Chemical Engineering Science 63 (2008) 4924–4934www.elsevier.com/locate/ces

Approximate design of loop reactors

Olga Nekhamkina∗, Moshe SheintuchDepartment of Chemical Engineering, Technion-I.I.T., Technion City, Haifa 32 000, Israel

Received 2 May 2007; accepted 11 September 2007Available online 21 September 2007

Abstract

A loop reactor (LR) is an N-unit system composed of a loop with gradually shifted inlet/outlet ports. This system was shown in our previousstudy [Sheintuch M., Nekhamkina O., 2005. The asymptotes of loop reactors. A.I.Ch.E. Journal 51, 224–234] to admit an asymptotic modelfor a loop of a fixed length with N → ∞. Both the finite-unit and the asymptotic model exhibit a quasi-frozen or a frozen rotating pulse (FP)solution, respectively, within a certain domain of parameters that becomes narrower as feed concentration declines.

In the present paper we derive approximate solutions of the ignited pulse properties in an LR as a function of the external forcing (switching)rate. Analysis of these solutions enable us to determine the maximal temperature in the system, as well as the boundaries of the FP domain.For the optimal solution we determine the maximal temperature and conversion dependencies on the reactor length and on N. The approximatesolutions are verified by comparison with direct simulations of the asymptotic model and a good agreement was found. The obtained resultscan be successfully used for prediction of the finite unit LR.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Chemical reactors; Packed bed; Mathematical modeling; Pulse and front velocity; Model reduction

1. Introduction

The search for a technological solution for low-concentrationVOC combustion has led to several conceptual solutions thatcombine a catalytic packed bed with enthalpy recuperation.The reverse-flow reactor (RFR), the best known example in thisclass, has been investigated for the past two decades and hasbeen applied commercially. The loop reactor (LR), in the formof several units with gradual feed switching between them, hasbeen conceptually proposed by Matros (1989) as one of the tran-sient schemes that follow front propagation within the system.Such a reactor was numerically simulated for the case of twounits (Haynes and Caram, 1994) or for three units (Brinkmannet al., 1999), for certain applications like VOC abatementand exothermic reversible reaction. The concept of LR hasbeen extensively studied during the past five years focusingon the asymptotic solutions of its behavior (Sheintuch and

∗ Correspondign author. Tel.: +972 4 8293561.E-mail addresses: aermwon@tx.technion.ac.il (O. Nekhamkina),

cermsll@tx.technion.ac.il (M. Sheintuch).

0009-2509/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2007.09.019

Nekhamkina, 2005; Fissore et al., 2006a) as well as on itsstability and dynamics (Russo et al., 2002; Altimary et al.,2006). The first experimental implementation of the LR wasrecently reported for an almost isothermal process (selectivecatalytic reduction, NH3 + NO, Fissore et al., 2006b) and forthe VOC combustion concept (ethylene oxidation, Madai andSheintuch, in prep.).

In our recent study (Sheintuch and Nekhamkina, 2005) wedemonstrated that there exists an asymptotic analytical solutionby studying the LR of a fixed length (L) with increasing num-ber of units (N) and by deriving an asymptotic (continuous)model for the case of infinitely large N. This model impliesthat increasing N is accompanied by properly decreasing unitlength �L=L/N and switching time �sw, so that the switchingvelocity (Csw = �L/�sw, i.e. unit length divided by switchingtime) is preserved. In such a case the LR can be describedas a single unit of length L with a moving feed/exit port at aposition that follows �in = �0

in + Csw�. It was shown that theasymptotic model (Sheintuch and Nekhamkina, 2005) can ex-hibit a rotating pulse solution, propagating with velocity Csw,which is “frozen” in a rotating coordinate system within a cer-tain domain of parameters. The finite-unit model (Haynes and

O. Nekhamkina, M. Sheintuch / Chemical Engineering Science 63 (2008) 4924–4934 4925

0 0.5 1 1.5 2

0

2

4

6

8

10

Csw/Vth

B/B

0

Fig. 1. Typical domain of existence of “frozen” pulse solutions in the(B/B0, Csw/Vth) plane showing the domain for an asymptotic model (Eqs.(7)–(9), dots) and for a finite unit model (N = 4, circles). The parameters arelisted in the text (B0 corresponds to �Tad =50). Dashed-dotted line marks thedimensionless ideal front velocity (Vid/Vth) calculated from Eqs. (1) and (2).

Caram, 1994; Brinkmann et al., 1999, etc.), can exhibit onlyquasi-“frozen” solutions.

