Mass transport enhancement in annular-shaped lid-driven bioreactor

ORIGINAL PAPER

Mass transport enhancement in annular-shaped lid-drivenbioreactor

Mohammad Al-Shannag

Received: 28 October 2011 / Accepted: 16 December 2011 / Published online: 30 December 2011

� Springer-Verlag 2011

Abstract The current study investigated numerically the

two-dimensional (2D) incompressible flow and mass

transfer in a lid-driven cavity of annular geometry accom-

panied by enzymatic surface-reactions. The lid-driven bio-

reactor had a square cross-section of (H 9 H) and a radius

of curvature of rc. This flow configuration gives the

opportunity to evaluate effects of curvature as well as

operational parameters on the bioreactor performance. For

forced-convection, conservation equations were solved

numerically, using fourth-order finite volume schemes, to

identify the 2D flow structure and concentration distribution

of substrate within the bioreactor. For pure diffusion, ana-

lytical solution was obtained. Substrate transfer rates were

presented in terms of Sherwood number. While, effective-

ness factor was computed to evaluate the force-convection

contribution over pure molecular diffusion. Mass-transfer

against surface-reaction resistance was estimated via

Damkohler number. Results indicate the positive role of

increasing Peclet number, Reynolds number, and radius of

curvature in enhancing the substrate transport process.

Keywords Mass-transfer � Bioreactor � Newtonian fluid �Diffusion–reaction � Mixing � Bioprocess design

Introduction

In biotechnology, the performance of bioreactor can be

enhanced by focusing on mixing process necessary to

expose as much substrate possible to the active site of the

catalysts used [1, 2]. Most common types of bioreactors

adopted by industry are packed- or fluidized-bed reactors in

which solid particles with biocatalysts attached onto their

outer surface are used [3, 4]. Here, keep in mind that the

transfer rate of substrate to the active sites of catalyst is

reduced by both external and internal mass transfer resis-

tances [1, 4]. In such circumstances, mass transfer resis-

tances are strongly dependent on substrate diffusivity,

porosity of enzyme carrier, and reactor hydrodynamic

conditions [1, 3]. Al-Shannag et al. [5] proposed a straight

lid-driven cavity (LDC) as an alternative bioreactor system,

since mechanical stresses are low compared to traditional

stirred tank reactors that can cause cell damage, particu-

larly near the rotating blades.

In the lid-driven cavity, the motion of a Newtonian

fluid within a confined enclosure is maintained by the

continuous momentum diffusion from the moving wall.

From practical point of view, these lid-driven flows are

encountered in many engineering equipments such as heat

exchangers, dryers, evaporators, and chemical reactors [5–

10]. Moreover, the lid-driven cavity flow exhibits features

of more complex geometry flows and represents one of

the most important benchmarks for testing hydrodynamic

instability and various numerical Navier–Stokes (NS)

solvers [11–15]. Excellent reviews on LDC flows are

given by Al-Shannag et al. [5] and Albensoeder and

Kuhlmann [15].

Al-Shannag et al. [5] investigated numerically two-

dimensional incompressible flow and mass transfer in an

immobilized enzyme reactor. In their study, straight LDC

of a square cross-section was selected to analyze the effect

of forced convection on the movement of substrate from

the bulk solution to an immobilized enzyme surface.

Substrate was assumed to be consumed according to

M. Al-Shannag (&)

Department of Chemical Engineering, Faculty of Engineering

Technology, Al-Balqa Applied University, P.O. Box 15008,

Marka, 11134 Amman, Jordan

e-mail: [email protected]

123

Bioprocess Biosyst Eng (2012) 35:875–884

DOI 10.1007/s00449-011-0672-2

Michaelis–Menten kinetics. They reported the important

contribution of forced convection in the transport of a

substrate as compared to pure diffusion in a region

bounded by a sliding surface without mechanical agita-

tion. Still, no attempt has been cited in literature to

identify the curvature effect on the performance of such

bio-reacting systems. The present work discusses, for the

first time, the use of an annular-shaped lid-driven reactor

LDR with bioreacting system to take the curvature

effects into consideration; see Fig. 1. The performance of

such LDR was evaluated after solving the unsteady two-

dimensional momentum and substrate mass-transport

equation. Results were validated against the correspond-

ing ones reported in literature for the straight lid-driven

cavity (annular-shaped LDR with infinity radius of

curvature).

