1.2 Measurement of Rate of Reaction: Why? 1.3 Steps in Catalytic Reactions. 2. Kinetics in Relation to Thermodynamics and Transport Phenomena. 2.1Kinetics ...
Short Reference on
Kinetic Modeling of Catalytic Reactions
Tarek Moustafa, Ph.D. 2002
T. M. Moustafa
Contents 1. Introduction 1.1 Rate of Reaction: Basic Definition 1.2 Measurement of Rate of Reaction: Why? 1.3 Steps in Catalytic Reactions 2. Kinetics in Relation to Thermodynamics and Transport Phenomena 2.1Kinetics and Thermodynamics/Equilibrium 2.2 Kinetics and Transport Phenomena 3. Lab Reactors Used for Measuring Kinetics 3.1 Types of Reactors 3.1.1 Integral Reactors 3.1.2 Differential Reactors 3.1.3 Stirred Contained Solids Reactors (SCSR) 3.1.4 Straight-through Transport Reactors (STR) 3.2 Criteria for Choosing a Lab Reactor 3.3 Precautions in Kinetic Experiments 3.3.1 Radial Concentration and Temperature Gradients 3.3.2 Axial Concentration, Temperature and Pressure Gradients 3.3.3 Interfacial Concentration and Temperature Gradients 3.3.4 Intraparticle Concentration and Temperature Gradients 4. Kinetic Modeling 4.1 Pre-Experimental Phase 4.2 Calculation of Experimental Reaction Rates 4.3 Fitting of Kinetic Parameters 4.3.1 Non Linear Optimization Methods 4.3.2 Discrimination of Kinetic Models 4.4 Future Trends in Kinetic Modeling 5. Comprehensive Examples
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Summary This manual is a short reference dealing with the kinetic modeling of catalytic reactions. It highlights the most important issues concerning kinetic studies and kinetic modeling from a process development prospective. It is driven by the necessity to express quantitatively the chemical reactions in terms of process variables for the sake of reactor design and process optimization. Although each reaction system is unique in its components, mechanism, catalytic species etc., yet the overall approach to handle the kinetic modeling task had some similarities. The objective of this short reference is to unveil the main steps involved in this task. Software is also attached containing complete solution of a comprehensive example for a typical kinetic modeling problem.
1. Introduction In this section, the basic definition of the rate of reaction and the need for having a rate law is introduced. A brief description of transport, adsorption and reaction steps taking place in catalytic systems is also given. The approach taken in this reference is to start with the basic element of kinetics and chemical reaction engineering and to highlight on the important aspects in kinetic modeling for process development purpose. 1.1 Rate of Reaction: Basic Definition The extensive rate of reaction with respect to a species A, rA is the observed rate of formation of A referred to a specified normalizing quantity such as volume of reacting system or volume/mass of catalyst: rA =
moles A formed (unit time)(unit volume)
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The rate of reaction is defined for a species involved in a reacting system either as a reactant or as a product. This rate is positive if A is produced and negative if A is consumed. 1.2 Measurement of Rate of Reaction: Why? The primary use of chemical kinetics in reaction engineering is the development of a rate law (for a simple system) or a set of rate laws (for a kinetic scheme in a complex system). This requires experimental measurement of rate of reaction and its dependence on concentration, temperature, etc. The development of such rate law (which is often referred to as “kinetic modeling’) is the first step to design a reactor for a certain process or to optimize an existing one. Having a rate law, one is able to quantify the effect of various process variables on the reactor performance and consequently on the overall process. 1.3 Steps in Catalytic Reactions The overall process by which heterogeneous catalytic reactions proceed can be broken down into a sequence of individual steps. These are shown in Table 1. The overall rate of reaction is equal to the rate of the slowest step in the mechanism. When the diffusion steps (1, 2, 6 and 7) are very fast compared with the adsorption and reaction steps (3, 4 and 5), the concentrations in the immediate vicinity of the catalyst active sites are the same as the bulk fluid. In this situation, the transport or diffusion steps do not affect the overall rate of reaction. In other situations, if the reaction steps are very fast compared with the diffusion steps, mass transport does affect the reaction rate. As for the surface reaction step, when a reactant (or more) has been adsorbed onto the catalyst active site, it is capable of reacting in different ways. By way of example, this surface reaction may be a single-site mechanism in which only the site where the reactant is adsorbed is involved. The surface reaction may be a dual-site mechanism in which the adsorbed reactant, A·S, interacts with another 4
