Chapter 2 Dynamic Properties of Contact Apparatus

87

Chapter 2

DYNMilIC PROPERTIES OF CONTACT APPARATUS

An heterogeneous catalytic reactor represents a complex system comprising a number of elementary items. A detailed investigation of the inner structure of the reactor, encompassing the most important factors determining the technological regime and a representation of the processes at their elementary steps in the form of a mathematical description, enables the creation of a model describing only the essential features of the reactor behaviour as a whole. Investigation of the reactor behaviour on the basis of the analysis of its mathematical model allows for the construction of an optimum industrial apparatus functioning under forced non-steady-state conditions and avoids the expensive and time-consuming stage of successive process development.

2.1

CONSTRUCTION AND ANALYSIS OF MATHEMATICAL MODELS OF REACTORS

2.1.1 General principles (ref. 1) The mathematical models of complex activities created on the basis of classification approach are always hierarchic. The upper level, i.e., the sixth level of the reactor model with a fixed catalyst bed includes a mathematical description of the chemical section or the assembly regarded as the large-scale system consisting of a great many interrelated processes realized in different apparatus. The fifth level, i.e., the mathematical model of processes in the reactor, is a part of the mathematical model of the whole assembly. In spite of wide diversity in the construction patterns of the contact apparatus, they have one thing in common, the bed of catalyst that makes up the fourth level of modelling. Naturally, the mathematical description of the catalyst bed is included in the model of the reactor as a component. Various heat-exchanging devices, recovery boilers, mixers, distributors, etc., are other components of the reactor model. The arrangement of the catalyst beds, application

88

of a recycling stage and/or presence of the heat inside the contact section should be taken into account when constructing the mathematical model of the reactor. The mathematical model of the third level, i.e., of the fixed catalyst bed invariably enters as a component of the model of the catalyst bed as a whole. On this level, the inlet and outlet conditions, distributive removal or allocation of heat to/from the catalyst bed, the random character of the internal bed structure, the presence of space inhomogeneities on the boundaries and inside the catalyst bed, etc., must be taken into account. The mathematical description of the processes on one porous grain of the catalyst represents the second level of the reactor modeL It comprises the model of the steady and non-steady (as the limited case) processes on the internal surface of the catalyst, with consideration of the influence of the reaction medium on the catalyst composition, structure and properties. As discussed in Chapter 1, the mathematical model of the such non-steady-state process is nothing but a system of algebraic, differential and integro-differential equations reflecting the state of the catalyst at any moment of time relative to the time-varied composition, temperature and pressure in the gas phase. In the long run it also determines the observable rates of consumption and formation of different components of the gas phase. The method of levels in the construction of a mathematical model of the reactor proceeds on the assumption that the modelling of a given level is based on a profound study and experimental verification of all essential chemical and physical laws which determine the properties of this level. If that be the case, these laws acqUire the predictive ability of physical laws: they are invariant in space and independent of time. This means that the laws of the process proceeding in a component of the model for the given level and also the laws of interaction of these components are expressed in a form independent of the scale of the level or of the moment of time. Specific structural elements of the mathematical model of the reactor, for example, the internal catalyst surface, isolated grain of the catalyst,

89

the void volume between the grains, etc., can be regarded as the elementary dynamic units or groups of units. Every unit has its own internal properties determining the variation of its state in time in the case of quantitative alterations of its outer and inner links. The characteristic time of the non-steady-state process or in other words the time-scale, M, can serve as the quantitative standard. The size of this time-scale can be evaluated as the ratio of the unit capacity to the intensity of its outer link. The characteristic time of the reactor model component is determined by the time-scale of the units comprised by the links between them. The relationships between the units are often of both distributed and rpversed character. That is why the time-scale has a complex dependence on the time-scales of all units. The construction of an essential mathematical model requires the investigation of this dependence since it allows summarization of the basic properties of those elements which alone produce a decisive effect on both the static and dynamic characteristics of the whole reactor. The characteristic times of various processes in the reactor may differ from one another greatly, as is clear from Fig. 2.1. This means that the rapid and slow processes occur independently and there is no need to consider these processes together. In other words, a separation between the "quick" and "slow" components of the reactor model takes place and we cRn flay that the model of the non-steady-state process is split. Then, in the invpstigation of the processes in the interval of time of the order of the time-scale of the "qui-ck" element one can regard the process in the "slow" element as invariable. In the investigation of the processes in the interval of time of the "slow" element, the process in the "quick" element is auasi-steady. Let there be a broad mathematical model, i.e., differential equations, describing the unsteady process in any chosen component or in the reactor as a whole and consisting of n elements. Let the function UlQ = U(t,[", cX,f' ..• , ai' ... , Ci. ru ) (where a i is the coefficient proportional to the element's capacity; t is the time; I is the coordinate vector) be the solution of this system of equations i"or the preset initial and

90

conditions. The process in the ith element is quasi-steady in relation to the complete system if in any moment of time t with the exception of the small enough intervals called zones of the boundary layer the following inequality is fulfilled bound~y

where a is the maximum allowed relative difference between the solutions (in the sense of some normal) which correspond to the broad mathematical model Uur and to the narrow model UTl/ =..uWI~i=O = = U (t, f, d.,f, ••. , 0, ...• et ru ) where the sluggishness of the ith element is set to zero. The value 0" can be expressed, for example, as a relative error allowing for an experimental meassurement of the function Uw• The absolute value of If can be set in the range between 0.01 and 0.10. The coefficient ~i in expression 2.1 is small.

hydrodinamics chemical transformation

OJ QJ

OJ OJ

mass transfer

QJ

o

o

heat transfer

H p.

selectivity change

time scale (s)

Fig. 2.1. The regions of possible variations of the time-scales of the particular processes for an element of the catalyst bed.

91

Similarly to expression 2.1, another inequality can be written, the validity of which shows the region of insufficient influence of this or that factor, for example, the influence of the effective diffusion or heat conductivity inside the granular bed of the catalyst upon the non-steady and, in a particular case, the steady regime. As for the investigation of proximity of solutions Uw and U~ for singularly disturbed systems in the vicinity of initial points, one can see that selection of the initial conditions which are the solution of the steady-state problem allows to disregard the temporary boundary layer as well as convergence of both external and internal asymptotic disintegration. Evaluations of the regions of strong and weak influence of different parameters or a set of parameters on the dynamic properties of the reactor as a whole can be performed via analysis of the time-scale value which is often determined as
~ M=

[ u(=) - U(t)]

dt

o U(=) -U(o)

where u(t) is the solution of the system of differential equations which comprises the model of the non-steady-state process after the jump-like disturbance at time t = o. u(O) and u(oo) are the values of uf t ) at time t = 0 and t - 00 • So, the fulfilment of the inequality

IM(etf, ... ,rLi, ..• ,rLn,J-M{
~

f:'

u

means that the parameter a.i evidently has a weak influence on the dynamic properties of the reactor and its value in the mathematical model can be regarded as null.

