Qualitative analysis of the behaviour of nonlinear parabolic equations—II

Chemical Engineering Science, 1971, Vol. 26, pp. 1743-1752.

Pergamon Press.

Printed in Great Britain

Qualitative analysis of the behaviour of nonlinear parabolic equations- II Application of the methods for the estimation of domains of multiplicity V L A D I M I R HLAVA(~EK, M I L A N K U B i C E K and J A N C A H A Department of Chemical Engineering, Institute of Chemical Technology, Prague 6, Technickfi Str. 1903, Czechoslovakia

(Received 20 February 1971 ) Abstract-The application of linearization, orthogonal collocation and difference method is discussed for various reaction engineering problems. Following cases are considered: (a) heat transfer and exothermic chemical reaction of zero order (b) mass transfer in a porous catalyst accompanied by chemical reaction with Langmuir-Hinshelwood reaction rate expression (c) heat and mass transfer in a porous catalyst for (a) slab and sphere in a non-uniform external field (fl).reaction with a volumechange (d) heat and mass transfer in recycle-reactor.

1. I N T R O D U C T I O N

subject to the boundary conditions

IN THE preceding communication[l] applicability of three methods-linearization, orthogonal collocation and high order difference t e c h n i q u e - h a s been discussed. These methods make it possible to convert a set of nonlinear parabolic differential equations into a set of ordinary differential equations which can be analysed more easily. In this paper we wish to illustrate the possibility of application of all these methods for studying various problems describing heat and mass transfer and chemical reaction. 2. A P P L I C A T I O N S O F T H E A P P R O X I M A T E TECHNIQUES

(a) Heat transfer and chemical reaction of zero order The explosion of solid materials may often be supposed to be a zero order reaction having a high activation energy. Heat transfer may then be described by Eq. (1): d20 a d0 --+ . . . . dx2 x dx

8e °

d0 x = 0: ~ = 0 x=l:

(2)

0=0.

Equation (1) for 8 > 8crit. has no solution for boundary conditions (2). The magnitude of the parameter 8cra. can be determined analytically (for a = 0 and a = l) or numerically ( a = 2 ) . Let us try to estimate the value ~crit. by means of linearization and orthogonal collocation as well. The Laplace operator may be approximated at some mesh point by the relation d~O

a dO

dx 2 ~ x - ~

~ CaO

(3)

where the value of constant C1 may be evaluated by taking use of linearization or orthogonal collocation. After combination of Eq. (3) and Eq. (1) a nonlinear algebraic equation results

(1) 1743

C 1 0 - 8 e ° = F(O) = O.

(4)

V. H L A V A ~ E K , M. K U B [ ~ E K and J. C A H A

The parameter Befit" may be established when [3] F' = F = 0

(--1)" cos-( 2 n + l ) z r x 0 = C [ 1 - - x ~ - - ~32~=o = (~-h-~i]a

(5) × exp[7-- (2n-~ 1 ) 2rr=t] }.

Eq. (5) yields 0 = 1, i.e.

C1 ~crit. = 2"718"

(6)

A comparison of both exact and approximate values of 8crit. may be found in Table 1. Further, in Table 2, the results obtained for a plate for various types of the orthogonal polynomials are summarized. In accordance with Villadsen and

Let us truncate the series (8) after the first term. After differentiation with respect to t we may write ----~Ox

0 1 2

Accurate 8cm. value lineaxiz, 0.897 2.00 3.32

020

Jacobi collocation

zO20~D-~'-~4-O OX

Accurate value Jacobi Legendre Tschebyshev

0.897 0.920 1.10 1.47

D =-

Stewart's results[4] the Jacobi polynomials seem to be superior to other types of orthogonal polynomials considered. For a slab (a = 0) the linearization technique, after omitting the nonlinear term Be e in Eq. (1) yields C1 = rr2/4. N o w let us approximate the nonlinear term 8e ° by a nonzero constant C, i.e.

Ot

020 -Ox - +2C

(11)

C [ 8~----(x~ - 1 ) + 1].

