CHE3044F : Reactor Design 1 Klaus Möller Department of Chemical Engineering University of Cape Town March 8, 2012
Contents 1
Mole (or mass) balances 1.1 Chemical Reaction Engineering . . . . . . . . . . . . . . . . . 1.1.1 Generic flow diagram of a chemical process . . . . . . 1.1.2 What do we need to know . . . . . . . . . . . . . . . 1.1.3 Industrial processes . . . . . . . . . . . . . . . . . . . 1.1.4 The Chemical Feed stocks . . . . . . . . . . . . . . . 1.2 Rate of a reaction . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Defining when a chemical reaction takes place . . . . 1.2.2 Defining the rate of reaction . . . . . . . . . . . . . . 1.2.3 The volume in the reaction rate and its relationship to reacting system. . . . . . . . . . . . . . . . . . . . . 1.2.4 The rate equation . . . . . . . . . . . . . . . . . . . 1.3 The general mole balance . . . . . . . . . . . . . . . . . . . . 1.4 Mole balance for batch reactors . . . . . . . . . . . . . . . . 1.5 Mole balance for continuous stirred tank reactor (CSTR) . . . 1.6 Mole balance for a homogeneous plug flow reactor (PFR) . . 1.7 Mole balance for a heterogeneous fixed bed reactor (FBR) . . 1.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 6 6 8 9 10 12 12 13 14 14 15 16 18 19 22 23
2
Reactor sizing for single reactions using conversion
3
Properties and definitions of the reaction rate equation (reaction rate law) (Fogler chapter 3) 47
4
Reaction stoichiometry and the reaction rate law (Fogler chapter 3)
60
5
Isothermal reactor design for single reactions (Fogler chapter 4)
78
6
Isothermal reactor design for multiple reactions (Fogler chapter 6)
79
7
Collection and data analysis (Fogler chapter 5) 7.1 Algorithm . . . . . . . . . . . . . . . . . . . 7.1.1 Limitations on data collected . . . . . 7.1.2 Postulate a rate law . . . . . . . . . 7.1.2.1 Homogeneous reactions : .
80 81 81 82 82
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CONTENTS
CONTENTS 7.1.2.2
7.2
Heterogeneous reactions (those with a catalyst): . . . . . . .
82
7.1.3
Choose the same reactor type that matches the data collected
. . . .
82
7.1.4
Write the rate equation in terms of the data that has been collected .
83
7.1.5 7.1.6
Make simplifications based on good chemical reaction engineering principles 83 Differential analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.1.7
Integral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
7.1.8
Goodness of fit and variance in model parameters . . . . . . . . . . . .
85
batch reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Method of excess . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86 86
7.2.2
Example : Differential method . . . . . . . . . . . . . . . . . . . . . .
86
7.2.3
Example : Integral method . . . . . . . . . . . . . . . . . . . . . . . .
91
7.2.3.1 7.2.3.2
The standard (linearised) analyses . . . . . . . . . . . . . . . Example 5.2 (Fogler) . . . . . . . . . . . . . . . . . . . . .
91 93
Example : Non-linear regression . . . . . . . . . . . . . . . . . . . . .
94
7.2.4 7.3
Method of initial rates
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
7.4
7.3.1 Example (Fogler 5-4) . . . . . . . . . . . . . . . . . . . . . . . . . . . Method of half lives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96 98
7.5
Differential reactors (including CSTR and recycle reactors) . . . . . . . . . . .
99
7.5.1
Mole balances : CSTR (and recycle reactor with rapid recycle) . . . . 100
7.5.2 7.5.3
Mole balances : Tubular and packed bed systems . . . . . . . . . . . . 101 Example : Fogler 5-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8
Developing rate laws from reaction mechanisms and reaction pathways
107
9
Bio-reactor engineering mechanisms
108
10 Bio-reactor design
109
11 Analysis of reactor flow patterns on reactor performance (Fogler chapter 13)
110
2
List of Figures 1.1
Generic flow diagram of a process. (adapted from Schmidt, 2005)
. . . . . .
6
1.2
Phthalic anhydride process. (Fogler, 2006)
. . . . . . . . . . . . . . . . . . .
6
1.3 1.4
The face of chemical reaction engineering. (Fogler, 2006) . . . . . . . . . . . The multidisciplinary nature of chemical reaction engineering. (Schmidt, 2005)
7 8
1.5
The reaction rate in a reactor.
1.6
General mole balance.
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13
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15
1.7 1.8
Batch reactor. (Fogler, 2006) . . . . . . . . . . . . . . . . . . . . . . . . . . Change of moles with time in a batch reactor. . . . . . . . . . . . . . . . . .
16 17
1.9
Continuous stirred tank reactor, abbreviated as CSTR. . . . . . . . . . . . . .
18
1.10 Diagrams of typical plug flow reactors commonly known as PFR’s.
. . . . . .
19
1.11 The PFR mole balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Arbitrary shaped PFR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20 21
1.13 Molar flow rate profiles across the PFR.
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22
1.14 Tubular fixed bed reactor. 7.1
Cubic and polynomial fit to the data.
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89
7.2
Comparing the linear data fit with the non-linear data fit. The green line represents the non-linear fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
Batch reactor integral analysis linear plots for zero, first and second order kinetics using concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
7.3 7.4 7.5
Standard regression plot for the second order reaction. . . . . . . . . . . . . Predicted and experimental data when using the non-linear leasts squares method applied to concentration data. . . . . . . . . . . . . . . . . . . . . . . . . . .
94
7.6
Data for example (Fogler5-4). . . . . . . . . . . . . . . . . . . . . . . . . . .
97
7.7 7.8
Linearised initial rate plot for the dissolution process. . . . . . . . . . . . . . Half-life analysis plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98 99
7.9
Differential reactor operation : concepts. . . . . . . . . . . . . . . . . . . . . 100
7.10 Regression plot for the −rCO dependence on PCO .
96
. . . . . . . . . . . . . . . 104
7.11 Plotting the −rCO dependence on PH2 . . . . . . . . . . . . . . . . . . . . . . . 104 7.12 The prediction of the reaction rate when regressing for all the rate law constants.106 7.13 The prediction of the reaction rate when regressing for k, KH2 in the rate law. . 106
3
List of Tables 1.1
Summary of reactor mole balances.
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7.1 7.2
Reaction rate as a function of concentration using cubic spline and polynomial fitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Experimental reaction data for the conversion of CO to CH4 . . . . . . . . . . 102
7.3
Processed reaction data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4
23
Chapter 1 Mole (or mass) balances
5
1.1. CHEMICAL REACTION ENGINEERINGCHAPTER 1. MOLE (OR MASS) BALANCES
1.1 1.1.1
Chemical Reaction Engineering Generic flow diagram of a chemical process
Figure 1.1: Generic flow diagram of a process. (adapted from Schmidt, 2005)
Figure 1.2: Phthalic anhydride process. (Fogler, 2006) • Reactors are the key unit operation on plant. A plant is typically designed around a reactor. The operation of the reactor determines the operability and performance of the plant. See Fig’s 1.1 and 1.2. • Almost all the worlds commodities and essential items are made using a reactive process. • The reactors can be as simple as a long hollow tube or a complex reaction vessel many 10’s of meters tall with many reactions occurring in multiple phases and with the aid of a catalyst. • It is their knowledge of chemical reaction engineering that makes chemical engineers unique Reactors come in all shapes and sizes and arise in many interesting places, see Fig 1.3
6
1.1. CHEMICAL REACTION ENGINEERINGCHAPTER 1. MOLE (OR MASS) BALANCES
Figure 1.3: The face of chemical reaction engineering. (Fogler, 2006)
7
1.1. CHEMICAL REACTION ENGINEERINGCHAPTER 1. MOLE (OR MASS) BALANCES
1.1.2
What do we need to know
Some facts about practicing chemical engineers • Single reactions never happen. Reactors are complex unit operations with many simultaneous chemical reactions occuring in multiple phases. The engineer needs to know enough about the reactions, the flow patterns in the vessel, the interaction of heat and mass transfer in order to assemble the basic understanding of the reactor operation. • The kinetic rate equation almost never exists. Rate equations must be estimated, approximated from reaction engineering fundamental knowledge and from experimental or plant data. The rate equation indicates how fast a reaction proceeds and thus determines both the size, productivity and selectivity of the reacctor. • Industrial processes are often severely limited by heat and mass transfer. The engineer must know how to integrate these into the reactor design to yield its actual performance. Reactor design typically proceeds according to a well defined path Bench Scale reactor (batch, continuous) → pilot plant → operating plant Very few engineers will have the honour of following this process. Practicing engineers will most likely encounter and old reactor that has been modified many times which the engineer must now operate in such a way to double the through-put, improve selectivity and reduce the waste. Alternatively, reactor has failed and the engineer must understand the reasons of failure and re-commission the reactor to operate better without failing. The typical tasks are • Maintain and operate a process, • Fix a problem, • Increase the capacity or selectivity at minimum cost. The task must be solved as quickly as possible,for there are many more tasks that need doing. Often, gaining a deeper understanding is often not possible due to the presssure of other more pressing tasks. Besides being competent at “back of the envelope” calculations and having a good intuitive feel for reaction engineering, the engineer also needs to be able to build quick practical models of the reaction process by integrating all the other engineering disciplines, see Fig 1.4.
Figure 1.4: The multidisciplinary nature of chemical reaction engineering. (Schmidt, 2005)
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1.1. CHEMICAL REACTION ENGINEERINGCHAPTER 1. MOLE (OR MASS) BALANCES This course is the first course of two in reaction engineering and will aim to develop good reactor engineering skills for optimal homogeneous, isothermal reactor design by combining the knowledge of • kinetic models and their structure, • elementary reactor design equations, • various type of reactors and configuration of multiple reactor sequences, • multiple reactions and nature of the reactions, • experimental data to determine the rate equation. There will essentially be two types of problems 1. The ones that can be solved in 3 lines often without a calculator. These will build good reaction engineering intuition; 2. The ones that require more rigorous calculation and in many cases a computer. These are the design type problems that will exercise the procedural skills and study the “what if” scenario’s.