92

2.1.2 Evaluation of the regions of strong and weak influence of various parameters (ref. 2) First let us touch upon the estimate of the weak influence of the dynamic properties of the catalyst surface which were discussed in Chapter 1. Of course, the smaller the value of the time-scale of the non-steady-state process on the catalyst surface, Ms, and the slower the variation with time of the gas phase condition, the less is the difference observed between the rate of chemical transformation, W, in the dynamic regime and the rate, r, given by the kinetic model of the steady-state or the quasi-steady-state. The quasi-steady character of the surface process in relation to a varied composition of the gas phase can be expressed as Ms

I d, tn W [ c(t), di

T (t) ]

I

~ 0

where c and T are respectively the concentration vector and the temperature in the gas phase. The left-hand part of expression 2.4 conveys the relationship between the characteristic times of transient regimes on the catalyst surface and the times of variation of the gas phase state. If this relationship is small, the prosess on the surface of the grain is quasi-steady. An isothermal catalyst grain comprises two elements functioning in parallel, i.e., the internal surface and the void volume, through which the substance from the external surface is transported. The parameter Ms does not noticeably affect the dynamic properties of the grain as a unit if

where

e~

is the porosity of the catalyst pellet.

Concentration gradients in the dead-end or connecting pores of length Lmi c and diameter dmi c for the bidispersed structures in the steady-state regime are seldom large. Large gradients may only occur when

93 2

L m ic r

d

. D.

mLC

C

TnLC

:5 0.1

where Dmi c is the diffusion coefficient in these pores, c is the concentration of the reacting component at the pore's mouth. Inequality 2.6 relative to the non-steady-state regime is only the condition required to maintain the quasi-steady character of the processes in the dead-end pores in respect of the processes in the transport pores. Fulfilment of the following condition is also necessary 2

DmicL max D

max

L

2

mic

~

to

7

where Dmax is the diffusion coefficient in the transport pore of length Qualitative and quantitative analysis made it possible to evaluate the regions of insufficient influence of various parameters determining the processes on the catalyst grain. The internal heat transfer had no dramatic influence on the transient regimes in the catalyst grain when the following conditions were fulfilled simultaneously

Lmax'

B i rru ~ O. 3

ry:5 o. 5;

where 1jF = R Vr/D ef c: Bi m = fi R..f/De; ,j3 is the coefficient of mass exchange between the grain surface and the flow, R~ is the radius of the grain, De; is the effective coefficient of diffusion. By analogy with the above, we can ~ake the catalyst grain as the volume of the perfect mixing because of the high intensity of the internal heat transfer if

where

2

7JF.,. "" V

2

.1 Bad (jJ

94

2=

.3 B im, (ry cth ry -I) yr

2(

B i Tn-I +

1jf

diu yr)

Q is the heat effect of the reaction, E is the activation energy, Cp is the heat capacity of the reaction mixture, R is the universal gas constant, To is the temperature in the flow around the grain, )., ef is the effective coefficient of heat conductivity inside the catalyst grain. In practice, the external mass exchange does not influence either the steady-state or the non-steady-state processes unless its intensity is far greater than that of the internal mass transfer and than the rate of chemical transformation. It may take place with

Similarly, we may not take into account the external heat transfer if

To complete the analysis of the processes on the grain, let us have a look at the quasi-steady character for processes utilizing mass transport while investigating the heat fields

and also for processes utilizing the heat transfer while investigating the concentration fields M?T/Mgmi~O.03

where

(2.13)

I

MgT = (c? +c? Cp )R; ).,ej

Mgm = ((;9+j)R/IDe/

C; is the heat capacity of the catalyst; V is the ratio of the quantity of substance which can exist on the internal surface of the catalyst to the quantity of substance which can occupy the volume equal to the size of the pellet.

95

In the investigation of the concentration fields the inequality 2.12 is also the condition of splitting of the model of the processes on the grain, whereas inequality 2.13 is the condition of splitting in the investigation of the heat fields. Expotential dependence of the reaction rate on the temperature may result in the appearance of several regimes on the grain even at identical temperatures and compositions of the reaction mixture in the flow. The expression

can serve as an estimate of the stability of the steady-state regime and, in particular. of the transition of the process to the region of external diffusion. Violation of inequality 2.14 in the low temperature region demonstrates the possibility to initiate a process on the grain when the exchange coefficients are similar over the entire external area. However, the surface of the catalyst grain is not equally accessible. This feature causes an earlier ignition in practice and a shorter ignition time than in the case of the grain with ~ equal access to its surface. Moreover, considerable overheatings of sites with good exchange conditions are possible. A different picture is observed in the steady-state regime: the sites with poorer exchange conditions are always found to be overheated. That is why it is important to know when the external surface of the grains can be regarded as practically equally accessible. This may take place in the case of weak external diffusion resistance, i.e., when inequalities 2.10 and 2.11 are valid. Criteria Bi=and BiT in these inequalities are determined in conformity with the mean exchange coefficients. The external surface of the grain may also be regarded equally accessible in the region of external diffusion which is estimated by inequality 2.9b and by

Let us take for further consideration a group of parameters characterizing an element of the fixed bed. Experimental research on the hydrodynamic situation has revealed two

96

contrasting regions in the void volume of the bed: a flow zone with mixing and disintegrating jets and a non-flow zone located in the vicinity of contact points of the pellets. In the range of criterion Re between 200 and 1000 taken for the system "gas-solid" (GSS), an eddy was observed in the non-flow zone, twirling and pulsating with high intensity. A characteristic space dimension in Reynolds criterion is the diameter of the catalyst grain, d g• The frequency of pulsations,~, is directly proportional to the linear velocity, u, relative to the void volume of the bed and in inverse proportion to the grain size, dg, so that the Strukhal criterion Sh =wdg/U = 0.5. A border is established between the zones due to the presence of a free boundary layer. In the system "liquid-solid" (LSS) the non-flow zone is fixed with Re = 1 - 100. If Re ~ 100, there is a similar situation as in the case with the GSS at the mouth of the non-flow zone and only if Re ~ 800, turbulent pulsations can spread over its entire volume. Such a difference in the behaviours of the GSS and LSS can be explained by the fact that with the increase in the flow density the turbulence degree is increased, which typically is determined by a number of various interfering factors. The non-flow's share is dependent on the flow velocity and on the shape of the catalyst pellets and lies within the range 0.1 - 0.3. Taking account of a discrete character of the exchange between the zones may prove necessary when the following inequality is fulfilled r(c,T)dgcU ~

(2.16)

o.s

which is the case only for rapid processes. In other cases a continuous mass-exchange model is used with the exchange coefficient between zones fl derived from the experimental dependence fldfl/ll "" 0.18. The size of the volume of the non-flow zone does not affect the steady-state regime of the reactor. In the non-steady-state regime this effect is not pronounced unless &Ul-

~

0.1& (1-&) (Bi i + BiuSu/Si) ~ 2lj/2(ytanh lj/!13i-rtV+1) x

x

(2 'f/ tanh lj//B~ + 1-2lj/cosech 2 r)(1+2lj/GDSech2lj/)Bi
(2.17)

97

where Suz' S is the fraction of the volume of the non-flow zone andSu/~is the fraction of the void volume between the grains of the whole volume of the bed, Bij" Bill. are Bio criteria calculated by taking account of the coefficients of the external mass-exchange surface of the grain washed by the flow and the non-flow zones respectively. The temperature in the non-flow zone is almost identical to that of the grain surface. That is why one of the heat elements of the bed model is the so-called carcass or frame of the bed comprising the grains and the non-flow zones. The value of the effective heat conductivity coefficient, J f , is determined via the expression ).; ;"AJm,+ O.85RePr},rru whereAJ""is the heat conductivity of the unblo\Vll bed, Pr is Prandtl criterion, Re is Reynolds criterion related to the effective size of the catalyst grain, J l7U is the coefficient of molecular heat conductivity, A = constant. For most catalytic processes carried out under invariable conditions at the inlet of the apparatus there is no need to take into account the longitudinal heat and mass transfer caused by the molecular and eddy diffusion (Dm and Dv)' heat conductivity Urn and lv) in the void volume of the bed and by the heat transfer along the catalyst carcass ,AI. The processes in which the inequalities r(c, T) . IJ/TII + Du < C u2 - 0.03

(2.18)

Ll 8CL~ r( C. T)

c

(where u is the reaction mixture velocity related to the whole cross-section of the catalyst bed) are violated in any cross-section of the bed represent an exception. In the non-steady-state regime these inequalities are complemented by the following conditions: Re~20

98

(2.21)

The mathematical analysis of the reactor model as a whole is significantly simplified if the time-scales of variation in the concentration fields, temperature and catalyst activity differ by as much as 10-20 times. The mathematical model in this case is split (see Fig. 2.1).

