Reactors in the modelling of Combustion Processes

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

[

]

 In the reactor, an irreversible chemical reaction R → P occurs, with reaction rate r

[

]

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 concentration C , multiplied by the volume V of the reactor, gives the amount of reactant (in moles) 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:

V dC

d ^t =−Vr (1.1)

where ^t is the dimensional time variable. Per unit volume, we have:

dC d ^t=−r

associated with the initial condition C(^t0)^=C0 .

r(C, T) C, T

For a reaction of the first order, we have dC

d ^t=−kC (1.2)

that can be easily solved by separating the variables:

dC

C =−kd ^t ln (C )=−k ^t +c C ( ^t )=e^{−k ^t +c}=e^ce^{−k ^t}

The integration constant c is found by using the initial condition:

C₀=C (0 )=e^ce⁰→ e^c=C₀ and thus the solution is:

C ( ^t )=C₀e^{−k ^t} (1.3)

The following figure reports C ( ^t ) , ^C=1 for ^t =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).

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

−r=−d C_R

d ^t =k C_RC_P

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

C₀=C_{R 0}+C_{P 0}=C_R+C_P=constant

and the rate equation becomes

−r=−d C_R

d ^t =k C_R(^C0−C_R)

which, rearranging and breaking into partial fractions, yields:

−d C_R C_R(^C0−C_R)⁼

−1

C₀

(

)

and, integrating:

lnC_{R 0}(^C0−C_R)

C_R(^C0−C_R) ^=ln

C_R/C_{R 0}

C_P/C_{P 0}=C₀k ^t=(^CR 0−C_{P 0})^{k ^t}

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. If the system at the beginning contains a very small concentration of P , it is seen that the reaction rate increases as P is formed. In the 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.

1.2 Non-isothermal reactor

The positive feedback mechanism may also exists 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 R → P : d C_R

d ^t =−r(^CR,T)^{=−k (T ) C}R (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 =δQ (1.4)

where δQ 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 follows that

U=H − pV

which, differentiated for a constant volume system, yields

dU =dH−Vdp− pdV . (1.5)

On the other hand

pV =N_totRT (1.6)

For simplicity, we assume that the reaction occurs with no change in the total number of moles, i.e. dNtot=0 . By substituting, we have

dU =dH−N_totRdT

and, remembering Mayer’s formula, ^R=´cp−´c_V , we write

dU =dH−N_tot(^c´p−´c_V)dT (1.7)

where ^c´p e ^c´V are the molar heat capacities. The change of absolute enthalpy dH can be decomposed into two contributes, namely:

1. The change ^dHc due to the change in the chemical composition, given by d H_c=(^{−∆ H}r)^{d N}R=(^{−∆ H}r)^{Vd C}R

where ∆ Hr is the enthalpy of reaction per mole of reactant;

2. The change d Hs in the thermal agitation of the molecules (sensible heat), given by d H_s=N_totc´_pdT

where for simplicity we assume constant molar heat capacity ^c´p .

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, CR 0 in general will be different (smaller) from ^Ctot .

By observing that, from (1.2’), d CR=−rd ^t , we have:

dH=dH_c+d H_s=(^{−∆ H}r)^{d C}R+N_tot´c_pdT =(^{−∆ H}r)^{V(−rd ^t)+N}totc´_pdT hence, by substituting into (1.7), we have:

dU =(^{−∆ H}r)^{V(−rd ^t)+N}tot´c_pdT −N_tot(^´cp−´c_V)^dT (1.7’) and then, substituting into (1.5) and simplifying:

∆ H_rVrd ^t+N_totc´_VdT =δQ (1.5’)

Divide now by Vd ^t

, let ^Q=´ δQ

d ^t , observe that ^Ntot=VCtot and, for a 1st order reaction, substitute r=k (T ) C and write:

C_totc´_V dT

d ^t=−∆ H_rk (T ) C+Q´

V (1.8)

Let us make the equations dimensionless. Define the conversion degree x=C_{R 0}−C_R

C_{R 0} →C_R=C_{R 0}(1−x )

and introduce the two dimensionless parameters

∆ T_max=−∆ H_r Ctotc´V

C_{R 0} (1.10)

β=∆ T_max

T0 , (1.11)

then define the nondimensional temperature as θ=T −T₀

∆T_max→T =T₀+∆T_maxθ=T₀(1+ βθ) (1.12)

By substituting into (1.8) and simplifying, we find dθ

d ^t=(1−x ) k (T )+ Q´

V C_totc´_V∆ T_max (1.8’)

in which, if the reaction follows Arrhenius’ law for the dependence of k by the temperature, it is

k (T )=k₀e

−Ea

RT . (1.13)