Very much like the RFR the LR can sustain high tempera-tures, significantly beyond the adiabatic temperature rise. Themain drawback of the LR is the narrow domain of switchingvelocities over which such a solution can be sustain. Typicalbifurcation diagram showing the domains of “frozen” patterns(FP) of the asymptotic LR model and of a four-unit system inthe (B, Csw) plane are shown in Fig. 1. These domains expandwith increasing exothermicity (or feed concentration, B) form-ing a cusp-like structure. The finite unit FP domain is enclosedwithin the corresponding limit ones and tends to the asymptoticsolution with increasing N. A proper design should map thewidth of this domain of switching velocities. An even better ap-proach should use control for that purpose. In our recent study(Sheintuch and Nekhamkina, 2005) we have suggested that theFP domain is approximately bounded by the “ideal” combus-tion front (Vid, a front forming in an infinitely long packedbed) on the left and by the heat front (Vth) propagation velocityon the right. These estimations are not satisfactory in predict-ing the real boundaries (Fig. 1). The present work improvessignificantly these predictions and uses suggests a systematicapproach for designing a LR by predicting the boundaries ofFP solutions using the asymptotic LR model to derive approx-imate relations between the main parameters of “frozen” solu-tions (maximal temperature rise, conversion) and the switchingvelocity.

As it follows from the conducted analysis, the front velocity(V), as well as the maximal temperature (Tm) behind the frontare essential parameters that define the reactor regime. In cat-alytic systems the moving pulse temperature can be increasedsignificantly by pushing the front. An ideal front admits thefollowing relations for the case of a single irreversible reaction(Wicke and Vortmeyer, 1959):

�Tm = Tm − T0 = �Tadv − Vid

v − LeV id,

Vid = v�Tm − �Tad

Le�Tm − �Tad. (1)

Obviously, the front velocity cannot exceed the thermal frontvelocity Vth = v/Le, which corresponds to an infinitely largetemperature rise in the ideal front. Relatively simple integralbalance relations were derived to determine Tm for a gas-lesscombustion (Zeldovich et al., 1959; Puszynski et al., 1987).Several attempts were made to account for the effect of con-vection in a catalytic bed (i.e., Le?1): Kiselev et al. (1980)and Kiselev (1993) using the narrow reaction zone assumptionproposed by Frank-Kamenetski (1955) derived an approximaterelation between Tm and Vid in the following dimensionlessform:

BPey(u − Vid)2

(1 + ym/�)2Da exp[ym/(1 + ym/�)] = 1 + �(ym, Vid), (2)

where �(ym, Vid) is a small term which can be neglected. Eqs.(1) and (2) allow to approximate the maximal temperature riseof an ideal front and its velocity. Kiselev’s approach (which hasreceived little attention in the west) was successfully verifiednumerically (Burghardt et al., 1999).

In the present study we derived two approximate relationsbetween the maximal temperature rise and the propagation ve-locity of a non-ideal front in a finite-length LR operating in the“frozen” mode following Kiselev’s approach and using the ap-propriate form of boundary conditions (BC). The first approxi-mation is based on the assumption of the complete conversionat a point where the temperature achieves its maximum value(i.e. xm(�m) = 1 if T (�m) = Tm), and differs from relation (2)only by additional terms on the right-hand side (RHS) that ac-count the effect of BC. The second approach accounts for the in-complete conversion and includes an additional approximationwith respect to xm. The obtained approximations are verifiedby comparison with direct simulations of the asymptotic LRmodel within a wide domain of operation conditions. In Sec-tion 5 we analyze the optimal solution, that is with Csw = Vth,that assures the highest temperatures and derive the relation be-tween the maximal temperature and the exit conversion with thereactor length and the number of units. Thus, these approxima-tions provide an easy approach for crude design, which shouldbe fine-tuned by direct simulations.

The structure of this work is the following: in the next sectionwe review the asymptotic LR model formulated in our previousstudy (Sheintuch and Nekhamkina, 2005). In the third sectionthe approximate relations showing the effect of the switchingvelocity on the maximal temperature rise and the boundariesof the slow-switching domain are derived. These relations areverified in the fourth section by comparison with the directsimulation results of the asymptotic LR model within a widedomain of operation conditions following by brief recommen-dation concerning their implementation.

2. Reactor models

We limit our study to generic first-order activated andexothermic reactions with the Arrhenius kinetics (r =A exp(−E/RT )C). To a first approximation the transport co-efficients (ke, Df ) and thermodynamic parameters (�, cp) areassumed to be constant. With these assumptions the enthalpy

4926 O. Nekhamkina, M. Sheintuch / Chemical Engineering Science 63 (2008) 4924–4934

and mass balances may be written in the following dimension-less form:

Le�y

��+ v

�y

��− 1

Pey

�2y

��2= BDa(1 − x) exp

(�y

� + y

)= B(1 − x)f (y), (3)

�x

��+ v

�x

��− 1

Pex

�2x

��2= Da(1 − x) exp

(�y

� + y

)= (1 − x)f (y) (4)

with appropriate initial and boundary conditions. In the caseof a once-through operation Danckwerts’ BC are typicallyemployed

� = �in,1

Pey

�y

��= v(y − yin),

1

Pex

�x

��= v(x − xin), (5)

� = �out,�y

��= 0,

�x

��= 0.

