Chemical Reaction Engineering - Notes, Exercises, Questions for the exam

ADIABATIC REACTORS

Reactors are considered adiabatic if:

●​ no heat is removed or released to the reactor → for example by thermally insulating

its walls

●​ the heat removed/released is negligible compared to the heat that is released/absorbed

by the reaction

The heat that is generated by the mixture, Q , remains inside the mixture determining its final

R

temperature T .

ADIAB

ADIABATIC PFR and BATCH

The energy balance, for a constant pressure batch reactor or a PFR reactor, under adiabatic

conditions in which a single reaction occurs is:

The mass balance, assuming a constant mixture density, is: = = ν

τ 1

It is therefore possible to combine the equations by inserting the expression in

=

τ

the energy balance →

Multiplying by the infinitesimal variation of time and expressing the derivative between the

two dependent variables we obtain:

Where:

●​ is a constant

i

●​ the concentration of the mixture and the specific heat of the mixture depend slightly

on the temperature but the dependence is usually weak so they can be considered

constants

●​ the heat of reaction depends slightly on temperature but the dependence is

∆

usually weak so it can be considered a constant

By enclosing all the constants in a single parameter it is possible to write .

= β

By separating the variables and integrating from the initial condition to the final condition

[ ]

→

∫ = β ∫ (τ) − (0) = β (τ) − (0)

0

0 84

The sign of is given by the sign of the constants that constitute it:

β

●​ can be either positive (i is a product) or negative (i is a reactant)

i

●​ concentration and specific heat are always positive

●​ can be:

∆

○​ negative for exothermic reactions

○​ positive for endothermic reactions

Generally, in a c -T phase plot, the graph is represented by a straight line:

i

In adiabatic reactors, the temperature variation is due to the reaction occurring. The phase

plot is constructed where the degree of progress is represented by the shift in operating

conditions on the lines.

If i is a reactant:

●​ and the reaction is exothermic → blue line

●​ and the reaction is endothermic → red line

If i is a product:

●​ and the reaction is exothermic → green line

●​ and the reaction is endothermic → yellow line

The phase plot T - X is:

A 85

●​ if the reaction is exothermic → green line

●​ if the reaction is endothermic → blue line

●​ if the reaction is carried out at constant temperature (isothermal case) → orange line

If the maximum conversion value is it means that the reaction is practically irreversible.

∽1

ADIABATIC CSTR

The energy balance for a CSTR under adiabatic conditions where a single reaction occurs is:

The mass balance, assuming a constant density of the mixture, for a single reaction is:

∆

= = ν

.

ϑ

It is possible to combine the equations by inserting the expression of R=cii in the energy

balance: .

By multiplying by and bringing the deltas to the left it is possible to obtain the following

relationship

(which when explained becomes:) ( )

0 .

= + β −

The case of a process running adiabatically is explained through the same formal expression

for all reactor types (CSTR, PFR and batch): ( )

0

= + β −

For simplicity, we consider a process in which a single, irreversible elementary reaction

∆

occurs. The mass balance is: . The balance solution appears to

→ =−

ϑ

be, looking from the "Simple solutions" file, and is replaced within the energy

= 1+()·ϑ

balance:

●​ the function to the left hand side is a linear function of temperature 86

●​ the function on the right hand side is a nonlinear (exponential) function of

temperature

→ the equation is not linear.

The graphs of the two functions are such that, in a region of the ϑ domain, there exist 3

solutions (3 values ​

of T that verify the equation.

To obtain these solutions it is possible to use the "fsolve" or "vpasolve" function of Matlab

(these solvers are used and not "ode" because the balances of matter and energy are two

algebraic equations).

Based on the starting guess (whether you start from small ϑ or high ϑ) Matlab is able to

identify only two sections of the complete graph:

●​ a region of high activity, where the c is very low (so the reactor had a high

A

conversion)

●​ a region of low activity, where c is close to the initial value (so conversion is low)

A

There is a region of ϑ in which the solutions overlap, but the points of the graph do not

connect. Considering for example an exothermic reaction (which causes an increase in

temperature:

By applying this approach it is therefore not possible to obtain the 3 solutions of the equation,

but only the two extreme ones. However, if we consider the residence time ϑ as the dependent

variable, there is no overlapping of solutions, so it is possible to obtain the 3 solutions with

Matlab:

The presence of a region in which there are 3 solutions indicates the existence of at least 3

different types of steady states for the CSTR (green, blue, red in the figure). 87

RIGOROUS ANALYSIS of the STABILITY of

STEADY STATES in an ADIABATIC CSTR

The choice of the best steady state to use is also based on the analysis of the stability of these

steady states: stability is a measure of the tendency of a reactor to remain in a certain

steady state.