Introduce now the dimensionless parameter γ= E_a

R T0 (1.14)

and the dimensionless time variable

t=k₀t^ (1.15)

We can apply the chain differentiation rule d

d ^t (∙)=dt

d ^t

[

]

to write k₀ d

dtθ=(1−x) k₀e

−γ 1+ βθ

+ Q´

V C_totc´_V∆ T_max e.g.

d

dtθ=(1−x ) e

−γ 1+ βθ

+ Q´

k₀V C_tot´c_V∆ T_max (1.8”)

Now imagine that the thermal flow from the environment to the reactor, ´Q , is expressed as a linear term in the equation, proportional to the surface area S of the reactor and to the temperature difference between the environment and the reactor, ^T∞−T , by means of a global thermal exchange coefficient U :

Q=−US(T −T´ _∞) (1.16)

From (1.12) we have T −T_∞

∆ T_max=θ−θ_∞ hence (^{T −T∞})^{=∆ T}max(^θ−θ∞)

and then the heat transfer term becomes

−US ∆T_max(^θ−θ∞)

k₀V C_totc´_V∆ T_max

Introduce the dimensionless global heat transfer coefficient as

ϕ= US

k₀V C_tot´c_V (1.17)

The energy balance equation takes now its final form:

dθ

dt=(1−x ) e

−γ 1 +βθ

−ϕ(^θ−θ∞) ^(1.18)

Equation (1.18) is a nonlinear differential equation and contains two unknown variables, θ and x .

If we go back for a moment to the material balance equation (1.4) and substitute the dimensionless expressions introduced earlier for the dependent variables and the constants, we can easily write

dx

dt=(1−x )e

−γ 1+ βθ

(1.19) In their dimensional form, the material (1.4) and energy (1.8) balances were associated to the initial conditions:

C (0 )=C₀

T (0 )=T₀

which, by substitution of expressions (1.9) and (1.12), become

x (0 )=θ (0)=0 (1.20)

In summary, the dimensionless governing equations set is:

DISCUSSION OF PARAMETERS

The above-introduced parameters have the following physical interpretation:

 ^{∆ T}max=−∆ H_r

C_totc´_V C₀ is the maximum temperature increase obtained by total conversion of the reactant, present in concentration C0 , to the benefit of the whole system, characterized by total concentration ^Ctot , with no heat transfer to/from the environment. The ratio Ctot/C₀ is named dilution ratio; more diluted mixtures produce a smaller temperature increase.

 β=∆ T_max

T₀ is a measure of the thermal content of the reaction in terms of effects on the temperature: the larger β , the larger the temperature increase will be.

 γ= E_a

R T₀ is a measure of the activation energy of the reaction: the larger γ , the faster the transition from slow to fast reaction will be, and the worst consequences will follow from the explosion.

 ϕ= US

k₀V C_tot´c_V is the dimensionless global heat transfer coefficient. It depends on, among other things, the geometry of the reactor. Suppose we have a spherical reactor: we can write

S=π D²;V =π D³

6 → S

V=6 D and equation (1.17) becomes

ϕ= 6 U D k₀C_tot´c_V

For a given set of physical and chemical parameters, if we only change the size of the reactor, e.g. if D→ ∞ , then ϕ= 6 U

D k₀C_tot´c_V→ 0 . The risk of explosion is higher for reactors of larger size.

dx

dt=(1−x )e

−γ

1+ βθ (1.19)

dθ

dt =(1−x ) e

−γ 1 +βθ

−ϕ(^θ−θ∞) ^(1.18)

x(0)=θ(0)=0 (1.20)

1.2.1 Non isothermal reactor – adiabatic case

For this limit case, in the model we set ϕ=0 , and the equations become:

dx

dt=(1−x )e

−γ 1+ βθ

dθ

dt=(1−x ) e

−γ 1 +βθ

formally identical and associated to identical initial conditions. It appears that, in this case, θ ≡ x , and that the system is fully described by any of the two equations, equivalently:

dθ

dt=(1−θ )e

−γ 1+ βθ

(1.21) with the initial condition

θ (0)=0

Equation (1.21) is nonlinear and can be solved numerically, for example as described in Appendix A. Two solutions are shown in the figure, for β=8 and for γ =8 and γ =11 from left to right. It is seen that, with the growth of γ , the induction time increases, while the rate of growth of temperature does not change significantly from what is observed for lower values of γ .

0 500 1000 1500 2000 2500 3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 teta(t)

Systems that have slow-to-fast transitions are more dangerous, as the system seems quiet for a long time and then explodes unexpectedly causing damage, injuries, and even loss of human lives.

