The drug loaded within the bead is the source for mass transfer, whereas the fluid surrounding the bead is the sink for mass transfer. This is

Ao. The drug loaded

within the bead is the source for mass transfer, whereas the fluid surrounding the bead is the sink for

mass transfer.This is an unsteady-state process, as the source formass transfer is contained within the

diffusion control volume itself.

Consider a limiting case where the resistance to film mass transfer of the drug through the liquid

boundary layer surrounding the capsule surface to the bulk surrounding the fluid is negligible.

Furthermore, assume that the drug is immediately consumed or swept away once it reaches the bulk

solution so that in essence the surrounding fluid is an infinite sink. In this particular limiting case, cAs is equal to zero, so at a long time the entire amount of drug initially loaded into the bead will be

depleted. If radial symmetry is assumed, then the concentration profile is only a function of the r

direction (Figure 27.4).

It is desired to design a spherical capsule for the timed release of the drug dimenhydrinate,

commonly called Dramamine, which is used to treat motion sickness. A conservative total dosage for

one capsule is 10 mg, where 50% of the drug must be released to the body within 3 h. Determine the

size of the bead and the initial concentration of Dramamine in the bead necessary to achieve this

dosage. The diffusion coefficient of Dramamine (species A) in the gel matrix (species B) is 3 � 10�7 cm2/s at a body temperature of 378C. The solubility limit of Dramamine in the gel is 100 mg/cm3, whereas the solubility of Dramamine in water is only 3 mg/cm3.

The model must predict the amount of drug released vs. time, bead diameter, initial concentra-

tion of the drug within the bead, and the diffusion coefficient of the drug within the gel matrix. The

physical system possesses spherical geometry. The development of the differential material balance

model and the assumptions associated with it follow the approach presented in Section 25.4.

One way to deliver a timed dosage of a drug within the human body is to ingest a capsule and allow it to settle in the gastrointestinal system. Once inside the body, the capsule slowly releases the drug by a

diffusion-limited process. A suitable drug carrier is a spherical bead of a nontoxic gelatinous material

that can pass through the gastrointestinal system without disintegrating. A water-soluble drug

(solute A) is uniformly dissolved within the gel and has an initial concentrationC

The general differential equation for mass transfer reduces to the following partial differential

equation for the one-dimensional unsteady-state concentration profile cA(r; t):

@cA @t

¼ DAB @2cA @r2

þ 2 r

@cA @r

� � (27-18)

Key assumptions include radial symmetry, dilute solution of the drug dissolved in the gel matrix, and

no degradation of the drug inside the bead (RA ¼ 0). The boundary conditions at the center (r ¼ 0) and the surface (r ¼ R) of the bead are

r ¼ 0; @cA @r

¼ 0; t � 0 r ¼ R; cA ¼ cAs ¼ 0; t > 0

At the center of the bead, we note the condition of symmetry where the flux NA(0; t) is equal to zero. The initial condition is

t ¼ 0; cA ¼ cAo; 0 � r � R The analytical solution for the unsteady-state concentration profile cA(r; t) is obtained by separation- of-variables technique described earlier. The details of the analytical solution in spherical coordi-

nates are provided by Crank. The result is

Y ¼ cA � cAo cAs � cAo

¼ 1 þ 2R pr

X1 n¼1

ð�1Þn n

sin npr

R

� � e�DABn

2p2t=R2; r 6¼ 0; n ¼ 1; 2; 3; : : : (27-19)

At the center of the spherical bead (r ¼ 0), the concentration is

Y ¼ cA � cAo cAs � cAo

¼ 1 þ 2 X1 n¼1

(� 1)ne�DABn2p2t=R2; r ¼ 0; n ¼ 1; 2; 3; : : : (27-20)

Once the analytical solution for the concentration profile is known, calculations of engineering

interest can be performed, including the rate of drug release and the cumulative amount of drug

release over time. The rate of drug release, WA, is the product of the flux at the surface of the bead

(r ¼ R) and the surface area of the spherical bead

WA(t) ¼ 4pR2NAr ¼ 4pR2 �DAB @cA(R, t)

@r

� � (27-21)

It is not so difficult to differentiate the concentration profile, cA(r; t), with respect to radial coordinate r, set r ¼ R, and then insert back into the above expression for WA(t) to ultimately obtain

WA(t) ¼ 8pRcAoDAB X1 n¼1

e�DABn 2p2t=R2 (27-22)

r = 0

r

cA(r, 0) = cAo

∂cA ∂r

Bulk fluid

cA = 0

NA(R, t)

t = 0

= 0

Drug in gel bead

r = R cA⎯ = 0

r

cA(r, 0) cAo

Bulk fluid

cA = 0

t > 0

∞∞

Figure 27.4 Drug release from a spherical gel bead.

