Thermal theory: the batch reactor model.
DISCONTINUOUS (BATCH) CONSTANT VOLUME REACTOR
Explosions are dynamic phenomena, that is, system parameters change rapidly with time. To study explosions we must abandon the idea of static equilibrium or quasi-static transformations and introduce time-dependent models of our physical systems.
In this ideal reactor, the reactant R is loaded and let react for a certain amount of time, to obtain some amount of product P.
Composition of the reactant C [mole/m3] and temperature T [K] are uniform in the reactor volume V [m3] and change with time, with rate r [mole/(m3s)] depending on C, T
The degree of conversion depends on the reaction time
The degree of conversion does not depend on the reactor volume
ISOTHERMAL REACTOR
Let us first treat temperature T as a parameter and keep it constant with time. At any time, the content of reactant in the reactor is given by the volume of the reactor V multiplied by the concentration C. As the reaction proceeds at rate r, the concentration C decreases and so does the content of reactant, according to the equation
V dC V r dt that is, per unit volume:
dC r
dt
that must be solved together with the initial condition C t 0 C0. Let us assume that the reaction kinetic be first order, that is
r kC The simple differential equation is then
dC kC
dt
C, T r(C, T)
that can be easily solved by separating the variables:
dC kdt C
ln C kt const
exp exp exp
C t kt const const kt
We can now use the initial condition to determine the integration constant:
0 0 exp exp 0 exp 0
C C const const C and the solution is:
0exp C C kt
The figure reports C(t), starting from C = 1 at t = 0, for three different values of the reaction rate constant k, namely 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 quicker the decay. Higher temperatures mean higher values of k.
NON-ISOTHERMAL REACTOR
Explosions occur when a system is evolving slowly, then evolution suddenly accelerates. This is clearly possible when the system is not isothermal and the reaction is exothermic. Then, evolution of reaction leads to increase in temperature, and this accelerates the reaction rate with a positive feedback mechanism. To illustrate this mechanism, we now derive equations for a non-isothermal reactor.
Since now the temperature is no longer constant in our model, we write, for a first order kinetic:
dC
k T C dt
We also need an equation to describe the evolution of temperature T with time. The equation is the energy balance, i.e. the first law of thermodynamics, written for a closed system over an infinitesimal time:
dU Q
where Q is the heat transferred to the system from the environment. If we express the change in internal energy dU as a combination of the change in internal energy due to the composition change, –Hr(VdC) where Hr is the enthalpy of reaction, negative for exothermic reactions, and of the change in internal energy due to thermal agitation of molecules (sensible heat), Vc dTV , by observing that dC = – rdt, we have:
r V
V H rdt Vc dTQ
Divide by Vdt and introduce Q Q
dt
, and substitute r k T C
to obtain:
r
V V
dT H Q
k T C
dt c Vc
If the reaction follows the Arrhenius law for temperature dependence, we have
0
Ea
k T k eRT
ADIABATIC REACTOR
We first consider adiabatic systems, i.e. Q 0. We have
0 Ea
r RT
V
dT H
k e C dt c
with initial condition T 0 T0. This differential equation is nonlinear, and contains two variables, T and C. Thus the problem is described by two coupled nonlinear differential equations:
0
0
a
a
E RT
E
r RT
V
dC k e C
dt
dT H
k e C dt c
with initial conditions
0 0; 0 0
C C T T Now we introduce the following constants:
0
max r ; a a
V
H C E
T T
c R
The two parameters have the following physical interpretation: Tmax is the maximum temperature increase that can be obtained for complete reaction and no heat losses from the reactor; Ta is a measure of the activation energy of the reaction: the higher Ta, the sharper the transition from slow reaction to fast reaction. We can now make nondimensional the dependent variables, as
0 0
0 max
C C; T T
x C T
then scale and make nondimensional the independent variable as k0t
and substitute, so that the equations become
0 max
0 max
1 1
Ta
T T
Ta
T T
dx x e
d
d x e
d
with initial conditions
0
0 0x
.It is self-evident that, in this case, x and that the system corresponds to the single nonlinear equation:
1
T0 TTamaxd e
d
with initial condition
0 0
.The system can be solved numerically. A MATLAB sample code is reported in Appendix A.
Solutions for Tmax= 2000 K and three different values of Ta (from the leftmost to the rightmost:
1000, 1600 and 2200) are reported in the following figure.
0 20 40 60 80 100
0 500 1000 1500 2000 2500
It is seen that for lower values of Ta the induction time becomes larger, while the rate of temperature rise during explosion is still high and does not change significantly with respect to what observed for lower values of Ta.
Systems showing transition slowfast are more dangerous because the system seems not to increase its temperature for a long time, and then suddenly explodes, creating unexpected damage, possible injuries and loss of lives.
It also seems that, if we could prevent the temperature to achieve too high a value, we could prevent explosion. In order to investigate this, we must describe the non-adiabatic reactor.
NON-ADIABATIC REACTOR
Going back to the dimensional energy balance equation, and assuming Arrhenius temperature dependence of the reaction rate constant:
0 Ea
r RT
V V
dT H Q
k e C
dt c Vc
We imagine that that the heat removal from the system be a linear term, proportional to the exposed surface of the reactor vessel and to the temperature difference between the vessel and the surrounding environment, via a global heat transfer coefficient h:
Q hS T T For a spherical vessel:
3 2;
6 SD V D and the equation becomes
0
6
Ea
r RT
V V
dT H h
k e C T T
dt c Dc
Again, we use the following constants:
max r 0; a a
V
H C E
T T
c R
make nondimensional the dependent variables, as
0 0
0 max
C C; T T
x C T
then scale and make nondimensional the independent variable as
k0t
and introduce the nondimensional heat transfer coefficient
0
6
V
h
Dk c
to write
0 max
0 max
1 1
Ta
T T
Ta
T T
dx x e
d
d x e
d
with initial conditions
0
0 0x
.The following figure reports temperature as a function of time for Tmax= 2000 K , Ta = 1600 and three different values of , from left to right: 0, 0.01 and 0.05.
0 20 40 60 80 100
0 500 1000 1500 2000 2500
For 0 the reactor is adiabatic. For 0.01 the heat removal is insufficient to avoid explosion.
For 0.05 explosion will not occur.
Remarks
There exists a critical value for , depending on the system parameters.
Note that, for the same chemical and physical parameters, if only the size of the reactor is increased (D ) then
0
6 0
V
h
Dk c . The risk of explosion increases for larger vessels.
Appendix A – MATLAB code
% 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)')