Apparently, however, it seems that if we could prevent the increase in temperature beyond a certain limit, we could prevent the explosion. To study this possibility, we need to resume the reactor model developed for the general, non-adiabatic case.

1.2.2 Non isothermal reactor – non adiabatic case

Let us go back to the general model:

dx

dt=(1−x )e

−γ

1+ βθ (1.19)

dθ

dt=(1−x ) e

−γ

1 +βθ−ϕ(^θ−θ∞) ^(1.18)

x (0 )=θ (0)=0 (1.20)

The following figure reports the two state variables as a function of time, for β=3 , γ =8 , and three different values of ^ϕ , from left to right: 0.001, 0.01 e 0.1:

0 500 1000 1500 2000 2500 3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 x(t) et teta(t)

For ϕ=0.001 the heat removed is insufficient to avoid the explosion; for ϕ=0.01 the behavior maintains an explosive character; for ϕ=0.1 the explosion is not observed.

Remarks:

 There exists a critical value of ^ϕ , depending on the system parameters;

 For any given set of parameter values, if only the size of the reactor is increased, i.e. if D→ ∞ , then ϕ= 6U

D k₀c_V→ 0 . The risk of explosion is larger for larger reactors.

Appendix A – Non-adiabatic BATCH reactor The equations to integrate numerically are:

dx

dt=(1−x )e

−γ 1+ βθ

(1.19) dθ

dt=(1−x ) e

−γ 1 +βθ

−ϕ(^θ−θ∞) ^(1.18)

x (0 )=θ (0)=0 (1.20)

If we write semi-implicit finite-difference schemes, we have:

x^{n +1}−xⁿ

∆ t =(1−x^{n +1})e

−γ 1+β θⁿ

θⁿ⁺¹−θⁿ

∆ t =(1−xⁿ⁺¹)e

−γ 1+ β θⁿ

−ϕ(^θn+1−θ∞)

hence:

xⁿ⁺¹=xⁿ+e

−γ 1 +β θⁿ∆ t 1+e

−γ 1 +β θⁿ∆t θ^{n +1}=θⁿ+(1−xⁿ⁺¹)e

−γ

1+ β θⁿ∆ t+ϕθ_∞∆ t 1+ϕ ∆ t

to be solved in the order, first xⁿ⁺¹ and then θ^{n +1} as they appear above¹. A sample MATLAB code follows:

Appendix A – MATLAB code

% BATCH chemical reactor, non-isothermal, non-adiabatic

% first-order reaction, Arrhenius kinetics

% A -> B

%

clear all;

close all;

%

% Parameters of the reactor

%

beta=1; % Dimensionless reaction enthalpy [default=1]

gamma=12; % Dimensionless activation energy, Ea/R*T_rif [default=10]

1It is not so evident but, having chosen a semi implicit formula, the two equations must be solved in the order, at each time step, x first and then θ . The reason being that right hand sides must be evaluated at the same time level.

If we computed θ^{n +1} first and then xⁿ⁺¹ , we would be forced to evaluate the source term in the θ

equation by using (1−x ) at time level n – because we have not computed xⁿ⁺¹ yet – whereas we would be computing the conversion change with the term (1−x ) evaluated at time n+1 . The conversion change would be misaligned time wise, with respect to the temperature change, and this would introduce a systematic error.

On the other hand, if we computed the change in x with (1−x ) evaluated at time level n , we would be using ax explicit formula, numerically unstable al large values of ∆ t .

phi=0.00001; % Dimensionless global heat transfer coefficient (default : 0.0001, 0.001)

%

teta_8=1; % Dimensionless external temperature (0)

%

% Initial conditions

%

x_0=0; % conversion at t=0 teta_0=0; % temperature at t=0

%

% Definition of the extremes of integration a=0; b=3000;

%

m=2000; % number of integration intervals

dt=(b-a)/m; % amplitude of the integration interval

%

t(1)=0;

x(1)=x_0; % initial conversion, dimensionless

teta(1)=teta_0; % initial temperature, dimensionless

%

for i=1:(m-1)

% Semi-implicit method t(i+1)=t(i)+dt;

k1=exp(-gamma/(1+beta*teta(i)));

x(i+1)=(x(i)+k1*dt)/(1+k1*dt);

teta(i+1)=(teta(i)+(1-x(i+1))*k1*dt+phi*teta_8*dt)/(1+phi*dt);

end hold on

% plot(t,teta,'r') plot(t,x,'g') plot(t,teta,'r')

title('x(t) and teta(t)')