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vacant site, S, or with a different adsorbed species, B·S to form products C·S. Another example is the reaction of two species on different types of sites, S and S’. These types of reactions are referred to as following Langmuir-Hinshelwood kinetics, and are schematically represented in Fig 1. Another mechanism may involve the reaction between an adsorbed molecule and a molecule in the gas phase, which we refer to as an Eley-Rideal mechanism and is depicted in Fig 2. Table 1. Steps in a Catalytic Reaction No
Step
1
Mass transfer (diffusion) of the reactant(s) (e.g. species A) from the bulk fluid to the external surface of the catalyst pellet
2
Diffusion of the reactant from the pore mouth through the catalyst pores to the immediate vicinity of the internal catalytic surface
3
Adsorption of reactant A onto the catalytic surface
4
Reaction on the surface of the catalyst (e.g. A = B)
5
Desorption of the product(s) from the surface
6
Diffusion of the product(s) from the interior of the pellet to the pore mouth at the external surface
7
Mass transfer of the products from the external pellet surface to the bulk fluid
2. Kinetics in Relation to Thermodynamics and Transport Phenomena To understand a chemical reaction system, it is important to be aware of thermodynamics/equilibrium
and/or
the
transport
phenomena
involved.
Thermodynamics can answer a question such as: what is the maximum conversion for a certain reaction system. Diffusion and mass transfer of reactants in a liquid solvent or solid catalyst may also affect (and sometimes controlling) the overall performance of the system. Heat transfer is also important as chemical reactions are usually accompanied by heat consumption or production. A typical example is oxidation reactions of hydrocarbons.
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B
A
B
A
Single Site Mechanism: A·S ? B·S
A
B
Dual Site Mechanism: A·S + S ? B·S + S
D
C
A
Dual Site Mechanism: A·S + B·S ? C·S + D·S
B
C
D
Dual Site Mechanism: A·S’ + B·S ? C·S’ + D·S
Fig 1. Langmuir-Hinshelwood kinetics
D
B A
C
A·S + B(g) ? C·S + D(g)
Fig 2. Eley-Rideal mechanism
2.1Kinetics and Thermodynamics/Equilibrium Kinetics and thermodynamics address different kinds of questions about a reacting system. Differences between chemical kinetics and chemical thermodynamics can explain the domain of each as follows: (a) Time is a variable in kinetics but not in thermodynamics, e.g. equilibrium is a time-independent state 6
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(b) One is able to get information about mechanism of chemical reaction from kinetic study but not from thermodynamics. Kinetics is concerned with the path of chemical change while thermodynamics is concerned with the “state” and the “change of state” of a system. (c) The Gibbs energy ( G) of reaction, which is a thermodynamic property, is a measure of the tendency for reaction to occur, but it tells nothing about how fast this reaction is. (d) Chemical kinetics is concerned with the rate of reaction and factors affecting rate, while chemical thermodynamics is concerned with the position of equilibrium and factors affecting equilibrium. Nevertheless, equilibrium can be important aspect of kinetics, because it imposes limits on the extent of chemical change (conversion). 2.2 Kinetics and Transport Phenomena Mass and heat transfer may take place in chemical reaction systems. In solid catalytic reaction system, there is an external mass transfer between the bulk gas and the catalyst pellet. There is a difference between the rate by which a chemical component reacts at the surface of the catalyst, ( rA ), and the rate by which it is S
diffused from the bulk gas to this catalyst surface, ( rA ). If it happens that rA > D
S
rAD , we call the situation in this case: “An external mass transfer controlled”, as
the rate of diffusion is a slower step. If the situation is the opposite, it is a surface reaction controlled case. This mass transport may be also internal in the pores of the catalyst pellets. In some cases, the rate of diffusion inside the pellet becomes controlling when it is slower than the rate of surface reaction, i.e. The rate by which the reactant transfers from the gas phase to the inner part of the catalyst pellet is slower than the rate by which it reacts. Typical industrial examples for this case are methane steam reforming and dehydrogenation of ethylbenzene to styrene.