2.1.) Mathematical models of the non-steady-state

processes in the reactor Easy arithmetic shows that there are hundreds of possible models of the processes which can occur in a fixed catalyst bed. With the use of the above inequalities which point out the most important factors and determine the behaviour of the temperature and concentration fields in the reactor, it is easy to construct an essential model of a process as a whole. For example, for the processes of oxidation of S02 to so) in a reactor with adiabatic catalyst beds, the process in the first bed is described by a model taking account of the temperature and concentration gradients inside the catalyst grain. The process in the grain in the following catalyst beds may be represented by the model of isothermic regime inside the catalyst grain. The steady-state regimes in all beds can be satisfactorily described by the model of ideal substitution (piston flow). The steady-state regime for vinyl chloride synthesis in a tube reactor is described by the quasi-homogeneous model with account taken of the temperature change along the radius of the tube, whereas for the non-steadystate processes in the reactor one should not ignore the temperature changes inside the grain. Let us go through some examples of the mathematical descriptions of the processes in a granular catalyst bed. When only one exothermic reaction without variation of its volume is carried out in the reactor, then, provided the state of the catalyst is quasi-steady, the non-steady-state regimes in the adiabatic bed can be described by the following system of differential equations

99

(2.22a) O~r:{R

[J

O~7:~7:L

dT

C dt

c;

·.A m +AV dET _ dT + dre2 are C u 2

=

dx at

p

=

u.; -ti; u

d 2 .x;

dre 2

2

0.. Ssp

C p

(

Tg


;

(2.22b)

(2.22c)

T

ax - - + J3S (V-X) dre sp

(2.22d)

(2.22e)

r

=R~

;

O"'7:~7:L

dTf!=

. Am+Azr dT

..J

0

d Tf]

= aT = ctx

~7:

t

v

Cp

u

=

dljd.r

=

2

as:

d t:

(£7;

d Tf! d.r

I

0

dn:

=

=

_

.

T To,

Dm+Dzr dx=x-x u,2. dt: 0

0}

}

Il : x=u=1; (j

J

t: =

R~

7;=0

(2.22f)

j

(2.22g)

(2.22h)

(2.22k) r~-- T -- T iro

(2.221)

where Tg, T, Tin' To are the temperatures in the catalyst grain, in the void volume of the bed, the initial (at t = 0) and the inlet temperature respectively; x and yare the extents of conversion of the initial substance in the void volume and in the grain; Co is the initial concentration of the original substance; Ssp is the specific external surface area of the catalyst grains related to unit bed volume, for example, for a bed consisting of spheres, Ssp = 3(1 - 6)/Rg; L is the length of the catalyst bed; Af is the effective coefficient of the

100

longitudinal heat conductivity in the carcass of the bed; & is the porosity of the catalyst bed; ~ = L/u is the conventional time of contact; ~L = L/u; 1 is the current length of the bed; r is the current coordinate of the catalyst grain. Eqns. 2.22a and 2.22b describe the non-steady-state processes in a porous catalyst grain: the first conveys the temperature field and the second the concentration field. Eqns. 2.22c and 2.22d describe the transport processes in the void volume of the bed, whereas eqns. 2.22e and 2.22f are boundary conditions (at r = Rg ) and prescribe the filtration of flows through the external surface of the grain. The flows of heat and mass through the cross-section of the bed at the inlet (~= 0) are respectively preset by the second and third equations of the system 2.22g. The fact that by use of eqn. 2.22e the relationship between the processes of heat transfer both inside the catalyst grain and between the neighbouring grains is taken into account is the distinctive feature of this model. The derivation of this equation is based on the notion of the carcass (or frame) of the bed and on the processes of heat transfer through this carcass. The carcass of the bed is the assembly of the external surfaces of all the particles and of the non-flow zones between them. The heat transfer along the carcass is performed with the help of vortexes, which are formed in the stagnant zones in the vicinity of the points of contact between the grains. If the grains are small enough compared to the characteristic dimensions of the reaction zone and of the bed as a whole, then one can consider the integrity of the averaged continuous fixed and solid phase,i.e.,the carcass of the bed rather than the grains and the non-flow zones between them taken separately. A complex mechanism of heat transfer from the surface of one catalyst grain to that of its neighbour can be described by the effective heat conductivity along the carcass of the bed. The heat balance in the bed carcass is a combination of the heat flow across the carcass described by the process of effective heat conductivity, the heat exchange with the gas phase (proportional to the total external surface of the catalyst grains per unit layer volume, Ssp) and the source of

101

the heat energy in the carcass of the catalyst bed. Heat flowing from the catalyst grain through its surface in the direction of the normal (also proportional to Ssp) is the source of the heat energy in the bed. It can easily be demonstrated that the mathematical model obtained of the description of the dynamic processes in an adiabatic bed of non-isothermal catalyst grains coincides in all rational extremities with the known and approved models. The present model allows for generalization in the case of several reactions in the catalyst grain carried out simultaneously and accompanied by a change in the volume of the initial mixture. The mathematical description in the dimensionless form is always convenient for the calculation of definite chemical processes, for which all parameters are quantitatively determined. It is also expedient to use such a model for investigation of the general properties of the system, which might be, for example, connected with the static and dynamic characteristics or with the plurality of steady-state regimes and their stability. For example, with consideration of the previously mentioned assumptions determining the applicability of model 2.22 in a tube reactor where only one first-order reaction is carried out and the temperature of the coolant between the tubes is identical at every point, we can derive the following system of equations in the dimensionless form

O~jJ~1;

(2.2)a)

o ~ .5 ~ JVj o « J!;.~M

~~ dd'~~ -- 0

--r : r LU

}

.P =0;

0 '"/

5 ""./ JV; Ar

0 '"/

z ,,; M

(2.2.3b)

(2.2)c)

102

-B't- ( lJ-X ) m

(2.23d)

= -dlj

dfJ

ij

jJ=f;

(2.23e)

O~.5~.N';

O~z';;M

(2.23f)

(2.23g)

z=O:

dx dz

(2.23h)

(2.23k)

5=0: _1_ dx =x-x . _1_ d8 =8-8 . dBf/=o Pen

t=J{: :>

ds

dx

as

=

0'

dB df,

PeT d

s

d

0'

s

d8~ =D

=

d

r

e

d =2R ; ~

g

(2.231)

(2.23m)

s

where

»:»:' srs:: R~ Ij

P = 1;

rf;

,,= d.- "J

N

~

103

Pe

]) =

d g Ue

nilerru

6

--"-':-7.--=---

H=;;:e ; 9

Tg, T, To' To are the temperatures inside the grain, in the void volume of the bed, around the inlet and of the coolant respectively; Tb is the base temperature; r t is the radial coordinate of the tube; dRt is the radius of the tube; N, Mare the amounts of catalyst grains, which can be placed along the height and the radius of the tube; P = 0, 1, 2 depending on the form of the grain (plate, cylinders, spheres); D~m D:m are the effective coefficients of the longitudinal and radial diffusion in the void volume of the bed; are the effective coefficients of the longitudinal and radial heat conductivity in the void volume of the bed; Qf'~2 are the coefficients of the heat exchange between the walls of the tube and the carcass of the catalyst bed and between the walls and the reaction mixture, respectively.