Here conventional notation is used

� = z

z0, � = tu0

z0, y = �

T − T0

T0, x = 1 − C

C0,

v = u

u0, L = L̃

z0,

� = E

RT 0, B = (−�H)C0

(�Cp)f

�

T0, Da = z0

u0A exp(−�),

Le = (�cp)e

(�cp)f,

Pey = (�cp)f z0u0

ke

, P ex = (�cp)f z0u0

D.

Note that we use arbitrary values for the velocity and thelength scales (u0, z0) in order to investigate the effect of lengthand velocity (L, v) as independent parameters. Typically inour simulations, as well as in practical situations, Le?1 andPey, P ex?1.

2.1. Multi-port reactor

Consider an LR of N identical units (each of length �L =L/N ) with gradual switching of the inlet/outlet ports over eachtime interval �sw. Such a reactor can be described by a contin-uous model (3)–(5) if we ignore the end-effects of the inter-mediate sections and apply the boundary conditions at the feedand exit positions. These positions vary in time as stepwisefunctions with a total period � = N�sw:

�in = (n − 1)�L, �out = L − (n − 1)�L,

� ∈ [(n − 1)�sw, n�sw] + k�,

n = 1, N, k = 0, 1, . . . . (6)

Here n specifies the unit number and k is the cycle number.

2.2. Asymptotic continuous model (infinite port model)

In the limiting case of an infinite-port LR (N → ∞) thestepwise functions �in(�), �out(�) can be replaced by continu-ously varying feed and exit positions described by the functions(Sheintuch and Nekhamkina, 2005)

�in(�) = �0in − Csw�, �out(�) = �0

out − Csw�,

where the switching velocity (Csw) is defined as Csw =�L/�sw.In such a case it is convenient to transform the governing equa-tions (3) and (4) using a moving coordinate system (�′ = �, �=� − Csw�) with fixed positions of the inlet and outlet:

Le�y

��+ (v − LeCsw)

�y

��− 1

Pey

�2y

��2

= BDa(1 − x) exp

(�y

� + y

)= B(1 − x)f (y), (7)

�x

��+ (v − Csw)

�x

��− 1

Pex

�2x

��2

= Da(1 − x) exp

(�y

� + y

)= (1 − x)f (y). (8)

Danckwerts’ BC conditions (see Eq. (5)) can be applied for themass balance (8). For the energy balance in the case of slowswitching the appropriate boundary conditions were shown(Sheintuch and Nekhamkina, 2005) to be

1

Pey

(�y

��

∣∣∣∣in

−�y

��

∣∣∣∣out

)= v(y − yin), y|in = y|out. (9)

Note that the first of conditions (9) was derived as a simplecombination of Danckwerts’ BC, while the second conditionenable the formation of a pulse solution and was verified in ourprevious work by studying the convergence of the multi-portreactor solution to the asymptotic one with N → ∞.

3. Pulse characteristics

We derive now approximations for the maximal temperaturerise (ym) of the LR. Assuming a “frozen” solution in a movingcoordinate system and ignoring the mass dispersion term (i.e.Pex → ∞), we can rewrite Eqs. (7) and (8) in the followingform:

1

Pey

�2y

��2− (v − LeCsw)

�y

��= −B(1 − x)f (y), (10)

(v − Csw)�x

��= (1 − x)f (y), (11)

y(0) = y(L), x(0) = 0,1

Pey

[y�(0) − y�(L)] = v[y(0) − yin]. (12)

Combining Eqs. (10) and (11) results in

1

Pey

�2y

��2− (v − LeCsw)

�y

��+ B(v − Csw)

�x

��= 0. (13)

O. Nekhamkina, M. Sheintuch / Chemical Engineering Science 63 (2008) 4924–4934 4927

Integrating this equation by � from 0 until L yields

1

Pey

[y�(L) − y�(0)] + B(v − Csw)xout = 0, (14)

where xout denotes conversion at the reactor exit (xout =x(L)).Comparing Eq. (14) and BC (12) we obtain

y(0) = yin + B

(1 − Csw

v

)xout. (15)

Integrating Eq. (13) by � from 0 until � we get

1

Pey

y� − (v − LeCsw)y + B(v − Csw)x = −S0, (16)

where

S0 = (v − LeCsw)y(0) − 1

Pey

y�(0). (17)

Applying Eq. (16) at point � = �m corresponding to y = ym

(y′m = 0) with x = xm (we do not specify xm here) we obtain

−(v − LeCsw)ym + B(v − Csw)xm = −S0. (18)

It is also useful to write

y′(0) = Pey[B(v − Csw)xm − (v − LeCsw)(ym − y(0))] (19)

and

y′(L) = y′(0) − PeyB(v − Csw)xout

= Pey[B(v − c)(xm − xout) − (v − LeCsw)(ym − y(0))].(20)

Using an auxiliary relation

B(v − Csw)x = x

xm

[−S0 + (v − LeCsw)ym]

we can rewrite Eq. (16) as

1

Pey

y� − (v − LeCsw)y + x

xm

(v − LeCsw)ym − x

xm

S0 = −S0

or, with definition of a new variable w = y/ym as

1

Pey

w′ − (v − LeCsw)

(w − x

xm

)= −S0

(1 − x

xm

)1

ym

.

