Analysis and Design of Chemical Reactors I
By DEAR
email: d34r123@yahoo.co.id
KOMUNITAS BLOGGER UNIVERSITAS SRIWIJAYA
Analysis and Design of Chemical Reactors I
1. Introduction
Chemical reactors are the main components of chemical plants. They are employed in most
processes in petroleum, petrochemical, chemical, biochemical, pharmaceutical and
electronic industries. The most frequently used reactor is the catalytic converter. The
objective of the analysis and design of chemical reactors is to achieve several critical
measures of performance:
¨ Selective production of desired products at maximum productivity
¨ Process and product safety
¨ Environmentally compatible processes and products
¨ Minimization of waste generation
¨ Minimization of investment
¨ Minimization of energy consumption
¨ Operability and control
Design of reactors require detailed consideration of various topics:
¨ Stoichiometry of reactions
¨ Thermodynamics – calculation of chemical equilibrium, heat of reaction, phase
equilibrium, physical and thermodynamic properties
¨ Transport phenomena – calculation of fluid flow, pressure drop, heat transfer,
mass transfer between phases and diffusion
¨ Kinetics – expressions of the rates of reaction as a function of concentration
and temperature
The scale of chemical reactors in the framework of related topics is described in Figure 1.
Figure 1. Length and time scale of chemical reactors
Stoichiometry
Stoichiometric coefficients are determined arbitrarily as long as the chemical reactions are
balanced. Usually, the coefficient of the limiting reactant is normalized to unity. Lets take the
following example:
C2H4 + 0.5O2 à C2H4O (1.1)
C2H4 + 3O2 à 2CO2 + 2H2O (1.2)
Those reactions can be expressed as:
The stoichiometric coefficients in this example are given in the matrix:
ij
C2H4, 1
O2, 2
C2H4O, 3
CO2, 4
H2O, 5
1
-1
-0.5
1
2
-1
-3
2
2
Parameters that are used to quantify the progress of the reaction, extent of reaction xi and
conversion xj are defined as:
¨
¨
Nij is the number of moles. The extent of reaction is identical for all components in a specific
reaction, while conversion is defined only for the reactants. Similar definitions are given for
continuous reactors in terms of the molar flowrate Fij. The definition in equations (1.4) and
(1.5) are used to calculate the number of moles or the molar rate of component j:
Types of reactors
Chemical reactors are classified in terms of the type of flow:
¨ mixed or axial flow
and residence of reactants and products:
¨ batch, semibatch, continuous.
The flow characteristics are complex. Two extreme flow regimes are defined:
· Well-mixed – temperature and concentration is identical everywhere in the
reactor. This is accomplished with a good mixing system that provides a volumetric flowrate
in the reactor significantly higher than the inlet volumetric flowrate. Batch reactors (BR),
semibatch (SBR) and continuous stirred tank reactors (CSTR) are included in this category.
· Plug-flow – turbulent flow with no axial mixing in tubular reactor (PFR).
Temperature and concentration change in axial direction.
Kinetics
Definition of the rates of reaction is essential for analysis and design of reactors. The rate of
reaction is defined as the number of moles reacted (reactant) or produced (product) per unit
time and per extensive unit (volume, mass). The rate depends of the concentration of
reactants, products and catalyst (in case it is used) and temperature. The mathematical
formulation of the reaction rate is defined in kinetic models. Three types of kinetic
expressions are identified:
¨ Theoretical, based on mechanism, like the production of HBr from bromine and
hydrogen:
¨ Semi-empirical, partially based on theoretical model and partially empirical
¨ Power-law, using reaction order, for example the reaction:
The reaction order should be assigned so as to satisfy the chemical equilibrium expression.
2. Mass balances in isothermal reactors
Four types of reactors were defined:
¨ Continuous PFR and CSTR
¨ Semibatch well-mixed reactor
¨ Batch well-mixed reactor
The need to establish quantitative links between reactor performance and input and
operating variables is essential in optimizing reactor operation, proper reactor selection and
scale-up and interpretation of data from bench and pilot plants. The general formulation of a
reactor mass balance is expressed as:
Rate of input – Rate of output + Rate of generation = Rate of accumulation
The selection of the control volume in critical. Only single-phase reactors will be defined. The
control volume is normally the volume of the phase where the reaction takes place. The
general mass balance for a well-mixed reactor is expressed as:
where Rj is the rate of change defined as:
Then for a batch reactor:
For a single reaction, considering the limiting reactant k with ak = - 1 and introducing the
definition of conversion yields:
Separation of variables and integration gives an expression for the time of reaction as a
function of conversion:
Normally the volume is constant. Then:
The general mass balance for a CSTR at steady-state is expressed as:
Repeating the same derivation using conversion and the relationship:
yields:
(2.8)
The concentration changes as a function of axial in PFR. Therefore, the control volume is DV
to give:
at steady-state:
Repeating the same procedure using the conversion gives:
The similarity between expression (2.12) for PFR and (2.6) for BR is evident. t is called the
space time. If the volumetric flowrate is constant (constant density) then this ratio is the
residence time in the reactor, q.