The above equation shows that the rate of drug release will decrease as time increases until all of the

drug initially loaded into the bead is depleted, at which point WA will go to zero. Initially, the drug is

uniformly loaded into the bead. The initial amount of drug loaded in the bead is the product of the

initial concentration and the volume of the spherical bead

mAo ¼ cAoV ¼ cAo 4

3 pR3

The cumulative amount of drug release from the bead over time is the integral of the drug release rate

over time

mAo � mAðtÞ ¼ Zt 0

WA(t) dt

After some effort, the result is

mA(t)

mAo ¼ 6

p2

X1 n¼1

1

n2 e�DABn

2p2t=R2 (27-23)

The analytical solution is expressed as an infinite series summation that converges as ‘‘n’’ goes

to infinity. In practice, convergence to a single numerical value can be attained by carrying the series

summation out to only a few terms, especially if the dimensionless parameter DAB t/R 2 is relatively

large. It is a straightforward task to implement the infinite series summation on a spreadsheet

program such as Excel (Microsoft Corporation).

Table 27.1 Excel spreadsheet for drug�release profile, Example 2 mAo ¼ 10 mg DAB ¼ 3:00E � 07 cm2/s

R ¼ 0:326 cm CAo ¼ 68:9 mg/cm3

Time, t (s) 0.0 18 180 1800 3600 7200 10800 14400 18000 21600

Time, t (h) 0.0 0.005 0.05 0.50 1.00 2.00 3.00 4.00 5.00 6.00

mA(t)/mAo 1 � mA(t)/mAo

1.0 0.964 0.925 0.774 0.689 0.578 0.500 0.439 0.389 0.347

0.0 0.036 0.075 0.226 0.311 0.422 0.500 0.561 0.611 0.653

Series term n ¼ 1 9.99E�01 9.95E�01 9.51E�01 9.05E�01 8.18E�01 7.40E�01 6.70E�01 6.06E�01 5.48E�01 2 2.49E�01 2.45E�01 2.05E�01 1.67E�01 1.12E�01 7.50E�02 5.02E�02 3.36E�02 2.25E�02 3 1.11E�01 1.06E�01 7.08E�02 4.51E�02 1.83E�02 7.41E�03 3.00E�03 1.22E�03 4.94E�04 4 6.20E�02 5.77E�02 2.80E�02 1.26E�02 2.52E�03 5.07E�04 1.02E�04 2.05E�05 4.11E�06 5 3.95E�02 3.53E�02 1.14E�02 3.26E�03 2.66E�04 2.16E�05 1.76E�06 1.44E�07 1.17E�08 6 2.73E�02 2.32E�02 4.57E�03 7.51E�04 2.03E�05 5.49E�07 1.48E�08 4.01E�10 1.08E�11 7 1.99E�02 1.60E�02 1.75E�03 1.50E�04 1.10E�06 8.07E�09 5.92E�11 4.34E�13 3.19E�15 8 1.51E�02 1.13E�02 6.31E�04 2.55E�05 4.15E�08 6.77E�11 1.10E�13 1.80E�16 2.93E�19 9 1.19E�02 8.22E�03 2.13E�04 3.66E�06 1.08E�09 3.21E�13 9.52E�17 2.82E�20 8.36E�24 10 9.51E�03 6.06E�03 6.64E�05 4.41E�07 1.94E�11 8.56E�16 3.77E�20 1.66E�24 7.33E�29 11 7.78E�03 4.50E�03 1.91E�05 4.43E�08 2.38E�13 1.27E�18 6.84E�24 3.67E�29 1.97E�34 12 6.46E�03 3.37E�03 5.07E�06 3.71E�09 1.98E�15 1.06E�21 5.65E�28 3.02E�34 1.61E�40 13 5.44E�03 2.54E�03 1.23E�06 2.57E�10 1.12E�17 4.87E�25 2.12E�32 9.23E�40 4.02E�47 14 4.62E�03 1.91E�03 2.75E�07 1.48E�11 4.29E�20 1.25E�28 3.61E�37 1.05E�45 3.04E�54 15 3.97E�03 1.44E�03 5.59E�08 7.03E�13 1.11E�22 1.76E�32 2.79E�42 4.41E�52 6.98E�62 16 3.44E�03 1.08E�03 1.04E�08 2.76E�14 1.95E�25 1.38E�36 9.72E�48 6.87E�59 4.85E�70 17 2.99E�03 8.12E�04 1.76E�09 8.93E�16 2.30E�28 5.94E�41 1.53E�53 3.95E�66 1.02E�78 18 2.62E�03 6.08E�04 2.71E�10 2.38E�17 1.83E�31 1.41E�45 1.09E�59 8.41E�74 6.48E�88 19 2.31E�03 4.53E�04 3.80E�11 5.22E�19 9.85E�35 1.86E�50 3.50E�66 6.60E�82 1.24E�97 20 2.05E�03 3.36E�04 4.86E�12 9.43E�21 3.56E�38 1.34E�55 5.06E�73 1.91E�90 7.20E�108