1.1.3
Industrial processes
The following table shows the interesting contrasts between the large volume producers and high value chemicals produced by the pharmaceutical companies.
(Schmidt, 2005)
9
1.1. CHEMICAL REACTION ENGINEERINGCHAPTER 1. MOLE (OR MASS) BALANCES
1.1.4
The Chemical Feed stocks
The following tables shows the usage of chemicals in the USA in 1994. This ranking of these chemicals isn constantly changing. It provides an interesting incetive for the development of alternative processes.
(Schmidt, 2005) This is in contrast to the prices of chemicals shown in the following table. Note the low value of the crude oil in contrast to solvents and more so pharmaceuticals (see insulin).
10
1.1. CHEMICAL REACTION ENGINEERINGCHAPTER 1. MOLE (OR MASS) BALANCES
11
1.2. RATE OF A REACTION
CHAPTER 1. MOLE (OR MASS) BALANCES
(Schmidt, 2005)
1.2
Rate of a reaction
1.2.1
Defining when a chemical reaction takes place
Chemical species are defined by the type, the number and arrangement of the atoms. For example isomers differ only by the arrangement of the chemical species
These, isomers, although they have the same type and number of atoms, the different arrangement of the atoms results in these molecules having different physical properties and are thus chemically distinct from each other. A chemical reaction has taken place when one of more of the chemical species in the system of study are transformed by an infinitesimal amount into (an)other chemical species. This transformation can be one of the following • Change the arrangement of the atoms : double bond isomerisation :
skeletal isomerisation :
• Decomposition :
12
1.2. RATE OF A REACTION
CHAPTER 1. MOLE (OR MASS) BALANCES
• disproportionation :
• Addition :
• and many more other types of reactions (see organic chemistry hand book!) Often it is simpler to work with letters such as A, B, C, D for chemical species. Thus the reaction of methanol to dimethyl-ether and water written as 2CH3 OH → CH3 OCH3 + H2 O can be written as 2A → D + W or A → 12 D + 12 W
1.2.2
(1.1)
Defining the rate of reaction
Consider that A is reacting in the vessel in Fig 1.5. −rA represents the consumption of A at some position in the reactor vessel and it is defined as follows.
Figure 1.5: The reaction rate in a reactor. Definition of reaction rate : The number of moles of A that are being reacted (consumed or produced) per unit volume per unit time [mol.m−3 .s −1 ]. 13
1.2. RATE OF A REACTION
CHAPTER 1. MOLE (OR MASS) BALANCES
Definition of rA : rA represents the rate of formation of species A. Thus −rA represents the rate of consumption of A. Thus rA is positive for a product and negative for a reactant. Reaction stoichiometry : In reaction 1.1, the consumption of one mole of A will yield 0.5 moles of D and 0.5 moles of W. For example if the rate of consumption of A is -5 mol.m−3 .s −1 then rA = −5 or −rA = 5 and rD = 2.5 and rW = 2.5. Similarly the rate equations for A, rD rW D and W are related by the reaction stoichiometry: −rA = 2rD = 2rW = 1 = 1 . /2 /2 It is good practice to decide on a convention and stick to it. This course will work with the definition of rA as given above.
1.2.3
The volume in the reaction rate and its relationship to the mass of the reacting system.
Homogeneous reactors well mixed : When the vessel is well mixed such that the concentration of A is equal trhough the reacting vessel, the volume represented by the m3 term refers to the vessel volume. In gas phase systems the actual volume of the vessel is used. In liquid phase systems, if the vessel is not completely filled, the liquid volume is used. Homogeneous reactors with concentration gradient : When the species concentrations vary throughout the reactor volume, the volume refered to in the rate equation is that of a differential element in which it can be assumed that the concentration is uniform for all species (i.e. well mixed). Heterogeneous reactors : In these reactors the reaction takes place on a catalyst surface (or a solid surface or within the catalyst pores). The volume refered to in the rate equation is then the volume of the catalyst. Since the density of a catalyst is constant, it is easy to convert the rate equation into a mass basis using the catalyst density and then the rate equation is defined as moles of A reacting per mass of catalyst per time [mol.kg −1 .s −1 ] and is denoted as rA0 . (Note that the activity per unit surface area can similarly be converted into a mass basis by using the catalyst properties which define the surface area to mass ratio. Multi-phase reactions : If the reaction contains a vapour and a liquid phase, then there will be a rate equation for each phase, rAv ap and rAliq and the volume refered to will be the vapour phase volume and the liquid phase volume respectively. Similarly, for more phase including solids and catalysts.
1.2.4
The rate equation
The rate equation rA = f uncti on(Ci , T, P, Ccataly st ) is a function of the concentration (activities) of all or some of the chemical species in the reacting system, or the temperature and pressure of the system and the properties of the catalyst. The rate equation is an algebraic expression defining how the reaction rate varies with the chemical and physical properties of the reacting system. NOTE : The rate equation is independent of the reactor in which the reaction is being carried out in. 14
1.3. THE GENERAL MOLE BALANCE
CHAPTER 1. MOLE (OR MASS) BALANCES
Examples of possible rate equations for the reaction A → P r oducts • Linear variation of reaction rate with concentration : rA = −kCA • Quadratic variation of the reaction rate with concentration : rA = −kCA2 • Saturation of the reaction rate : rA =
−k1 CA (1 + k2 CA )2
dCA NOTE : Although rA = is true from the definition of the units of the reaction rate, this dt equation is actually the design equation for a constant volume batch reactor, which will be developed later in this course. It is not the definition of the rate equation.
1.3
The general mole balance
Consider a mole balance of species j in a arbritary shaped reaction vessel (or control volume) in Fig 1.6. , For example in the reaction A → pr oducts A would represent j. It is not necessary the know what the other products are at this stage. Also species j can be a reactant of a product (i.e. sometimes products are fed to the reactor as a result of recycle)
Figure 1.6: General mole balance. The mole balance then can be written as
rate of flow of jINTO the system − [mol.s −1 ]
IN − OUT + Gener ati on = Accumulati on rate of flow rate of generation rate of accumulation of jOUT of of jby chemical of jwithin = + the system reaction in the system the system [mol.s −1 ] [mol.s −1 ] [mol.s −1 ] 15
1.4. MOLE BALANCE FOR BATCH REACTORS CHAPTER 1. MOLE (OR MASS) BALANCES and in symbols this is dNj (1.2) dt where Nj is the total number of moles of j in the system. Gj represents the total transformation of j that has occurred when passing through this reaction vessel. Gj can be related to the reaction rate for two special cases Fj,0 − Fj + Gj =
1. When the concentration of all species is constant throughout the entire system volume : Then the reaction rate rj is the same throught the whole reaction vessel and Gj = rj V where V is the volume of the vessel. The general mole balance, equa 1.2 then becomes dNj Fj,0 − Fj + rj V = (1.3) dt 2. When the concentration of all species varies throughout the reaction volume : Then the reaction rate will be different at all locations. Dividing the system volume into differential elements (see Fig 1.6), ∆Vi and assuming that the concentration of all species is constant in each differential element, means that the reaction rate is constant on each differential element. Thus the rate of generation of species j in each element is given by rj,i ∆Vi . The rate of generation of species j in the system is thus the sum of all the differential elements M X : Gj = rj,1 ∆V1 + rj,2 ∆V2 + rj,3 ∆V3 · · · = rj,i ∆Vj . In the differential limit, as ∆V → 0 and i=1
M → ∞ then Gj =
RV
rj dV . The general mole balance in this case becomes ZV Fj,0 − Fj +
rj dV =
dNj dt
(1.4)
Equation 1.4 is the GENERAL DESIGN EQUATION for chemical reaction engineering. When used on its own, it is applicable to all ISOTHERMAL reaction systems. For NON-ISOTHERMAL reaction systems an ENERGY BALANCE will also be needed in order to solve equation 1.4.
1.4
Mole balance for batch reactors
Figure 1.7: Batch reactor. (Fogler, 2006) 16
1.4. MOLE BALANCE FOR BATCH REACTORS CHAPTER 1. MOLE (OR MASS) BALANCES Fig 1.7(A) shows a schematic diagram of a industrial batch reactor. Its a vessel with a lid so that it can be opened for cleaning. It has ports for adding and removing chemicals. It almost always has a stirrer (agitator) and they are almost always used for liquid phase operations. There are coils or a jacket around the outside of the vessel for heating and cooling. Some vessels have coils internally, others use evaporation cooling or direct steam heating. The vessel is always raised off the ground. Fig 1.7(B) shows the text book schematic of a batch reactor as a vessel filled with liquid and a stirrer to provide perfect mixing, such that the concentration off all species in the reactor are the same everywhere in the vessel. The reactor volume is that of the reactive mixture, in this case the liquid volume and NOT the volume of the vessel. A batch reactor is operated by filling the vessel with the required reactants. The reactor is then heated to the desired reaction temperature and the reaction is allowed to proceed for a predetermined amount of time, called the reaction time. During this time there is no flow into or out of the reactor. An the end of the the reaction time the mixture is removed (pumped...) out of the vessel. The reactor is then cleaned and the cycle starts again with fresh reactants. Since there is no flow into or out of the vessel, the general mole balance on species j becomes ZV rj dV =
dNj dt
Assuming that the system is well mixed, the reaction rate equation will be the same through the whole vessel, thus the design equation is dNj = rj V dt
(1.5)
Notice that the general equation, equa 1.5 is based on the number of moles in the vessel and not the concentration. More on this later. For a reaction A → B the number of moles of A decrease with time while the number of moles of B increase with time, as shown in Fig 1.8.
Figure 1.8: Change of moles with time in a batch reactor.