3. Design and analysis of isothermal reactors
The solution of the mass balances requires the expression of the reaction rate in terms of
concentrations and temperature, namely the kinetic expression. The simplest kinetic
expression that fits many practical systems is first order:
For constant density systems, the concentration is a simple function of conversion:
Substituting eqn. (3.2) in the mass balance for batch reactors yields:
Similar expressions are derived for PFR and CSTR, respectively:
The similarity between BR and PFR is evident. At equal time and residence time, the
conversion in BR and PFR is identical, higher than in CSTR.
Reactors arranged in series are analyzed relating the conversion to the inlet of the system.
Let consider a first-order reaction carried out in N CSTRs in series, each one with a volume
V/N. The mass balance in the first reactor is expressed as:
The mass balance for the second reactor i
Repeating the same procedure for the N reactors yields:
Increasing the number of reactors at constant total residence time increases conversion. At
sufficiently large value of N (the required value of N increases with increasing kq) the
conversion in N CSTR in series approaches the PFR conversion.
A similar behavior is reached in PFR with recycle. In the recycle system several parameters
are defined. The recycle ratio is given by the expression:
The single pass and overall conversion are defined as:
The relationship between xs and xo is developed as follows:
Substituting eqns. (3.14) and (3.15) into eqn. (3.13) yields:
As R increases, the single-pass conversion decreases. At the limit, the recycle reactor
approaches CSTR.
The mass balance for the key component, assuming constant density:
Substituting the definition of conversion and integrating gives:
At R à 0, eqn. (3.18) converges to the PFR expression. At the other extreme, R
à ∞ yields xs à 0. Eqn. (3.18) becomes:
Since R is dominant at the denominator, eqn. (3.19) converges to eqn. (3.4). Therefore, the
performance of the two models, PRF with recycle and N CSTRs in series provides the link
between PFR and CSTR.
4. Expressions for the concentration
Gas phase systems with changing density in continuous reactors require the derivation of
the concentration. A simple expression is:
Application of eqn. (4.1) means that the mole flowrate of all components need to be
calculated. However, using the conversion, only the key component needs to be calculated,
assuming ak = -1:
Substituting Ft in eqn. (4.1) gives:
The mass balance for PFR for a first order reaction:
If one of the components is condensable, the concentration should be adjusted accordingly.
First the conditions of condensation are determined, namely the mole fraction of the
condensable component at the condensation point. Then the conversion is calculated:
The molar flowrate of the condensable component remains constant after condensation
starts. Then the total molar flowrate and the concentration of each component are calculated
accordingly. An example will be solved to demonstrate the procedure.
5. Pressure drop in tubular reactors
The general momentum balance for calculating the pressure drop for gases flowing through
pipes is expressed as:
The mass velocity is constant in the reactor. Combining this expression with the
relationships:
and the mass balance in eqns (2.7) provides a model of PFR.
The pressure drop in a packed (fixed) bed reactor is calculated from the Ergun equation:
6. Unsteady-state operation of stirred reactors
1. Startup of a CSTR
A simple case is selected to illustrate the procedure to solve this type of problems: first-order
reaction in constant volume reactor. Eqn. (2.1) becomes:
with the initial condition: t = 0 Cj = 0.
Applying the integrating factor method, the solution of eqn. (6.1) is:
The time needed to reach 99% of the steady state value is calculated using the relationship:
99 (6.3)
that yields:
2. Semi-batch reactors
The system analyzed here consists of two reactants, A is present in the reactor and B flows
in at the molar rate FB0. Two cases will be considered:
2.1 . Case I – first-order with respect to A
The mass balance for A:
The volume changes with time: V = V0 + Qt
The solution of eqn. (6.5) given that Na = CAV
2.2 . Case II – first-order with respect to B
Mass balance for component B
Substituting the derivation of NB = CBV
Substituting the relationship q = q0 + t and rearrangement of terms in eqn. (6.9) yields:
The solution of eqn. (6.10) with t = 0 CB = CBi is:
7. Heat balances in chemical reactors
The energy balance in the batch reactor is:
The total energy in the system is expressed as:
Differentiating eqn. (7.2) and substituting the pertinent expressions:
Rearrangement of eqn. (7.3) yields:
where:
A similar derivation for CSTR:
Combination with the mass balance yields:
The differential heat balance in a PFR:
yields the heat balance:
A similar derivation including the differentiation of the term in the right hand side of eqn. (7.8)
and substitution of the mass balance gives the transient form of the heat balance:
The last two terms on the right hand side of eqn. (7.9) are normally negligible.