Problem formulation

Model system, with boundary conditions, and cylindrical

coordinates are presented in Fig. 1. The present work

considers a (H 9 H) square bioreactor with a radius of

curvature of rc [ (ro - ri)/2. As rc ? ?, the reactor

approaches a straight geometry, while, a cylindrical config-

uration is obtained at rc = (ro - ri)/2 since H = (ro - ri).

Initially, the cavity is filled with liquid solvent at rest. Fluid

motion is then induced by sliding the top wall, at z = H, with

a radial velocity, u, which is inversely proportional to the

radial distance according to [16],

uðrÞ ¼ a=r ð1Þ

The constant a is correlated to the average velocity

value, �u; at z = H, using,

�u ¼R r¼ro

r¼ria dr=r

ro � ri

¼ a lnðro=riÞro � ri

ð2Þ

and, hence, the radial velocity profile along the top moving

wall is,

uðrÞ ¼ �uro � rið Þ

lnðro=riÞrð3Þ

On the other hand, liquid substrate S is introduced

continuously into the reactor via the top moving wall where

its concentration assumes the prescribed surface value co.

The other three walls (bottom, external, and internal walls)

are fixed and impermeable. For such enzymatic surface-

reaction systems, Michaelis–Menten kinetics could be used

to describe the substrate consumption at the bottom wall,

S ? P, to yield product P [1]. However, the substrate

concentration in this problem is assumed to be low enough

to approximate the consumption rate, and the macroscopic

vertical gradient in substrate concentration, c, by first-order

kinetics [1]:

Ns ¼ Ds

dc

dz¼ kc ð4Þ

where Ns is the substrate surface-reaction rate, Ds is the

molecular diffusivity of the substrate, c is the substrate

concentration, and k is the first-order reaction constant.

Dimensionless form of governing equations

In order to formulate the mathematical equations

describing the physical model, the following assumptions

were made [5]: (i) constant physical properties; (ii) tem-

perature is constant; (iii) the substrate diffusion can be

modeled by Fick’s law; (iv) the fluid mixture is incom-

pressible and Newtonian; (v) enzyme activity is uniform

over the entire impermeable bottom wall and no enzyme

deactivation occurs; (vi) zero intraparticle diffusion

resistance; (vii) both substrate S and product P are passive

and have no effect on fluid motion, density, or concen-

tration. Also, the dimensionless form of the governing

equations was obtained by appropriately scaling the

dimensional variables involved. Lengths were scaled by

the reactor’s height/width, H, velocities by the average

velocity of the top wall, �u, and the substrate concentration

by the top wall value, co, and combinations of �u, H, and

fluid mixture viscosity, l, were used to scale time and

pressure,

r

z

r = ri

r = rc= ri+H/2

H

H

Bottom stagnant wall v = u = 0; Ds dc/dz = k c

Top moving wall u = a/r; v = 0; c = co

External stagnant w

all u =

v = dc/dr =

0

u = v =

dc/dr = 0

Internal stagnant wall

r = ro

Fig. 1 Schematic diagram of the annular-shaped lid-driven reactor

(LDR) with boundary conditions and coordinate system

876 Bioprocess Biosyst Eng (2012) 35:875–884

123

t ¼ �u t�=H; U ¼ u=�u; V ¼ v=�u; Z ¼ z=H; R ¼ r=H;

Ri ¼ ri=H; Ro ¼ ro=H; Rc ¼ rc=H; P ¼ pH=l�u; C ¼ c=co

ð5Þ

where u and v are the dimensional velocity components in

the radial, r, and axial, z, coordinate directions respec-

tively, p is the dimensional pressure, and t* is the dimen-

sional time, c is the dimensional substrate concentration.