17
1.5. MOLE BALANCE FOR CONTINUOUS CHAPTER STIRRED TANK 1. MOLE REACTOR (OR MASS) (CSTR) BALANCES The time t1 that is required to consume (NA0 − NA ) moles can be obtained by integrating equa 1.5: Z NA1 dNA t1 = (1.6) NA0 rA V where NA0 are the number of moles at the start of the reaction, t = 0 and NA1 are the number fo moles of A that are remaining after a reaction time of t1 .
1.5
Mole balance for continuous stirred tank reactor (CSTR)
(A)
(B) Figure 1.9: Continuous stirred tank reactor, abbreviated as CSTR.
Fig 1.9(A) shows a schematic diagram of a continuous stirred tank reactor (CSTR). These reactors always have a stirrer or some form of vigorous agitation to ensure that the composition of all species is uniform throughout the reaction system. They also have a heating/cooling jacket of coils to control the temperature in the vessel. They can be closed vessels or open vessels. They can be filled to capacity or have partially filled with or without two phase operation. These reactors are most often used for liquid applications involving poorly miscible streams and/or solids. They are also used for systems which require the additiion of another phase e.g. like bubbling a reactive gas through a liquid reactant. Fig 1.9(B) shows the diagramatic way in which a CSTR is represented for calculation purposes. The mole balance for species j across this vessel is ZV Fj,0 − Fj +
rj dV =
dNj dt
If the reactor is operated at steady state, the there is no variation with time of all species on dNj the reactor, thus, = 0 and the mole balance becomes; dt ZV Fj,0 − Fj +
18
rj dV = 0
1.6. MOLE BALANCE FOR A HOMOGENEOUS CHAPTER PLUG1.FLOW MOLE REACTOR (OR MASS) (PFR) BALANCES In addition, since there are no spatial variations in the concentrations as the contents of the vessel is well mixed, the reaction rate, rj is constant throughout the whole vessel, which means RV that rj dV = rj V and the mole balance then becomes; Fj,0 − Fj + rj V = 0 Fj0 − Fj V = rj
(1.7)
This is the final design equation for the CSTR. It is important to note that the concentration of the species in the reactor have the same concentration as the species leaving the reactor in the exit flow.
1.6
Mole balance for a homogeneous plug flow reactor (PFR)
Figure 1.10: Diagrams of typical plug flow reactors commonly known as PFR’s. A plug flow reactor, Fig 1.10 is basically an empty tube through which a reactive fluid flows. The reactive streams are mixed at the reactor inlet. The reactor can be heated or cooled depending on the reaction requirements by using a jacket, by making steam on the outside of the tubes or coils in the vessel. The important aspect of these reactors is that they are extremely long in relation to their diameter, as it is best to have turbulent flow. This is to promote plug flow. Plug flow is defined as flow in which there is no axial mixing (no mixing along the length) by good radial mixing (mixing perpendicular to the flow). This means that all the species that enter the reactor spend exactly the same amount of time in the vessel. Species interact only radially and not axially. Plug flow reactor can be mounted inside furnaces for very high temperature operation.
19
1.6. MOLE BALANCE FOR A HOMOGENEOUS CHAPTER PLUG1.FLOW MOLE REACTOR (OR MASS) (PFR) BALANCES
Figure 1.11: The PFR mole balance. In developing the general mole balance for a PFR, consider the schematic in Fig 1.11. The general mole balance can be applied to the whole reaction vessel, thus for species j it is ZV Fj,0 − Fj +
rj dV =
dNj dt
Since the species concentrations (or molar flow rates) are not constant, it is not possible to simplify the integral expression. Furthermore, when more than one reaction is taking place, the reaction rate term, rj does not only depend on species j but can also depend on the other species in the reaction vessel, preventing the integration from being carried out. Thus in this case the general mole balance in this form is not particularly useful. Assuming steady state dNj = 0. Then differentiating with respect to the reactor volume yields makes dt dFj,0 dFj − + rj = 0 dV dV and since the feed does not depend on the reactor volume, thus dFj = rj dV
dFj,0 = 0 and thus dV (1.8)
which is the general mole balance for a PFR. Note that this is a differential equation. The reactor mole balance can be obtained in a more intuitive way by considering a differential element, in which it can be assumed that the species concentrations are constant and thus the reaction rate is also constant. The differential mole balance over the differential element is then
molar flow rate of species j − IN at V −1 [mol.s ]
IN − OUT + Gener ati on = Accumulati on Molar rate of rate of accumulation molar flow rate of species j of species j + generation of = within ∆V OUT at V + ∆V species jin ∆V −1 −1 [mol.s −1 ] [mol.s ] [mol.s ]
which in symbols, for a reactor at steady state becomes Fj |V − Fj |V +∆V + rj ∆V = 0 Dividing through by ∆V and re-arranging yields Fj |V +∆V − Fj |V = rj ∆V 20
1.6. MOLE BALANCE FOR A HOMOGENEOUS CHAPTER PLUG1.FLOW MOLE REACTOR (OR MASS) (PFR) BALANCES Fj |V +∆V − Fj |V Since the flow rate us a function of V in the sense that F = f (V ) the term ∆V represents an approximation to the dereivative of F with respect to V . In the limit as ∆V → ∞ the PFR mole balance becomes dFj = rj dV as before. IMPORTANT NOTE : for plug flow conditions, the shape of the reactor does not matter and is arbitrary (in the derivation volume was used with no mention of the shape of the vessel). Thus Fig 1.12 could also have been used for the derivation of the mole balance. This is no longer true once other factors are accounted for, on in particular is pressure drop, which depends on the velocity, which would vary with cross-setional area.
Figure 1.12: Arbitrary shaped PFR. Consider again the reaction A → B but this time in a PFR. The molar flow rate profiles across the reactor as shown in Fig 1.13. These are similar to the batch reactor, except that volume has replaced time.
Figure 1.13: Molar flow rate profiles across the PFR. The design equation for this reaction is dFA = rA dV
21
(1.9)
1.7. MOLE BALANCE FOR A HETEROGENEOUS CHAPTER FIXED 1. MOLE BED REACTOR (OR MASS) (FBR) BALANCES To determine the volume of reactor required to achieve the exit flow rates FA1 and FB1 in Fig 1.13 can be obtained by integrating equation 1.9 from the beginning of the reactor where V = 0 and FA − FA0 until V1 where FA = FA1 ; Z V1 Z FA1 dFA dV = V1 = rA 0 FA0
1.7
(a)
Mole balance for a heterogeneous fixed bed reactor (FBR)
(b) Figure 1.14: Tubular fixed bed reactor.
Fig 1.14(a) shows a fixed bed reactor with the catalyst packed in the tubes and coolant flowing around it (sometimes steam is made in this way). This configuration is used if the reaction is very exothermic (produces a lot of heat) and/or the reactants or products are heat sensitive and/or the catalyst deactivates at higher temperatures and/or the catalyst selectivity deteriorates and it is crucial to keep the reactor as close to isothermal operation as possible. For other reactions where the exothermicity can be tolerated or the reaction is near thermo-neutral, the vessel can simply be packed with catalyst particles as a uniform bed. The particle size and the bed diameter are key design parameters. The isothermal vessels are typically long and thin with a relatively small vessel : particle diameter ratios (20-40). Adiabatic vessels are typically short and squat with large vessel : particle diameter ratios (>>100). Mass and heat transfer in the catalyst particle to/from the active site becomes more difficult as the particle size increases. In contrast, the pressure drop across the bed increases as the particle size decreases. These opposing effects need to be optimised during the reactor design phase. The interplay of heat and mass transfer in catalytic reactions is a leads to exciting reaction systems. More of this will be the topic of reactor design 2. In this first level mole balance, it will be assumed that heat and mass can be neglected and that pressure drop can be ignored and that the flow in the vessel is perfect plug flow. The reaction rate is thus based on the catalyst mass (or catalyst volume), since it is the catalyst that is responsible for the reaction that is taking place. (Vessel volume and catalyst volume can easily be used or converted to, through the density of the catalyst particles). Thus the reaction rate 22
1.8. SUMMARY
CHAPTER 1. MOLE (OR MASS) BALANCES
is defined as −1 rA0 = [(mol A r eacted).s −1 .gcataly st ]
Fig 1.14(b) shows the differential element used in the mole balance; IN − OUT + Gener ati on = Accumulati on Fj |W − Fj |W +∆W + rj0 ∆W = 0 Dividing through by ∆V and re-arranging yields Fj |W +∆W − Fj |W = rj0 ∆W Fj |W +∆W − Fj |W Since the flow rate us a function of W in the sense that F = f (W ) the term ∆W represents an approximation to the dereivative of F with respect to W . In the limit as ∆W → ∞ the PFR mole balance becomes dFj = rj0 (1.10) dW and the catalyst weight required to achieve a exit floe rate of FA is given by Z FA dFA W = 0 FA0 rA
1.8
Summary
Table shows summary of all the mole balances carried out in this chapter. Table 1.1: Summary of reactor mole balances. reactor Comment Mole balance differential algebraic integral Z NA dNA dNA = rA V t= Batch no spatial variation dt NA0 rA V FA0 − FA CSTR no spatial variation, steady state V = −rA Z FA dFA dFA PFR steady state = rA V = dV r ZFA0FA A dFA dF A FBR steady state = rA0 W = dW FA0 rA (Fogler, 2006) These represent the definite equations of the reactor design 1 course and for that matter all reaction engineering design problems can be solved using these equations. The rest of this course is on the application of these equations to specific problems.