J..:'m, J..;m

If, for example, the transport of mass and heat inside the porous catalyst grain is rapid enough so that the presence of gradients inside the grain can be ignored or, in addition, the external heat and mass exchanges are intense enough which may take place in the case of the inequality

(2.24) the following relatively simple expressions can be written instead of the system 2.22

104

where PeD

, Pe T

are the diffusion and the heat criteria

nil Pe =d1J

~U

j

r,:

e)J

[ 1+(1-(;) (5 .:e + L e ecp

D" is the coefficient of the longitudinal effective diffusion.

2.1.4 Dynamic characteristics Due to external influences or/and variations of the internal properties of both the catalytic process and the reactor, the temperature and concentration fields in the catalyst bed change with time. It has already been pointed out that parameters which might well have omitted from consideration of a steady-state regime often prove essential for a non-steady-state one. The effective diffusion of the substance along the catalyst bed, the mass and heat capacity of the bed, the different accessibilities of the external surface of the catalyst grain, the mass and heat exchange through the surface of the grain may be such parameters. In the steady-state regime a large number of factors independently influence the state of the system and are often additive. This allows the application of more simple models and effective parameters reflecting the combined impact of these factors. The influence of these factors on a non-steady-state regime may be somewhat different and may be strongly dependent on the state of the system. That is why the effect of each factor should be considered separately. For example, the heat dispersion along an adiabatically operating catalyst bed in the steady-state regime can be sufficiently represented by the coefficient of effective longitudinal heat conductivity, while in the non-steady-state regime this is not possible. The heat

105

transfer through the carcass of the catalyst, the heat exchange between the reaction mixture and the external surface of the grain and sometimes the heat transfer inside a porous grain of the catalyst must also be taken (certainly separately) into account. Because of the inertial properties, the non-steady-state regime has larger temperature and concentration gradients on the grain and in the catalyst bed than the steady-state regime. This results, for example, in the absence of a dependence between the temperature and the extent of conversion, a short-time but still noticeable overheating in the region near the surface of the grain having the best exchange conditions. The phase shift of the temperature and concentration fields in relation to one another can sometimes bring about the appearance of oscillative transient regimes and even the stable limited cycles. This is likely to occur in an heterogeneous catalytic reactor where all processes are satisfactorily described by the perfect mixing model related to the heat and by the model of ideal substitution related to the mass. A fixed granular catalyst bed possesses a very interesting feature. It allows for initiation of the so-called dynamic overshoot when after slight and at the same time rapid inlet perturbations of some parts of the catalyst bed, the temperature considerably deviates from its steady-state value.

2.2

THE FRONT OF AN EXOTHERMIC REACTION IN A FIXED CATALYST BED

Presently, various non-steady-state methods are suggested to organize chemical processes in an adiabatic catalyst beds (ref. J). A greater part of them arise from the formation and propagation of a comparatively narrow high-temperature reaction zone along the catalyst bed. The peculiarity of this zone lies in its ability for self-regulation: at low inlet temperatures when the rate of the chemical transformation can be neglected, temperature and concentration fronts of an exothermic chemical reaction can be formed in an sufficiently extensive bed of the catalyst. The fronts move parallel to one another without distortion. The non-steady-state temperature and concentration fields which are thus created in the bed also possess a number

106

of peculiarities of practical importance. Now not only theoretical but also experimental data have demonstrated the existence of a high-potency heat wave spontaneously propagated along the fixed bed of a granular catalyst where an exothermic chemical reaction including filtration of the fresh cool reaction mixture through the bed is carried out (refs.3-8). The temperature and concentration profiles experimentally observed are characterized by a striking constancy of their values before and behind the front and also in a relatively narrow zone by a considerable differential between these values. In the next paragraph some physico-chemical and mathematical aspects of the notion of the heat front of an exothermic reaction in a fixed catalyst bed will be discussed.

2.2.1 Physical aspects Let a steady-state regime be established in a sufficiently extensive fixed bed of catalyst with coordinates 0 ~.5 ~ 1 where one reversible exothermic reaction is carried out. Further, an equilibrium of the extent of conversion degree is established at the bed outlet. This can be provided by the relatively high temperature of the reaction mixture at the bed inlet, eo = 2

=[(T-To)£J /RTo which allows for the chemical transformation to be performed at a noticeable rate. At the outlet of the catalyst bed in the steady-state regime the temperature 9 (1) = 9 0 + +~ 9adXeq is established where x e q is the equilibrium of the extent of conversion at temperature 9 (1). Fig. 2.2 shows the temperature fields and the extent of conversion in the bed calculated at different moments of formation and distortionless propagation (creeping) of the temperature and concentration profiles along the bed. The initial temperature distribution, e (S ~t=D(profile 1), corresponds to the solution of the steady-state problem at eo = 0.5.

At time t = 0 the initial temperature was decreased stepwise to eo = -6.5 and then maintained • It is assumed that the value of the rate of chemical transformation at this temperature is

107

negligibly small. It is clear from the scheme that cooling of the inlet-adjacent catalyst area is the result of the temperature decrease of the reaction mixture at the inlet. This occurs due to the existence of the inter-phase heat exchange mechanism. At the same time, with the cooling of the catalyst bed due to the action of the inter-phase heat exchange mechanism, there is an heating of the mixture which has not yet participated in the reaction. The rich reaction mixture warmed to the temperature at which a noticeable rate of chemical reaction can be attained is exposed to the grains of the catalyst where it is additionally heated. The mechanism of the longitudinal heat conductivity in the catalyst bed tends to decrease the temperature difference which is manifested through organization of the heat flow against the flow of cool reaction mixture.

8 2 1 0

x a~

a~

au 0

Fig. 2.2. Formation of the heat and concentration fields in the catalyst bed as the result of an exothermic reaction. The steady-state regime (1,1); temperature (g) and the extent of conversion (x) profiles along the catalyst bed (2-6, 2·-6) respectively at various moments of time (2, J, 4, 5, 6).

108

As is seen from Fig. 2.2, the temperature gradient in the developing front is larger than the steady-state one, but at t > t it is practically invariable. It is possible to accept 4 that the front is formed. Now the temperature and concentration profiles (fronts), 9 (S' t) and x (5' t), propagate (creep) at a constant rate along the catalyst bed. These profiles remain unaffected in the system of coordinates r = I - vt (where 1 is the length of the catalyst bed, v is the propagation rate of the front). The temperature and concentration profiles in the established steady-state front do not depend on the initial distributions in the bed. The very slow propagation of the front is accounted for by the large heat sluggishness of the bed in relation to the sluggishness of the gas phase as the value of the front creep rate is an hundred times smaller than the rate of the gas flow. The distance covered by the reaction zone in the process of formation of the steady-state heat front is dependent on the initial conditions. At the initial temperature close to the front's maximum temperature, the steady-state profile is formed at a distance of the order of the width of the reaction zone in the front. Later we shall consider the strikingly peculiar properties of the heat front of an heterogeneous exothermic chemical reaction. Among them, for example, is the considerable difference between the maximum attainable temperature in the front, 9ma x' and the inlet temperature of the reaction mixture which can be many times greater than69 a d x eq (9ma x) where x eq (9ma x) is the equilibrium conversion extent at the maximum temperature in the front. Let us point out two pairs of processes competing with one another and which are responsible for the main characteristics of the heat front. First, a chemical transformation accompanied by a reaction heat release leads to a temperature increase in the catalyst bed. The intensity of the chemical transformation is determined by the rate of this reaction, the dependence of this rate on the temperature and composition of the reaction mixture and by the heat effect of the reaction. The temperature gradients are proportional to the activation energy of the adiabatic heat buid-up at the equilibrium state of the

