed of the drum (s − 1) generic expression for mass transfer area available per unit volume (Eq. (1)) mass transfer coefficient used by Blumberg and Schlu¨ nder coefficient for fitted correlation of Blumberg and Schlu¨ nder diffusivity of water in air at the particle surface (m2 s − 1) dynamic angle of repose of bed (°) generic expression for mass transfer coefficient (Eq. (1)) generic expression for time term (Eq. (1)) generic concentration term (Eq. (1)) concentration term in Eq. (1) concentration term in Eq. (1) factor in Eq. (35)
1. Introduction Rotating drum bioreactors (RDBs) have potential to provide better heat and mass transfer
characteristics than solid state fermentation (SSF) bioreactors with static beds, while providing gentler agitation than bioreactors with internal stirrers. Furthermore, the absence of internal moving
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
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 motion 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 widespread 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 increase in scale evaporative cooling becomes more important 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 generalised models are comparatively recent (Riquelme and Navarro, 1986). There have been few attempts to quantify these effects for SSF. In the current work, a series of experiments was performed 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%(V1 − V2) (~
91
(1)
where V is a concentration of the component in question (e.g. mass water per unit mass dry air), V1 − V2 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 transport occurs 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 number of residence times). For simplicity it is commonly assumed in rotating drums that the mass transfer coefficient is constant over the length of the drum (Blumberg and Schlu¨ nder, 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 the headspace to estimate the concentration of the species being transported as a function of position. 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 controlled by water sprays (Barstow et al., 1988). This means that the driving force for evaporation depends only on the gas phase water concentration 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 diffusion 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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M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
drum operating in the slumping and tumbling regimes during a constant-rate drying period. It compares three gas flow patterns; plug flow, perfect mixing and the Central Jet residence time distribution (RTD), derived from earlier experiments in our laboratory (Hardin et al., 2001) with an empirical correlation from literature (Blumberg and Schlu¨ nder, 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 the drum. 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-
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.
age at 4 °C, was autoclaved at 121 °C for 1 h to simulate the treatment the substrate normally undergoes. 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 to each 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 periodically 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 particle cohesiveness and is difficult to predict. Flow regimes beyond tumbling were not observed in the rotational speed range of these experiments.
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
93
tion of radial gradients. The thickness of the mobile layer depends on the material, fill depth and the rotation speed. Blumberg and Schlu¨ nder (1996) studied mass transfer in rotating drums and developed an implicit equation for the thickness (s) of this mobile layer: pz[Ddrum(h−s)− (h− s)2]− Ks 5/2 + 2pzs(Ddrum/2−h +s)=0
Fig. 2. Flow regimes in rotating drums showing direction of rotation (anti-clockwise) and the direction of motion for the substrate bed.
The maximum rotation rate used in the experiments 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. Conversely, particles deeper in the bed have little contact with the gas. Movement of particles within the bed ensures that, ultimately, the entire bed is exposed to the air and minimises the forma-
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 gsink 3 C Vd 2 ×
1/2
3 12 18 12 3 − N+ N 2 − N 3 + N 4 5 11 17 23 29
(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 viscosity (0.6 for most materials), g is the acceleration due to gravity (ms − 2), k is the dynamic angle of repose of the bed and d is the particle diameter in metres (Savage, 1979). Blumberg and Schlu¨ nder (1996) give the full derivation of these equations. Note that Eq. (4) is different from the equation given in Blumberg and Schlu¨ nder due to a typographical error in that paper. In the current work the dynamic angle of repose was measured as 379 2° through the Perspex end of the drum. Blumberg and Schlu¨ nder (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
Fig. 3. Detail of drum in the tumbling regime showing the direction of travel for the mobile layer as well as the nomenclature for equations for calculating the mobile layer thickness.