23
Chapter 2 Reactor sizing for single reactions using conversion
24
Chapter 3 Properties and definitions of the reaction rate equation (reaction rate law) (Fogler chapter 3)
47
Chapter 4 Reaction stoichiometry and the reaction rate law (Fogler chapter 3)
60
Chapter 5 Isothermal reactor design for single reactions (Fogler chapter 4)
78
Chapter 6 Isothermal reactor design for multiple reactions (Fogler chapter 6)
79
Chapter 7 Collection and data analysis (Fogler chapter 5)
80
7.1. ALGORITHM CHAPTER 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) This chapter develops some of the techniques that are used to obtain rate laws for chemical reactions, which are then ultimately used in the design of a pilot plant and ultimately a commercial unit. THE KNOWLEDGE AND TOOLS GAINED HERE ARE EXTREMELY IMPORTANT. The purpose of this chapter is to develop the following knowledge 1. To be able to estimate the reaction rate constant, the reaction order and other constants in the rate law from appropriately collected experimental data. 2. To be able to study experimental data and decide how it should be analysed 3. To be able to develop the appropriate reactor design equation and rate law that will describe the experimental data 4. To be able to formulate the design equation in the appropriate form such that it is suitable for regression with experimental data to determine the rate law parameters 5. To be able to formulate the objective function and use appropriate computer code to carry out the non-linear regression 6. To be able to select the best model based on good statistical methods.
7.1
Algorithm
The following steps are general and apply to all the examples and reactor types that follow.
7.1.1
Limitations on data collected
The following are guidelines when collecting data or looking at data to be analysed 1. Make every effort to ensure that the data is collected under isothermal and isobaric conditions 2. Make sure that the data are collected at steady state and that the MASS BALANCE IS OBEYED. 3. Make every effort to ensure that the data collected is free of transport limitations such as film resistance, pore diffusion. (see later chapters) 4. Make every effort to ensure that the idealisations of the assumed reactor type are obeyed. For example, if a plug flow reactor is assumed, the reactor configuration choosen must behave like a plug flow reactor or else the estimated rate constant will depend on the shape of the reactor and CANNOT BE USED FOR SCALE-UP. For example in a two phase batch reactor (gas bubbles in the liquid), it is necessary that the stirring is fast enough such that the bubbles are small enough to ensure that there are no mass transfer limitations. 5. Make every effort to ensure that there are no unexpected concentration and temperature gradients. For example, in a flow reaction bomb (CSTR) in which a combustion reaction is taking place, gfeat care needs to be taken to ensure that the mixing (stirring) is fas enough to ensure that there are NO temoperature and concentration gradients in the vessel. The same applies to fast liquid phase reactions in a batch reactor, the stirring must be fast enough to ensure complete mixing at all times during the reaction. 81
7.1. ALGORITHM CHAPTER 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) 6. There are ways to analyse non-isothermal and non-isobaric data, but it is considerably more difficult to get estimates of the kinetic constants that are reliable. It is also beyond thre scope of this course. Similarly, non-ideal reactor behaviour can be accounted for, but this also leads to less accurate estimations of the rate constants.
7.1.2
Postulate a rate law
The design and development of rate laws is the topic of the next section (Fogler!!!!). In many cases the rate law needs to be developed in conjunction with the data analysis and experimental observations. Good places to start 7.1.2.1
Homogeneous reactions : mono-molecular : −rA = kCAn = k 0 PAα bi-molecular : −rA =
kCAα CBβ
=k
reversible : −rA = k CA CB −
0
(7.1) PAα PBβ CC CD Ke
(7.2) (7.3)
In particular for reversible reactions it is important that the rate law is able to describe the chemical equilibrium of the reaction(s) i.e. for reaction 7.3 the equilibrium limit is Ke =
CCe CDe CAe CBe
where the Cie are the equilibrium concentrations when rA = 0, and obtained from thermodynamics. 7.1.2.2
Heterogeneous reactions (those with a catalyst): kPA 1 + K A PA PB k PA − Ke LH-reversible : −rA = 1 + K A PA + K B PB
Langmiur-Hinshilwood (LH) : −rA =
(7.4)
(7.5)
Also here the rate law must obey the thermodynamic limitations of the reaction thermodynamics.
7.1.3
Choose the same reactor type that matches the data collected
Batch reactor : −rA =
1 d(NA ) dCA dCA 1 dPA = = = V dt dt dt RT dt
(7.6)
dFA dXA = −FA0 dV dV
(7.7)
FA0 − FA FA0 = XA V V
(7.8)
PFR : −rA = CSTR : −rA =
82
7.1. ALGORITHM CHAPTER 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5)
7.1.4
Write the rate equation in terms of the data that has been collected
e.g. PA v s t, NA v s t, XA v s W etc.
7.1.5
Make simplifications based on good chemical reaction engineering principles
Some typical examples are; 1. Reactants in excess : This means that for example that when the concentration of B exceeds the concentration of A by more that a factor of 100, then the second order reaction −rA = kCAα CBβ can be approximated by −rA = k 0 CAα where k 0 = kCBβ . NOTE that in this csae the rate constant k 0 is a function of the concentration of B (CB ). NOTE also that this rate law cannot be used for conditions in which the concentration of B is of the same order of magnitude as A. For situations like this new experimental data will have to be collected. 2. Reactants are dilute in a reaction mixture : If for example the mole fractions of A, B in a reacting mixture are less than 1%, then any volume expansion, pressure change, flow rate change, due to the change the number of moles, can be neglected (i.e. ε = 0). Similarly for liquid phase reactions. 3. Fast reactions can be assumed to be at chemical equilibrium, which simplifies the rate equations.
7.1.6
Differential analysis
This method of analysis takes the approach that the data will be in the form REACTION RATE v s RATE LAW dCA dFA dPA FA0 −rA = − , ,− , XA , etc v s kCAα CBβ dt dV dt V
(7.9)
FA0 XA . However, for In a CSTR, the data automatically comes in this form i.e. −rA = V all other reactors this provides an approximation to the data collected. All other reactors will collect primarily C v s t or F v s V data which has to be converted into rate data by numerical differentiation. Numerical differentiation increases the scatter (error) in the data and leads to poor data analysis and this method is not recommended, although its use will be quickly demonsrated as is done in Fogler. Often, data is collected in what is known as a differential reactor, in which the feed entering is already partially converted (i.e. Xin > 0) and products leaving only increase the conversion by a few percent (i.e. Xout = Xin + ≤ 0.05), then the reactor can be treated similarly to a CSTR, Xin − Xout viz. −rA = −FA0 . ∆V In all cases the data analysis normally follows the following approach: −rA = kCAα CBβ
83
(7.10)
7.1. ALGORITHM CHAPTER 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) taking logs on both sides:
dCA ln(−rA ) = ln − dt
= ln(k) + αln(CA ) + βln(CB )
(7.11)
which is a linear equation in (ln(k), α, β) which can then be found by linear regression. This is a good approach when the reaction order are unknown and are needed to be estimated, since the equation is linear in the reaction orders. NOTE : this methods uses log weighting of the data, and this will have slightly different answers than the direct non-linear regression, which can be formulated as : n X β 2 α mi n [f (k, α, β)] = −rA,i − kCA,i CB,i ≈0 i=1
where −rA,i represents the measured data data at each experimental value i of the concentrations CA,i , CB,i in the reaction system when a total number of n experiments have been carried out. The objective function f (k, α, β) must be minimised by adjusting k, α, β until the least squared error is as close to zero as possible (using excel’s solver or scilab’s lsqrsolve).
7.1.7
Integral analysis
This analysis is more applicable to standard reactors which generate data in the form: CA , FA , PA , XA
vs
t, V, W, τ,
W FA0
(7.12)
In all these cases the design equation must be integrated. Note that integration is a smoothing process and reduces the errors in the estimated rate constants as opposed to the differential method. For the simple αth order reaction −rA = kCAα in a liquid phase batch reactor this yields: CA = CA0 exp(−kt)
α=1
(7.13)
(1−α) CA(1−α) = −k(1 − α)t + CA0
(7.14)
For the first equation (α = 1) the linear plot ln(CA ) vs t will yield rate constant from the slope. However, for the second equation, the linear plot of CA(1−α) vs t will require an estimate of α. Thus the linear regression requires repeated guesses of α, which is silly. Thus this type of data can only be analysed using non-linear analysis except in some special cases. (typically when the reaction order is given, linearisation is often possible). Non-linear regression in this case yields: mi n [f (k, α)] =
n h X
(1−α) CA,i
+ k(1 − α)ti −
(1−α) CA0
i2
≈0
(7.15)
1 2 1 − α ≈ 0
(7.16)
i=1
or in terms of the measured variable, CA it is
mi n [f (k, α)] =
n X
CA,i − −k(1 − α)ti − C (1−α) A0
i=1
There are many situations where integration is not possible. For example, consider the second order non-equi-molar reaction with the rate law : −rA = kCAα CBβ . Writing in terms of conversions 84
7.1. ALGORITHM CHAPTER 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) for a constant flow rate system yields, the PFR design equation is: β CB0 dXA α+β α FA0 = kCA0 (1 − XA ) − XA dV CA0 ZXA V 1 α+β = kCA0 β dXA FA0 CB0 α 0 (1 − XA ) − XA CA0 which can be formulated as an objective function ZXA,i n X V α+β min [f (k, α, β)] = kCA0 − F A0,i i=1 0
(7.17)
(7.18)
2
1 dX β A ≈ 0 CB0 (1 − XA )α − XA CA0
(7.19)
where the different conversion are obtained by varying the flow rate (this is the most common way, changing V is the other way). In this formulation the integral must be evaluated each time for each value of XA,i . All this is not a problem on a computer. (NOTE, that if CA0 = CB0 then it is no longer possible to distinguish α from β. The advantage of this approach is that it is general, requires no guessing of α, β and has no limitations of these values either. The disadvantage is that it needs some computer code to be written! It is possible to have multiple reactions with multiple rate constants, where integration using an ODE solver is needed each time is also easily done on a computer.
7.1.8
Goodness of fit and variance in model parameters
The goodness of fit can be obtained in two ways (i) A correlation coefficient can be obtained between the experimental and the predicted data e.g plotting (−rA )exp versus (−rA )model and estimating the mean and variance. v n uX 2 u (−rA )exp − (−rA )model u t i=1 (ii) looking at the least squares error : where n is he number n−m of data points and m is the number of estimated parameters (3 in the case shown here) The variance in the modelparameters can be obtained by studying the Jacobian matrix of the ∂f (p)j objective function : Ji,j = where p is the vector p = [k, α, β] and f (p) is evaluated ∂pi at each data point j, and thus J is a m × n matrix. Then the variance in the parameters p is given by v h i u u f (p) × J T J −1 t i,i pi = p i ± σi,i ; σi,i = (7.20) n−m h −1 i T where J J refers to the diagonals of the matrix. When σi,i 0.2p i then the parameter i,i
is a poor estimated and the model is most likely over parameterised and that means that there are most likely too many model parameters or there is not sufficient data.