(21)

Combining this equation with Eq. (18) results in

1

Pey

w′ = Bxm

ym

(v − Csw)

(w − x

xm

)− S0

ym

(1 − w). (22)

The maximal temperature of the forced front can be determinedfrom Eqs. (11) and (22).

3.1. Complete combustion case xm = xout = 1

Following Kiselev’s approach (Kiselev, 1993) we can divideEq. (11) by Eq. (22) resulting in

dx

dw= ym(1 − x)f (ymu)

(v − Csw)P ey[Bxm(v − Csw)(w − x/xm) − S0(1 − w)] .

(23)

Eq. (23) possesses a singularity at point w=1, x=xm. However,this expression becomes regular (see below) when xm = xout =1 (the numerator and denominator tend to 0 simultaneously).Note, that for the “ideal” front this assumption follows fromthe narrow reaction zone approach (Frank-Kamenetski, 1955)employed for an infinitely long system (Kiselev, 1993).

Assuming xm = 1 we obtain

dx

dw= p(w)g(w, x), (24)

where

p(w) = ymf (ymw)

PeyB(v − Csw)2, g(w, x) = 1 − x

(w − x) − S̃0(1 − w),

S̃0 = S0

B(v − Csw). (25)

Now function g(w, x) has a singularity if (x, w) → 1, howeverthe limit g(w, x) as well as dx/dw are finite

limw,x→1

x′w = p(1) lim g(w, x) = p(1)

−x′w

1 − x′w + S̃0

So

limw,x→1

x′w = 1 + p(1) + S̃0. (26)

Now we can estimate g(w, x) around w = 1 (i.e. y = ym),approximating x(w) � 1 − x′

w(1 − w)

g(w, x) � x′w(1 − w)

w − 1 + x′w(1 − w) − S̃0(1 − w)

= 1 + 1 + S̃0

p(1).

Integrating Eq. (24) by dw∫ 1w(0)

and using the narrow zoneassumption we obtain

1 �[

1 + 1 + S̃0

p(1)

] ∫ 1

w(0)

p(w) dw

=[

1 + 1 + S̃0

p(1)

]ymDa

PeyB(v − Csw)2

×∫ 1

0exp

�ymw

� + ymwdw. (27)

To estimate the integral in the RHS of Eq. (27) consider theapproximation of the power (�ymw)/(� + ymw) in the vicinityw = 1

ymw

1 + wym/�� ym

1 + ym/�+ ym(w − 1)

(1 + ym/�)2+ O((w − 1)2).

So we have∫ 1

0exp

�ymw

� + ymwdw �

exp

[ym

1 + ym/�

](1 + ym/�)2

ym

×[

1 − exp

(−ym[1 − w(0)]

(1 + ym/�)2

)](28)

4928 O. Nekhamkina, M. Sheintuch / Chemical Engineering Science 63 (2008) 4924–4934

and for large ym the last term in the square brackets can beneglected in comparison with 1. Substituting this relation intoEq. (27) we obtain

BPey(v − Csw)2

(1 + ym/�)2Da exp[ym/(1 + ym/�)] = 1 + (ym, Csw), (29)

where

(ym, Csw) = 1 + S̃0

p(1)= Pey(v − Csw)(v − LeCsw)

Da exp[ym/(1 + ym/�)] (30)

is a correction due to inhomogeneous BC. Note that Eq. (29)is reduced to the “ideal” front relation (2) if we set � = = 0and replace Csw by Vid.

3.2. Incomplete combustion case

Define conversion at the front (y = ym) as xm ≡ 1 − , > 0.In this case it is helpful to consider the inverse of Eq. (23), i.e.

dw

dx= q(w, x)

p(w), (31)

where p(w) is defined by Eq. (25) and

q(w, x) = xm(w − x/xm) − S̃0(1 − w)

1 − x, (32)

which can be reduced to 1/g(w, x) with xm = 1. Now w′x = 0

at � = �m and w(x)-profile around this point approximatelyfollows:

w � wm + 12w′′

m(x − xm)2 + O((x − xm)3). (33)

Estimating w′′m using Eqs. (31) and (32) we obtain

w′′m = 1

p(w)

dq(w, x)

dx

∣∣∣∣w=1,x=1−

= − 1

p(1).