A representative spreadsheet solution is provided in Table 27.1. Note that in Table 27.1 the terms

within the series summation rapidly decay to zero after a few terms. The cumulative drug release vs.

time profile is shown in Figure 27.5. The drug-release profile is affected by the dimensionless

parameter DAB t=R 2. If the diffusion coefficient DAB is fixed for a given drug and gel matrix, then the

critical engineering-design parameter we can manipulate is the bead radius R. As R increases, the rate

of drug release decreases; if it is desired to release 50% of Dramamine from a gel bead within 3 h, a

bead radius of 0.326 cm (3.26 mm) is required, as shown in Figure 27.5. Once the bead radius R is

specified, the initial concentration of Dramamine required in the bead can be backed out

cAo ¼ mAo

V ¼ 3mAo

4pR3 ¼ 3ð10 mgÞ

4p(0:326 cm)3 ¼ 68:9 mg

cm3

In summary, a 6.52-mm-diameter bead with an initial concentration of 68:9 mg=cm3 will dose out the required 5 mg of Dramamine within 3 h. The concentration profile along the r direction

at different points in time is provided in Figure 27.6. The concentration profile was calculated by

spreadsheet similar to the format given in Table 27.1. The concentration profile decreases as time

increases and then flattens out to zero after the drug is completely released from the bead.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

D ru

g c

o n ce

n tr

a tio

n ,

c A (r

, t)

( m

g /c

m 3 )

Radial position, r (cm)

3 h 12 h 24 h

0

20

40

60

80

100

Figure 27.6 Concentration

profile of a 0.326-cm-radius

bead after 3, 12, and 24 h.

0

F ra

ct io

n o

f d ru

g r

e le

a se

d , 1 –

m A (t

)/ m

A O

Time, t (h) 1 2

0.326 cm 0.163 cm

3 4 5 6 0

0.2

0.4

0.6

0.8

1.0

Figure 27.5 Fractional drug

release vs. time profiles.

Chemical Engineering Journal 104 (2004) 35–43

Modeling of counter current moving bed gas-solid reactor used in direct reduction of iron ore

Daniel R. Parisi∗, Miguel A. Laborde Departamento de Ingenier´ıa Quı́mica, Facultad de Ingenier´ıa, Universidad de Buenos Aires, Pabell´on de Industrias,

Ciudad Universitaria, 1428 Buenos Aires, Argentina

Received 11 November 2003; received in revised form 6 August 2004; accepted 9 August 2004

Abstract

In this work, the shaft furnace reactor of the MIDREX® process is simulated. This is a counter current gas-solid reactor, which transforms iron ore pellets into sponge iron.

Simultaneous mass and energy balance along the reactor leads to a set of ordinary differential equation with two points boundary conditions. The iron ore reduction kinetics was modelated with the unreacted shrinking core model. Solving the ODE system allows to know the c

tion ( e used for d ©

K

1

g t f n o f

w [ a m v p

m

dy- lance dy-

tor same nking

but

ctor d the oth

. c- he nsfer con- olid is tive

1 d

oncentration and temperature profiles of all species within the reactor. The model was able to satisfactorily reproduce the data of two MIDREX® plants: Siderca (ARGENTINA) and Gilmore Steel Corpora

U.S.A.). Also, it was used to explore the performance of the reactor under different operating conditions. This capacity could b esign and control purpose. 2004 Elsevier B.V. All rights reserved.

eywords: Shaft furnace simulation; Gas-solid reactor; Moving bed; Direct reduction.

. Introduction

Direct reduction of iron ore is today’s major process for enerating metallic iron, necessary in the iron and steel indus-

ry. World production of direct reduce iron (DRI) has grown rom near zero in 1970 to 45.1 Mt in 2002. MIDREX® Tech- ology is the most important one, responsible for the 66.6% f the world total DRI production. Its main reactor (the shaft

urnace) is a moving bed reactor. The first studies related to moving bed solid-gas reactors

ere performed by Munro and Amundson[1], Amundson 2] and Siegmund et al.[3]. In these works, the authors used linear function of the solid temperature in order to approxi- ate the reaction rate and to obtain analytical solutions. The

alidity of this solution is limited to a narrow range of tem- eratures.

∗ Corresponding author. Tel.: +54 11 4576 3240; fax: +54 11 4576 3241. E-mail addresses:[email protected] (D.R. Parisi),

[email protected] (M.A. Laborde).

Schaefer et al.[4] studied the heat generation in a stea state reactor. They used a step function for the heat ba and the results show the existence of multiplicity of stea states.