85
7.2. BATCH CHAPTER REACTORS 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5)
7.2 7.2.1
batch reactors Method of excess
The idea of this method is to reduce the dependence of the rate law on multiple concentrations into one concentration. This method is particularlygood for getting first estimates of the reaction order with respect to the key component. For example, the reaction : A + B → pr oducts with the rate law −rA = kCAα CBβ
(7.21)
is difficult to analyse when both α, β are unknown. The idea then is to carry out the reation in two ways (i) Using excess B such that CB = CB0 and does not change significantly during the reaction. Then β CAα = k 0 CAα (7.22) −rA = kCAα CBβ = kCB0 which can then be used to find α. (ii) Using excess A such that CA = CA0 and remains effectively constant during the reaction, similarly α −rA = kCAα CBβ = kCA0 CBβ = k 00 CBβ (7.23) which can be used to find β. (iii) Once α, β have been found, then normal concentration data can be analysed for k.
7.2.2
Example : Differential method
(Fogler, ex 5.1, p260) The reaction of triphenyl methyl chloride (A) with methanol(B) viz (C6 H5 )3 CCl + CH3 OH → (C6 H5 )3 COCl + HCl A+B → C+D is carried out in a batch reactor in a solution of benzene and pyridine at 25°C. HCl reacts with the pyridine, which then precipitates making the reaction irreversible. The concentration time data is collected time (min) 0 50 100 150 200 250 300 −1 Concentration of A, CA [mmol.L ] 50 38 30.6 25.6 22.2 19.5 17.4 The initial concentration of methanol was 500 mmol.L−1 . 1. determine the reaction order with respect to triphenyl-methyl-chloride 2. the reaction order of methanol was given as 1, determine the reaction rate constant
SOLUTION PART 1 Looking at the data the time vector is given by : [t] = [t1 t2 t3 ...tn ] the concentration vector is given by : [CA ] = [CA,1 CA,2 CA,3 ...CA,n ] 86
7.2. BATCH CHAPTER REACTORS 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) where n is the number of data points. This also means that you cannot estimate more than n model parameters. Proposed rate law : −rA = kCAα CBβ now, CB0 = 10CA0 thus the it may be assumed that CB = CB0 ≈ constant over the range of the data collected (well, not really that constant, by the change is less than 5%) β Then −rA = k 0 CAα where k 0 = kCB0 dNA = rA V Mole balance : dt stoichiometry : NA = NB = −NC = −ND Since this is dilute liquid phase system, the density (volume) of the system is constant, thus the design equation becomes; dCA − = k 0 CAα (7.24) dt differential analysis then requires; dCA ln − = ln(k 0 ) + αln(CA,i ) (7.25) dt i where the i values run over all data points. Thus it is necessary to find the reaction rate as a function of time. The best way to do this is to fit a cubic-spline to the data //fit a cubic-spline to the data d = splin(tt,Ca); function Cat=spln(t) Cat=interp(t,tt,Ca,d) endfunction Cas=interp(t,tt,Ca,d) //spline interpolated values and then to differentiate the cubic-spline //get the gradient ttt=tt;ttt(n)=ttt(n)-0.1; //takingcare of the last point dCa_dt_s=diag(numdiff(spln,ttt)); disp(’-dCa/dt=’+string(-dCa_dt_s)) . This yields the following data in table 7.1. An alternative is to fit a polynomial to the data. This is done as follows; The error between each data point and the polynomial is given by εi = CA,i − (a1 + a2 t + a3 t 2 + a4 t 3 + a5 t 4 ...)
(7.26)
which can be vectorised as follows since it is a linear operation with basis sets εi = CA,i − [p]i [a]T [p]i =
[1 ti ti2 ti3 ti4 ...]
[a] = [a1 a2 a3 a4 a5 ]
(7.27) (7.28)
Here the vectot [p] represents the basis sets. Note that these can be any set of independent functions e.g. [p] = [1 t si n(t) exp(t) t 3 ]. This can be expanded for every data point n : 1 1 1 1 1 ε1 cA,1 a1 ε2 CA,2 t1 t2 · · · · · · · · · a2 .. ... ε C 2 a 2 3 A,3 3 . = − t1 t2 = [ε] = [CA ] − [p][a]T ≈ 0 (7.29) .. .. .. . . . .. . . t3 . . 1 εn CA,n am t1n · · · · · · · · · tnn 87
7.2. BATCH CHAPTER REACTORS 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) Note that m is the number of parameters [a] which in this case is 5. Note also that m ≤ n for a meaningful solution. When m = n then the solution is unique, one parameter for every equation. However, as with most data, n > m and the solution will never be exactly zero, rather the objective function needs to be minimized, thus min Ψ = [ε]T [ε]
(7.30)
Matlab’s and Scilab’s backslash operator (\) does exactly that (i.e. the overdetermined system, Ax = b yields the least squares solution x = A\b). Applying this concept here yields the parameters of the polynomial that fits the CA v s t data; [a]T = [p]\[CA ] = [50 − 0.2978 1.34E − 03 − 3.485E − 06 3.67E − 09] The code that does this is and the data is given in the table. //polynomial fit to the data using a non-linear leasts squares solver (the long way) function f=polfit(a,n) tx=[ones(1:n); tt; tt^2; tt^3; tt^4]’; //define the vector of polynomial basis sets f=Ca’-tx*a’ //calculate the error at each point endfunction a0=[1 1 1 1 1]; //initial guess a=lsqrsolve(a0,polfit,n) //solve the non-linear problem disp(’a_cubic=[’+string(a)+’]’) //display tx=[ones(1:nt); t; t^2; t^3; t^4]’; Cap=tx*a’; //calculate the polynomial fitted values function f=CaPol(t) //define a function in terms of t tx=[ones(1:n)’ t t^2 t^3 t^4]; //changes since t is a column vector f=tx*a’ //the value of Ca endfunction //polynomial regression using linear leasts squares (the short but abstract way) txx=[ones(1:n); tt; tt^2; tt^3; tt^4]; aa=txx’\Ca’ disp(’a_poly=[’+string(a)+’]’)
Table 7.1: Reaction rate as a function of concentration using cubic spline and polynomial fitting. CA dCA − dt spline dCA − dt poly
50
38
30.6
25.6
22.2
19.5
17.4
0.303
0.185 0.119 0.0814 0.0590 0.0485 0.0351
0.298
0.188 0.119 0.0801 0.0603 0.0485
The fitted data is shown in Fig 7.1.
88
.0334
7.2. BATCH CHAPTER REACTORS 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5)
concentration; CA [mmol:L¡1]
50 45 40 35 30 25 20 15
0
50
100
150
200
250
300
time; t [s] Figure 7.1: Cubic and polynomial fit to the data. The linear regression using equa 7.25 is //linear regression for the rate constant and reaction order //using cubic spline data aa_s=regress(log(Ca’),log(-dCa_dt_s)); //such that log(-dCa_dt)=aa(1)+aa(2)*log(Ca) kr=exp(aa_s(1)) alpha=aa_s(2) disp(’k=’+string(kr)+’ alpha=’+string(alpha)); //using polynomial data aa_p=regress(log(Ca’),log(-dCa_dt_p)); //such that log(-dCa_dt)=aa(1)+aa(2)*log(Ca) kr=exp(aa_p(1)) alpha=aa_p(2) disp(’k=’+string(kr)+’ alpha=’+string(alpha)); Linear regression of the spline fit data this data yields : ln(k) = −9.116;
k 0 = 0.0001099 L.mmol−1 .min−1 and α = 2.035 and lsqr Er r r or = 0.0136
Linear regression of the poly fit data this data yields : ln(k) = −9.116; k 0 = 0.0001046 L.mmol−1 .min−1 and α = 2.048 and lsqr Er r r or = 0.0199 Direct non-linear regression of the cubic spline data uses this equation n X dCA 0 α − k CA,i mi n f (k , α) = − dt i i=1 0
and the following code
89
7.2. BATCH CHAPTER REACTORS 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) //direct non-linear regression function f=EE(b,n) k=b(1);alpha=b(2) f=-dCa_dt_s-k*(Ca^alpha)’ endfunction b0=[1 1]; b=lsqrsolve(b0,EE,n) disp(’k=’+string(b(1))+’ alpha=’+string(b(2))); this data yields : k 0 = 0.0001467 L.mmol−1 .min−1 and α = 1.954 and lsqr Er r r or = 0.00902 So why are these values different : Figure 7.2 shows the two different plots which effectively represent the two different methods. 0
0.35
dCA [mmol:L¡1:min¡1] dt
-1
¡
10
¡
dCA [mmol:L¡1:min¡1] dt
10
-2
10
1
0.25 0.20 0.15 0.10 0.05 0.00 15
2
10
0.30
10
20
25
30
35
40
45
50
CA [mmol:L¡1]
¡1
CA [mmol:L ]
(a) linear-fit
(b) non-linear-fit
Figure 7.2: Comparing the linear data fit with the non-linear data fit. The green line represents the non-linear fit. The linear regression transforms the data into a log-log scale. This means that the error for the small values of the concentrations and small rates are weighted much more heavily than the values at the large concentrations. However, typically, small concentrations are more difficult to measure and thus also have the largest experimental error. This error is magnified by the log-log scaling. The non-linear fitting, on the other hand, leaves the data in its original for and weights all errors on this basis. Figure 7.2(a) shows the log-log plot is not able to distinguish between the different regressions. On the other hand Fig 7.2(b) shows that the non-linear regression (green) shows a superior fit to the rate data at high concentrations. The non-linear fitting thus provides a more representative set of rate constants. This deviation is more pronounced in the rate constant. PART 2 Since β = 1 then the true rate constant k can eb calculated: k=
k0 = 2.92E − 7 CB0
[L.mol−1 .min−1 ]2
Thus the rate law is given by −rA = 0.292CA2 CB 90
[mol.L−1 .s−1 ]
7.2. BATCH CHAPTER REACTORS 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) It is possible to fit for all 3 parameters k α β if the concentration of B can be found from the concentration of A. (normally B should also be measured). This would give an idea of the errors made in assuming that the concentration of B is constant. From the reaction stoichiometry CA = CA0 (1 − X);
X =1−
CA CA0
CB = CB0 − CA0 X = CB0 − CA0 + CA Then the rate law becomes −rA = kCAα (CB0 − CA0 + CA )β which can be used for dirrect non-linear regression or linear regression using the log-log approach non-linear n X 2 α (CB0 − CA0 + CA,i )β mi n f (k, α, β) = (−rA,i ) − kCA,i i=1
which can be solved by a non-linear leasts squares, Scilab’s “lsqrsolve” [k, α, β] = [180, 2.13, −2.37] with lsqr Er r r or = 0.00712. This has the smallest error of them all. Interesting, WHY Linear log-log: ln(−rA )i = ln(k) + αln(CA,i ) + βln(CB0 − CA0 + CA,i ) ln(−rA ) = p ∗ a p=
1 ln(CA ) ln(CB0 − CA0 + CA,i ) .. .. 1 . . .. .. .. . . . .. .. 1 . .