Substituting this expression into Eq. (33) we obtain

w(x) � 1 − 1

2p(1)(x − xm)2. (34)

Now we can estimate q(w, x) around � = �m

q(w, x) = xm − x − (1 − u)(xm + S̃0)

1 − x

� (xm − x)[1 − (xm − x)(xm + S̃0)/2p(1)]xm − x +

or

q(w, x) = s(1 − as)

s + = q̂(s), (35)

where

s = xm − x,

a = xm + S̃0

2p(1)= (v − LeCsw)(v − Csw)

2Da/Pey exp[ym/(1 + ym/�)] . (36)

Integrating Eq. (31) and using the narrow zone assumption weobtain:

JL =∫ 1

w(0)

p(w) dw =∫ xm

0q(w(x), x) dx

�∫ xm

0q̃(s) ds = JR . (37)

The integral JL in the LHS of Eq. (37) with the incorporationof Eq. (28) is

JL � (1 + ym/�)2Da exp[ym/(1 + ym/�)]BPey(v − Csw)2

, (38)

while integral JR in the RHS of Eq. (37) accounts forcorrections due to incomplete combustion = 1 and non-homogeneous BC

JR =∫ xm

0q̃(z)dz

= xm + axm(2 − xm)

2+ (1 + a) ln . (39)

Now incorporating Eqs. (38) and (39) we find

(1 + ym/�)2Da exp[ym/(1 + ym/�)]BPey(v − Csw)2

= xm + �(xm, ym, Csw), (40)

where

�(xm, ym, Csw) = axm(2 − xm)

2+ (1 + a) ln , (41)

which differs from the corresponding relation for a completecombustion case (Eq. (29)) by RHs.

To determine the unknown xm we can multiply both sides ofEq. (13) by �y/�� and integrate by � from 0 until �m yielding

1

2Pey

∫ �m

0

d

d�(y2

� ) d� − (v − LeCsw)

×∫ �m

0y2� d� + B(v − Csw)

∫ �m

0y�x� d� = 0 (42)

or

− 1

2Pey

[y�(0)]2 − (v − LeCsw)

∫ �m

0y2� d�

+ B(v − Csw)

∫ �m

0y� x�d� = 0. (43)

To estimate the integral (J1) in the second term of Eq. (43)we can assume a parabolic profile y = y(�) within interval � ∈[0 �m]

y(�) = ym − A1(� − �m)2, �m = 2ym − y(0)

y�(0),

A1 = y�(0)

2�m

. (44)

O. Nekhamkina, M. Sheintuch / Chemical Engineering Science 63 (2008) 4924–4934 4929

We checked the parabolic profile approximation Eq. (44) andfound a good agreement with the simulations (Fig. 4c). UsingEq. (44) we obtain

J1 =∫ �m

0y2� d� = 4A2

1

∫ �m

0(� − �m)2 d� = 4

3A21�

3m

= 23 [ym − y(0)]y�(0). (45)

The integral (J2) in the third term of Eq. (43) after substitutingof Eqs. (16) and (17) is reduced to

J2 =∫ xm

0y� dx

= Pey

[B(v − Csw)

∫ xm

0(xm − x) dx.

−(v − LeCsw)

∫ xm

0(ym − y) dx

]. (46)

An inspection of the simulation results shows that the last termin the RHS of Eq. (46) is significantly smaller than the firstone. Nevertheless, as is shown below, it is useful to take itinto account using approximation (34), which results a smallcorrection of y� over the whole domain � ∈ [0 �m]. Thus,

J2 = Pey

[B(v − Csw)

x2m

2− (v − LeCsw)

ymx3m

6p(1)

]. (47)

Substituting Eqs. (45) and (47) into Eq. (43) we obtain

− 1

2Pey

[y�(0)]2 − 2

3(v − LeCsw)[ym − y(0)]y�(0)

+ Pey

2

[B2(v − Csw)2x2

m

−PeyB2(v − Csw)3(v − LeCsw)x3

m

3Da exp[ym/(1 + ym/�)]

]= 0 (48)

and upon accounting for Eq. (19) the last equation is reduced to

[(ym − y(0)][(v − LeCsw)(ym − y(0)) + 2B(v − Csw)xm]= PeyB

2(v − Csw)3x3m

Da exp(ym/(1 + ym/�). (49)

The set of Eqs. (40) and (49) forms a system with respect totwo unknown parameters ym, xm as functions of the switchingvelocity Csw.

As was pointed in Introduction, the “frozen” rotating solutioncan be maintained over a certain domain of switching velocities

Cmnsw < Csw < Cmx

sw . (50)

The lower limit Cmnsw can be estimated using the physically

reasonable condition for y�(0) to be positive. Following Eq.(19) the necessary condition can be rewritten as

B(v − Csw) − (v − LeCsw)

[ym − yin − B(1 − Csw

v)

]> 0.(51)

Assuming Csw>v (recall Csw ∼ O(v/Le) or 10−3v in orderof magnitude) this condition can be reduced to the following

simplified form:

Csw >v

Le

ym − yin − 2B

ym − yin − B= Cmn

sw . (52)

The upper limit Cmxsw can be estimated using the other physically

reasonable condition for y�(L) to be negative. An inspectionof Eq. (20) shows that

Cmxsw = Vth = v/Le if xout = xm

and

Cmxsw > Vth if xout > xm.