(12)

The relation (11) is similar to the approximation (3). For D = 0, i.e. for x = 0-4352, the approximation (11) is identical with the approximation (3). This observation is important and explains why the neglecting of the source term yields satisfactory results. Finally, let us note that this co-ordinate is almost identical with the node x = 0.4472 of the Jacobi polynomial P1 = 1 -- 5x 2. Further, let us expand the nonlinear term 8e ° in a Taylor series. After omitting the higher order terms it may be written for a slab: 00

O0

(10)

where

Table 2. A comparison of 8mr. for different orthogonal polynomials. Plate 8rot.

-~-t . (9)

which may be rearranged to

0.920 2-21 3"87

Polynomial

• exp

a o+ :~-c(1-~)-c

Ox2

~crlt.

0-908 2-13 3-63

0---[ ~ - 4 " - 2 " cos

The following combination of Eq. (9) with the truncated series (8) yields the approximation

Table 1. A comparison of accurate and approximate values of 8rot.

a

(8)

020

.

~7=-~-t-A

.

+ BO

(13)

(7)

subject to the boundary conditions (2) and the initial condition t = 0 : 0 = 0. We may write the solution of Eq. (7) in the form

subject to the same initial and boundary conditions. Also in this case an analytical solution can be found [5], therefore, the linearization may be used. The infinite series after truncation

1744

Qualitative analysis of the behaviour of nonlinear parabolic equations- I 1

The first derivative of Eq. (19) gives

yields ~x

_ A[cos xX/-d 0 - BL cos ~

1] ~ 16A cos-y ,r(4B zr~) -

8e ° = -~-.

(20)

-

With regard to Eq. (5) we arrive at × exp[--l('na -- 4B) t]

= Fo(x) +F,(x) exp(--h,2t) 1 where we have denoted Xl2 = 4 ( ~ - 4B).

After differentiation of Eq. (14) with respect to t we can write

O0 Ot

F,(x)X12exp(_X12t)"

(15)

A comparison of Eqs. (13) and (15) yields 00_ Ot

030

AI'[O--Fo(x)] ~-~q-A-q-BO. (16)

By rearranging Eq. (16) the approximation for the second space derivative can be developed

o~

4

L ~-

° = ~ t - - q y L ~os W

(14)

L~s-~

1]}÷,

We can insert Eq. (21) into Eq. (20) in order to calculate Beret.- The location of the mesh point x = 0 . 4 4 7 2 (zero of the Jacobi polynomial) seems to be most convenient. (b) Mass transfer and chemical reaction of the Langmuir-Hinshelwood mechanism in a porous catalyst Schneider and Mitschka[6] have pointed out that the chemical reaction with the LangmuirHinshelwood reaction rate expression and mass transfer in a porous catalyst may lead sometimes to multiple steady states. This is true for such a Langmuir-Hinshelwood mechanism which exhibits an increase in the reaction rate as a result of increasing conversion. Let us study the case described by Schneider and Mitschka[6] for a slab:

(17)

dZy The similarity with the previous results is obvious. N o w the question of the selection of both constants A and B arises. Frank-Kameneckij [2] has shown that the ignition temperature is low and, therefore, the expansion into the Taylor series in the vicinity of the point 0----0 can be considered. This yields A=8,

B=&

0+sft -48rc°sxv 48 L c o s ~

4

x= 0:y' = 0

]]

1

+Be ° = F = 0 . (19)

(22) (23)

x=l:y=l. Let us use the approximation

(18)

After inserting relations (18) and Eq. (17) in Eq. (1) we can write

_z[l+B\ 2 [y+C\

y " - - X,2(1--y)

(24)

in order to study the branching points of Eqs. (22) and (23). The value of the coefficient )k1 and the corresponding coordinate Xl where the differential operator is approximated can be found in Table 3. The substitution of Eq. (24)

1745

V. HLAV.&(~EK, M. KUBi(~EK and J. C A H A

tive may then be replaced by the formula

Table 3. The values of coordinates and coefficients in the discrete approximation of the Laplace operator d2/dx 2+ a[x d/dx Lineafization a 0 1 2

(x) 0.4352 0.5554 0.6261

d2y

- ~ x=xi ~ Bilyl + Bt2Y2+ BIay(1).

