Reactors in the modelling of Combustion Processes
1. Thermal Theory of Explosions: The non-isothermal BATCH reactor.
Explosions are dynamical phenomena, i.e. phenomena in which the system state variables change very rapidly in time. In order to study the explosions we must abandon the idea of static equilibrium or quasi-static transformations, and introduce dynamical models of physical systems.
In this schematic BATCH reactor, the reactant R is loaded and let react for a certain amount of time, to obtain a determined amount of the product P. Model assumptions are:
The concentration of the reactant, C [mol/m3] and the temperature T [K] are uniform across the reactor volume V [m3], i.e. C and T only change with time
In the reactor, an irreversible chemical reaction RP occurs, with reaction rate r [mole/
(m3s)] which depends on C and T.
1.1 Isothermal reactor
Let us begin by treating the temperature T as a parameter, by keeping it constant with time. At each instant, the volume V of the reactor, multiplied by the concentration C, gives the amount of reactant contained in the reactor. As the reaction advances at speed r, the concentration C decreases and so does the total amount of reactant, according to the mass balance equation:
ˆ
V dC V r dt
(1.1) that is, per unit volume:
ˆ
dC r
dt
associated to the initial condition
0 0
C t C .
For a reaction of the first order, we have
r(C, T) C, T
ˆ
dC kC
dt
(1.2) that can be easily solved by separating the variables:
dC ˆ C kdt
ˆ
ln C kt c
ˆ exp ˆ exp exp ˆ
C t kt c c kt
The integration constant c is found by using the initial condition:
0 0 exp exp 0 exp 0
C C c c C
and thus the solution is:
0exp ˆ
C C kt
(1.3) The following figure reports C(ˆt), beginning from C = 1 for ˆt = 0 (sample initial condition), for three different values of the rate constant k, from left to right: k = 10, 1, 0.1:
0 2 4 6 8 10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The higher the rate constant k, the faster the decay. Higher temperatures correspond to higher reaction rates (-> faster reaction).
1.2 Non-isothermal reactor
We speak of explosions when, for a system, we observe a slow evolution that, all of a sudden, accelerates. This is possible for example with isothermal autocatalytic reactions, such as for example:
R P r P P
for which the rate equation is ˆR
R R P
r dC kC C
dt
In this case, the advancement of the reaction leads to an increasing reaction rate, with a positive feedback mechanism. We will explain why this happens. Because the total number of moles of R and P remain unchanged as R is consumed, we may write at any time
0 R0 P0 R P constant C C C C C
and the rate equation becomes
0
ˆR
R R R
r dC kC C C
dt
which, rearranging and breaking into partial fractions, yields:
0 0 0
1 ˆ
R R R
R R R R
dC dC dC
C C C C C C C kdt
and, integrating:
0 0 0
0 0 0
0 0 0
ˆ ˆ
ln R R ln R R R P
R R P P
C C C C C
C kt C C kt
C C C C C
Conversion-time and rate-concentration curves for autocatalytic reaction. This shape is typical for this type of reaction [from O. Levenspiel, Chemical reaction Engineering].
For an autocatalytic reaction to take place in a BATCH reactor, some minimal amount of product P must be present to make sure that the reaction rate is not zero. Starting with a very small concentration of P, it is seen that the reaction rate will rise as P is formed. At the other end, when R is totally consumed, the rate becomes zero. This is seen in the above figure, which shows the rate following a parabola, with a maximum where the concentrations of R and P are equal.
Look also at the shape of the curve in the conversion degree versus time diagram (at left in the figure). It shows a slow increase, then a faster increase, then a slower increase again. What is important is the transition slow -> fast (Low rate – High rate) that gives the system the explosive character.
The positive feedback mechanism may also exist, as an effect of the increase of the system temperature, when the system is non-isothermal and the reaction is exothermic. The effect is even more important, due to the nonlinear dependence of the reaction rate constant from the system temperature.
To study this phenomenon, we must derive the equations for a non-isothermal BATCH reactor.