Thus, assuming xout > xm (xm < 1), and using Eqs. (40), (49)and (20) we can derive the upper switching velocity boundaryof the “frozen” pattern domain.

4. Comparison of analytical predictions and simulationresults

To verify the approximations derived above we compare thepredictions based on the assumptions of complete (Eq. (29),referred below as the first approach) and incomplete (Eqs.(40) and (49), the second approach) conversion with numeri-cal simulations results for a set of parameters used earlier ina study of a flow reversal reactor (Eigenberger and Nieken,1988): C0 = 1.21 × 10−4 kmol/m3, E/R = 8000 K, −�H =206 000 kJ/kmol, A = 29 732 s−1, u0 = 1 m/s, z0 = 1 m, Tin =293 K, D=5×10−3 m2/s, ke=2.06×10−3 kW/m/K, (�Cp)g=0.5 kJ/m3 K; (�Cp)e = 400 kJ/m3 K, which result in �T 0

ad =C0(−�H)/(�Cp)g = 50 K, Pex = 200; Pey = 194; Le = 800and B = B0 = 0.523 (calculated with T0 = 873 K). For this setthe system attains an extinguished solution for the whole showndomain and we wish to predict the domain of ignited “frozen”pulse solutions.

4.1. Bifurcation diagram

We study the effects of switching velocity (Csw), of feedconcentration (B) and of convective velocity (v) on the sustainedpatterns.

Typical pattern transformation upon varying the switchingvelocity within FP domain is illustrated in Fig. 2 showing thetemperature and conversion spatial profiles simulated with theasymptotic model in the moving coordinate system. The highestmaximal temperatures are achieved around Csw =Vth (=v/Le)

for which the spatial profile (Figs. 2a and c) is linear both down-stream and upstream of the reaction zone (peak point) sincethe heat is lost by conduction only (defined as “convection-lesscase” by Haynes and Caram, 1994). With increasing Csw whileCsw < Vth the ignited front is shifted toward the reactor inlet(in a moving frame) and becomes steeper (Figs. 2a and b). Thisprocess is accompanied by increasing maximal temperature Tm

with practically complete conversion (at �=�m defined as pointwhere T = 0.995Tm we have xm � xout = 1). The opposite

4930 O. Nekhamkina, M. Sheintuch / Chemical Engineering Science 63 (2008) 4924–4934

0 0.5 1

0

500

1000

1500

T, C

0 0.5 1

0

0.5

1

x

0 0.25 0.5

0

500

1000

1500

T, C

0 0.25 0.5

0

0.5

1

ξ

x

1.0

1.04

0.8

0.8

0.91.0

1.0

0.9

ξ

ξ

ξ

1.08

1.0

1.04

1.08

Fig. 2. Typical temperature (T, plates a, c) and conversion (x, plates b, d)spatial profiles transformation upon varying the switching velocity (Csw).Plates (a), (b) and (c), and (d) correspond subdomains of Csw < Vth andCsw > Vth, respectively. Numbers show the ratio Csw/Vth. B/B0 = 4, otherparameters as in Fig. 1.

tendencies were detected in simulations with Csw > Vth: withincreasing Csw the “frozen” spatial profiles become smoother,the front (inflection point) is shifted downstream with respectto the reactor inlet (Figs. 2c and d) and Tm decreases. Theconversion also declines and the temperature- and conversion-maximum locations (the last one can be defined as a point,where x = 0.99xout) diverge. Note that for most simulationsconducted with Csw > Vth we get a sufficiently high exit con-version (xout > 0.95) while xm (at point Tm) can be significantlyless than the maximal value.

Following the asymptotic model simulation results we canexpect that approximations based on the assumption of com-plete conversion (xm = 1, Eq. (29)) are suitable to predict solu-tions for Csw < Vth while the second approach (Eqs. (40) and(49)) is more suitable for Csw > Vth. The analytically predictedlower switching boundary of the FP domain (Cmn

sw (Eqs. (29)and (52)) is in an excellent agreement with numerical simula-tion results (Figs. 3a and b). Velocity Cmn

sw gradually decreaseswith increasing B and intersects the ordinate-axis at B=Bcr (�13.8B0 for the parameters of Fig. 3a), i.e. beyond Bcr patternscan emerge in an LR without any forced switching (Csw = 0)and such a reactor can be constructed as a single section withproperly matched inlet and outlet. (Actually a simple bed isignited under these conditions.)

To calculate the upper boundary of the FP domain we setxout = 1 (Eq. (20)) following the simulation results. The agree-ment between the asymptotic model results and Cmx

sw is satis-factory for low B (B �4 in Fig. 3a), for larger B the resultsdiverge.