Jacobi orthogonal collocation hi 2

x

hi 2

2.4674 5.7840 9.8696

0.4472 0.5773 0.6547

2.5 6 10.5

(28)

After inserting approximations (28) in Eq. (22) two nonlinear algebraic equations result .2[ I + B \z

G,(y,, Y2) = BilYl q- Bi2Y2+ B,aY3--O ~ l--+~i) into Eq. (22) yields a nonlinear algebraic equation -2f I + B ~2 {y+C'~ G(y) = h12(1 -- y) --0 ~]--~-~] Y ~ - - ~ ] = O. (25)

{y~ + C'X

Xy,~]

O_GIOGI\

I -G2 aG2l

(26)

Equation (26) can be rearranged into a cubic equation

ByZ+ (2BC-- 1)y2+ ( 2 - B C ) y + C = O.

OGxOG2 OG2OGI =o" Oyl Oyz Oyl Oyz

1 100 10 100

Yl 0"0104 0.0102

Y2

Linearization 41 42

Table 5. The estimation of critical values of the parameter 4' from a two-point collocation. Jacobi polynomials. Langmuir-Hinsheiwood kinetics (C = 1, B = 100)

Exact 41 42

0 " 3 9 9 6 0 " 4 3 5 0"933 0 " 5 9 0"89 0.4773 0.324 0.813 0.50 0.78

For getting better accuracy a two-point collocation method can be used. The second deriva-

(31)

From nonlinear Eqs. (29) and (31) two pairs of roots y o) and y2°) and yl t2) and y2t2~ may be calculated which yield the corresponding values of the Thiele parameter ~bl and ~b2. All results are summarized in Table 5.

Table 4. A comparison of critical values of Thiele parameter 4 obtained by the numerical integration of Eq. (22) and from the linearization

B

(30)

i.e.

(27)

Equation (27) may have one or three real roots. In the region y e (0, 1) either no or two real roots of Eq. (27) exist. For the parameters B = 100 and C = 10 as well a s B = 1 0 0 a n d C = 1 the concentrations y calculated are summarized in Table 4. Both critical values of the parameter ~b may be evaluated from Eq. (25) after inserting both roots y~ and Y2. A comparison of exact and approximate values of the Thiele parameter ~bcan be found in Table 4.

C

(29)

The necessary condition for branching Eq. (29 is the singularity of the Jacobian matrix J

For the branching condition Eq. (26) holds

G(y) = G'(y) = 0.

= 0.

Yl

Y2

4

0"0103 0'3738

0"6350 0"7345

0"45 0"907

The calculation shows that the value ~b2 may be determined fairly accurately because for this condition the concentration profile within the particle can be well approximated by a simple polynomial. On the other hand, the value ~bl is

1746

Qualitative analysisof the behaviour of nonlinearparabolicequations- I I still inaccurate because of a flat shape of the profile in the central part of the pellet. It is obvious that the location of the collocation points ought to consider the steep decrease of concentration near the surface of the catalyst. Therefore, other orthogonal polynomials having mesh points near at the surface (e.g. Chebyshev polynomials) yield better results than the Jacobi polynomials. The coefficients Blz as well as the co-ordinates of the mesh points are summarized in Table 6 for various orthogonal polynomials.

root of the Chebyshev polynomial, better results may be calculated taking advantage of the high order difference method. (c) Heat and mass transfer and chemical reaction in a slab in a nonuniform external field This case has been investigated in order to estimate in a simple way the effects of external gradients of temperature and concentration on the effectiveness factor[7]. The balance equations may be written for steady state conditions in the form [7]:

Table 6. Coordinates and coefficients in the orthogonal collocation for the operator d2/dx~ Polynomial Jacobi Legendre Tschebyshev

xl

d2Y 4)2Yexp[ Tfl(1--Y) dx 2 ---[1 + fl(1 -

(32)

Y(1)=Yx=I--2A.