Under the aforementioned assumptions and using the

dimensionless variables defined in Eq. 5, the dimensionless

form of unsteady two-dimensional equations for the con-

servation of total mass, momentum, and mass of substrate

inside the bioreactor can be written as [17],

Continuity:oU

oRþ oV

oZþ U

R¼ 0 ð6Þ

z-momentum: ReoV

otþ U

oV

oRþ V

oV

oZ

� �

¼ � oP

oZþ 1

R

o

oRR

oV

oR

� �

þ o2V

oZ2ð7Þ

r-momentum: ReoU

otþ U

oU

oRþ V

oU

oZ

� �

¼ � oP

oRþ o

oR

1

R

oRU

oR

� �

þ o2U

oZ2ð8Þ

Substrate mass transport:

PeoC

otþ U

oC

oRþ V

oC

oZ

� �

¼ 1

R

o

oRR

oC

oR

� �

þ o2C

oZ2ð9Þ

where Re and Pe are Reynolds and Peclet number,

respectively, defined as [17],

Re ¼ q�uH=l; Pe ¼ ReSc ¼ �uH=Ds ð10Þ

where q is the fluid mixture density and Sc is Schmidt

number defined as [17],

Sc ¼ lqDs

: ð11Þ

The Peclet number, which expresses the relative

importance of forced-convection over diffusion, was used

in the current investigation instead of Sc. Both Re and Pe

were treated as independent variables so that Sc is not

fixed.

Dimensionless form of initial and boundary conditions

Initial and boundary conditions applied to solve the

mathematical model can be summarized as follows:

Initial conditions

U ¼ V ¼ C ¼ 0 at t ¼ 0 ð12Þ

Boundary conditions at t [ 0

U ¼ V ¼ 0;oC

oZ¼ KC; at Z ¼ 0 for Ri�R�Ro

UðRÞ ¼ 1

lnðRo=RiÞR; V ¼ 0; C ¼ 1; at Z ¼ 1

for Ri�R�Ro

U ¼ V ¼ 0; C ¼ 1; at R ¼ Ri or Ro for 0\Z\1

ð13Þ

where K is the dimensionless reaction constant defined as,

K ¼ Hk=Ds: ð14Þ

External effectiveness factor, Sherwood number,

and Damkohler number

The mass-transfer rate of substrate was expressed in terms

of Sherwood number, Sh, defined as [17],

Sh ¼ �jH

Ds

ð15Þ

where �j is the average mass-transfer coefficient which can be

determined by equating the substrate convective flux with

the corresponding diffusive one, at either top or bottom wall.

For example, applying this fact at the bottom wall gives,

Ns ¼ �j co �R r¼ro

r¼ric dr

H

!

¼ Ds

R r¼ro

r¼ridc=dzjz¼0dr

Hð16Þ

Using the corresponding dimensionless variables of

Eq. 5, Eq. 16 can be rearranged as,

�j ¼ Ds

R R¼Ro

R¼RidC=dZjz¼0dR

H 1�R R¼Ro

R¼RiC dR

� � ð17Þ

The combination of Eqs. 15 and 17, yields,

Sh ¼R R¼Ro

R¼RidC=dZjz¼0dR

1�R R¼Ro

R¼RiC dR

� � ð18Þ

The external effectiveness factor, g, measures the reduction

in the overall reaction rate of the spatial model due to the effect

of diffusion. It is defined as the ratio between reaction rates for

the spatial and the well-mixed model [1],

g ¼ Nsjactual

Nsjwell�mixed

ð19Þ

In the current study, the reaction rate for the well-mixed

model is given by,

Nsjwell�mixed¼ kco ð20Þ

and the actual reaction rate for the spatial model is given by

Eq. 16. By combining Eqs. 5, 16, 19, and 20, the

effectiveness factor can be obtained as,

Bioprocess Biosyst Eng (2012) 35:875–884 877

123

g ¼ Nsjactual

Nsjwell�mixed

¼Ds

R r¼ro

r¼ridc=dzjz¼0 dr

Hðkc0Þ

¼R R¼Ro

R¼RidC=dZjZ¼0 dR

Kð21Þ

In the absence of diffusion limitations, g = 1; otherwise

g\ 1. In order to evaluate the mass-transfer against

surface-reaction resistances, the Damkohler number, Da,

which is defined as the ratio of the maximum reaction rate

to the maximum mass-transfer rate was computed from [1],

Da ¼ k

�j: ð22Þ

Using Eqs. 14 and 15, it can be verified that the

Damkohler number is ratio of the dimensionless reaction

constant, K, to the Sherwood number, Sh,

Da ¼ K

Sh: ð23Þ

The transport process is mass-transfer limited for

Da � 1. Otherwise, for Da � 1 the mass-transfer

resistance is low and the mass transport process is

reaction limited.