With increasing convective velocity (v) the boundaries ofFP domain defined via a normalized switching velocity csw =Csw/Vth gradually approach each other (Fig. 3b) forming acusp-like structure of the inverse form. Note that decreasing ofthe FP domain (�csw) leads to drastic decrease of the switchingtime operation window

��sw ∼(

1

Cmxsw

− 1

Cmnsw

)� 1

v

�csw

〈c2sw〉

where 〈•〉 denotes an averaged value over the switching do-main).

4.2. Maximal temperature rise

The effect of the switching velocity Csw on the maximaltemperature rise is illustrated in Figs. 4 and 5 showing two setsof analytical predictions, based on the assumption of eithercomplete (T (1)

m , Eq. (29), dashed lines) or incomplete ( T(2)m ,

Eqs. (40) and (49), solid lines) conversion. Direct numericalsimulations (symbols) are presented also. Following the firstapproach the maximal temperature increases monotonicallywithin the whole domain of Csw > 0. Maximal temperature cal-culated by the second approach (T (2)

m ) exhibits a non-monotonicfunction of the switching velocity with maximum Tm beyondCsw = Vth. For all sets of parameters being considered wefound that T

(1)m > T

(2)m . The asymptotic model simulations

exhibit maximal temperatures (Tm) that exceed T(1)m within

subdomain Csw < Vth and fall below T(2)m with Csw > Vth. The

agreement between the asymptotic model and analytical pre-dictions of Tm is satisfactory for low B and becomes poor withincreasing B.

To elucidate the reason of such distinctions we compare con-version xm calculated by asymptotic model (7)–(9) with thatof the incomplete combustion approach (Eqs. (40) and (49)).The “exact” (numerical) simulations revealed that xm = 1 formost of the left part of the FP subdomain (i.e. with Csw < Vth)and sharply decreases with increasing Csw above Vth (Figs. 4b

O. Nekhamkina, M. Sheintuch / Chemical Engineering Science 63 (2008) 4924–4934 4931

0 0.5 1 1.5

0

2

4

6

8

10

B/B

0

Csw/Vth

0.9 0.95 1 1.05

0

2

4

6

8

10

Csw/Vth

v

Fig. 3. Comparison of the analytically (lines) and numerically (points) calculated boundaries of the FP domain in the Csw/Vth vs B/B0 (a, v = 1) and theCsw/Vth vs v (b, B = 1) planes. The lower switching boundary (Cmn

sw , solid lines) is calculated using Eq. (29); the high switching boundary (Cmxsw , dashed

lines) is calculated using Eqs. (20), (40) and (49) with xout = 1. Other parameters as in Fig. 1.

0.5 1 1.5

500

1000

1500

2000

Csw/Vth

Tm

, C

0.5 1 1.5

0.6

0.7

0.8

0.9

1

Csw/Vth

xm

B=1

B=1

4

10

10

4

Fig. 4. Comparison between the approximated (using Eq. (29) with xm = 1, dashed lines or using Eqs. (40) and (49) with xm < 1, solid lines) and “exact”numerical solutions (signs) calculated by asymptotic model (7) and (9). Plates (a) and (b) show the maximal temperature (Tm) and the correspondingconversion (xm) as functions of dimensionless switching velocity (Csw/Vth). Solutions of system (40) and (49) are shown for Vth < Csw < Cmx

sw . Numbersdenote B/B0-values, v = 1, other parameters as in Fig. 1.

0.96 0.98 1 1.02

500

1000

1500

2000

Csw/Vth

Tm

, C

0.96 0.98 1 1.02

0.5

0.6

0.7

0.8

0.9

1

Csw/Vth

xm

6

4

v=1

1

2

6

4

v=1

2

4

6

2

Fig. 5. Comparison between the approximated (using Eq. (29), dashed lines; using Eqs. (40) and (49), solid lines) and “exact” solutions (signs) calculatedby asymptotic model (7)–(9). Plate (a) shows the maximal temperature (Tm) of the FP as functions of varying dimensionless switching velocity (Csw/Vth).Solutions of system (40) and (49) are shown for Vth < Csw < Cmx

sw . Numbers show v-values, B = B0 = 1, other parameters as in Fig. 1.

4932 O. Nekhamkina, M. Sheintuch / Chemical Engineering Science 63 (2008) 4924–4934

and 5b). These results justify our previous assumption con-cerning preferential use of the first approach within subdomainCsw < Vth, and of the second one within subdomain Csw > Vth.Note, that conversion xm calculated with the incomplete con-version assumption (49) exhibits a non-monotonic function ofthe switching velocity with maximum xm around Csw = Vth.The simulations are in a good agreement with the approxima-tions near the boundary of the FP domain and becomes poorwith decreasing Csw.

4.3. Summary

Summarizing this comparison we can conclude that thetwo proposed analytical approaches assuming complete andincomplete conversion can be successfully used for predic-tion of the ”frozen” rotating patterns within subdomains ofCmn

sw < Csw < Vth and Vth < Csw < Cmxsw , respectively. Deriva-

tion of a unique approach within the whole domain of theswitching velocity will be a subject of the following study.