(32a)

B~2

0.4472 0.5774 0.7071

with the boundary conditions

2.5 3 4

Y(-1)=I;

The calculated values of (~)1 are summarized in Table 7 for different types of orthogonal polynomials (one-point collocation). The value ~bl calculated by means of the Chebyshev polynomial is the best approximation. Hence we may draw some general conclusions on the basis of this example. The value ~b2, i.e. the "ignition point" may be calculated in systems where transport phenomena and exothermic reaction take place either on the basis of the linearization or orthogonal collocation, on the other hand, the value 61, i.e. the "extinction point", is to be evaluated from orthogonal polynomials having zeros near at the coordinate x---- 1, e.g. Chebyshev polynomials. For a mesh point which is located closer to the coordinate x = 1 than the Table 7. A comparison of ~bl evaluated for various orthogonal polynomials.LangmuirHinshelwood kinetics (C= 10,B = lOO)

Accurate Jacobi Legendre Tschebyshev

]

Y)J

0.50 0.325 0.357 0.413

At first, let us study the linear transient diffusion equation without chemical reaction OY 02Y a---~-= a----~

(33)

subject to the boundary conditions (32a). The initial condition may be considered in the form: t = 0, Y = 0. Under these conditions the solution of Eq. (33) may be written [5] Y = 1-- ( x + 1 ) A + ~ Ft(hi, x) exp(--hizt) i=l

(34)

where Fi(hi, x) = 2 (1 --2A) cos ilr . irr(1 + x ) 7r i sm 2

(35)

and Ai2= 4 "

(36)

In an analogous way as we have described in the previous section we may arrive at the approximation 02y Ox2 ~--Xl2[y - 1 + ( x + l ) A ] (37) 1747

V. HLAVfi.(~EK, M. KUB[(~EK and J. CAHA /~1 = ¢r/2. After inserting Eq. (37) in Eq. (32) a nonlinear algebraic equation results where

I- ~//3(1--Y) 1 haS[Y - 1 + (1 + x ) A ] + ~b2y exp [1 +/3(1 -- Y)J =F=O.

(38)

Y/3 < (3/3)* unicity always is assured. Table 8 reports different values of the parameter (3/3)* calculated from equations discussed. This table indicates that the Jacobi mesh point gives better results in comparison with the choice x = 0. As we may expect the choice x = 1 appears to be the worst one. F o r A----0 the critical value (3//3)* = 5.2 has been found [8].

H o w e v e r , Eq. (38) contains a hitherto undetermined coordinate x. T h e simplest choice x = 1 or x = 0 yields Eq. (39) and Eq. (40) respectively

Table 8. Evaluation of (yfl)* (3' = 20, A = 0.2) (y#)*

X,2[y(1)--Y]--4~Yexp [ Tfl(l-Y) ] [1 +/3(1-- Y)J = F = 0 (39)

Eq. (39) Eq. (40) Eq. (41) Eq. (42) correct value*

hlZ(1 -- A-- Y ) - ~b2y exp [1Y+fl(l(?5)) ] L ,. , = F=0. *Estimated from a graphical dependence.

(40) T h e Villadsen-Stewarts orthogonal collocation cannot be used owing to the nonsymmetry of the problem. Let us approximate the derivative in Eq. (32) from three points Y ( - - 1 ) , Y and Y(1). T h e coordinate of the point Y may be e.g. x = 0 or x = 0.1056. T h e latter value locates the x coordinate as the root of the Jacobi polynomial in the region ( - 1 , 1). F o r these mesh points we may write Eq. (41) and Eq. (42) respectively

15.5 7.4 7.4 7.7 8.4

(d) H e a t and m a s s transfer and chemical reaction in a sphere with a volume-change T h e transport equation can also be nonlinear in the differential operator if the volume-change occurs. This case has been discussed in detail by Weekman[9]. T h e steady state transport equation for a sphere may be written:

d2 Y dx 2

[ yfl(1-Y) ] 2(1 - - A - - Y) -- 4~Y exp L1 -+-/3(1 - Y)J = F = 0

o,+/3T v . 2'{dY~ 2_t 2 d Y 1+o'Y\dx}

x dx

2 l+o-Y

1

(41) where

2-0226(1 -- Y) + 1.1181A-- ~b2y exn [ 7/3(1 -- Y) ] x rLl+fl(l_Y)j=F=O.