Solution methods of governing equations

While analytical method can be applied to solve the

mathematical model of pure-diffusion (Pe = 0), numerical

method is mandatory in forced-convection (Pe = 0). The

numerical procedure adopted to solve the unsteady trans-

port equations in terms of the primitive variables (U, V, P,

and C) is the SIMPLE finite control-volume algorithm with

a staggered-grid arrangement [18]. Numerical calculations

were performed using Computing Unsteady Three-dimen-

sional Elliptic Flows (CUTEFLOWS-II) code which uses

fourth-order accurate discretization schemes for both space

and time. This numerical code was successfully applied to

many flow configurations, in particular, lid-driven cavity

flows [5, 11, 16] and flows between corotating disks [19].

The equations of motion were solved first, and the resulting

velocity field was used in the unsteady substrate-transport

equation, which was then solved by using the same algo-

rithm. A central difference scheme was used to approxi-

mate pressure and diffusion terms, whereas a quadratic

upstream-weighted scheme was used for the convection

terms. Both the discrete Poisson equation resulting from

enforcing the continuity equation and the discrete substrate

mass transport were solved iteratively by means of the

conjugate gradient method [20]. An explicit fourth-order

Runge–Kutta method was used to integrate the discrete

conservation equations in time. The dimensionless time

step was selected to be small enough in all the calculations

to guarantee convergence and accuracy of the results.

The following enforced condition was employed as a

convergence criterion to recognize steady state condi-

tion and to terminate the corresponding calculation:

Kjþ1 � K j��

�� Kjþ1��

�� 10�6, where K denotes a depen-

dent variable of Eqs. 6–9 and j is the time index.

Results and discussion

This section analyzes both analytical and numerical results

obtained for various geometrical, kinetic, and operational

parameters. Influence of 2D grid size upon numerical

results was considered. Numerical results were also vali-

dated against the corresponding ones found in literature for

straight LDR arrangements.

Analytical results for diffusion–reaction (Pe = 0)

Analytical solutions of the substrate mass-transport equa-

tion, Eq. 9, can be obtained when Pe = 0. Since the bio-

reactor walls are impermeable, the equation simplifies to be

one-dimensional,

d2C

dZ2¼ 0 ð24Þ

with the boundary conditions,

dC

dZ¼ KC at Z ¼ 0

C ¼ 1 at Z ¼ 1

ð25Þ

The analytical solution of Eqs. 24 and 25 is,

C ¼ K

K þ 1Z; Pe ¼ 0 ð26Þ

Substituting the above substrate concentration profile

into Eqs. 18, 21, and 23, gives a Sherwood number of

Sh = 1, and the following external effectiveness factor and

Damkohler number:

g ¼ 1

K þ 1; Pe ¼ 0 ð27Þ

Da ¼ K; Pe ¼ 0: ð28Þ

In order to utilize the maximum capacity of the catalyst,

the bioreactor should operate at the highest value of

effectiveness factor, implying a lower value of K; see

Eq. 27. As H value increases, the size of bioreactor

enlarges and K value increases resulting in poor utilization

of the catalyst capacity. On the other hand, lower values of

H will lead to reactors with smaller volume but with better

utilization of enzyme.


123

Numerical results for reaction–diffusion-convection

(Pe = 0)

Numerical solutions were obtained for parameters in the

ranges Re B 400, Pe B 1,000, K B 4, and Rc C 1. The

upper limit for Reynolds number was set to 400 because

transition from 2D to 3D flow occurs at higher values [14].

For numerical convenience, non-zero average velocity

values at the top wall were considered even for

Re = Pe = 0. Thus, Re = 0 implies infinite kinematic

viscosity and Pe = 0 implies infinity diffusivity. When

Pe B 1, current results show that mass transfer rates, rep-

resented by Sherwood number, are not affected signifi-

cantly by Re, K, and Rc, and a pure diffusion is obtained in

which Sh = 1. In contrast, even when Pe B 1, the effec-

tiveness factor and Damkohler number are affected sig-

nificantly by the dimensionless reaction constant.

Grid test and validation of the numerical method

The influence of grid refinement upon numerical results

was assessed at Pe = 1,000, Re = 400, K = 4, and for

radius of curvature in the range Rc C 1. Under those

conditions, a sufficiently large number of grid points is

required to properly solve the momentum and concentra-

tion boundary layers developed in the bioreactor. Three

different uniform grids of nz 9 nr = 21 9 21, 81 9 81,

and 201 9 201 points were used to solve the mathematical

model. Where, nz and nr indicate the number of grid points

in axial and radial directions, respectively. Comparing the

results obtained from these three grids indicates that further

refinement from 81 9 81 grid is not necessary since sim-

ilar findings are noticed. Thus, a 81 9 81 grid was adopted

to calculate the flow and substrate transport responses over

the range of operational parameters considered.

