Fundamentals of reaction engineering worked exam

Homogenepus reactions - Isothermal reactorsWorked Examples - Chapter I Homogeneous reactions - Isothermal reactors Problem 1.1 reactor, starting with the mass balance equation and assu

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Rafael Kandiyoti

Fundamentals of Reaction Engineering

Worked Examples

Fundamentals of Reaction Engineering – Worked Examples

© 2009 Rafael Kandiyoti & Ventus Publishing ApS

ISBN 978-87-7681-512-7

Fundamentals of Reaction Engineering – Worked Examples

Contents

Contents

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Fundamentals of Reaction Engineering – Worked Examples I Homogenepus reactions - Isothermal reactors

Worked Examples - Chapter I Homogeneous reactions - Isothermal reactors

Problem 1.1

reactor, starting with the mass balance equation and assuming plug flow, derive an expression for calculating the reactor volume in terms of the molar flow rate of reactant ‘A’

Solution to Problem 1.1a: Assuming plug flow, the mass balance over a differential volume element of

a tubular reactor is written as:

Taking the limit as 'V R 0

A A R

A n

dn V

r

 ³

s-1, calculate the reactor volume and the residence time required to achieve a fractional conversion of 0.95.The rate constant k=0.6 s-1

Solution to Problem 1.1b: The molar flow rate of reactant “A” is given as,

negligible, calculate the volume of reactor required for a fractional conversion of 0.85

Fundamentals of Reaction Engineering – Worked Examples I Homogenepus reactions - Isothermal reactors

Data:

R, the gas constant : 8.314 J mol-1 K-1 = 0.08314 bar m3 kmol-1 K-1

P the total pressure : 2 bar

n T0 total molar flow rate at the inlet : 1000 moles s-1= 1 kmol s-1

k, reaction rate constant : 10 s-1

Let us see by how much the total volumetric flow rate changes between the inlet and exit of this reactor First let us review how the total molar flow rate changes with conversion:

n I = n I0 for the inert component

n A = n A0 – n A0 x A for the reactant

n S = n S0 + 2 n A0 x A for the product

The difference is far from negligible! In dealing with gas phase reactions, rates are often expressed in terms of

partial pressures: r A = kp A, as we will see in the next example

Fundamentals of Reaction Engineering – Worked Examples I Homogenepus reactions - Isothermal reactors

Problem 1.3

The gas phase thermal decomposition of component A proceeds according to the chemical reaction

1 2

k k

with the reaction rate r A defined as the rate of disappearance of A: r A k C 1 Ak C C 2 B C.The reaction will be carried out in an isothermal continuous stirred tank reactor (CSTR) at a total pressure of 1.5 bara and a temperature of 700 K The required conversion is 70 % Calculate the volume of the reactor necessary for a

ctant feed rate of 4,000 kmol hr-1 Ideal gas behavior may be assumed

Fundamentals of Reaction Engineering – Worked Examples I Homogenepus reactions - Isothermal reactors

Solution to Problem 1.3 Using the reaction rate expression given above, the “design equation” (i.e isothermal mass balance) for the CSTR may be written as:

n C

We next write the mole balance for the reaction mixture:

0 0

x x

2 0

Fundamentals of Reaction Engineering – Worked Examples I Homogenepus reactions - Isothermal reactors

Problem 1.4

The liquid phase reaction for the formation of compound C

C B

with reaction rater A1 k C C1 A B, is accompanied by the undesirable side reaction

D B

components are preheated separately to the reactor temperature, before being fed in The pressure drop across the reactor may be neglected The volumetric flow rate may be assumed remain constant at 22 m3 hr-1

The desired conversion of A is 85 % However, no more than 20 % of the amount of “A” reacted may be lost through the undesirable side reaction Find the volume of the smallest isothermally operated tubular reactor, which can satisfy these conditions

Data

k 1 = 2.09u1012 e-13,500/T m3s-1kmol-1 ; (T in K)

k 2 = 1017 e-18,000/T m3s-1kmol-1 ; (T in K)

Solution to Problem 1.4

reacted through Reaction 2 Not allowing more than 20 % loss through formation of by-product “D” implies:

4 e

10

10 09

¯ ¿ Solving,

4500

370 12.16

Fundamentals of Reaction Engineering – Worked Examples I Homogenepus reactions - Isothermal reactors