109

conversion extent. The longitudinal heat transfer and inter-phase heat exchange redistribute the liberated reaction heat throughout the bed, thereby smoothing out the gradients and diminishing the maximum temperature. Secondly, a flow of cool inlet reaction mixture through the preliminarily warmed up catalyst bed together with the mechanism of the inter-phase heat exchange cause an heat extraction from the bed in the direction of the gas flow. On the other hand, the longitudinal heat conductivity of the bed allocates the heat energy of the hot reaction zone to the cooled portions of the catalyst bed. The direction of this heat flow is opposite to the flow of gas. The reaction zone may propagate in this or that direction depending on the intensity of this or that process. If the reaction zone propagates in the direction coinciding with that of the gas, the cooling of the warmed up catalyst bed is somewhat retarded. So, the zone of elevated temperatures is continuously supplied with the reaction mixture rich with fresh (not react,ed) substance and the temperature differential in the reaction zone appears to be higher than the adiabatic heating of the mixture. If the reaction zone propagates against the gas flow, then the warming of the cooled catalyst bed lags behind. That is why in this case the temperature differential in the reaction zone is smaller than the adiabatic heating of the mixture. The first report on the heat waves in a fixed catalyst bed evidently belongs to D.A. Frank-Kamenetsky and dates back to 1947 (ref. 9). However, he only reported migration of the reaction zone in the oxidation of isopropyl alcohol on a copper catalyst. Wicke and Vortmyer and co-workers (refs. 4-7, 10-12) studied the catalysed combustion reaction in a granular bed consisting of activated coal through which was blown a mixture of nitrogen and oxygen. It was found that with an increase in the filtration rate, the rate of the front propagation is reduced to its negative maximum and then rises again. The value w also increases with increasing oxygen concentration in the mixture and after elevation of the initial mixture temperature. The maximum front temperature increased with increasing filtration velocity. With the support of the experimental data obtained, an empirical dependence of the heat propagation rate

110

on the conditions of the experiment was suggested. Refs. 4 and 5 report an experimental investigation of CO oxidation on a platinum catalyst (Pt/A1 20 A certain critical 3). value of the inlet temperature of the initial gas mixture was found. An high-temperature steady state could not be attained below that value, instead, drifting of the narrow reaction zone was observed. This zone crept in the direction of gas filtration, provided the rate of filtration was high. If not, the propagation of the heat front was directed against the flow of gas. There was also an intermediate value for the rate of filtration under which a standing-wave regime was realized. The results of a study (ref. 6) on CO oxidation on Pt/A1 20 were 3 critical in many respects for the use of a two-phase model in the mathematical description of processes in an adiabatic catalyst bed. The dissimilarity of the temperature and concentration fields was reported: the temperature front curve appeared to be more linear than might be explained by the heat conductivity of the catalyst bed frame and radiation. Moreover, a considerable temperature difference was observed in the flow and on the surface of the catalyst grain. At elevated temperatures this temperature difference was registered between the centre and periphery of the grain.

2.2.2 A mathematical model of the front of the

chemical reaction To describe the processes in an adiabatic catalyst bed a widely used two-phase model was selected as the starting point. This model can be represented in the following form

(2.26)

G

dB

-

dt

d., 2 8

Aj e d7: 2

dx

6 db

=.fllt

d2.x d'G 2

dB

-+a; (6J -B)

d/t:

0

g

dx d7: + W(B.y>' x)

(2.28)

111

with boundary conditions

and with initial conditions

where rY

=

7!C (t

x=o.£L..£ c ' o

c) ,

r

p

e

7: = U

Co, C are the reagent concentrations at the bed inlet and in the gas flow; Sg' S are the dimensionless temperatures of the dense phase of the gas; 1, L are the longitudinal coordinate and the total Langtih of the catalyst bed; 7;' ,7:L are the current and the general time of contact in a bed with length L; u is the linear velocity of the gas flow; x is the extent of substance conversion in the gas flow. The mathematical description of the processes in an adiabatic catalyst bed takes the form of eqns. 2.26-2.31 only i~ the following assumptions are valid: (a) the temperature gradients inside the catalyst pellets are small; (b) the chemical processes on the internal surface of the catalyst pellet and the diffusion processes inside the porous catalyst pellet are quasi-steady in relation to the transfer processes of the gas phase; (c) only one exothermic reaction of the A~ B type is carried out in the reactor with no change in the reaction volume. With these suppositions, the function CoW(Sc,x) has the sense o of the observable rate of the chemical reaction, taking account

112

of the external and internal diffusive resistance. In the case of a reversible reaction of the first order, the observable rate of the chemical reaction is represented by

=

where

(2.32)

FnJ B } [1- x(t -t- l-U e?){ 81 )J ff

f~~B J

lU( 8f) = l-U(T8 ) exp [

direct reaction;

lie? (Bfj)

equilibrium constant;

is the rate constant of the

fj

=

- t-1J

lieF expl8(f+-881)

]

is the

'2=E_/E, E_ is the activation energy of

the reversed reaction; l-U(TgJ--=lUoexp(-f '18), 'if,(8fjJ'=(f-c;)lU(B?), flo""J3 ~[),.f; is the coefficient of the mass exchange between the

R2

r

flow of gas and the catalyst pellet; yr2(8 J = ~ rL((Jfj)(f+-lUeg(8gJ).

i

The function ~~(y) is determined by the shape of the pellet and equals ~o (r) = tp cocan. 1j/ for a pellet in the shape of an endless plate; ~1(vr)=f(.la(vr)/J1(vr)) for a pellet in the shape of an I

endless cylinder; !t'z(r) = 3"

lj/

ytanh

r

y _ can.ti.

lj/

for a spherical

pellet. Here J o and J, are the modified Bessel functions of zero and of first order. It should be noted that if Def - 00 then 0/" 0 while 1fJTl/ (r)" 1 • We shall further assume that the influence of the conductive transfer is insignificant, that is, :Aft ~ 0; DIt ":: 0 .