In order to check the validity of the numerical approach,

present numerical solutions were compared against

benchmark results documented by Ghia et al. [12] on

steady 2D lid-driven flows within straight LDR with

Rc ? ?. For a purpose of better comparison between the

corresponding results, the dimensionless radial distance, R,

was normalized as (R - Ri)/(Ro - Ri). Figure 2 shows the

distributions of the radial and axial velocities along the

axial and normalized radial centerlines, respectively, for

Re = 100 and 400 and with different Rc values. For

Rc ? ?, as can be seen from the figure, the results of the

(a) Re = 100

Z

U

0 0.25 0.5 0.75 1-0.5

-0.25

0

0.25

0.5

0.75

1

Rc = 1 ; Present workRc = 4 ; Present workRc ≥ 20 ; Present workRc = ∞ ; Ghia et al. [12]Rc = ∞ ; Al-Shannag et al. [5]

(R-Ri)/Ro-Ri)

V

0 0.25 0.5 0.75 1-0.5

-0.25

0

0.25

0.5


(b)Re = 400

Z

U

0 0.25 0.5 0.75 1-0.5

-0.25

0

0.25

0.5

0.75

1


(R-Ri)/Ro-Ri)

V

0 0.25 0.5 0.75 1-0.5

-0.25

0

0.25

0.5


Fig. 2 Schematic diagram of

the square lid-driven cavity

reactor with boundary

conditions and coordinate

system


123

present work are in excellent agreement with the corre-

sponding benchmark data of Ghia et al. [12] obtained using

two uniform grids of 129 9 129 and 257 9 257 points for

Re = 100, 400, respectively. Figure 2 also illustrates the

significant effects of Rc and Re on the momentum boundary

layers developed near the walls. It is clear in Fig. 2 that

increasing both bioreactor curvature, by reducing Rc, and

Reynolds number reduces the momentum boundary layers.

To further check the accuracy of the present results, the

numerical results were also validated by comparison with

the analytical solutions (Pe ? 0). An excellent agreement

was found between the analytical and numerically calcu-

lated effectiveness factors, Sherwood number, and Dam-

kohler number, where discrepancies do not exceed 0.10%.

Two-dimensional flow structure in the bioreactor

The present study captures the well-known features of the

flow structure in a square LDR for moderate Reynolds

number. For all values of radius of curvature considered,

steady 2D flows were obtained for Reynolds number up to

Re = 400. Figure 3 presents the streamline patterns at

different values of Reynolds number and dimensionless

radius of curvature in the range Re B 400 and Rc C 1,

respectively. For Re B 1 and Rc B 20, identical flow

structures were observed. They are characterized by a large

primary recirculation zone which occupies the bulk of the

cavity with a pair of small circulation cells of much smaller

strength located in the lower corners of the cavity, namely

(a) Re ≤ 1; Rc ≥ 20 = 100; Rc c = 4

(R-Ri)/(Ro-Ri)

Z

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

(R-Ri)/(Ro-Ri)

Z

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

(R-Ri)/(Ro-Ri)

Z0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1

c c c = 4

(R-Ri)/(Ro-Ri)

Z

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

(R-Ri)/(Ro-Ri)

Z

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

(R-Ri)/(Ro-Ri)

Z

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

(b)Re ≥ 20 (c)Re = 100; R

(d)Re = 100; R = 1 (e)Re = 400; R ≥ 20 (f) Re=400; R

(g)Re = 400; Rc = 1

(R-Ri)/(Ro-Ri)

Z

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Fig. 3 Streamline flow patterns at different dimensionless radius of curvature, Rc, and Reynolds numbers, Re


123

the downstream secondary eddy (DSE) near the bottom-

right corner, and the upstream secondary eddy (USE) near

the bottom-left corner; see Fig. 3a. The center of the pri-

mary vortex (BV) is located near the axial centerline in a

region between the radial centerline and the top wall.