Problem 1.5

A first order, liquid phase, irreversible chemical reaction is carried out in an isothermal continuous stirred tank

reactor Ao products The reaction takes place according to the rate expression r A = k C A In this expression,

r A is defined as the rate of reaction of A (kmol m-3 s-1), k as the reaction rate constant (with units of s-1) and C A

as the concentration of A (kmol m-3) Initially, the reactor may be considered at steady state At t = 0, a step increase takes place in the inlet molar flow rate, from n A0,ss to n A0 Assuming V R (the reactor volume) and v T

(total volumetric flow rate) to remain constant with time,

a Show that the unsteady state mass balance equation for this CSTR may be expressed as:

A0 A

b Solve the differential equation above [Part a], to derive an expression showing how the outlet

molar flow rate of the reactant changes with time

c How long would it take for the exit flow rate of the reactant A to achieve 80 % of the change between the original and new steady state values (i.e from nA,ss to nA) ?

n A0,ss steady state inlet molar flow rate of reactant A before the change : 0.1 kmol s-1

n A0 new value of the inlet molar flow rate of reactant A : 0.15 kmol s-1

V R reactor volume : 2 m3

k reaction rate constant : 0.2 s-1

n A,ss steady state molar flow rate of reactant A exiting from the reactor, before the change,

kmol s-1

n A the molar flow rate of reactant A exiting from the reactor at time t, kmol s-1

t time, s

W average residence time, V R /vT , s

r A rate of reaction of A , kmol m-3 s-1

Fundamentals of Reaction Engineering – Worked Examples I Homogenepus reactions - Isothermal reactors

º and write the complementary and particular solutions for the first order ordinary

differential equation derived above

n A,complementary = C 1 e -E t ; n A,particular = C 2 (const)

where the solution is the sum of the two solutions: nA (t) = nA,c (t)+ nA,p (t).We substitute the particular solution into differential equation to evaluate the constants:

0 2

An C

E W .

EW

initial condition: n A =n A,ss at t=0, into the general solution:

Fundamentals of Reaction Engineering – Worked Examples I Homogenepus reactions - Isothermal reactors

The relationship linking t, nA,ss and nA is derived from the solution of the differential equation

For 80 % of the change to be accomplished: nA = nA,ss + 0.8 (nA,final - nA,ss)

nA= 0.8 nA,final + 0.2 nA,ss = 0.8(0.03) + 0.2 (0.02) = 0.028 kmol s-1

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Fundamentals of Reaction Engineering – Worked Examples II Homogeneous reactions - Non-isothermal reactors

Worked Examples - Chapter II Homogeneous reactions – Non-isothermal reactors

2nd term: heat generation/absorption by the reaction(s)

3rd term: sensible heat exchange Steady state: no accumulation

Fundamentals of Reaction Engineering – Worked Examples II Homogeneous reactions - Non-isothermal reactors

the adiabatic temperature rise for 100 % conversion

The mole fraction of A in the input stream yA0 = (nA0/nT0) = 1

The total heat capacity, C p = 80 J mol -1 K -1

r 0

p

80 C

'



Problem 2.2

in an adiabatic CSTR; (ii) for an exothermic reaction carried out in an adiabatic CSTR

Solution to Problem 2.2a:

Fundamentals of Reaction Engineering – Worked Examples II Homogeneous reactions - Non-isothermal reactors

Problem 2.3

The gas phase reaction

1 2

k k

with reaction rate expression r A k p 1 Ak p 2 Bwill be carried out in a continuous stirred tank reactor The feed

is pure A The required conversion of x A is 0.62 Calculate

a) the minimum reactor size in which the required conversion can be achieved, and,

b) the amount of heat to be supplied or removed for steady state operation

Ideal gas behaviour may be assumed

DATA:

k 1= 103exp{-41570 / RT} kmol m-3 s-1 bar-1

k 2 = 106exp{-83140 / RT} kmol m-3 s-1 bar-1

Activation energies are given in kJ kmol-1

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Fundamentals of Reaction Engineering – Worked Examples II Homogeneous reactions - Non-isothermal reactors

Solution to Problem 2.3 The minimum reactor size for achieving a given conversion implies operating at the temperature where the reaction rate is a maximum The reaction rate expression is written as:

dr

0 dT

1 x

dk / dT 1

3 1

e 2000 1

1 1 x

1 ln

5000 T

Fundamentals of Reaction Engineering – Worked Examples II Homogeneous reactions - Non-isothermal reactors

Problem 2.4

A continuous stirred tank reactor is used for carrying out the liquid phase reaction:

The reaction rate expression is r A = k(T) C A C B kmol h-1 m-3 The total molar flow rate at the inlet is 3 kmol h

-1 The inlet flow is equimolar in A and B The total volumetric flow rate (vT) is: 1 m3 h-1 Calculate the operating temperature required for a fractional conversion of 0.7 Also calculate how much heat must be added to the reactor to maintain the system at steady state, if the feed is introduced at the reactor operating temperature

The vessel is designed to withstand the vapour pressure of the reaction mixture No reaction takes place in the vapour phase Density changes due to (i) the chemical reaction and (ii) changes in temperature may be ignored Data:

Reaction rate constant, k(T), : 1015 e-13000/T m3 kmol-1 h-1 (T in K)

1

Fundamentals of Reaction Engineering – Worked Examples II Homogeneous reactions - Non-isothermal reactors

Problem 2.5

rate expression is given by the equation: r A = k(T) p A, where k(T) is the reaction rate constant and pA the partial pressure of the reactant A The feed stream is pre-heated to 600 K; it contains an equimolar amount of reactant A and inert component D and The total pressure is 1 bar; it may be assumed constant throughout the system Calculate the steady state reactor operating temperature and the conversion of the reactant

(Note: There are two equally valid solutions to this problem One of the two solutions will suffice as an

adequate answer.) Ideal gas behaviour may be assumed You may also assume total heat capacities of the feed and product streams to be equal

Data:

Reactor volume, VR : 10 m3

Inlet total molar flow rate, nT0 : 10 kmol s-1

Reaction rate constant, k(T) : 1.0 x 1020 e-30,000/T kmol bar-1 m-3 s-1 (T in K)

Heat of reaction, ' Hr : 48 000 kJ (kmol A reacted)-1

Heat capacity of the feed stream, C p : 200 kJ kmol-1 K-1

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Fundamentals of Reaction Engineering – Worked Examples II Homogeneous reactions - Non-isothermal reactors

Solution to Problem 2.5 The mass and energy balances must be solved simultaneously The energy balance equation:

n T0 : total inlet molar flow rate

From the energy balance equation

k p .Next we need to calculate nT as a function of xA.

Fundamentals of Reaction Engineering – Worked Examples II Homogeneous reactions - Non-isothermal reactors

The rate of the first order irreversible gas-phase reaction, A o B is given by rA = k pA The reaction will be

carried out in a tubular adiabatic reactor The reaction is exothermic Calculate the reactor volume, V R (m3),

Definitions & data:

Inlet mole fractions: y A0 = 0.20; y I0 (inerts) = 0.8

p

p A partial pressure of A, bar

n A0 molar flow rate of A, kmol s-1

Note 1 The operating line for a tubular reactor assumed to operate in plug flow is given by:

A ,exit

n A R

dn V

r

 ³

Fundamentals of Reaction Engineering – Worked Examples II Homogeneous reactions - Non-isothermal reactors

wherer A kp A k( 1 x ) p A A0 , V R is the reactor volume (m 3 ), n A the molar flow rate of “A” (kmol m-3 s-1)

and p A0 the inlet partial pressure of “A” (bar)

plug flow [i.e Q=0]:

Fundamentals of Reaction Engineering – Worked Examples III Catalytic reactions - Isothermal reactors

Worked Examples - Chapter III Catalytic reactions – Isothermal reactors

Problem 3.1

The first order irreversible gas phase reaction

is to be carried out in an isothermal tubular fixed bed catalytic reactor operating at atmospheric pressure The

intrinsic reaction rate expression is r A k C A Pure “A” will be fed to the reactor at a rate of 0.13 kmol s-1

a Compare the relative magnitudes of the rates of external mass transport and chemical reaction

b Calculate the mass of catalyst required for a conversion of 80 %

reactor Plug flow and ideal gas behaviour may be assumed The pressure drop along the length of the reactor may be neglected

Data:

Reaction rate constant, k: 0.042 m3 (kg-cat s)-1

Average gas density, Ug: 1.0 kg m-3

Average gas viscosity, μg : 5.0 x 10-5 kg m-1 s-1

Average diffusivity of “A”, D A : 5.0 x 10-5 m2 s-1

External surface area

of catalyst pellets, am: 0.667 m2 kg-1

Bed void fraction, HB: 0.5

Mass flow rate, G : 8.3 kg m-2 s-1

R (gas constant) = 8.314 kJ kmol-1 K-1 = 0.08314 bar m3 K-1 kmol-1

Solution to Problem 3.1