(43)

.B, l+oz = 1 -t- ~'-~ inl + o.Y

(44)

(42) subject to boundary conditions

We can note after comparison of Eqs. (40) and (41) a satisfactory agreement between both coefficients because h l 2 ~--- ¢rZ/4 "-- 2.46. T h e sufficient condition for the branching of the solution requires F----F'----0. An additional condition F " = 0 must be used for determination of the critical value (y/3)*, i.e. for a such value that for

dY x = 0 : - ~ - = 0;

x=l:Y=l.

(45)

It is obvious that the linearization method cannot be utilized with regard to the nonlinear operator. H o w e v e r , the orthogonal collocation can be

1748

Qualitative analysisof the behaviourof nonlinearparabolicequations- I 1 successfully used. The derivatives in Eq. (43) may be replaced by [4]: d2Y 2 dY dx----~ + x d x

10.5(1--Y)

(46)

dY ~ - - 2.2913(1 -- Y).

(e) Heat and mass transfer in recycle reactors The steady state conditions can be described by two nonlinear first order differential equations [10]:

dxdY= Da(1-- y) exP( l +--~) exp (l

(48)

+-~)--fl(O--Oc)

(1 --/~)0(1) = 0(0).

= y(1) --y(0).

The nonlinearity can be expressed e.g. at the point x---- 1. In this case the tubular reactor is approximated by a stirred tank. A better approximation will be obtained by choosing a point near at the middle of the interval. Taking Eq. (51) into account we can rewrite Eqs. (48) and (49) for x-l: 0(1) ] exp 1+0(1)/3~

by(l) =

Da[1--y(1)]

/.t0(1) =

Da. B[1 - - y ( 1 ) ]

o(1)]

-/3(0(1)-0c). (53) A combination of these equations yields: (54)

(50)

Eq. (54) can be inserted into Eq. (52) and an algebraic equation similar to the C S T R description results. The values of D a m k r h l e r number at branching points are summarized in Table 10 where exact values are presented too. The results which have been obtained by using three mesh points are presented in Table 11. Different values of Da*~, obtained for the

Table 9. A comparison of critical values of Thiele parameter. Reaction with a volumechange Accurate* 0-4 0.4 0.6

(52)

exp 1 qg0-(-1)/TJ

y(1) = ~--~X{/310(1) - Oc] + / z 0 ( l ) }

With respect to the nonsymmetry of the problem the orthogonal collocation cannot be used. The high order difference procedure appears

--0.75 -- 0-50 --0.50

(51)

(49,

subject to boundary conditions ( 1 - - k ) y ( 1 ) = y(0);

y'(x)

(47)

After inserting Eqs. (46) and (47) into Eq. (43) a nonlinear algebraic equation results. The branching points can be determined in an analogous way. However, the resulting equation has to be solved numerically, e.g. by Newton method. The results are summarized in Table 9. The comparison with the exact Weekman's results indicates a rather good agreement.

-~= dO DAB(1 -y)

to be a convenient approach. In the simplest case two points, the inlet (xl = 0) and the outlet (xz = 1) of the reactor, can be considered. The first derivative can be then approximated at any point:

20.0 20.0 20.0

0.20 0.40 0.14

0.70 0.65 0.57

*Obtained from Weekman'sgraphs[9]. 1749

Approximation 0.176 0.723 0.390 0.537 0.132 0.554

V. HLAV,~t~EK, M. KUBI(~EK and J. CAHA Table 10. Recycle reactor. Critical values of Da (a) h =/~ = 0.35,/3 = 0, B = 6, y = 20

Dal Da~

Exact

Approximation

0.0305 0.0310

0.033 0.041

V2T = -

= Op2

0.020 0.022

inflexion

~ 0.021

(56)

where V 2 is the Laplace operator in spherical co-ordinates:

(b) X= p.=0.2, /3=0.1, Oc=O, B=8, 3' = 20 Dal ~ Da2J

~b2Ye x P [ T ( 1 - - 1 ) ]

pop

p2

sint~

. (57)

T h e boundary condition m a y be supposed in the f o r m

corresponding x3(xl = 0, x2 = 1), are c o m p a r e d with exact ones. T h e best agreement has been reached for the mesh point located in the vicinity of the Jacobi zero-point.

Y(1, v~) = 1 + X cos~9

(58)

T(1, t~) = 1 + r cosd.