Before deduction of the mathematical description of the front of a chemical reaction in a fixed catalyst bed which arises from the system 2.26-2.31, let us formulate some additional physicalchemical assumptions required to support this transition. The first thing to do is the idealization of the real finite catalyst bed by an endless bed. Of course, all the parts of the endless bed are supposed to possess similar properties, i.e., the equations and parameters included in these equations remain constant. Moreover, the properties of the processes are determined by the boundary conditions. For an endless catalyst bed, the character of the process performance should naturally

113

be preserved. But the conditions in this case are determined by the solutionIs behaviour in the infinity. Physically, the transition from a finite bed to an infinite one is reasonable because the ratio of the width of the reaction zone to its lenght is in practice very small. This situation allows for transition from the practically constant variables in the finite bed lengtili described by the system of eqns. 2.26-2.31 to the variables of the "ideal" heat (and concentration) front. The variables of this front are different via their ability to be transported along the infinite catalyst bed at a constant rate and with unchanged structure. This means that the structure of the ideal heat front is invariant relative to the group of all possible shifts along the bed

and is similar to the functions 9 and x from eqns.2.26-2.31. Here ~' is the coordinate connected with the moving front at the rate v, under the condition that v = uu>, u is the filtration velocity,w is the dimensionless velocity of the front migration. The initial conditions for the endless catalyst bed represent a smooth continuation of the initial conditions 2.31 infinity and must agree with conditions 2.29-2.30 at t-C'O and 7:-+ 0 0 respectively. Thus, in a new coordinate system we write the following system of equations for the reaction front (instead of e, g and x we shall write 9, 9 q and x a )

e

g

0

:A 8;' + JW8/ -

cL (

(f-ow)8'

o ( 8/l - B)

O

8~ -8) -1-.1 8tUt W( 8;, x )> 0

= cL

(1- ceo) x = W(B/l' x)

where (I) designates the differential with respect to ~. The solution of eqns. 2.34-2.36 belongs in the class of smooth limitative functions meeting the additional requirements posed by eqns. 2.35 and 2.36 which determine the behaviour of the solution at infinity

114

T' -

-

ex> :

B,y' = 8 = 80

j

x

=

0; 8/ = 8' =: x '= 0

T'-+oo.. e'=8'=x'==O ?

The system of eqns. 2.34-2.38 is a stationary limited variant of eqns. 2.26-2.31. It includes an unknown parameter which requires determination. From x - 0 at T - - 00 it follows that W(9 0 ' x)--O and the necessary condition of existence of the solution to the problem 2.34-2.38 lies in W(90 ' 0) ~ O. To make the existence of the problem solution independent of small variations of the temperature, 9 0 , the existence of such a value (9 > 9 0 ) of the temperature like that in the classical theory of combustion is assumed such that

Taking account of the invariant character of the solution in relation to the shift along the coordinate~, it follows that Bp I~=o = 8. The meaning of the necessary condition and the supposition that we have made lies in the fact that the formation and propagation of the wave with an unchanged profile is possible only when the temperature at the inlet of the catalyst bed is so low that one can ignore the rate of the chemical reaction in comparison with the rates in the region characterized by rapid conversion of the substance. This means that (in full agreement with the theory of combustion) the steady-state propagation of the reaction front only describes the process approximately and asymptotically. Now we should particularise upon the physical sense of the necessity to introduce the temperature of the "cut ll (6). Let us not forget that the structure of the heat fields close to that of the heat front in the endless catalyst bed is observed only when the temperature of the gas at the inlet is low enough, so that the rate constant of the chemical reaction is also very small. So,within any reasonable length of the catalyst bed where the temperatures are similar to the inlet temperature, the degree of conversion will appear to be negligibly small. With the transition to an endless bed, the degree of conversion will

115

appear finite eVen at very low rates of the chemical reaction. That is why the requirement for the rate of the chemical reaction carried out at the inlet temperature to be zero, seems quite natural. It reflects the real situation and compensates the transition from a finite bed to the endless one. Thus, to obtain an idealized description of the structure observed in the finite bed, one has to consider the problem of the endless catalyst bed and stipulate in the first place that the rate constants of the chemical reaction at the inlet temperature be zero.

2.2.3 Existence and unique character of the chemical reaction front (ref. 13) Conditions 2.37 allow derivation of the first integral of the system 2.34-2.36

and as the result the order of the system is decreased. It follows from assumption 2.39 that in the region [)~ ~ if xfr;') =0, while 9 g and 9 satisfy this linear system of equations jW

8=--(8-8)+

.i\.

~

?

0

(2.41)

the proper number of which are equal to

fl

1,2

= -1 [ -

where

rX,o

orlO) =

( d.o- +

JlaJ) ± V( -l-lw). rX,o- + raJ-J2+ 'I a:J.......Q O(w) 'J

!L.:.::....

1-8w.i\.

1-(C+cr)w

1

-&w

If

° which is possible when

o(w)~

1/(&+J} ~ W ~ t r e , then jUf
116

w < 1/(C+t) or w>I!t;, then !f(w»D and JU, < D < fl 2 the solution is determined by the proper number JU2 B (7:')=B +(fj-B)e

i

0

B= 8 +

o

rw

0

JU

2

"

Now

7:'

+ .AJU2. (Bfl-8 o )

/ - c,w

-

We can see from the above equation that at co > i/c , 8 < 8 0 ' which has no physical meaning. Thus, the allowable region of the values of parameter co is nothing but the interval (-00, t/(c+,y)) and eqn. 2.42 shows the temperature distributions in the zone where the chemical reaction does not occur. This zone represents the cooling zone if the front is propagating in the direction of gas filtration, but it can represent the heating zone if the direction is reversed. The behaviour of the solution of system 2.34-2.36 which may satisfy the conditions described by eqn. 2.37, that is of the system of aqns , 2.40, 2.35 and 2.36 at 7:'-+00, is determined by the conditions stated in eqn. 2.38. According to them, the solution at '7:'_+00 tends to the equilibrium state, 8*

8e? LI

=

B = B * < 00; W( 8 ~ x *)

Bact x* =otw ){ 8

=

0

* - 8o )

The proper numbers V"';2 and V,j in the system of equations 2.35, 2.36, 2.40 linearized in the vicinity of the equilibrium

state are real cv, :f;JU, < rN/(f-t;w) ~ Y2 ~ w..zJ8~ x*)/u-c;w) < < 0 < JU z ~ Y.j ; equality may take place in the case of a non-reversible reaction). The real character of the proper numbers excludes the possibility of the solution in the vicinity of the equilibrium state to be of an oscillative character. Evidently, the solution which converges to the equilibrium state at 7:'-+<:1:) is determined by the proper numbers V, and Pg. However, the direction of the vector field in the vicinity of the equilibrium state determined by the proper number v, is that d9/d9g < and dX/d9", > 0, whereas the proper number )/2 determines (in the vicinity of the equilibrium) such a direction of the vector

°

117

field for which d9/d9 ) 0 and dX/d9g > o. In reality, the 9 sought after solution in the vicinity of the equilibrium state is determined by the proper number v2 • The power balance of the dynamic regime of the reaction front propagation (eqn. 2.43b) represents a one-to-one correspondence of 9* andGU and gives the characteristic of the propagation process distinction for the heterogeneous, gaseous-homogeneous or condensed media where o(w) = 1, and, consequently, 9* = 9 0 + + A 9 adX*. For heterogeneous system this condition is fulfilled only in the case of a standing wave when G{) = o. If G{) > 0 then 9* > 9 0 +A 9 adx* and if G{)
8; -Bo ' V=8-8o

Then

u

ti + 1- f:,u) 1 r:£o

hi,

./-2

'

D < V(w) < i7

Let Pj ( U , x , w ) =- O(w ) 7.-U + IJBa,d, x , P2(u,V)=u-V and W(u,x) = W(Bo +u,x). Then, i f eqris , 2.40, 2.35, 2.36 are wri tten wi th the new variables, we can formulate the following problem: find the value of parameter G{) < 1/ tc +') with which the solution of the system of differential equations

118

f-Cw [

u

-).-

Pt(u,x,w) -

P2 (U, V)]

v'=~ f -(;UJ P2 (U, V) x'

t r sco W(U,x)

satisfies the conditions rc';O: U=U; 7:' -

+00;

U -

l/='[j(W); U *(W);

x=O

)

if --- U*(W)

X-X*(W)