ln(k) a= α β
which can be solved by the “backslash” (a = p\ln(−rA )). [k, α, β] = [9.44E13, 2.47, −6.93] with lsqr Er r r or = 0.067. This solution has failed. This is due to the linear dependence of the two concentration columns.
7.2.3
Example : Integral method
Contrary to what Fogler says, this is not a trial and error method. When the reaction orders are known, then there are many standard rate laws for which the design equations can be integrated and a appropriate regression can be carried out. If not, then they simply become partr of the regression procedure. 7.2.3.1
The standard (linearised) analyses
For the reaction A → pr oducts it is always useful to test the data with the standard cases, or make a standard plot. Assuming a constant volume batch reactor, the following analyses can be used for zero, first, second and nth order reactions. Fig 7.3 shows these plots on a concentration basis for a batch reactor.
91
7.2. BATCH CHAPTER REACTORS 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) 0.5
Zero order reaction CA = CA0 ¡ kt slope = ¡ k
0.7
¶ CA or log(1 ¡ X) CA0
0.8
0.6 0.5 0.4
0.2
µ
WARNING There is no natural limit on the smallest value of CA it can go negative
0.3
0.1 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
¯rst order reaction
0.0
log
concentration; CA [mol:m¡3]
1.0 0.9
0.8
-0.5
log
-1.0
-2.0 -2.5 -3.0 -3.5 -4.0 0.0
0.9
0.1 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
time; t [s]
4.0
-0.5
2nd order reaction;
kCA2
r no
1 1 = kt + CA CA0
de
r
<2
-1.0
CA CB0 ¡ CA0 + CA
¶
Rx
3.0
slope = k
1 CA
¶ CA = ¡kt CA0
slope = ¡ k
-1.5
time; t [s]
3.5
µ
2.5
no
r
r de
>2
Rx
-2.0 -2.5 -3.0
log
1.5
2nd order reaction; kCACB
-3.5
µ
2.0
-1.5
-4.0
log
µ
¶ µ ¶ CA CA0 = ¡k(CB0 ¡ CA0)t + log CB0 ¡ CA0 + CA CB0
-4.5 slope = ¡ k(C ¡ C ) A0 B0 1.0 0.0
0.1 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-5.0 0.0
1.0
0.1 0.2
0.3
time; t [s]
0.4
0.5
0.6
0.7
0.8
0.9
1.0
time; t [s]
Figure 7.3: Batch reactor integral analysis linear plots for zero, first and second order kinetics using concentrations. Zero order reaction :
dCA k = −k; CA = CA0 − kt; X = t (7.31) dt CA0 So plotting CA vs t is a linear plot with an intercept CA0 and slope −k, as shown in Fig k 7.3. (or plotting X vs t has a slope of . Note that it is not necessary to know the CA0 initial concentration CA0 when making the CA vs t plot, since this given my the intercept.
First order reaction : dCA = −kCA ; dt
CA = exp(−kt); CA0
1 − X = exp(−kt)
(7.32)
linearisation yields ln
CA CA0
= −kt;
ln(1 − X) = −kt
(7.33)
CA So when plotting ln vs t the slope of the line is −k, as shown in Fig 7.3 (similarly CA0 for plotting of ln(1 − X) vs t) Note that re-arrangement to ln(CA ) = ln(CA0 ) − kt
92
(7.34)
7.2. BATCH CHAPTER REACTORS 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) and plotting ln(CA ) vs t also does not require knowledge of the intial concentration. Furthermore, the reaction rate does not depend on the concentration, and thus it is not important to keep the initial concentration constant during experimentation. Second order reaction : dX dCA = −kCA2 ; = kCA0 (1 − X)2 Also for CA0 = CB0 (7.35) dt dt dCA dX CB0 (7.36) = −kCA CB = −kCA (CB0 − CA0 + CA ); = kCA0 (1 − X) −X dt dt CA0 and their integrated forms re-arranged into linear equations for CA0 = CB0 : 1 1 1 = kt + plot : vs t slope : k CA CA0 CA X X = kCA0 t plot : vs t 1−X 1−X
intercept :
1 CA0
slope: kCA0
(7.37) (7.38)
And for CA0 6= CB0 : CA CA0 ln = −k(CB0 − CA0 )t + ln (7.39) CB0 − CA0 + CA CB0 CA CA0 plot : ln vs t slope : − k(CB0 − CA0 ) intercept : ln CB0 − CA0 + CA CB0 (7.40) CA0 CB0 X − 1 ln X − 1 = k(CB0 − CA0 )t
(7.41)
CA0 CB0 X − 1 plot : ln X − 1 vs t
slope : k(CB0 − CA0 )
(7.42)
1 1 = kt + CA CA0 it is necessary to know the concentrations of A and B at the start of the reaction. Some of these standard plots are made in Fig 7.3. In all cases, except for
qth order reaction : dCA = −kCAq ; dt
1 CAq−1
dX q−1 = kCA0 (1 − X)q dt 1 1 − (1 − X)q−1 = k(1 − q)t = −k(1 − q)t + q−1 ; (1 − x)q−1 CA0
(7.43) (7.44)
In these linearised equations it is necessary to know the reaction order q. (see Fig 7.3) 7.2.3.2
Example 5.2 (Fogler)
Continuation of example 5.1 : The purpose now is to regress the concentration data directly without finding the reaction rate. Thus, assuming that teh reaction is second order; −
dCA = k 0 CA2 ; dt
1 1 = k 0t + CA CA0 93
(7.45)
7.2. BATCH CHAPTER REACTORS 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) Regression can be done by noting that the integrated equation is related to y = a + bx when 1 y= and x = t. The the solution of this equation by for example Scilab’s “regress” routine, CA 1 yields b = k 0 and a = . The value obtained for a provides a check on the validity of the CA0 model, since CA0 is already known. So k 0 = 0.0001248 and ”CA0 ” = 49.7. So the model fits the data well as shown in Fig 7.4.. 0.060 0.055
1 [L:mol¡1] CA
0.050 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010
0
50
100
150
200
250
300
350
time; t [min] Figure 7.4: Standard regression plot for the second order reaction.