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Heat transport is due to the difference in temperature between the bulk gas and the catalyst pellets or within the pellet itself. This temperature difference is due to heat of reaction, which is generated or consumed at the catalyst surface. Again, it is important to consider this in some cases where heat of reaction is considerably high and/or if the thermal conductivity of the catalyst pellet is low. A clear illustration is given in Fig 3, showing the possible profiles of concentration and temperature in the bulk gas and the catalyst pellet for transport limited cases.
Gas film Catalyst particle surface (s)
Bulk gas R
Catalyst particle interior
Exothermic reaction
Ts
CAg CA
CAs
Tg
Endothermic reaction
T
Fig 3. Concentration (CA) and temperature (T) gradients in a porous catalyst pellet
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3. Lab Reactors Used for Measuring Kinetics 3.1 Types of Reactors There are a lot of options while choosing a reactor for kinetic study. Criteria like ease of construction, sampling, analysis of data, isothermality, cost etc. may lead to the superiority of one reactor configuration over the other. For continuous systems, there are mainly four types of reactors used to perform a kinetic study for a given catalyst, and these are given below. 3.1.1 Integral Reactors This is a typical fixed bed reactor type. A schematic is given in Fig 4. It is easy to construct and to operate near the targeted plant conditions. Care should be taken while designing this type of lab reactor to avoid possibilities of transport limitations during the reaction. The possibility of operating such reactor at high conversions makes it attractive option for studying the effect of product inhibition on reaction rates. Having said that, this reactor configuration may have poor performance for certain catalytic systems. By way of example, achieving isothermality may be a problem in this reactor along the bed length. 3.1.2 Differential Reactors This reactor consists of a tube containing a very small amount of catalyst. A typical schematic is given in Fig 5. Because of this small amount of catalyst the conversion of the reactants is extremely small and consequently the change in the concentration of the reactants through the bed. As a result the concentration of the reactants are assumed to be constant and the reaction rate is considered constant through the bed. This makes the data analysis simple if compared to the integral reactor case. Owing to low conversion, this reactor can operate essentially in an isothermal manner but care should be taken to avoid channeling or bypass of reactants. Ensuring sampling accuracy may be also difficult for multicomponent systems due 9
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to low conversion. The effect of product inhibition is not also felt in the reaction rate, as the products’ concentration is always very low due to low conversion. This drawback can be overcome by introducing products in the feed stream.
Inert material CA
CAo
CA
CAo Catalyst
Fig 4. Integral Reactor
Catalyst
Fig 5. Differential Reactor
3.1.3 Stirred Contained Solids Reactors (SCSR) A typical design for this reactor is shown in Fig 6. Catalyst particles are contained in paddles that rotate at sufficiently high speed to minimize external mass transfer effects and to keep the fluid contents well mixed as well. Isothermal condition can be maintained and data can be analyzed based on outlet concentrations measured. If the catalyst particle size is small, difficulties could be encountered containing the particles in the paddle screens. 3.1.4 Straight-through Transport Reactors (STR) This type of reactor configuration is used to study kinetics when there is a considerable decay in the catalyst activity with time. One example of this catalyst deactivation is coke formation as in zeolite catalysts. A schematic of this configuration is given in Fig 7.
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CA
Catalyst
Catalyst Feed
CA CA
Fig 6. Stirred Contained Solids Reactor
Fig 7. Straight-Through Reactor
3.2 Criteria for Choosing a Lab Reactor The ratings of various reactors discussed above are summarized in Table 2. Depending on the catalyst and the reaction system, one can determine the appropriate configuration. In some cases more than one reactor type can be used. Table 2 Summary of Reactor Ratings a Reactor Type Integral
Sampling & Analysis G
Differential SCSR STR a
P-F
Fluid-Solid Contact F
Decaying Catalyst P
Ease of Construction G
P-F
F-G
F
P
G
G
G
F-G
P
F-G
F-G
P-F
F-G
G
F-G
Isothermality
G, good; F, fair; P, poor.