Because the temperature is not constant, we write, for a first order reaction:
ˆ
dC k T C
dt
(1.2’) We need a second equation to describe the evolution of the temperature with time. The equation is the energy balance, i.e. the first law of thermodynamics written for a closed, constant volume system, no work exchange with the surroundings, for an infinitesimal time:
dU Q (1.4)
where Q is the heat transferred from the environment to the system. To evaluate the variation of the temperature, it is necessary to distinguish in the internal energy U the two contributions: the one related to changes in the system composition, and the one related to changes in the system temperature (sensible heat). To this aim, it is convenient to express the energy balance in terms of enthalpy. From the definition of enthalpy it is
U H pV
which, differentiated for a constant volume system, yields dU dH Vdp .
On the other hand pV N RTtot
(1.5) For simplicity we assume that the reaction occurs with no change in the total number of moles, i.e.
tot 0 dN
. By substituting we have
dU dH N RdT tot (1.6)
and, remembering Mayer’s formula, Rcp cV, we write
tot p V
dU dHN c c dT
The change of absolute enthalpy, dH, can be decomposed into two contributes:
1. The change due to the change in the chemical composition, given by
H dNr reag H VdCr reag
where Hr is the enthalpy of reaction per mole of reactant;
2. The change in the thermal agitation of the molecules (sensible heat) given by
tot p
N c dT
where for simplicity we assume constant molar specific heat capacity cp.
This formulation includes the case in which the reactant R is not the only chemical species in the initial mixture: we distinguish NR
from Ntot and, therefore, CR0 in general will be different (lower) from Ctot.
By observing that dC rdtˆ we have
ˆ
r tot p
dH H V rdt N c dT hence
r ˆ tot V
dU V H rdt N c dT Q
(1.7)
Divide by Vdtˆ, let Q Q dtˆ and, for a 1st order reaction, substitute r k T C and Ctot NVtot :
ˆ r
tot V tot V
dT H Q
k T C
dt C c VC c
to obtain ˆ
tot V r
dT Q
C c H k T C
dt V
(1.8) in which, if the reaction follows Arrhenius’ law for the dependence of k by the temperature, it is
0 Ea
k T k eRT
.
1.2.1 Non isothermal reactor – adiabatic case
Let us first consider adiabatic systems, i.e. systems for which Q 0 : ˆ 0
Ea
tot V r RT
C c dT H k e C dt
(1.9) with the initial condition
0 0
T T .
This differential equation is nonlinear and contains two variables, T e C. Thus the system is described by the following two nonlinear coupled equations:
0
0
ˆ ˆ
a
a
E RT
E
r RT
tot V
dC k e C
dt
dT H
k e C dt C c
(1.10)
with associated initial conditions:
0 0
0 0
C C
T T
Introduce now the two constants:
max r 0
tot V a
a
T H C
C c T E
R
(1.11) which have the following physical interpretation:
1) Tmax
is the maximum increase in temperature that can be obtained when there is no heat transfer to/from the environment, and the reactant is completely consumed to form products. The reactant is initially available at concentration C0, whereas the system is characterised by a total concentration Ctot. Here we note that, for a highly diluted system, i.e. C0 Ctot, the expected temperature increase will be small (most of the initial mixture is inert), while for concentrated systems i.e. C0 Ctot the initial mixture is mostly made of the reactant, and thus the temperature increase will be large.
2) Ta is a measure of the activation energy of the mixture: the higher Ta, the faster the transition from slow to fast evolution, and the more disruptive will be the explosion.
We can now define the following nondimensional dependent variables:
0 0
0 max
C C; T T
x C T
(1.12)
We can also scale and make nondimensional the independent variable:
0ˆ tk t
(1.13) and, by applying the chain derivation rule:
ˆ ˆ 0
d dt d d
dt dt dt k dt
after substitutions the equations become:
0 max
0 max
1 1
Ta
T T
Ta
T T
dx x e
dt
d x e
dt
(1.14)
with initial conditions
0 0 0
x .
In this case, it is evident that in the solution of our system , and the system can be describedx by a single nonlinear equation:
1 T0 TTamax
d e
dt
(1.15) Define now the two nondimensional parameters:
max
0 0
; a
T T
T T
to write
1 1
d e
dt
(1.15’) with the initial condition
0 0
.
Equation (1.15’) is nonlinear and can be solved numerically, as shown in the subsequent lecture.