5. Optimal solution

Obviously, the optimal conditions are at Csw = Vth = v/Le,which is within the FP domain and assures the highest temper-ature and a sufficiently high exit conversion. As was discussedbelow, in such a case the spatial temperature profile is linearboth downstream and upstream of the reaction zone (which isnarrow with respect to the reactor length). That allows to pro-pose a relatively simple reactor design based on the piece-wiselinear approximation of y(�). Assuming xout = 1 we can ap-proximate the total balance equation (14) as

ym − y(0)

�m

− y(0) − ym

L − �m

= BPeyv

or

ym − y(0) = BPeyv�m

(1 − �m

L

). (53)

For the infinitely long system (L → ∞) this relation is reducedto

y∞m − y(0) = BPeyv�∞

m . (54)

Noting that �m varies only slightly with L we set it constant at�m = �∞

m , yielding

y∞m − ym = BPeyv

�∞m

L. (55)

Similarly, we can estimate xm using the approximation of Eq.(20) and assuming xout = 1

y′(L) � −ym − y(0)

L − �∞m

� BPeyv(xm − 1),

or

xm � 1 − y∞m − y(0)

P eBv

1

L − �∞m

. (56)

0 0.5 1 1.5 2

0

500

1000

T, C

0 0.1 0.2

0

0.5

1

x

0 0.25 0.5 0.75 1

0.9

0.95

1

1/L

x

0 0.25 0.5 0.75 1

950

1000

1050

Tm

, C

z, m

z, m

1/L

L=1 m

2

48

L=1 m

2

4

8

Fig. 6. Effect of the reactor length (L) on the optimal solution (Csw = Vth)showing spatial temperature (a) and conversion (b) profiles, and the maximaltemperature (c) and the corresponding conversion (d) as functions of 1/L.The star in (c) marks T ∞

m obtained by linear extrapolation. B = B0 = 1,v = 1, the other parameters as in Fig. 1.

To verify the obtained approximations we simulated the modelwith Csw = Vth for varying reactor length (Fig. 6a and b). Themaximal temperature declines with 1/L (Fig. 6c) as predictedby Eq. (55). Linear extrapolation of the obtained results to1/L = 0 yields T ∞

m (marked by star in Fig. 6c). Using relation(56) and T ∞

m (y∞m ) we can estimate the slope (AL) of xm de-

pendence on 1/L. For the set of parameters employed in Fig. 6we obtained AL = −0.10 which is in a good agreement withthe numerically calculated value Anum

L = −0.093.

O. Nekhamkina, M. Sheintuch / Chemical Engineering Science 63 (2008) 4924–4934 4933

6. Concluding remarks

We have suggested an approximate procedure for designingan LR for simple irreversible exothermic reactions. It involvesthe following steps:

1. Determine the maximal operation temperature allowed inthe system, and find the corresponding operating conditions(flow rate) that will maintain this temperature in the asymp-totic (N → ∞) long reactor solution using Eq. (29) withCsw = Vth = v/Le.

2. Find the windows of switching velocities corresponding tothese conditions from Eq. (50) and determine if this domainis too narrow or comfortable. In the former case, incorporatea control procedure to maintain it at the optimal value.Such control was suggested by Barresi et al. (1999) and itswitched the port position as a feedback to the difference ofthe temperature at a certain position from its setpoint value.

3. Find the maximal temperature and the corresponding con-version dependencies on L (Eqs. (54)–(56)) and determinean acceptable reactor length.

4. Find the maximal temperature and the exit conversion de-pendencies on N and determine an acceptable number ofunits.

5. Proceed with detailed simulations of the reactor dynamics.

Notation

a parameter defined by Eq. (36)A rate constantB dimensionless exothermicitycp volume-specific heat capacityC key component concentrationCp heat capacityCsw switching velocityD axial dispersion coefficientDa Damkohler numberE activation energyg function defined by Eq. (25)�H reaction enthalpyke effective conductivity�L, L dimensionless length of a single unit and total

reactor lengthLe Lewis numberN number of the reactor unitsp function defined by Eq. (25)Pey, P ex Peclet numbers of heat- and mass dispersionq function defined by Eq. (32)r reaction rateS0 parameter defined by Eq. (17)t timeT temperatureu, v dimension and dimensionless fluid velocitiesV front velocityw dimensionless temperaturex conversion

y dimensionless temperaturez spatial coordinate

Greek letters

� dimensionless activation energy� switching period 1 − xm

�, � dimensionless coordinate� density� dimensionless time

Subscripts

ad adiabatice effective valuef fluidin at the inletid idealm maximalout at the outletsw switching parameter0 reference value

Superscripts

in feed

Acknowledgments

Work supported by the US Israel Binational Scientific Foun-dation. M.S. is a member of the Minerva Center of NonlinearDynamics. O.N. is partially supported by the Center for Ab-sorption in Science, Ministry of Immigrant Absorption State ofIsrael.

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