3.3 Precautions in Kinetic Experiments In the design and/or selection of lab scale reactor to study kinetics, it is important to avoid transport resistances. Since the most frequently used in our labs in SABIC
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are tubular reactors (either differential or integral) it is important to highlight the precautions that should be taken to perform kinetic studies using these reactors and to satisfy the criteria for each precaution. Some of these precautions are also valid for other reactor types. Further, it is important to check these criteria (clarified in the following sections) in the given order, as they are physically dependent. 3.3.1 Radial Concentration and Temperature Gradients To avoid radial concentration gradient (wall effects), the ratio of the effective reactor tube diameter (accounting for the thermocouple if it is present inside the tube) to the pellet diameter should satisfy the following criterion. (dt
dTC ) f 10 2d p
Once this is satisfied, radial temperature gradient is avoided if the following criterion is satisfied: 0.4 RTmax 4 d p rmax E (1 + 8Bi w ) dt
er B
Tmax f1 H r d t2
3.3.2 Axial Concentration, Temperature and Pressure Gradients To avoid axial concentration gradient, the following relations should be satisfied for each reacting component: Peai ff
rmax
B
dp
u sCimax
Further, if the following L/dp ratio is satisfied, there will be no axial temperature gradient. L f1 30d p
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The Ergun equation is used to calculate the pressure drop along the reactor and the percentage drop
P P
should be minimum (preferably less than 3%) to avoid
variation of total pressure: P=
) 2 µL
aG (1 g
d p2
3
+
bG 2 (1 ) L 3 gdp
where a and b are constants and depend on the shape of the catalyst pellets. 3.3.3 Interfacial Concentration and Temperature Gradients The interfacial concentration gradient over the external film (between the catalyst and the bulk gas) should be examined. The J-factor correlation can be used to estimate the mass transport coefficient. The partial pressure gradient and consequently the percentage pressure drop along the external gas film for each component should be calculated. The later should be minimum. The equations are given as follows: For Reynolds number = dpG/µ > 350 JD =
k g i MP
µ
G
g
2/3
= 0.99
Di
d pG
0.41
µ
For Reynolds number = dpG/µ < 350 JD =
k g i MP
µ
G
g
Di
2/3
= 1.82
d pG
0.51
µ
and the pressure difference will be: pi =
rmax k g i am
For interfacial temperature gradient there is an analogous correlations. For external heat transfer calculation and for Re > 350:
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JH =
2/3
C pg µ
hf C pg G
= 0.99
d pG
0.41
µ
g
And for Re < 350, JH =
2/3
Cpg µ
hf Cpg G
= 1.82
d pG
g
0.51
µ
and the temperature difference will be: T =
rmax H r h f am
3.3.4 Intraparticle Concentration and Temperature Gradients The Weisz-Proter correlation should be used to examine the existence of intraparticle mass transfer limitations. To ensure that the measured kinetics is truly intrinsic and free from internal mass transfer limitations, the following parameter should be less than 1.0: CWP =
rmax RTmax
p
d p2
4 Deff i pi s
To ensure that there is no significant temperature gradient inside the pellet, the criterion given by Parter for the temperature difference within a catalyst pellet (between the center and the surface) should be minimum: Tparticle =
De pi s H r RTmax
s
Note: Temperature gradient is less likely to occur for unsupported metal and metal oxide catalysts due to high thermal conductivity. 4. Kinetic Modeling 4.1 Pre-Experimental Phase To perform kinetic modeling for a catalytic reaction, it is very important to start by (1) the deep understanding of the underlying chemistry. This is an initial and 14
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important step. (2) Writing proper mechanism(s) based on detailed elementary steps taking place in the catalytic system should be the following step in the kinetic modeling task. The exact mechanism taking place on the catalyst surface is not usually known. This is due to the complexity of the catalytic reactions and the fact that it is very hard (or even impossible) to trace out the adsorbed intermediates and/or activated complexes formed during the reaction. Also the nature and type of active sites are not really known particularly for complex catalysts’ systems composed of different species. This is why one usually has more than one possible mechanism and consequently more than one rate law. It is then the task to discriminate between these mechanisms as will be described in the coming sections. The following step is (3) the design of experiment. This includes (a) the choice of reactor type suitable for the reaction system to be investigated. It also includes (b) the determination of the range of variables to be studied e.g. partial pressures, temperature, and finally (c) the number of experiments needed to be performed including repetitions. It is advisable that the variation of variables should be around the typical industrial conditions. Also, the number of experiments should be large enough to ensure the statistical significance of the estimated kinetic parameters on a later stage. It is worth noting that simplified power law can fit the kinetic data in a certain range, yet they don’t reflect the real physical situation of the chemical system. Further, power law forms can fail in case of reaction rates having non-monotonic behavior, which may result form different reasons one of which is product inhibition. Examples reported in the literature for non-monotonic rate behavior are in aromatics hydrogenation, steam reforming, ethylene partial oxidation to ethylene oxide, etc.