Moreover, u* (w) and x*(w) are determined by the relationship

W (u*(w) , x*(w) =

a

The result of the solution of this problem can be formulated by the following. Theorem 1 (ref. 13): let some conditions and assumption 2.39 in relation to the function W(9~, x) be valid. If with all 9 , o g ~o + W x(9g, x) > 0, there exist only one! value of the parameter w 0, this theorem also proves valid, at least with rY)& • This limitation is, however, insignificant, as for heterogeneous systems, in practice rY ~ [, • The theorem is also valid in the situation where cx,o = 00 (a quasi-homogeneous model). However, with .A = a (a two-phase model of perfect substitution), the following result was reached. Theorem 2. Let some conditions and assumption 2.39 in relation to the function W(9 g, x) be fulfilled, then there exist only one 1 value of the parameter co E- (0, I/(;y+£)) with which problem 2.24-2.46 (with ~ = 0) has its solution in the

119

rigorous case when with all U to: ca. urco» !:La U > {j Bad W(u, (9). This solution is monotonous and lies in the region [(U,V,X): P/u,x,w) > P2(u,V»O, w(u,x) >0, x>O}

Thus, if the influence of the longitudinal heat conductivity through the carcass of the catalyst bed is insignificant, the heat front, nevertheless, can propagate only in the direction of gas filtration and only if the condition ocou >,1 Bad W(u,O) is valid. It should be noted that violation of this condition coincides with the condition of plurality of the stationary solutions of problem 2.26-2.31 with ) = A ej =Dft =0. If at .A = 0 there still exists a solution, then c(,o+ + W'(u,x(u,w»>O for all u. E(lZ,u'*(U))) , i.e., this condition is not the limitation of the problem (unlike in Theorem 1) but follows from it and is the limitation (when ~o< fi o ) on the value B*(w), and on the value of parameterw, respectively. I believe that the following conclusion is essential: when the system is described by a quasi-homogeneous model ( 0 (or :A> 0), there exist only one! value :A> 0 (or Bad> 0) with which problem 2.44-2.46 can be solved with oco=oo, w= O. This solution is monotonous and lies in the region

u u., X):

U>,1 []ad X,

W (B, x ) > 0, X > 0 ]

All results mentioned above were obtained with the assumption 2.39 and, consequently, the rate of the front propagation,GO, depends, generally, on the "cut" temperature value B. With a quasi-homogeneous model (cX.o = 00) it is easy to illustrate that the dependence of the front propagation rate on the "cut" temperature is monotoneous and so there is a one-to-one correspondence between them. Consequently, the opposite problem can also be solved: for every value of the parameter w~ 1/(/ +e) there exists such a temperature which may be accepted as the standard for determination of the "cut" temperature. The

120

dependence of the maximum temperature, g*, on the "cut" temperature is also of a monotonously increasing kind. So, within the prescribed accuracy of the maximum temperature determination, one can approximately identify such tolerant interval for the "cut" temperature which would allow for the corresponding maximum temperature be varied within this accuracy. The lower boundary of this interval is rigorously larger than the inlet temperature. Its comparison with the corresponding interval of the "cut" temperatures in condensed combustion shows that in an heterogeneous catalytic process, which differs from condensed combustion by the presence of an additional parameter, J (that is the phase heat capacity ratio), the allowed "cut" temperature interval expands sidewards. The condition of absence of such an interval is known in the theory of combustion as the condition of the heat wave degeneration. In an heterogeneous catalytic system this condition can be qUalitatively characterized as the condition at which the reactor performance approaches that of a reactor with ideal mixing or as the condition of low intensity of the inter-phase heat exchange or, as the condition of low activation energy of the chemical reaction. The last case is the most significant because the peak sensitivity of the above-mentioned condition is dependent upon the activation energy change rather than the other factors mentioned.

2.2.4 Estimates of the major characteristics of the exothermic reaction front The determination of the approximate value of the maximum temperature in the reaction front involves the method of narrow reaction zones (ref. 14), according to which the greater part of the conversion in the front is realized at temperatures close to the maximum. When there is no internal diffusion resistance in the porous catalyst pellet, the maximum temperature in the reaction front is derived from the following equation (ref. 13)

[X,0

121

which can be considerably simplified if the maximum temperature is selected to be the base temperature, Tb• Then e* = o. Another extreme case allowing estimation of the maximum temperature corresponds to the supposition ~o = 00 8*

~ FnJB)dB

"'-'

B

If r = 0, then e* = So + n ead' but then it follows from eqn, 2.48 that

This formula coincides with the estimate of the gas combustion rate in condenced media (ref. 15), iflJ e; =00, 130 ==, &= 1. Eqn. 2.48 may be interpreted as follows. The total amount of heat energy abstracted in the course of the reaction is directly proportional to ;1 Sad while the density of the distribution of the heat source among the temperatures is proportional to F~(B). So, the higher the density of the heat source distribution in the region of low temperatures and the higher the heat conductivity at a given temperature, tha lower is the maximum temperature. In the case of a reversible reaction, eqn. 2.48 should be replaced by

e* (t-&w)Z

SFn(B)(f+/i.ec/B))r£B

s

~

Here under the integral sign is placed the sum of the observable constants of the rates of the direct and the reverse reactions. Of course, the presence of the reverse reaction lowers the maximum temperature. If fi o is large enough, then, supposing that /i.(B) rv e e , one can obtain a more general estimate

122

where 8*

~ FnJBJd8

8

sn. v: yr

J7U (0)=/; The value of co which is entered into eqn. 2.49 is determined via e* from eqn. 2.43 {uJ

f _ tJ8ad

=

l-ouJ f - ouJ

8*-8o

=

[

f + -[, ( I- Ll Bad

f

8*-80

)J

-1

This shows that the dependence of the maximum temperature upon parameters [; and is of the order o/f, that is, it is negligibly small for heterogeneous systems. However, once the maximum temperature is known, the rate of the front propagation can be derived from eqn. 2.43b. In the real region of the parameters' values when the front is propagating in the direction of gas filtration, the 3rd summand proportional to ~/~o in eqn. 2.49 may not, as a rule, be taken into account, so, in fact, two mechanisms of heat transport (inter-phase heat exchange and heat conductivity along the catalyst bed) produce an additive effect upon the front's characteristics. It follows from eqn. 2.49 that in the absence of the longitudinal heat conductivity in the catalyst bed (A= 0), the front can propagate only in the direction of gas filtration (uJ> 0). This deduction is in accord with the statement of Theorem 2.

r

123

Evaluation of the width of the chemical reaction zone in which almost complete conversion is attained and where in the case of a reversible reaction the equilibrium is practically reached, may also be of interest from a technological point of view. It sounds quite reasonable to determine the width of the chemical reaction zone as the ratio of the maximum temperature jump to the maximum temperature gradient in this zone. In other terms, z-*= (8*-80 ) / B'max. To be more exact, we have determined the time of contact required for the complete (equilibrium) transformation of the substance. The width of the chemical reaction zone is equal to e* = 'iFC *. It follows from eqns , 2.34 and 2.40 and the condition B" = 0 that

.1 + I-cw

I d-o

~w f-E;u)

118ad - :

In addition, if we suppose that in some part of the catalyst bed wi th the maximum temperature gradient a maximum gradient of the extent of conversion is realized (x" = 0), then for a reaction of the first order we can derive a better estimation than that of eqn. 2.50

==(t-c,w)(e*-8) (- IJ?nJ1f/(Bi ) ] '

o

Wi th De; =

00

'r: *<>f,(t-(;w)(8

when

l

1f.,{Bj(I-C)

CfTl/ (r) -=

*- ( 0 )

B/J

I

8g =8

(2.51)

*"

1 this yields the simple expression

1f.:( 8 *)

tt

2(

B * )(f - e )

Though in eqns. 2.51 and 2.52, strictly speaking, one should make use on the temperature values in the part of the maximum gradient, these estimates will do.