(59)

Aris and C o p e l o w i c z [11] have found an invari-

Table 11. Recycle reactor. A comparison of regions of multiplicity. Effect of the coordinate of the interior point (h = p. = 0.265,/3 =0, B = 1 0 , y = 20) x Dal Da2

0.3

0.5

0.6

0.8

Accurate

0.00391 0.00398 0.00402 0.00413 0.00415 0.01370 0.01361 0.01352 0.01346 0-01355

(f) H e a t and mass transfer and chemical reaction in a sphere p l a c e d in a nonuniform external field Hitherto we have dealt with one-dimensional problems only. H o w e v e r , a lot of multidimensional problems exist in chemical engineering. We shall show the application of qualitative methods to a two-dimensional problem of elliptic nature: heat and mass transfer and chemical exothermic reaction in a sphere placed in a nonuniform external field. T h e b e h a v i o u r of both concentration and t e m p e r a t u r e fields within a porous particle m a y be described b y a set of elliptic partial differential equations with two space variables. L e t us consider the external gradient only in the flowdirection, then the equations are in the form V2y = 6 2 Y e x p [ T ( 1 - - 1 ) ]

(55)

ant of Eqs. (55) and (56) subject to b o u n d a r y conditions (58) and (59), i.e. these equations m a y be rewritten V2T = ~b2[1 + f l - - T +

(K+flX)pcos0]

ex+(, 1)]

,60,

W e can expect that the application of the linearization m a y be complicated b e c a u s e of the dependence of the b o u n d a r y conditions on d. T h e two-dimensional variant of the orthogonal collocation cannot be used with regard to the n o n s y m m e t r y of the problem. Let us try to utilize the difference method. With respect to experience obtained with Jacobi polynomial we can choose both coordinates, i.e. the radius p and the angle d, to be the roots of the Jacobi polynomial. T h e r e f o r e the approximation will

1750

Qualitative analysis of the behaviour of nonlinear parabolic equations - 11

Table 12. A comparison of regions of multiplicity.Sphere

be taken at p = 0.654 and at O = 62°. For comparison of the results we may use also the angle 180°-62 °. The mesh points are depicted in Fig. 1. Further, for the derivatives in the O-direction we can assume only a weak dependence on p especially near at the surface. This assumption is in accordance with the observation of Aris and Copelowicz [ l l] and authors of this paper[12]. Both derivatives can be therefore approximated from boundary conditions. We may write 1 0 |- - / / s i~n T O\ - - - - K - cosO. p~sinO 00 \ O0/ p~ 7÷h"

7 -X

(61)

7+X

7 -M

Fig. I. Location of mesh points in a sphere.

The second part of the Laplace operator can be approximated

02T 20T 0 7 + p Op

Co+ CIT(p, O)

(62)

where we have for O = 62° and O = 118°, Co = 10.0364, C1 = - 10.5000 and Co = 10.7457 and C1 = - 10.5000 respectively. The analysis of branching conditions can be performed in an analogous way as in the previous paragraphs. All results are summarized in Table 12 where also exact results[Ill are presented. 3. CONCLUSIONS AND DISCUSSION The methods suggested may be successfully used for studying transport phenomena. The lumping of the differential problem leads either to a simple nonlinear algebraic equation or to a

in a nonuniform external field (K=0'I, h=--0"25, 7=20, fl = 0.4)

O = 62 ° t9 = 118° Exact*

0.66 0.93 0.63

* F r o m A r i s - C o p e l o w i c z [ l 1] graphs.

set o f equations. The relations obtained allow one to determine analytically the branching points, the domain of instability etc. as a function of the governing parameters. For one interior point and symmetrical problems a remarkable qualitative agreement between orthogonal collocation and linearization exists. For nonsymmetrical problems with linear differential operator of second order the linearization may be used, supposing that the analytical solution of the "linearized problem" can be constructed. The difference method seems to be the most general approach. This procedure can also be used for nonsymmetrical problems. Moreover it may be utilized for problems with nonlinear differential operator. Attention has only been payed to one internal mesh point because the algebraic equation developed can be readily handled. We have dealt with estimation of the branching points as well as domains of multiplicity, however, these methods may be used for studying stability of steady states, trajectories of transient equations etc. [ 12]. NOTATION

a shape coefficient A, B linearization constant, see Eq. (13) B dimensionless adiabatic temperature rise Bo collocation constant B,C constant in Langmuir-Hinshelwood reaction rate expression, see Eq. (22) C, C1 constants F, G functions t time T temperature