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4.2 Calculation of Experimental Reaction Rates The value of reaction rates can be computed based on the lab reactor configuration used to perform kinetic study. The experimental rate of formation of a certain product in a differential reactor is calculated by dividing the outlet flow rate of this product by the catalyst weight in the reactor bed. The same approach for rate calculation is applied in case of SCSR. Both types usually called “one point reactor” as the rate is not changing with the reactor dimension. In case of integral reactor, the rate cannot be directly evaluated since it is changing with the reactor length. The material balance (ODE, ordinary differential equation) should be written for the component(s) and hence the rate could be calculated by integration either analytically or numerically depending on the form of the rate law. The same approach is applied for the STR, but the material balance in this case should account for the slip between the gas and the solid catalyst (due to difference in flow velocity of gas and solid). 4.3 Fitting of Kinetic Parameters According to the reactor used for kinetic study, the algorithm for estimating the rate parameters may change. In all cases, one should start the fitting task by formulating the objective function. The value of this function needs to be minimized and physically it reflects the difference between the experimental and the calculated rates. For n experiments, the objective function (also called sum of residuals) can be written as:
S=
n
(ri
rˆi )
2
i =1
where
rˆi )(ri i
i
is the measurement error (standard deviation) of the ith data point,
presumed to be known. If the measurements errors are not known, they may all be set to the constant value
i
= 1. As explained in the previous section, if the source
of kinetic data was a differential reactor or SCSR, then the values of rate (i.e. ri)
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are known and the algorithm for finding the kinetic parameters is as shown in Fig 8. If the data source was from integral reactor or STR, then the algorithm should include in integration step and instead of comparing the rates, the modified objective function will compare the outlet composition of certain component(s) or reaction conversion (xi) etc. The algorithm for this case is shown in Fig 9. The objective function will be written as:
S=
n
( xi
i =1
xˆi )( xi
xˆi )
2 i
4.3.1 Non Linear Optimization Methods
Rate laws of catalytic reactions are usually non linear. In rare cases (like when power law forms are used), it is possible to linearize these rate forms and linear regression method (often called least square method) can be used. In most cases linearization is not possible and efficient non-linear optimization method should be used to estimate the kinetic parameters. The non-linear optimization methods can be categorized into two main branches, namely the gradient and the nongradient methods. As appearing from the name, the gradient methods involve the calculation of the partial derivatives of the objective function with respect to the parameters. This is usually done numerically at a certain set of values for these parameters and the result is called the gradient matrix. The matrix of the second derivative with respect to the parameters is called the Hessian matrix, H. S (k p ) Gradient =
k1 . . S (k p )
, p=1,m
km
and
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2
S (k p )
...
2 1
k ... ... 2 S (k p ) ... km k1
H =
S (k p )
k1 km ... 2 S (k p )
, p=1,m
km2
The most famous gradient method used is the Marquardt-Levenberg. This elegant method is based on a combination of the so-called inverse-Hessian method and the steepest descent method. Mathematically, formulae of this method to estimate m parameters is given as follows: m p =1
lp
kp =
S , l=1,m kl
and ll
lp
= =
2
S
kl
2
(1 + )
2
S for (l kl k p
p)
The non-gradient methods (often called search methods) involve less numerical calculations and less preparation of the problem. On the other hand, they are in general converging much slower to the optimum value (minimum) compared with gradient methods. One famous method is Rosenbrock, which is based on iterative procedure. The objective function is calculated every time after changing the value of each parameter. The direction of changing the parameter value is accordingly based on weather the objective function is decreasing (success) or increasing (failure). Both gradient and non-gradient methods are sensitive to initial estimates for the parameters especially for high dimensional problems.