Admitting the possibility that a stationary state can change is equivalent to taking into

account the variation over time of the variables c , T etc... → the material and mass balances

i

are carried out considering the partial derivatives with respect to time, for example:

From a practical point of view, after having resolved the balances, the value of the flow rate

supplied to the reactor is set and the characteristics of the steady state achieved are identified.

Subsequently it is possible to change the flow rate value and two behaviors can be observed:

●​ the oscillations due to the change in the steady state weaken over time until a new

stable steady state is reached

●​ the oscillations continue to increase → the steady state is not stable

INTUITIVE ANALYSIS of the STABILITY of

STEADY STATE in ADIABATIC CSTR

The analysis is extended to a CSTR reactor in which there is a flow rate of heat

absorbed/released to the outside. The energy balance is:

By bringing to the left what depends linearly on T and to the right what depends on T in a

non-linear way, assuming that an irreversible and elementary reaction (A → products) is

taking place in the reactor then it is possible to obtain an expression of using the

(ϑ)

'Simple solution': 88

●​ the function to the left of the equal is a straight line and represents the dissipated heat

Q DISS IN EXT

○​ the intercept depends on T and T

○​ consists of:

■​ convective heat exchange (with V )

FLOW

■​ heat exchange with the wall namely conduction (with u)

●​ the function on the right is a nonlinear function of temperature and represents the

heat generated by the Q reaction

R

The temperature at which the mass balance occurs is the one in which the two functions

intersect: (the figure shows the case in which the reaction is exothermic)

It is possible to have conditions such that there are 3 solutions (exactly as described by the

resolution using matlab). The three sections are identified:

IN

●​ low activity section → obtained from very low T

●​ intermediate section IN

●​ high activity section → obtained from very high T

It is assumed to have a positive and very small fluctuation in T inlet temperature.

IN

Depending on the section in which the CSTR is located it will react differently:

●​ low/high activity section:

1.​ by slightly increasing the temperature, the Q curve exceeds Q → more

DISS R

heat is dissipated than the reaction produces → the temperature decreases.

2.​ Once the temperature decreases the Q curve is above Q → the reaction

R DISS

creates more heat than the system can dissipate → the temperature increases

3.​ This process continues to occur unless corrections are applied from the

outside. What is certain is that the system will remain in the section it is in,

oscillating around its equilibrium point → stable steady state 89

●​ intermediate section

1.​ by slightly increasing the temperature, the Q curve exceeds that of Q →

R DISS

more heat is produced by the reaction than can be dissipated → the

temperature increases

2.​ the increase in temperature causes the condition to always be Q >Q →

R DISS

continuously increases the temperature → unstable steady state

3.​ the system stabilizes when the stable section is reached, in this case the one

with high activity

The stability condition is then summarized in the definition: a steady state is stable if

For a number of reactions > 1 the intersections between the dissipated heat and the reaction

heat can be more than 3, and the same thing happens for a reversible reaction. 90

QUALITATIVE ANALYSIS of DIFFERENT

OPERATIVE CONDITIONS

In the case of a reversible exothermic reaction, through adiabatic processes the conversion

has a lower maximum value compared to the isothermal process, since, thermodynamically,

an exothermic reaction is disadvantaged by high temperatures:

NB: the phase plot curves cannot cross the equilibrium conversion curve (thermodynamic

limit).

In the case of an endothermic reaction, the adiabatic process can reach a maximum

conversion of a lower value than the isothermal process, since, thermodynamically, an

endothermic reaction is unfavored by low temperatures:

Adiabatic processes are therefore less efficient from the point of view of the maximum

achievable conversion but have positive sides:

●​ From a plant point of view, the construction of an adiabatic reactor is simpler

●​ in the case of exothermic reactions, the reaction occurs faster thanks to the increase