There are two types of reactors:
¨ adiabatic - where there is no heat exchange with the surroundings
¨ non-adiabatic – where there is heat exchange with the surroundings
Isothermal operation of reactors is an ideal case of the latter type. CSTR can be operated
isothermally. Since the rate changes with conversion, it varies with time (batch) or place
(PFR). Thus the rate of heat generation changes, rendering controlling the temperature
constant very difficult.
The preferred method of operation is adiabatic since it does not need heat-exchanging area.
The heat balance for the batch adiabatic reactor and a single reaction system is:
Substituting the mass balance:
yields
Eqn. (7.12) is rearranged to give the expression for the temperature as a function of
conversion:
The term that multiplies the conversion is called the adiabatic temperature rise and could be
assumed to be essentially constant:
Similar expressions are derived for CSTR:
and PFR:
8. Performance of exothermic reversible systems
The analysis of exothermic reversible systems is demonstrated with a simple elementary
reaction A B. The rate is then expressed as:
The conversion at equilibrium decreases with temperature. Therefore this conversion at a
given temperature is the maximum possible value. If we take a certain conversion well below
its maximum value, and increase the temperature at constant conversion, the reaction rates
should increase. However, as equilibrium is approached, the rate should decrease.
Therefore, there is a point, close to the equilibrium curve that determines the maximum value
of the reaction rate.
The derivation of the maximum rate at constant conversion is done as follows:
where the two derivatives are expressed as:
Substitution into eqn. (8.2) yields:
Rearranging this eqn. gives:
Operation on the maximum rate curve should yield the minimum reactor volume. Design of
an optimal CSTR is feasible since it runs at one operating set of conditions. However if the
designed conversion is relatively high, a number of adiabatic CSTRs would provide a more
economical proposition.
Design of adiabatic PFRs in series is the best method in many industrial processes, for
example methanol synthesis or sulfur trioxide production. Inter-cooling is the accepted
procedure to adjust the inlet temperature. This is done with a heat exchanger or by
quenching with one of the reactants.
9. Stability of CSTR operation
The heat balance of CSTR can be divided into rate of heat generated:
and the rate of heat removed:
The expression for the rate of heat generated in case of a first-order reaction is given as:
The curve that describes Qg as a function of T is S-shape with a plateau at high values of T.
Qr as a function of T is given by a straight line expressed as:
Three possibilities of the simultaneous solution of mass and heat balances were identified:
¨ single stable state so that generation of minor perturbations in the operating conditions
do not cause changes in the steady-state condition
¨ two states, one stable and the other unstable
¨ three states, two stable and one, between the two, unstable
The necessary condition for absolute stability of the system is given by the expression:
10. Multiple reaction systems
Most organic systems consist of multiple reactions. Some of them are extremely complex.
Oxidation of organic molecules is a good example:
C7H8 + O2 C7H6O + H2O
C7H8 + O2 C6H6 + CO + H2O
C6H6 + 7.5O2 6CO2 + 3H2O
C7H8 + 9O2 7CO2 + 4H2O
Lets consider a simple system of two reactions in parallel:
A D r1 = k1CAa1
A U r2 = k2CAa2
Several definitions were adopted:
· point selectivity
· overall selectivity
· yield
In the case of two parallel reactions, the point selectivity is expressed as:
If a1 > a2 , the selectivity increases with increasing concentration of reactant A.
Therefore, the overall selectivity in PFR and BR would be higher than in CSTR. In the
opposite case, a1 < a2 , CSTR is the preferred choice.
Lets consider another simple system of two reactions in parallel:
A + B D r1 = k1CAa1CBb1
A + B U r2 = k2CAa2CBb2
The selectivity is defined as:
The most interesting case is a1 > a2 , b1 < b2 .A special reactor system would be
needed to maximize selectivity.
The temperature has a significant effect on selectivity through the dependency of k on
temperature determined by the activation energy. Going back to eqn. (3.4), if E1 > E2,
then the selectivity increases with temperature.
The general mass and energy balances have already been developed. In the case of a
single reaction, one mass balance was solved. If N components and M reactions are present
in the system, the maximum number of mass balances is equal to N. However this number
can be reduced to the number of independent equation in the system Mi. This number is
calculated from the matrix of the stoichiometric coefficients.
For example the reaction between CO, CO2 and H2:
CO + 2H2 CH3OH
CO2 + H2 CO + H2O
CO2 + 3H2 CH3OH + H2O
Clearly only two reactions are independent. Therefore only two mass balances are
necessary while all other components are expressed in terms of the two key components.