(3)
(5)
where up is the average flow velocity of the particles 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/2−h +s)
(6)
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
94
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 paper to determine an estimate for the mass transfer coefficient based on realistic RTDs. Blumberg and Schlu¨ nder (1996) developed other correlations that are based on the Reynolds number of the gas flow through the drum and over the particles. Their correlations are described in Appendix A. The overall mass transfer depends on the effective velocity of the gas over the particles, which is simply the vector sum of the gas velocity and the particle velocity (Eq. (5)). If plug flow is assumed, the two are at right angles to one another for a horizontal drum so the effective velocity is as given in Eq. (7): ueff = u 2g +u 2p
(7)
In Blumberg and Schlu¨ nder (1996), an effective Peclet number, Peeff, was calculated on the basis of the plug flow assumption, regardless of the real flow pattern. This simplification is helpful for scale-up and allows comparison with literature results. The effective velocity is combined with the diffusion coefficient (lg in m2 s − 1) 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 (representing 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-mixed flow patterns. Plug flow assumes a front of gas moving through the drum with an axial concentration 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 concentration of water in the space above the bran. For the plug flow analysis the concentration of water in the air above the bran (CAIR) was estimated as the log mean of the inlet and outlet driving forces. For the well-mixed case, the outlet 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 surrounding 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 1 ( 2cplug (cplug − =D 2 (1− Vdead) (x (t (x +
Qmix (c − cplug) (1−Vdead) dead
(cdead Q = − mix (cdead − cplug) (t Vdead
(9) (10)
where c refers to the dimensionless water concentration (Eq. (12)), D refers to the axial dispersion in the Central Jet, Vdead is the relative volume of the stagnant region and Qmix is the
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
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 description of the parameters is found in Hardin et al. (2001). If it is assumed that the bran lies entirely within the 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 Qmix (cdead −cplug) +ka(cbran −cdead) =− (t Vdead (11) Peeff =
uPd lg
(12)
The concentrations in the above differential equations are dimensionless. The dimensional concentrations (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=
CSAT − CAIR CSAT −CIN
(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 concentration 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.
95
( 2cplug 1 (cplug 0= D − 2 (x (1− Vdead) (x +
Qmix (c − cplug) (1−Vdead) dead
(14)
Qmix (cdead − cplug)+ ka(cbran − cdead) Vdead
(15)
0= −
Combining Eq. (14) and Eq. (15) to eliminate cdead gives: 1 (cplug ( 2cplug − 2 (1− Vdead) (x (x  ÁQmixcplug + Kc à bran Qmix à Vdead + à − cplugà (1−Vdead)à Qmix à +K Å Ä Vdead
0= D
(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
x= 1,
cplug =
CSAT − COUT CSAT − CIN
(18) (19)
Eq. (17) has the solution: cplug = Aexp(R1x)+ Bexp(R2x)
(20)
where A and B are constants and R1 and R2 are the roots of the equation y Qmix + (1− Vdead) (1−Vdead) ÁQmixy + kac  ÃV bran à dead à −yà = 0 à Qmix + ka Ã Ä V Å dead Dy 2 −
(21)
The second boundary condition is used to eliminate 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 R2 B 0, the final solution is: cplug = exp(R2x)
(22)
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96
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 represented as: cplug x = 1 = exp(R2)
(23)
Rearranging R2 allows the lumped mass transfer coefficient ka% to be calculated from: f NTU = Qmix 1−f
(24)
It can be shown that the NTU will be ln(c) for plug flow and (1/c − 1) for the well-mixed case (Levenspiel, 1999), where c is the water concentration as defined in Eqs. (5)–(13). To calculate ka% using the Blumberg and Schlu¨ nder (1996) correlation, the mass transfer coefficients derived from the Blumberg and Schlu¨ nder (1996) correlations were multiplied by the flat area of the bed surface and divided by the void space of the drum.
where f is given by the expression: f=
1 −Vdead 4QmixD −
1 −2DlnC* 1− Vdead
1 1−Vdead
n
3. Results
2
2
(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.