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4.3.2 Discrimination of Kinetic Models As discussed in the previous sections, fitting of experimental data to various available rate laws (obtained from the possible mechanisms) is generally performed using a non-linear optimization routine to minimize the residuals (objective function). There is a possibility after finishing this step that more than one rate law can give close values for the objective function, and in a sense these can all be regarded as “good fitting models”. There are other important criteria that should be satisfied before accepting certain model and not only the value of residuals’ sum. These criteria are thus used to discriminate between models in case there is more than one model well fitting the data. The first criterion is that the kinetic parameters estimated by data fitting should be all positive. Further they should be thermodynamically consistent (e.g. Arrhenius and Van’t Hoff relations should be satisfied) and their values should not violate the basics of the reaction chemistry. That is why it is sometimes advisable to perform a constraint minimization so that to ensure that this criterion is satisfied. The second criterion is the statistical significance of the estimated kinetic parameters. This is critical because usually the amount of uncertainty in the parameters for the hyperbolic models (rate laws having Langmuir-Hinshelwood forms) is typically high. Quantitative measure of the precision of parameter estimates is thus important. F-test and t-test should typically lie within the 95% confidence interval for the estimated parameters. 4.4 Future Trends in Kinetic Modeling Recently, more interest has been given to the quantitative estimates of kinetic parameters using Transition State Theory (TST) and Statistical Mechanics. The TST, which is sometimes referred to as Activated Complex Theory focuses on thermodynamics/statistical considerations and use this to predict how many combinations of reactants are in transition state configuration under reaction
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conditions. By way of example, let’s consider three atomic species A, B and C and reaction represented by: AB + C
A + BC
The TST considers the reaction to take place in the manner: AB + C
ABC‡
A + BC
According to TST, the intermediate is treated as unstable species in equilibrium with the reactants. Using statistical theories leads to the formulation of the reaction kinetic parameter as follows: k =
k BT K c‡ h
Kc‡ can be calculated using statistical mechanics as a function of the so called the partition function (Q‡) which is related to the modes of energy storage of the molecules and the activated complexes (e.g. transitional, rotational and vibrational). Present day quantum chemical packages (e.g. ab initio versions), lead to values for the energy level of the transition state which are in good agreement with the available experimental data. 5. Comprehensive Examples 5.1 The following data were obtained for a catalytic dehydrogenation reaction in a SCSR reactor at 260ºC. Partial pressures are determined at the outlet of the reactor. (i)
Show that the rate law having the following form is fitting the data.
rA =
k ( pA
pB pH 2
) K , with K(260ºC) =2.150 (1 + K A p A + K B pB ) 2
(ii) Estimate the parameters, specify their standard deviation and show their statistical significance. (iii) Plot the data on a parity plot.
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Exp. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
pA 3.487 4.255 4.302 4.315 5.214 5.205 5.185 5.026 5.042 6.205 60184 6.315 6.420 6.587 7.208 7.186 7.333 7.287
pB 0.603 0.652 0.721 0.792 0.686 0.742 0.782 0.963 1.128 0.759 0.952 1.328 1.528 2.050 0.956 1.385 1.489 1.662
PH2 0.603 0.652 0.601 0.502 0.686 0.618 0.537 0.412 0.385 0.759 0.621 0.537 0.621 0.324 0.956 0.648 0.544 0.502
rA 0.770 0.863 0.721 0.715 0.910 0.775 0.796 0.526 0.533 0.823 0.695 0.415 0.408 0.241 0.736 0.435 0.440 0.330
Solution: Description of the “Main Sheet” of the Excel file, Example 1: (a) The given data is tabulated in columns A to F. (b) Column G contains the calculated value for the rate law (Ybar) based on the estimated parameters k, KA and KB in column K. (c) The residuals are calculated in column H, and the sum is given in cell H21. The regression sum of squares is given in cell I21. (d) Keq, number of parameters and number of data points are given in column L, together with the calculated mean of the experimental reaction rates. (e) The square root of the variance of the error in the estimated parameters (T) is calculated in Cell M24. (f) The gradient matrix of the partial derivatives is placed in columns P,Q and R. The differentiation of the predicted rate is calculated numerically in a separate work sheet called “Differentiation”. 21
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(g) The covariance matrix, which is the inverse of the (XTX) is calculated in columns U,V and W. The diagonal elements of this matrix (Cjj) are variances of the estimated parameters. Description of the “Differentiation” of the Excel file, Example 1: (a) Partial differentiation of "Ybar"function with respect to the parameters, k, KA and KB was calculated numerically in columns M, N and O. Second derivatives are in columns P, Q and R Central formula was used for calculation: dy y1 y 1 = dx 2h 2 d y y1 2 yo + y = dx 2 h2
1
(b) Columns from W to AE contain the basic calculation for the Inverse Hessain method while columns AG to AK contain the MarquardtLevenberg method. Both are for demonstration and are not used to estimate the parameters. (c) Small values for “delta bj“ in both methods are indicative of how close the parameter values to the minimum. Description of the “Parity Plot & Results’ Summary” of the Excel file, Example 1: (a) The parity plot for the experimental and calculated rates is presented in the former worksheet. The dashed lines represent the 10% deviation lines. Almost all points are lying between these lines. (b) The Results’ Summary worksheet contains the estimates of the parameters with the upper and lower confidence limits. The t test is also given the same table showing acceptable significance for the estimates. The F-test was also acceptable as given in the second table, and the last table gives the degrees of freedom for regression and residuals respectively.