For slightly curved bioreactors, large Rc, an increase in

Reynolds number leads to a displacement of the primary

cell’s center towards the geometrical center of the reactor,

and to an increased size of the corner vortices; see Fig. 3a

and e. This is in excellent agreement with the results found

by Al-Shannag et al. [5] for straight LDR.

At low levels of Reynolds numbers, increasing the

radius of curvature of the reactor does not affect signifi-

cantly the 2D flow structure; see Fig. 3b–d. In contrast, at a

relatively higher Reynolds number, Re = 400, increasing

the radius of curvature affects significantly the 2D flow

structure, and it causes a migration of the primary cell’s

center towards the top-right corner, and an increase in the

Pe

Sh

100 101 102 103

1

2

3

4

5

6

Rc = 1Rc = 2Rc = 4Rc = 10Rc ≥ 20

Re ≤ 1

PeS

h100 101 102 103

1

2

3

4

5

6

Rc = 1Rc = 2Rc = 4Rc = 10Rc ≥ 20

Re = 400

Pe

η

100 101 102 1030

0.25

0.5

0.75

1

Rc = 1Rc = 2Rc = 4Rc = 10Rc ≥ 20

Re ≤ 1

Pe

η

100 101 102 3100

0.25

0.5

0.75

1

Rc = 1Rc = 2Rc = 4Rc = 10Rc ≥ 20

Re = 400

(a)

(b)

(c)

Pe

Da

100 101 102 103

0

1

2

3

4

5

Rc = 1Rc = 2Rc = 4Rc = 10Rc ≥ 20

Re ≤ 1

Pe

Da

100 101 102 103

0

1

2

3

4

5

Rc = 1Rc = 2Rc = 4Rc = 10Rc ≥ 20

Re = 400

Fig. 4 Variations of

a Sherwood number, b external

effectiveness factor, and

c Damkohler number with the

Peclet number at Re B 1 and

K = 4 and for different values

of dimensionless radius of

curvature Rc


123

size of the corner vortices, specially downstream secondary

eddy; see Fig. 3e and f.

Effect of Pe, Re, K, and Rc on bioreactor performance

Figure 4 illustrates the effect of varying Peclet number on

Sherwood number, effectiveness factor, and Damkohler

number for Rc C 1, K = 4, and Re B 400. The value of

Peclet number was adjusted between 1 and 1,000 in order

to cover a moderate range of operating conditions while

maintaining the accuracy of results. For Rc C 1 and when

Pe ? 0, Sherwood number has a value of Sh = 1 and both

g and Da are observed to approach the asymptotic values

shown in Fig. 4 and given by Eqs. 27 and 28. This basi-

cally implies that the transfer of substrate from the top to

the bottom wall occurs by pure molecular diffusion and

that the primary vortex observed in Fig. 3 does not play

any role in mass transport enhancement. Furthermore, the

results clearly indicate that an increase in Pe from 10 to

1,000 augments both Sherwood number and external

effectiveness factor, and diminishes the Damkohler num-

ber. This proves that forced convection takes over molec-

ular diffusion with increasing Peclet number. The increase

in convection activity increases the mass transfer rate; i.e.,

enhances reactor performance, due to the thinner boundary

layers that develop near the top and bottom walls; see

Fig. 5.

Since the flow structure shown in Fig. 3 does not vary

significantly with Rc at Re B 1, the reactor curvature has

nearly no effect on Sh, g, and Da as can be seen in

Fig. 4a–c. While at relatively higher values of Reynolds

number, Re = 400, the curvature affects significantly the

reactor performance. Namely, increasing reactor curva-

ture, by decreasing Rc, reduces both Sherwood number

and external effectiveness factor, and increases the

Damkohler number. The dependence of the 2D flow

structure on Rc at Re = 400, shown in Fig. 3, explains

this behavior. At Re = 400, as Rc increases the main

circulation cell reduces in size and the sizes of corner

vortices become larger; especially the downstream sec-

ondary eddy. With this flow structure, no direct transport

of liquid substrate from top wall to the bottom one is

performed by the main circulation cell over a significant

portion of the reactor. For this portion, the transport

process is achieved in two stages instead. First, the sub-

strate is driven by main circulation cell from the top wall

to the corner vortices. Second, the corner vortices will

deliver the substrate to the bottom wall. Obviously, the

two-stage transport process has a lower mass transfer rate

than the direct one. Table 1 summarizes numerical values

of Sh, g, and Da at Pe = 1,000, K = 4, Re = 1 and 400,

and with different values of Rc. It can be concluded from

Fig. 4a–c and Table 1 that the bioreactor performance is

improved largely in slightly curved enclosures and by

increasing both Reynolds and Peclet numbers.