Outlet humidities were measured for the drum operated under different conditions of aeration rate, rotation rate and fractional filling. The results 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 velocity 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.
Air rate (l min−1)
155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 92.6 118 137 155 174
Rotation speed (rpm)
0.9 1.8 3.6 5.4 7.2 9 0.9 1.8 3.6 5.4 7.2 9 0.9 1.8 3.6 5.4 7.2 9 5.4 5.4 5.4 5.4 5.4
30 30 30 30 30 30 22.5 22.5 22.5 22.5 22.5 22.5 15 15 15 15 15 15 30 30 30 30 30
% Filling
Table 1 Comparison of ka% derived from different methods
21.9 32.9 49.1 61.7 72.5 82.3 19.8 29.5 43.7 54.9 63.9 72.2 17.3 25.6 37.5 46.5 54.1 60.6 61.3 61.8 61.8 61.8 61.9
Peeff
0.0649 0.0940 0.1041 0.1124 0.1180 0.1248 0.0731 0.1016 0.1264 0.1314 0.1419 0.1569 0.0630 0.0722 0.0864 0.1011 0.1095 0.1225 0.1203 0.1562 0.1617 0.1707 0.1852
Well-mixed
Estimated ka%
0.0369 0.0460 0.0487 0.0507 0.0520 0.0536 0.0505 0.0616 0.0694 0.0708 0.0736 0.0777 0.0468 0.0511 0.0570 0.0625 0.0656 0.0698 0.0492 0.0635 0.0694 0.0760 0.0837
Plug flow
0.0472 0.0504 0.1323 0.1373 0.1252 0.2024 0.0538 0.0880 0.1337 0.1400 0.1742 0.1141 0.0449 0.0564 0.0746 0.0923 0.1164 0.1179 0.1423 0.1286 0.1746 0.1799 0.1692
Central Jet (Hardin et al., 2001) 0.0545 0.0822 0.1241 0.1584 0.1885 0.2156 0.0714 0.0861 0.1125 0.1351 0.1556 0.1741 0.0570 0.0666 0.0838 0.0985 0.1114 0.1232 0.1587 0.1587 0.1583 0.1584 0.1583
Blumberg and Schlu¨ nder (1996) correlation
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101 97
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M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
Estimating a ka% using either the simple wellmixed 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 Schlu¨ nder (1996). Blumberg and Schlu¨ nder (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 approach of Blumberg and Schlu¨ nder (1996). Blumberg and Schlu¨ nder validated their approach against a wide range of experimental data, and therefore, the numbers obtained using their approach should be reasonably close to the real values. Interestingly, the results of Blumberg and Schlu¨ nder (1996) were for smaller scales (maximum 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 Schlu¨ nder (1996). These deviations from plug flow tend to increase 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 the approach of Blumberg and Schlu¨ nder (1996). The method of Blumberg and Schlu¨ nder 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 drying studies.
The similarity, at small drum scales, between the values of the Central Jet ka% and ka% estimated using the Blumberg and Schlu¨ nder (1996) correlation 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-limiting step is diffusion of water from the saturated surface of the particle. The materials used by Blumberg and Schlu¨ nder 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 the Central Jet flow RTD (solid circles) and also using the Blumberg and Schlu¨ nder 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% = 02.32× 10 − 3Peeff. 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 Schlu¨ nder (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).
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
Peclet numbers with a reasonable correlation (R 2 =0.75).