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Nomenclature External surface are of catalyst per unit mass Catalyst particle diameter Reactor tube diameter Thermocouple diameter Average diffusivity of the mixture
(m2/kg cat) (m) (m) (m) (m2/s)
Deff i
Effective diffusivity of component i
(m2/s)
Di
Mean diffusivity of component i
(m2/s)
E G h hf Hr kB Kc‡ kgi kp 9kp L
Activation energy of the reaction Mass velocity Planks’ constant Interfacial heat transfer coefficient from gas to pellet Heat of reaction Boltzmann constant Equilibrium constant for activated complex formation Interfacial mass transfer coefficient of component i Kinetic parameter Change in kinetic parameter value Catalyst bed length Average molecular weight Total pressure Peclet number of component i in the axial direction (uiL/Dei)
(J/kmol) (kg/m2 s) (-) (J/s K m2) (J/kmol) (-)
(m) (kg/kgmol) (bar) (-)
Partial pressure of component I on the catalyst surface
(bar)
am dp dt dTC De
M
P
Peai pi s
(kmol/s Pa m2)
rˆi
Universal gas constant (J/kmol K) Pressure drop across the bed (bar) Pressure difference for component i between bulk gas and external (bar) catalyst surface Experimental reaction rate Calculated reactions rate
rmax T Tmax Tparticle
Maximum expected rate in the reactor Reaction temperature Maximum temperature inside the reactor Maximum temperature across the catalyst pellet
R P pi ri
(kmol/kg cat s) (K) (K) (K)
Greek letters
µ
B
Gas viscosity Porosity of the catalyst bed Shape factor for catalyst pellet Bulk density of the catalyst
(kg/m s) (-) (-) (kg/m3)
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s er g s
(kg/m3) (J/m s K) (J/m s K) (J/m s K) (-)
References X Fogler, S. (1992) Elements of Chemical Reaction Engineering, 2nd ed., Prentice Hall, New Jersy. X Froment, G. and Bischoff, K. (1990) Chemical Reactor Analysis and Design, 2nd ed., Wiley, New York. X Himmelblau, D.M., (1972) Applied Nonlinear Programming, Mc Graw Hill. X Kitrrel, J.R. (1970) Mathematical modeling of chemical reactions in Advances in Chemical Engineering, Vol 8. X Levenspiel, O. (1999) Chemical Reaction Engineering, 3rd ed., Wiley, New York. X Missen, R.W., Mims, C.A. and Saville, B.A. (1999) Chemical Reaction Engineering and Kinetics, 1st ed., Wiley, New York. X Moustafa, T.M. (1997) Ph.D. Thesis, Faculty of Engineering, Cairo University. X Perry, R.H. and Green, D. (1984) Perry’s Chemical Engineers Handbook, 6th ed., Mc Graw Hill. X Elnashaie, S. and Elshishini, S. (1993) Modelling, Simulation and Optimization of Industrial Fixed Bed Catalytic Reactors, 1st ed., Gordon and Breach. X Press, W., Teukolsky, S., Vetterling, W. and Flannery, B. (1992) Numerical Recipes, 2nd ed., Cambridge University Press.
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