Figure 6 shows the variations of Sherwood number,

effectiveness factor and Damkohler number with Peclet

number at Re = 400 and Rc C 20. As expected, Fig. 6a

confirms the irrelevance between Sh and K; See Eq. 18. It

Z

C

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Pure diffusion; Pe=0Rc = 1; Pe = 50Rc = 1; Pe = 1000Rc ≥ 20; Pe = 50Rc ≥ 20; Pe = 1000

(a) Re ≤ 1

(b)Re=400

Z

C

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Pure diffusion; Pe = 0Rc = 1; Pe = 50Rc = 1; Pe = 1000Rc ≥ 20; Pe = 50Rc ≥ 20; Pe = 1000

Fig. 5 Substrate concentration profiles at the cavity centerline, (R - Ri)/

(Ro - Ri) = 0.5, for different values of Rc, Re, and Pe and with K = 4

Table 1 Steady values of Sherwood number, effectiveness factor,

and Damkohler number resulted from 2D flow calculations at K = 4

and Pe = 1,000

Rc Re B 1 Re = 400

Sh g Da Sh g Da

1 4.930 0.552 0.811 2.831 0.414 1.413

2 5.052 0.558 0.792 3.828 0.489 1.045

4 5.081 0.560 0.787 4.924 0.552 0.812

10 5.090 0.560 0.786 5.623 0.584 0.711

C20 5.091 0.560 0.786 5.843 0.594 0.685

? Al-Shannag et al. [5]

– 0.561 0.785 – – –


123

can be also noticed that the bio-reaction system with higher

K value has lower effectiveness factor and larger Dam-

kohler number. This means that a bioreactor of higher

K requires larger Peclet and Reynolds numbers to improve

the performance. In this analysis, Peclet number is con-

sidered as an operational parameter which could be

adjusted by either changing the velocity of the top wall or

the height of the reactor. For fixed values of H, it should be

of interest to operate with the largest possible velocity of

the top wall in order to increase Pe, and hence, the effec-

tiveness factor. The upper limit of the top wall velocity

should be decided based on hydrodynamic considerations

in the reactor; particularly the shear stress levels.

Conclusions

The problem of annular-shaped lid-driven reactor with

surface enzymatic-reaction was considered. The charac-

teristics of the flow field and substrate concentration

boundary layer were analyzed under different operating

conditions. Pertinent dimensionless groups: Peclet number,

Reynolds number, and dimensionless reaction constant,

were utilized in the undergoing analysis. The overall

observation indicates that pure diffusion dominates the

transport process for Pe B 1 which results in low biore-

actor performance. With increasing Peclet number, forced

convection develops inside the bioreactor will improve the

performance notably. The results imply that increasing

Reynolds and Peclet numbers in slightly curved bioreactors

always induces flow activities causing an increase in the

substrate transfer rate by means of forced convection. For

strongly curved reactors, the increase in Reynolds number

up to moderate values, Re B 400, retards momentum and

substrate transport leading to low levels of Sherwood

number and effectiveness factor, and hence, larger values

of Damkohler number. Therefore, it is not recommended to

design lid-driven reactors with strong curvature for bio-

reaction systems that operate at Re B 400.

Acknowledgments This work has been carried out during sabbati-

cal leave granted to the author from Al-Balqa Applied University

(BAU) during the academic year 2010/2011. The author would like to

appreciate the support from both Al-Balqa Applied University and

King Khalid University at which the sabbatical leave has been spent.

Pe

Sh

100 101 102 1031

2

3

4

5

6

K=1K=2K=4

Pe

η

100 101 102 103

0

0.25

0.5

0.75

1

K = 1K = 2K = 4

(a) (b)

(c)

Pe

Da

100 101 102 1030

1

2

3

4

5

K = 1K = 2K = 4

Fig. 6 Variations of

a Sherwood number, b external

effectiveness factor, and

c Damkohler number with

Peclet number at Re = 400 and

Rc C 20 and for different values

of dimensionless reaction

constant K


123

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Mass transport enhancement in annular-shaped lid-driven bioreactor

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