7.2.4
Example : Non-linear regression
Non-linear regression requires the development of a objective function which is then used in a non-linear leasts squares solver (Scilab’s “lsqrsolve” is a particularly good one, EBE-matlab does not have such routines and the nextbest bet is EXCEL!!!) Continuation of Foglers example 5-1: exp Noting that k 0 and α are to be found and that the data is given in the form CA,i vs ti , it therefore makes sense to integrate a general α-order rate equation and its integrated form.;
1 1−α 1 − α = CA0 − (1 − α)kt
dCA = −k 0 CAα ; dt
CAmodel
(7.46)
This is the batch reactor design equation and will from now on be called the model. Now the purpose of this method is to minimise the error between the experimental concentrations and the model predicted concentrations, thus this will form the objective function; 0
mi n f (k , α) =
n X i=1
94
εi εi = εεT
(7.47)
7.3. METHOD CHAPTER OF INITIAL 7. COLLECTION RATES AND DATA ANALYSIS (FOGLER CHAPTER 5) where n is the number of experimental data points and εi is the error between the experimental and model concentrations; exp model − CA,i εi = CA,i
(7.48)
1 exp 1−α − CA0 − (1 − α)kti 1 − α εi = CA,i
(7.49)
where ε is a vector of values,one for each data point collected ε = ε1 ε2 . . . . . . ε n
(7.50)
The solution proceeds by developing a function that calculates the εi values at each data point and then call the scilab routine lsqrsolve to carry out the least squares minimisation. The scilab function is function f=model1(x,n) kr=x(1);alpha=x(2); //x is a vector of two parameters f=Ca-(Ca(1)^(1-alpha)-(1-alpha)*kr*tt)^(1/(1-alpha)) endfunction The calling of the least squares solver then has hte code //call the least squares solver x0=[0.0001 2]; //initial guess x=lsqrsolve(x0,model1,n) kr=x(1);alpha=x(2); disp(’kr=’+string(kr)+’ alpha=’+string(alpha)) Cam=(Ca(1)^(1-alpha)-(1-alpha)*kr*t)^(1/(1-alpha)) //model predicted values The model parameters obtained are : k 0 = 0.000111
α = 2.037
And the quality of the fata fit can be seen in Fig 7.5. It is possible to use the same code to regress with α = 2. When this is done then k 0 = 0.000126 similar to those obtained before. The values of k 0 are summarised here: method differential : Linear regression of the spline fit differential : Linear regression of the poly fit differential : non-linear regression Integral non-linear regression CA non-linear regression
7.3
k0 0.1109 0.1046 0.1467 0.1248 0.111 0.126
α 2.035 2.048 1.95 2.000 2.037 2.000
Method of initial rates
The idea here is to study the reaction at very short reactions times, or more precisely, at small conversions (say < 5% change in conversion over the reaction period). This avoids complications from any possible reversible reaction steps, deactivation as a result of by-products forming, etc. These experiments are carried out a various inlet concentrations (CA0 ) and/or various conversions at the inlet. Then the reaction rate at these conditions is estimated from data collected over a very small conversion range. Possible sequences might be 95
7.3. METHOD CHAPTER OF INITIAL 7. COLLECTION RATES AND DATA ANALYSIS (FOGLER CHAPTER 5) 55 50
CA [mol:L¡1]
45 40 35 30 25 20 15
0
50
100
150
200
250
300
time; t [min] Figure 7.5: Predicted and experimental data when using the non-linear leasts squares method applied to concentration data. CA0 CA t
20 19 10
10 5 2 1 9.5 4.75 1.9 0.95 15 12 10 8
∆CA CA0 − CA CA0 + CA Which yields the reaction rate = at the concentration from which ∆t t 2 the law parameters can be obtained. Or possibly Xin Xout FA0
0.1 0.11 10
0.2 0.21 15
0.3 0.31 20
0.4 0.41 25
0.5 0.51 30
From which the reaction rate can be obtained from a differential reactor balance (see later) Limitations of this method 1. Real reactions undergo deactivation. If deactivation is fast (FCC) then this method fails since it becomes impossible to esrimate the initial reaction rate without the influence of the deactivation. 2. Real reactors operate at very high conversions, contrary to the very low conversions used here. High conversions almost always produce side products which can influence the reaction kinetics, especially at high conversions, which are the conditions used under normal operation. These methods fail to analyse such reaction systems
7.3.1
Example (Fogler 5-4)
The dissolution kinetics Calcium-Magnesium-carbonate within HCl has been measured and is shown in the graphs below. The data in the tables have been extracted from the graphs. The reaction is 4HCl + CaMg(CO3 )2 → Mg 2+ + Ca2+ 4Cl − + 2CO2 + 2H2 O (7.51) 96
7.3. METHOD CHAPTER OF INITIAL 7. COLLECTION RATES AND DATA ANALYSIS (FOGLER CHAPTER 5) Determine the reaction order with respect to HCl? DATA: Run1, 4N HCl CHCl 4.0000 3.9993 3.9986 3.9980 3.9968 time [min] 0 2 4 6 8 Run2, 1N HCl CHCl 1 0.9996 0.9991 0.9986 0.9980 time [min] 0 2 4 6 8
Figure 7.6: Data for example (Fogler5-4).
SOLUTION Assume a rate law with reaction order q : q −rHCl = kCHCl
(7.52)
This is a batch reactor, so the design equation for a constant volume system is −
dCHCl q = kCHCl dt
(7.53)
Linearising:
dCHCl = ln(k) + qln(CHCl ) (7.54) ln − dt Thus need to estimate the reaction rate at t = 0 in each case. This can be done with a forward difference formula since the data is equally spaced; t=0 t=2 t=4 dCCCl −3CHCl + 4CHCl − CHCl = ; dt 2∆t
∆t = 2
(7.55)
which yields CHCl,0 [mol.L−1 ] dCHCl − [mol.L−1 .min−1 ] dt
1
4
2
1.74E-4 3.50E-4 2.49E-4
0.1
0.5
0.66E-4
1.36E-4
Regressing these to get the reaction order: q = 0.45 and Fig 7.7 shows that the model fits the data well. 97
7.4. METHOD CHAPTER OF HALF7.LIVES COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5) -3
-4
10
µ
¶ dCA ¡ [mol:L¡1:min¡1] dt 0
10
-5
10
-1
0
10
1
10
10
Figure 7.7: Linearised initial rate plot for the dissolution process.
7.4
Method of half lives
The half life of a reaction is the time it takes to reduce the conversion (concentration) by half the initial value. Thus a irreversible reaction, A → pr oducts with a q th order rate law in a batch reactor the design equation is; −rA = −kCAq
−
dCA = kCAq dt
(7.56)
Integration yields the reaction time to reach the concentration CA ; " # q−1 1 1 1 1 CA0 t= − q−1 = −1 q−1 k(q − 1) CAq−1 CA0 CA kCA0 (q − 1) 1 CA0 , thus 2 2q−1 − 1 1 = q−1 k(q − 1) CA0
(7.57)
Using the half-life concept : t = t1/2 then CA = t1/2
or using a general concept, the nth life time, such that t = t1/n when CA = t1/n
nq−1 − 1 = k(q − 1)
1
(7.58) 1 CA0 yields n
q−1 CA0
(7.59)
Linearising this equation ln(t1/2 ) = ln
2q−1 − 1 k(q − 1)
+ (1 − q)ln(CA )
The slope of the plot is (1 − q) from which the reaction order can be obtained as shown in Fig 7.8. 98
7.5. DIFFERENTIAL CHAPTER REACTORS 7. COLLECTION (INCLUDING AND CSTR DATA AND ANALYSIS RECYCLE (FOGLER REACTORS) CHAPTER 5)
Figure 7.8: Half-life analysis plot.
7.5
Differential reactors (including CSTR and recycle reactors)
The idea of these kinds of reactors is to directly calculate the reaction rate and not have to differentiate the concentration-time data. This reduces the error. These reactors also operate with no pressure drop and isothermally as a result of the reaction zone and conversion being small. There are essentially 2 types of differential reactors : Fixed bed (plug flow) : In these reactors the conversion in the reactive zone is small. The feed enters the reactor at various degrees of conversion which is achieved by dilution with product species. In this way the reaction rate can be estimated at different conversions. The flow rate of reactants to the reactor remains largely unchanged. CSTR’s and recycle reactors : The CSTR delivers the reaction rate at any concentration directly from the measured variables. The conversion is changed by changing the feed rate and not the feed conversion as is needed for the fixed bed reactor. A recycle reactor (no separation of the products must take place) can be assumed to have the same analysis as a CSTR when the recycle ratio is greater than 20, which translates into a conversion per pass of <5%. These type of experiments are not generally carried out in Batch type reactors, since it is difficult to precisely define the conditions at t = 0. These ideas can be viewed in Fig 7.9.
99
7.5. DIFFERENTIAL CHAPTER REACTORS 7. COLLECTION (INCLUDING AND CSTR DATA AND ANALYSIS RECYCLE (FOGLER REACTORS) CHAPTER 5)
Figure 7.9: Differential reactor operation : concepts.
7.5.1
Mole balances : CSTR (and recycle reactor with rapid recycle) FA0 − FA FA0 ν0 CA0 CA0 = X= X= X V V V τ FA0 − FA FA0 −rA [mol.kg−1 .s−1 ] = = X = (W HSV )CA0 W W −rA [mol.L−1 .s−1 ] =
(7.60) (7.61)
So CSTR’s yield the reaction rate directly at any concentration or conversion of the reactants or products (even for multiple reactions). This is the ideal reactor configuration to use for the development of rate laws. However, it is very difficult to achieve good CSTR operation, and great care must be taken to ensure that the mixing is adequate such that the CSTR assumptions are valid. The variation in reaction rate is obtained by changing the volume (V, W ), the flow rate (FA0 ) or more precisely the τ, W HSV . Fig 7.9 shows the concepts of how a recycle system relates to a real integral reactor. Essentially the catalytic zone (reactive zone) represents a slice of the normal reactor. When this reactor section of the reactor is placed within a recycle loop in which the recycle flow rate exceeds 20 times the feed flow rate, the reactor feed will no longer be pure feed, by will be mixed with products of the reaction and will thus enter partially converted. The high flow rates through the reactor section ensure that (i) the conversion is low <5% (ii) the film mass transfer limitations are negligible. It thus provides a ideal reactor for studying the rate law of reactions. 100
7.5. DIFFERENTIAL CHAPTER REACTORS 7. COLLECTION (INCLUDING AND CSTR DATA AND ANALYSIS RECYCLE (FOGLER REACTORS) CHAPTER 5)
7.5.2
Mole balances : Tubular and packed bed systems
Fig 7.9 also shows that the differential reactor is also a slice of the full scale reactor. However, in this reactor the feed must be pre-mixed with products to yield feeds that reproduce the conversion profile of the full-scale reactor. For complex reactions this proves to be impossible. In practice for complex reaction systems, the experimental system operates with two reactors in series, the first one large enough to generate the conversion desired to be used in the second differential reactor. These systems are not popular for the study of complex chemical reaction system kinetics. The mole balances are best viewed through the first order approximation to the PBR design equation, thus; FA |Z − FA |Z+∆Z − rA ∆W = 0;
dFA dCA = ν0 = −rA ; dW dW
FA0
dX = rA dW
(7.62)
Now making a first order approximation of the differential to yield the average reaction rate within the differential interval ∆W is −rA =
FA,out − FA,in CA,out − CA,in Xout − Xin = ν0 = FA0 ∆W ∆W ∆W
(7.63)
Here the conditions FA0 and CA0 are reserved for the flow rate and concentration of the unconverted feed and correspond to X = 0. The reaction rate so evaluated represents, to a FA,out + FA,in ; first approximation the reaction rate at the middle of the ∆W interval, i.e. at 2 CA,out + CA,in Xout + Xin , . It is not the reaction rate at the inlet to the differential reactor as 2 2 proposed by Fogler, i.e. draw a parallel with the trapazoidal integration where the integration uses the average value between the end points of the interval if the integration step.