Solutions for T0
= 300 K, Tmax
= 2000 K and for three different values of Ta
(from left to right:
1000, 1600 and 2200, corresponding to = 6.67 and = 3.33, 5.33 e 7.33 respectively) are reported in the following figure. It is seen that, as Ta
increases, the induction time increases as well, whereas the velocity of increase of the temperature is still high and does not change significantly with respect to what we compute for lower values of Ta
.
0 20 40 60 80 100
0 500 1000 1500 2000 2500
Systems presenting such slowfast transitions are the most dangerous, because the system looks
“calm” for a long time, and then it “explodes” unexpectedly causing damage, injuries and even loss of human lives. However it is reasonable to think that, if we can prevent too high an increase in temperature, we could prevent explosion. In order to study such possibility, we must describe the case in which the reactor is non adiabatic.
1.2.2 Non isothermal reactor – non adiabatic case
Let us go back for a moment to the dimensional form of the equations:
Ta=2200 K Ta
=1600 K Ta
=1000 K
ˆ 0
Ea
r RT
tot V tot V
dT H Q
k e C
dt C c VC c
(1.8’) Let us imagine that the removal of energy in the form of heat be represented by a linear term in the equation, proportional to the surface area of the reactor and to the difference between the reactor and the environment, through a global heat transfer coefficient U:
Q US T T
(1.16) For example, for a spherical reactor we have:
3 2;
6 SD V D
and the equation becomes
0
6 ˆ
Ea
r RT
tot V tot V
dT H U
k e C T T
dt C c DC c
(1.8”) Again, we use Tmax and T defined above, make nondimensional the independent variables, scalea and make nondimensional the independent variable, and introduce the nondimensional heat transfer coefficient:
0
6
V
U
Dk c
(1.17) to finally write:
0 max
0 max
1 1
Ta
T T
Ta
T T
dx x e
dt
d x e
dt
(1.18)
or equivalently, in the fully nondimensional form:
1
1
1 1
dx x e
dt
d x e
dt
(1.18’)
with the initial conditions
0 0 0
x .
The following figure reports the temperature-time history for Tmax
= 2000 K , Ta
= 1600 and three different values of , from left to right: 0 (the adiabatic case), 0.01 and 0.05.
0 20 40 60 80 100 0
500 1000 1500 2000 2500
For 0.01 the heat removed is not sufficient to avoid explosion: for 0.05 instead the explosion does not occur.
Note:
A critical value for exists, depending on the system parameters , and .
For the same set of physical and chemical parameters, if we only increase the size of the
reactor, i.e. if D , then 0
6 0
V
U
Dk c
. We conclude that the risk of explosion is higher for larger reactors.
Appendice A – programma MATLAB
% Réacteur chimique BATCH non isotherme non adiabatique
% réaction de première ordre, cinétique Arrhenius
% A -> B
% clear all;
close all;
%
% Paramètres du réacteur
%
beta=1; % Enthalpie de réaction sans dimension [default=1]
gamma=12; % Energie d'activation sans dimension, Ea/R*T_rif [default=10]
phi=0.00001; % coefficient d'échange de chaleur, sans dimension (default : 0.0001, 0.001)
%
teta_8=1; % température extérieure, sans dimension (0)
%
% Conditions initiales
%
x_0=1; % fraction molaire de A à t=0 teta_0=1; % température à t=0
%
% Définition des extrêmes d'intégration a=0; b=30000;
%
m=2000; % nombre des intervalles d'intégration dt=(b-a)/m; % amplitude de l'intervalle d'intégration x(1)=x_0; % Condition initiale
teta(1)=teta_0; % Condition initiale
gammabeta=gamma*beta;
%
for i=1:(m-1)
% Méthode semi-implicite k1=exp(-gamma/teta(i));
% x(i+1)=x_0;
x(i+1)=x(i)/(1+dt*k1);
teta(i+1)=(teta(i)+dt*(beta*k1*x(i+1)-phi*(-teta_8)))/(1+dt*phi);
end t=0:dt:b-dt;
subplot(211) plot(t,teta,'r') title('teta(t)') subplot (212) plot(t,x,'b') title('x(t)')