124

On the basis of the obtained dependences of the major characteristics of the chemical reaction front upon the system's parameters, one can think of some qualitative estimations about the behaviour of the reaction front. In this connection it is expedient to consider the following problem: find the value of this or that parameter with which a standing wave regime is realized (w= 0). From eqns. 2.49 and 2.43 it follows that in such a case (2.53)

Fig. 2.3 shows the corresponding qualitative dependences, derived from eqn. 2.53. In addition it should be noted that the maximum temperature and the propagation rate of the reaction front are monotonically dependent on the parameters cx,o' .A, nOaa and S : (B *);., > 0 ; (8*); <0; (8*)48 >0; sp , '" , ad, (8*)'Ssp <0-, w'>O; W >0; W <0; coSsp <0. d, .A t.8ad The dependence of the dimensional rate of the front propagation on the filtration velocity is not monotonous and has a negative minimum, whereas (e*)~ > o. With oc =
125

A

W
--------

o

Fig. 2.3. Regions of the values of the parameters determining the intensity of heat exchange, ~o, of the longitudinal heat conductivity, J,., and the adiabatic heating of the mixture, /1 9 d' in which the heat front is moving in the direction of the gasa fil tration (w > 0) and in the opposite direction (w < 0).

brings us to the important technological idea that in reactors wi th fixed catalyst beds under conditions of low inlet temperatures and weak adiabatic heatings of the reaction mixture, conditions can always be found which allow an high enough maximum temperature to be reached in the non-steady-state regime. This temperature, providing an higher rate of the chemical transformation, will be attained over a small area of the catalyst bed. Practical limitations of the maximum temperature are connected only with the value of the hydraulic resistance of the catalyst bed. The estimates of the maximum temperature can be used for the determination of the essential and insignificant influences of various factors on the dynamic regimes in the adiabatic catalyst bed. The optimum pellet size and some other characteristics can be also determined through the use of the maximum temperature. For example, from eqn. 2.47 one can obtain the relative difference of the maximum temperatures in the reaction front at cx,o = ex:> and o:a < 00

B*Jcx;o=oo- 8*jcx;
8*1a,==

+

F(8*1£x'
126

Then, the assumption that the relative temperature differential is insignificant yields the condition the validity of which permits one to ignore the inter-phase heat exchange f [

T

f-

o *1

] e + F( s * I
IiBad,

a,

<

00 )

0

where B determines the value of the admissible error (2 ~ 0,1). One can often make use of a more simple estimate if GO> 0

"»?

'2 J

[TV

!J Bad -~-(8--'-8--'-)-.1-0---(;-)

Similarly, as a condition which avoids taking account of the impact of the longitudinal heat conductivity through the catalyst bed carcass, the following simple estimate is convenient

It must be noted that the obtained evaluations of the major characteristics of the exothermic reaction front are approximations and so there are some drawbacks. Nevertheless, numerical investigations have revealed their ability to describe the process quantitatively over a wide range of values of the real parameters. When added to the theorems mentioned, a clear qualitative picture of the reaction front propagation along the fixed catalyst bed can be drawn. Once again, the major technological peculiarities of the heat front of a chemical reaction should be noted: (a) the front of the exothermic reaction exists at low temperatures of the inlet reaction mixture at which the rate of the chemical transformation can be ignored; (b) the difference between the

127

maximum temperature in the front and the inlet temperature of the reaction mixture may be many times greater than the value of the adiabatic heating of the mixture at complete or equilibrium (at maximum temperature) mixture conversion. With the assigned kinetic characteristics and initial concentration of the reaction mixture, the required temperature differential can be created through appropriate selection of the linear velocity and the size of the catalyst pellets which determine the conditions of both internal and external heat and mass exchange and also the value of the longitudinal heat conductivity; (c) the velocity of the heat front propagation is much smaller than the velocity of the reaction mixture migration in the zone of contact (the filtration velocity); (d) lower extents of internal and external heat exchange between the void volume of the bed and the catalyst pellet and also increased longitudinal heat conductivity result in a decrease in the maximum temperature and in the rate of the front propagation and at the same time lead to an expansion of the reaction zone; (e) an increase in the rate constant of the chemical reaction and a decrease in the activation energy lead to a reduced maximum temperature and to a decrease in the front propagation rate; (f) the increase in the initial concentration of the reactant in the inlet reaction mixture boosts the front's maximum temperature and considerably slows the rate of heat front propagation. Bringing this chapter to a close, it is expedient to do a retrospective survey of the development in the theoretical studies of the reaction zone propagation process in a fixed catalyst bed. The investigations can be roughly classified into two groups. The first group deals with a numerical analysis of the corresponding systems of differential equations. Positive results were obtained in refs. 11 and 12, with a cellular model. The authors investigated, in particular, the influence of radiation on the process of the reaction zone propagation. It was found that radiation could cause a widening of the reaction zone, while the front propagation rate and the maximum temperature were typically decreased. The reaction zone propagation on the basis of a two-phase model of the catalyst bed taking account of the longitudinal heat conductivity in the solid phase was investigated in ref. 16. However, it was only in

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ref. 17 that the two-phase model was provided with the components of the conductive transport in the gas phase and satisfactory numerical analysis was made possible. In particular, it was found that, in the region of the system's parameters such as linear velocity, the coefficient of the effective longitudinal heat conductivity in the solid phase, the inlet gas concentration and the temperature, there is a region of values where the front propagation rate is equal to zero. This phenomenon, however, has not received experimental confirmation. Ref. 17 which was published almost at the same time as refs. 18 and 19 is abundant in calculational results and supports the conclusions of the author of this book relating to the propagation of the reaction zone in the catalyst bed. A considerable amount of computation was done and reported in ref. 1. It has to be admitted that the analysis of the reaction front by means of numerical methods has been confined to a catalyst bed of a certain known length, whereas the notion of the reaction front has an asymptotic character and, strictly speaking, it must be regarded as existing only in an endless catalyst bed. That is why any conclusion on the reaction front behaviour solely on the basis of numerical calculation should be made with care. The second group of work shows some qualitative results of the analysis performed for the system of differential equations reflecting the non-steady processes in the fixed bed. These results are, in practice, limited to the determination of the one-to-one correspondence of the maximum temperature in the reaction front and the rate of its propagation. This connection was eVidently noticed first in ref. 4. The empirical formula for the reaction front propagation rate reported in refs. 11 and 12 cannot be applied for qualitative analysis. It has no physical meaning and has a restricted area of application. The theory of propagation of travelling waves was first propounded in combustion and biology. In heterogeneous catalysis this phenomenon was observed much later. The results laying the foundation for the development of the theory are found in the works of Zeldovich (ref. 14) and Kolmogorov et ale

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(ref. 20), which are rated today as classics. The theory of wave processes is rapidly developing nowadays in biology and in the theory of combustion. A substantial survey devoted to the present status of the mathematical theory of these processes is detailed in refs. 21 and 22. Application of the results of wave theory to increase the efficiency of similar processes in heterogeneous catalytic reactors does not seem possible, because the dynamic properties of a fixed catalyst bed are largely determined by the processes of the inter-phase heat and mass exchange, the vast difference in heat capacities between the solid and the gas phases and filtration of the reaction mixture through the catalyst bed. These factors taken together have nothing in common with the description of biological structures or in the theory of combustion.

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