1751 CES VoL 26, No. 10-Q

0.53 0.58 0.58

V~ HLAV,~(~EK, M. KUBICEK and J. CAHA X y Y Da

co-ordinate conversion dimensionless concentration Damk6hler number

0 0 K,~

angle c o - o r d i n a t e dimensionless temperature g r a d i e n t s of e x t e r n a l field, Eqs. a n d (54) ~,~ r o o t s of t r a n s c e n d e n t a l e q u a t i o n h,/z r e c y c l e coefficients tr coefficient o f v o l u m e c h a n g e

Greek symbols

/3

y 8 A

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

d i m e n s i o n l e s s a d i a b a t i c t e m p e r a t u r e rise in a p o r o u s c a t a l y s t [ E q . (32)], d i m e n sionless heat transfer coefficient [Eq. (48)] dimensionless activation energy Frank-Kameneckij parameter g r a d i e n t o f e x t e r n a l field, Eq. (33)

p dimensionless coordinate z dimensionless temperature ~b T h i e l e p a r a m e t e r crit. critical * critical c c o o l i n g side 1,2 e x t i n c t i o n , i g n i t i o n

REFERENCES HLAV,~(~EK V. and KUBJ(~EK M., Chem. Engng Sci. 1971 26 1737. FRANK-KAMENECKIJ D. A., Heat and Mass Transfer in Chemical Kinetics, 2nd Edn. Moscow 1967. H L A V , ~ E K Vl. and MAREK M., Chem. Engng Sci. 1968 23 865. VILLADSEN J. V. and STEWART W. E., Chem. Engng Sci. 1967 22 1483. CARSLAW H. S. and JAEGER J. C., Conduction of Heat in Solids. Clarendon Press, Oxford 1960. SCHNEIDER P. and MITSCHKA P., Colln Czech. Chem. Commun. 1966 31 685. HLAV,~(~EK V. and KUBI(~EK M., Chem. Engng Sci. 1970 25 1527. HLAV,/~t~EK V., MAREK M. and KUBICEK M., Chem. Engng Sci. 1968 23 1083. WEEKMAN V. W.,J. Catal. 1966 5 44. JELINEK J., M. S. Thesis, Inst. of Technology, Prague 1969. ARIS R. and COPELOWICZ I., Chem. Engng Sci. 1970 25 885. HLAV,/~t~EK V., KUBI(~EK M. and MAREK M.,J. Catal. 1969 15 17. R6sumk- L'application des m6thodes de lin6arisation, de collocation orthogonale et de diff6rence est discut6e pour divers probl~mes de r6action dans le g6nie chimique. Les cas suivants sont consid6r6s: (a) transfert de chaleur dans une r6action chimique exothermique d'ordre z6ro; (b) transfert de masse dans un catalyseur poreux accompagn6 d'une r6action chimique avec expression du taux de r6action de Langmuir-Hinshelwood; (c) transfert de chaleur et de masse dans un catalyseur poreux pour (a) une plaque et sphere dans un champ externe non uniforme, (fl) r6action avec changement de volume; (d) transfert de chaleur et de masse dans un r6acteur ~ recyclage. Zusammenfassuag-Die Anwendung der Linearisation, der orthogonalen kollokation und der Differenzmethode wird fiir verschiedene Probleme der Reaktionstechnik diss kutiert. Die folgenden F~ille werden behaudett. (a) die W~irmeiibertragungund eine exotherme chemische Reaktion nullter Ordnung, (b) der Stoffaustausch in einem portisen Katalysator begleitet durch chemische Reaktion mit Langmuir-Hinshelwoodschem Geschwindigkeitsausdruck, (c) W~irmeiibertragungund Stoffaustausch in einem por6sen Katalysator fiir (a) Platte und Kugel in einem uneinheitlichen ~iusserenFeld (fl) Reaktion mit einer Volumenver~inderung, (d) W~irmeiibertragungund Stoffaustausch in RecycleReaktor.

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