However writing five mass balances is correct, although it leads to a more complex system of
equations. The kinetic model may still require all three kinetic expressions.
Lets take a system of two first order reactions in series carried out in an isothermal batch
reactor:
A1 A2 A3
The two mass balances are expressed as:
The solution of eqn. (10.6) has been derived:
Substituting eqn.(10.8) into eqn. (10.7) and solving using the method of integrating factor
yields:
This expression gives a maximum selectivity to A2. The time at the maximum value of C2 is
calculated taking the derivative of eqn. (10.9):
The same expression is derived for a PFR. However a different expression for the maximum
is derived for a CSTR.
11. Kinetics of reactions
The rate of reaction is a basic constitutive relationship required for analysis and design of
reactors. Three general types of expressions have been mentioned: theoretical based on
mechanism, semi-empirical based kinetic models and empirical. One of the most important
types of kinetic mechanisms is the group of radical reactions that takes place in thermal
cracking reactions. Those mechanisms consist of a large number of reactions, included in
complex numerical codes. A typical example is the steam cracking for production of olefins.
Radical reactions are combined of three types of reactions. Each reaction is considered to be
elementary. Lets take the case of ethane cracking to illustrate the modeling of such systems:
1. initiation
2. propagation
3. termination
This system of reactions can be solved simultaneously, for a PFR, given all the rate
constants. However, this system can be greatly simplified if the assumption of negligible
rates of change of the intermediates is invoked:
Substituting the proper rates of reaction yields closed expressions for the intermediates:
Given the expressions in (11.7), the rates of change of the main components can be
introduced in the mass balances.
Another type of free radical reactions is polymerization, like polystyrene or polyvinylchloride
production:
The concentration of the initiator is calculated assuming that its rate of change is negligible:
Finally the rate of change of the monomers is:
Measurement and analysis of kinetic data are essential for derivation of kinetic expressions.
Data are measured in all types of reactors. Batch reactors are popular for liquid phase
reactions. However the rate changes with concentration. Therefore it changes with time.
Working with CSTR render direct measurement of rates feasible. PFR is used for gas-phase
reactions. In batch and PFR reactors types of measurements are possible:
· differential at low conversion levels so the rate can be considered approximately to
be constant
· integral at high conversion levels where analysis of the data is require to extract
the rate expressions
12. Residence time distribution
The analysis of chemical reactors has been based on ideal behavior: ideal mixing and plug
flow. This is an excellent first-order approximation for most cases of practical interest.
However, in many other cases, this approach is not satisfactory because of the type of flow
in the reactor, i.e. laminar flow, or problems in design and operation, i.e. flow bypass.
Therefore tools are needed to deal with those situations either to analyze such problems
(laminar flow) or to diagnose faulty operation. The most useful approach for solving those
problems is the application of principle of residence time distribution (RTD).
The flow in the reactors is assumed to consist of lumps of molecules called elements of flow.
Those elements are microunits of flow combined of a large number of molecules so
concentration and temperature are defined, but much smaller than reactor volume. The
simplest example in plug flow where all elements reside in the reactor identical time.
Measurements and analysis of RTD require the application of a tracer that defines the flow.
Its concentration is normally very small, it does not disturb the flow and can be clearly
identified. The tracer is injected at the inlet (only one) and is analyzed at the outlet (only one).
Two types of methods are normally used, although others are also employed:
· pulse function where the tracer is introduced in one shot at infinitely short time
defined by the Dirac function:
· step function where the tracer is injected at constant concentration::
Accordingly, two different functions are defined:
· residence time distribution function E(t)dt is the fraction of fluid elements at the
reactor outlet that resided in the reactor between time t and time t+dt.
· step function F(t) is the fraction of fluid elements at the reactor outlet that resided
in the reactor less than time t
There is a simple relationship between the two functions:
For PFR the functions can be defined:
The expression for E(t) is derived from a general mass balance on a pulse injection:
Since M is calculated from the mass balance, expression (6.6) becomes:
The expression for F(t) is derived from a general mass balance on a step injection:
Clearly each function can be calculated from the other function based on the relationship
(12.4).
Lets develop the two functions for CSTR using a step change:
It is important to develop the expression for the RTD functions:
Laminar flow is another case that can be solved using RTD. The derivation is based on the
parabolic velocity profile:
The function F(t) is defined as:
Finally substituting t for r yields:
Differentiating this expression yields
Calculation of conversion based on RTD requires the assumption of segregated flow
meaning no mass exchange between elements. Therefore each element is regarded as a
small batch reactor:
Is has already been demonstrated that a model of N CSTR in series covers the behavior
between CSTR and PFR. The derivation for this model in terms of RTD uses a pulse
function:
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