4. Discussion
4.1. Comparison of methods of estimation The Blumberg and Schlu¨ nder (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 assumptions are, firstly, that the area for mass transfer is simply the flat area of the bed surface, and secondly, 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 whole surface area available for mass transfer. The correlations 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. Stuart’s 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 between 1 and 5. Note that this assumption (a very thin layer) was explicit in the work of Blumberg and Schlu¨ nder (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 Central Jet RTD is there is currently a lack of information on how to predict Central Jet parameters for different drum geometries and sizes. This implies that each different reactor must be character-
99
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 conditions. Characterisation of the RTD using the Central Jet RTD allows a more accurate estimate of ka% when back mixing becomes more prevalent. Also, only one calculation is needed in our approach whereas, Blumberg and Schlu¨ nder require the worker to calculate four Sherwood numbers and combine these with a Reynolds-numberderived 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 processes in RDBs was developed by Stuart (1996). This model dealt with variations in rotation speed by a factor n, which varied the surface area available 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 operated 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). Sherwood numbers increased by a factor of approximately 4 over the range of rotation speeds from zero to 9 rpm. From a design perspective the correlations developed in this paper (and shown in Fig. 5) give a mass transfer coefficient for a given Peclet number. The effective Peclet number takes into account the fill depth, the gas flow rate and the
100
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
rotational speed. This figure allows the worker to estimate the ka% that corresponds to their calculated 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 RTD parameters. At the moment the correlations for Vdead, Qmix and D are valid only for the drum as described in Hardin et al. (2001). Different correlations 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 correlation. If these are found, scale-up on the basis of mass transfer phenomena within RDBs will be simplified.
5. Conclusion
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-section of the drum. These values were combined using: ig,eff = 1.8ig,P + (1− )ig,tube
(28)
where is evaluated from the fitted equation: u d = 0.0015 eff 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 = 2+ Sh 2g,lam + Sh 2g,turb lg
where Shg,lam = 0.664 Red(Scg)1/3
The Central Jet RTD correlation offers a simple yet accurate estimate of Sherwood numbers in RDBs for Peclet numbers up to 85. This will allow easy design of RDBs by improving estimates of evaporative cooling. Implicit in this correlation 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 constant-rate drying period.
(30)
Shg,turb = Red =
(31)
0.037Re 0.8 d Scg − 0.1 1+ 2.443Re d (Sc 2/3 g − 1)
(32)
ugd 6g
(33)
Mass transfer for forced convection in tubes: Shg,tube,lam =
ig,tubeDh lg
= 3.663 + 0.73 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 Schlu¨nder correlation The effective mass transfer coefficient, ig,eff estimated using the Blumberg and Schlu¨ nder correlation is calculated by combining the mass transfer
+
+ (1.615(ReDScgDh/L)1/3 − 0.7)3
2 1+ 22Scg
Shg,tube,turb =
n
1/6
( ReDScgDh/L)3
1/3
(34)
ig,tubeDh (x/8)(ReD − 1000)Scg = lg 1+ 12.7 x/8(Sc 2/3 g − 1)
n 1+
Dh L
2/3
(35)
where: x = (1.82logReD − 1.64) − 2 and
(36)
M.T. Hardin et al. / Journal of Biotechnology 97 (2002) 89–101
ReD =
ueffDh 6g
(37)
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Jauhari, R., Gray, M.R., Masliyah, J.H., 1998. Gas – solid mass transfer in a rotating drum. Can. J. Chem. Eng. 76, 224 – 232. Levenspiel, O., 1999. Chemical Reaction Engineering. Wiley, New York. McCabe, W.L., Smith, J.C., Harriott, P., 1985. Unit Operations of Chemical Engineering. McGraw-Hill, Sydney. Marsh, A.J., Stuart, D.M., Mitchell, D.A., Howes, T., 2000. Characterizing mixing in a rotating drum bioreactor for solid-state fermentation. Biotechnol. Lett. 22, 473 – 477. Mecklenburgh, J.C., Hartland, S., 1975. The Theory of Backmixing. Wiley. Riquelme, G.D., Navarro, G., 1986. Analysis and modelling of rotary dryer —drying of copper concentrate in drying of Solids— recent international developments. In: Mujumdar, A.S. (Ed.), Wiley Eastern, New Delhi. Savage, S.B., 1979. Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92, 53 – 96. Stuart, D.M., 1996. Solid state fermentation in rotating drum bioreactors. PhD Thesis, University of Queensland.