7.5.3
Example : Fogler 5-5
The formation of methane from carbon monoxide(A) and hydrogen(B) CO + 3H2 → CH4 + H2 O A + 3B → C + 2D is being studied at 260°C in a differential reactor where the concentration of methane leaving the reactor is measured. (a) From the data supplied determine the conversion for each experiment and the partial pressures of CO and H2 at the inlet and outlet of the reactor. Determine also the average reaction rate of CO across the reactor. (b) The reaction rate equation is proportional to the partial pressure of CO with the function f (CO) and proportional to the partial pressure of H2 with the function g(H2 ); −rCO = f (CO) · g(H2 ) Determine the reaction order with respect to PCO by assuming the rate equation; a −rCO = k1 PCO
101
7.5. DIFFERENTIAL CHAPTER REACTORS 7. COLLECTION (INCLUDING AND CSTR DATA AND ANALYSIS RECYCLE (FOGLER REACTORS) CHAPTER 5) (c) Show that the data suggests a rate law for the hydrogen dependence of −rCO =
k2 PHb12 1 + KH2 PHb22
(d) By combining the two rate laws, yields the overall rate equation −rCO =
kPCO PHb12 1 + KH2 PHb22
Regress this equation for the constants k, b1 , b2 , KH2 . DATA: Table 7.2: Experimental reaction data for the conversion of CO to CH4 . Run PCO [atm] PH2 [atm] CCH4 [mol.L−1 ]×104 1 1 1 1.73 2 1.18 1 4.40 3 4.08 1 10.0 4 1 0.1 1.65 5 1 0.5 2.47 6 1 4.0 1.75 The exit volumetric flow rate was maintained at 300 [L.min−1 ]. The catalyst bed contained 10 g of catalyst. PCO and PH2 were measured at the reactor inlet and CCH4 was measured at the reactor exit.
SOLUTION (a) Looking at the data (i) There are 2 sets of data, runs 1-3 are at constant PH2 and runs 1,4-6 are at constant PCO . Thus the CO and H2 dependence can be analysed separately (ii) The data required will require a stoichiometric table to relate all the quantities to each other. Make the usual assumptions : ideal gases, isothermal reactor, ... Design equation for the reaction rate of CO is −rCO = FA0
Xout ∝ f (PCO ) · g(PH2 ) ∆W
Thus it will be necessary to determine the conversion,X, the molar flow rate of CO, FA0 and the average partial pressures of CO and H2 across the reactor. Stoichiometric table gives
in
A FA0
out
FA0 (1 − X)
total
F0 = FA0 (1 + α)
B FB0 FB0 − 3X FA0 FA0 FT = FA0 (1 + α − X) 102
α=
C 0
D 0
FA0 X
2FA0 X
FB0 PB0 = FA0 PA0
7.5. DIFFERENTIAL CHAPTER REACTORS 7. COLLECTION (INCLUDING AND CSTR DATA AND ANALYSIS RECYCLE (FOGLER REACTORS) CHAPTER 5) From the data, the methane concentration gives the reaction rate, and since the exit volume flow rate (ν) is given, the molar flow rate of methane can be calculated from FC = νCCH4 = FA0 X The conversion can be obtained by looking at the methane concentration CCH4 =
FC P X = CT FT RT 1+α−X
Re-arrangement gives the conversion X=
CCH4 (1 + α) CCH4 + CT
Note that the total presssure P and thus the total concentration, CT is not constant since, P = PA0 + PB0 and varies from run to run. Thus FA0 can be obtained from the above equation. The partial pressures of CO and H2 at the reactor exit are given by PA =
1−X (PA0 + PB0 ); 1+α−X
PB =
α − 3X (PA0 − PB0 ) 1+α−X
and the reaction rate of CO can be obtained from the above equation. Making a new table 7.3.
Run PA,in 1 1.00 2 1.18 3 4.08 4 1.00 5 1.00 6 1.00
PB,in 1.00 1.00 1.00 0.10 0.50 4.00
α 1.000 0.847 0.245 0.100 0.500 4.000
Table P 2.00 2.18 5.08 1.10 1.50 5.00
7.3: Processed reaction data. CT X FA0 PA,out 0.0457 0.0075 6.88 0.996 0.0498 0.0161 8.16 1.171 0.116 0.0106 28.2 4.071 0.0251 0.0071 6.90 0.999 0.0342 0.0107 6.90 0.996 0.114 0.0076 6.86 0.993
PB,out 0.981 0.951 0.877 0.079 0.471 3.983
−rCO 0.0052 0.0132 0.0300 0.0049 0.0074 0.0052
PA,av e 0.998 1.175 4.075 0.999 0.998 0.996
PA is relatively constant, however the biggest variation (still not much) is in PB across the reactor even when PB0 is a constant, since 3 moles of B required for every mole of A that reacts away. (b) Regression for k1 , a : ln (−rCO ) = ln(k1 ) + a · ln(PCO ) thus k1 = 0.00745 and a = 1.03. Fig 7.10 shows that the data scatter is large and too few points!!
103
PB,av e 0.990 0.975 0.938 0.089 0.485 3.99
7.5. DIFFERENTIAL CHAPTER REACTORS 7. COLLECTION (INCLUDING AND CSTR DATA AND ANALYSIS RECYCLE (FOGLER REACTORS) CHAPTER 5) -1
ln(¡rCO) [mol:L¡1:min¡1]
10
-2
10
-3
10
-1
0
10
1
10
10
PCO [atm] Figure 7.10: Regression plot for the −rCO dependence on PCO . (c) Looking at the second table and the rate law −rCO =
k2 PHb12 1 + KH2 PHb22
and making a plot of the data, Fig 7.11.
ln(¡rCO) [mol:L¡1:min¡1]
0.0075 0.0070
0.0065 0.0060 0.0055
0.0050 0.0045 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
PH2 [atm] Figure 7.11: Plotting the −rCO dependence on PH2 . (i) plotting the data shows that at low P the rate is low,it then increase as PH2 increases, goes through a maximum and then decreases again. 104
7.5. DIFFERENTIAL CHAPTER REACTORS 7. COLLECTION (INCLUDING AND CSTR DATA AND ANALYSIS RECYCLE (FOGLER REACTORS) CHAPTER 5) (ii) At low PH2 : then 1 KH2 PHb22 and the rate law reduces to : −rCO = k2 PHb12 . Thus as PH2 increases, so does the reaction rate. k2 b1 −b2 (ii) At high PH2 : then 1 KH2 PHb22 and the rate law reduces to −rCO = P . If b1 −b2 < 0 KH2 H2 then increasing the pressure will decrease the reaction rate. The data is rather few, but the trend of the rate law obeys the rate data at least conceptually. (d) Regressing all the data,use a non-linear regression for all the constants ! 6 X kPCO,i PHb12 ,i mi n obj(k, b1 , b2 , KH2 ) = −rCO,i − 1 + KH2 PHb22 ,i i=1 In Scilab this looks like this //least squares regression for all the parameters function f=model(p,n) k=p(1);b1=p(2);b2=p(3);Kh2=p(4); //b1=0.5;b2=1; f=rate-k*Paa.*Pbb^b1./(1+Kh2*Pbb^b2) endfunction And calling the least squares solver with an initial guess p0=[0.02 0.5 1 1]; [p v info]=lsqrsolve(p0,model,n) disp(info,’info’); disp(norm(v),’lsqrE’); disp(p,’p’) k=p(1);b1=p(2);b2=p(3);Kh2=p(4); rate_m=k*Paa.*Pbb^b1./(1+Kh2*Pbb^b2) //predict the reaction rate from parameters The parameters are p = [k2 , b1 , b2 , KH2 ] = [0.01135, 0.33, 1.115, 0.510] with a LSQRerror of 0.0050 and a variance of σ = [0.79, 26, 46, 106] i.e p = p ± σ. The fit of the model to the reaction rate is shown inFig 7.12. It can be seen that the reaction rate at constant PH2 is not was well predicted as the variation with PH2 . This is already noted in the scatter when fitting for the reaction order of CO. Furthermore the variance in the model parameters is very high, in fact the model parameters are useless, since the variance exeeds the parameter value. The rate law then is 0.01135PCO PH0.33 2 −rCO = 1 + 0.510PH1.115 2 Also the reaction orders are non-integer values, which from mechanistic considerations is not realistic. Typically reaction orders are [0, 0.5, 1, 1.5, ...] With this in mind and the fact that there are too many parameters for such few data, rounding the orders of the reactions off to the nearest accepable value, yields −rCO
kPCO PH0.5 2 = 1 + KH2 PH2
Now regressing again yields k = 0.0287 and KH2 = 2.90 with a LSQRerror of 0.0050 and a variance σ = [0.0484, 3.79] which is still very high, but at least better then before. Fig 7.13 shows the fit of the reaction rate. 105
7.5. DIFFERENTIAL CHAPTER REACTORS 7. COLLECTION (INCLUDING AND CSTR DATA AND ANALYSIS RECYCLE (FOGLER REACTORS) CHAPTER 5) 0.035
¡rCO [mol:L¡1:min¡1]
0.030 0.025 0.020 0.015 0.010 0.005 0.000 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
PH2 [atm] Figure 7.12: The prediction of the reaction rate when regressing for all the rate law constants.
0.035
¡rCO [mol:L¡1:min¡1]
0.030 0.025 0.020 0.015 0.010 0.005 0.000 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
PH2 [atm] Figure 7.13: The prediction of the reaction rate when regressing for k, KH2 in the rate law. This yields the final rate law −rCO
0.0189PCO PH0.5 2 = 1 + 1.48PH2
The rate law is OK, just there are too few data points ot make these values reliable. Note, that the LSQRerror does not change much between the last 2 rate laws. There are not enough data points to accurately evaluate the reaction orders. The Scilab code “FoglerEx5-5.sce” does all the calculations. 106
Chapter 8 Developing rate laws from reaction mechanisms and reaction pathways
107
Chapter 9 Bio-reactor engineering mechanisms
108
Chapter 10 Bio-reactor design
109
Chapter 11 Analysis of reactor flow patterns on reactor performance (Fogler chapter 13)
110