Yoon et al.[5] developed a model for a Lurgi type reac used in the carbon gasification. They considered the temperature in both phases (solid and gas) and the shri unreacted core model for the solid particle.

Amundson and Arri[6] analyzed the same system considering different temperatures in both phases.

Arce et al.[7] studied a countercurrent moving bed rea using a heterogeneous model for the reactor design an shrinking core model for the solid particle. They applied b models to an irreversible first order exothermic reaction

Rao and Pichestapong[8] developed a model for a rea tor in which the reduction of iron mineral is carried out. T model considers that the controlling step is the mass tra of the gaseous reactants in the product solid layer. The centration of the gaseous species on the interface gas-s that of the equilibrium and it was evaluated using an itera

385-8947/$ – see front matter © 2004 Elsevier B.V. All rights reserved. oi:10.1016/j.cej.2004.08.001

36 D.R. Parisi, M.A. Laborde / Chemical Engineering Journal 104 (2004) 35–43

Nomenclature

Ap pellet external area (cm 2)

C reactor gas concentration (mol/cm3) D effective diffusion coefficient (cm2/s) Gm molar flow (mol/cm

2 s) H reaction enthalpy (cal/mol) h global heat transfer coefficient (pellets/gas)

(cal/s/cm2/K) k kinetics constant of the surface reaction (cm/s) kg external mass transfer coefficient (cm/s) L reduction zone length (cm) Mw molecular weight np number of pellets per unit volume (1/cm

3) R reaction rate (mol/cm3s) R̂ reaction rate per pellet (mol/s) r0 external radius of the pellet (cm) rc radius of the unreacted core (cm) T temperature (◦C) u gas velocity (cm/s) X extent of reaction/extent of reactant conversion

(mol/cm3) z space variable inside the reactor (cm)

Subscripts atm atmosphere i ith reaction in reactor inlet j jth reactant (gas or solid) n gaseous reactant rs reactive solid (Fe2O3) ps product solid (Fe) sol solid g gas

Greek letters α stoichiometric coefficient ρ density of the solid reactant (g/cm3)

method. As a consequence, the problem cannot be solved in terms of differential equations system. In this paper, the heat balance is avoided since the authors assumed a linear function of the temperature with the reactor length.

The aim of this work is to model and simulate a solid-gas countercurrent moving bed reactor in which the reduction of iron ore pellets is performed using CO and H2 as reducing gases.

In order to do that, mass and energy balance are taken into account simultaneously. This leads to a set of ordinary differential equation with two point boundary conditions.

The model is validated with data from two industrial plants. Also, it is used to explore the performance of the reactor under different operating conditions.

Fig. 1. Shaft furnace geometry.

1.1. Shaft furnace of the MIDREX® process

The main function of the shaft furnace is to generate sponge iron from iron ore. The solids flow downwards by gravity and the reducing gases flow upwards in counter cur- rent, while the corresponding chemical transformations oc- cur. Fig. 1 shows a scheme of the reactor.

As it can be observed, the furnace consists of a vertical cylindrical container, with a conic lower zone. The inner wall is covered with insulating materials resistant to erosion.

The reducing gases enter by the middle zone of the reactor through the bustle, which consists of a channel with approx- imately 70 nozzles that direct the gas towards the center of the solid bed.

Immediately underneath, the upper burdenfeeders are lo- cated. Following in descendent order one can found: the wind boxes (which take the cooling gas that circulates around the lower conical zone of the reactor), cooling gas distrib- utor or (“inverted Christmas tree”), which besides to inject cooling gases has the function to support most of the bed weight.

Gases going out from the shaft furnace are recycled into another reactor: the Reformer. This is a fixed bed catalytic reactor, which transforms the process gas (with addition of natural gas) into reducing gas again.

2

l- l

nre- ade

( film ith

. The model

In order to model the MIDREX® shaft furnace, the fo owing approximations are considered:

(a) The iron ore pellet consumption is governed by the u acted shrinking core model. This aproximation was m by several authors, see for instance[8,9].

b) Mass and heat transfer resistances through the around the solid particle are negligible comparing w

D.R. Parisi, M.A. Laborde / Chemical Engineering Journal 104 (2004) 35–43 37

diffusional resistance inside the porous solid (kg � D/2/r0).

(c) Only steady-state operating conditions will be consid- ered.

(d) Plug flow is assumed for gas and solid phase.

Due to high gas flow rate in the reactor, turbulence regime is reached. Under this situation, inertial effects are predom- inanant. So it will not be considered neither axial nor radial dispersion[10].

For the solid phase this hypothesis was verified through a previous work[11]. Distinct element method simulations (DEM) were performed in order to study the granular bed dynamics of a typical MIDREX® shaft furnace.

Only the global direct reduction reactions are taken into account. The area of interest

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