Mann U. Principles of Chemical Reactor Analysis and Design: New Tools for Industrial Chemical Reactor Operations

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ENERGY BALANCES

This chapter covers the fourth fundamental concept used in the analysis and design of

chemical rea ctors—conservation of energy (the ﬁrst law of thermodynamics). We use

energy balances to express the temperature variations during reactor operations.

Section 5.1 provides a brief overview of basic thermodynamic quantities and relations

used in reactor design. We deﬁne the heat of reaction and equilibrium constant of a

chemical reaction and discuss how they vary with temperature and pressure. In

Section 5.2, we apply the ﬁrs t law of thermodynamics to deriv e the energy balance

equation for closed and open systems. W e reduce these equations to dimensionless

forms and derive the energy balance equations for the three ideal reactorconﬁgurations:

ideal ba tch r ea ctor, plug-ﬂo w rea ctor, and continuous stirred-tank reactor (CSTR).

5.1 REVIEW OF THERMODYNAMIC RELATIONS

5.1.1 Heat of Reaction

The heat of reaction, or more accurately, the enthalpy change during a chemical

reaction, DH

, indicates the amount of energy being absorbed or released when a

chemical transformation takes place at given operating atmosphere and temperature

of 298 K. The standard molar heat of reaction of a chemical reaction (expressed in

energy per mole extent) is denoted by D

298

. It is calculated by

298

(5:1:1)

Principles of Chemical Reactor Analysis and Design, Second Edition. By Uzi Mann

131

where

is the standard molar heat of formation of species j at 298 K, and s

is the

stoichiometric coefﬁcient of species j. The superscript “o” indicates a standard

pressure of 1 atm. The values of

’s of many species are tabulated in thermodyn-

amic textbooks and handbooks. We adopt the common convention in applying the

ﬁrst law of thermodynamics—heat added to a system is positive and heat removed

from a system is negative. Therefore, for exothermic reactions, D

is negative,

and, for endothermic reactions, D

is positive.

The heat of reaction is a function of temperature and pressure. Since enthalpy is a

state quantity, we relate the heat of reaction at any temperature and the standard

pressure of 1 atm, to the standard heat of reaction at 298 K by

(T) ¼ D

298

dT (5:1:2)

where

is the speciﬁc molar heat capacity of species j. Note that Eq. 5.1.2 rep-

resents the calculation of D

by cooling the reactants from T to 298 K, carrying

out the reaction at 298 K, and then heating up the products from 298 K to T,as

shown schematically in Figure 5.1. Once the heat of reaction at temperature T

and at standard pressure (1 atm) is known, we can calculate the heat of reaction

at T and any pressure P by

(T, P) ¼ D

(T ) þ

 T





dP (5:1:3)

Figure 5.1 Determination of the standard heat of reaction at temperature T.

132 ENERGY BALANCES

where V

is the speciﬁc molar volume of species j. The term inside the integral indi-

cates the change in the enthalpy of species j due to the change in pressure at temp-

erature T. In most applications, this term is small, and, unless the reactor is operated

at a very high pressure (several hundred bars or higher), we can approximate the

heat of reaction by DH

(T ), the heat of reaction at 1 atm.

Example 5.1 Determine the heat of combustion of n-octane at 6208C and

1 atm. The heats of formations of the various species and their heat capacities

are given below. Data: The standard heats of formation of the species

(at 298 K) are

(g) ¼208:70 J=mol

(g) ¼393:509 J=mol

O(g) ¼241:818 J=mol

The speciﬁc molar heat capacities

(assumed constant) are

(g) ¼254 J=mol K

(g) ¼30:2J=mol K

(g) ¼40:3J=mol K

O(g) ¼34:0J=mol K

Solution The chemical reaction is

(g) þ 12:5O

(g) ! 8CO

(g) þ 9H

O(g)

and the stoichiometric coefﬁcients are

¼1 s

¼12:5 s

¼ 8 s

¼ 9 D ¼ 3:5

We determine the standard heat of reaction by Eq. 5.1.1, noting that the heat of

formation of an element is, by deﬁnition, zero:

298

¼ (1)

þ (12:5)

þ (8)

þ (9)

¼ (1)(208,750) þ (12:5)(0) þ (8)(393,509) þ (9)(241:818)

¼5115:68 kJ=mol (a)

5.1 REVIEW OF THERMODYNAMIC RELATIONS 133

Since DH

is negative, the reaction is exothermic. Next, we determine the heat of

reaction at 6208C by using (a) and Eq. 5.1.2. For this case, the individual species

heat capacities are constant; therefore, the term inside the integral is independent

of temperature:

¼ (1)(254) þ (12:5)(30:2) þ (8)(40:3) þ (9)(34:0)

¼3:10 J=mol K (b)

Substituting the values from (a) and (c) into Eq. 5.1.2,

(893 K) ¼5115:68  10

893

298

(3:10) dT ¼5117:52 kJ=mol (c)

Note that, in this case, because the difference between the heat capacity of the

reactants and the heat capacity of the products is small, the heat of reaction

changes very slightly with temperature.

5.1.2 Effect of Temperature on Reaction Equilibrium Constant

The equilibrium constant, K, of a chemical reaction relates to the heat of reaction by

d(ln K)

¼

(T)

(5:1:4)

To express how the reaction equilibrium constant varies with temperature, we

separate the variables and integrate:

K(T)

K(T

)



¼

(T)

dT (5:1:5)

In many instances, the heat of reaction is essentially constant, and Eq. 5.1.5

reduces to

K(T)

K(T

)



¼

)





(5:1:6)

The equilibrium constant, K, relates to the composition of the individual species at

equilibrium. For a general chemical reaction of the form bB þ cC ! rR þ sS,

K ¼

(5:1:7)

where a

is the activity coefﬁcient of species j.

134 ENERGY BALANCES

5.2 ENERGY BALANCES

The ﬁrst law of thermodynamics is concerned with the conservation of energy. In

its most general form, it is written as the following statement:

Rate

energy

enters

system by

streams

;

Rate heat

energy

enters

system

;

Rate

energy

leaves

system by

streams

;

Rate

work

done by

system

;

Rate

energy

accumulates

system

;

(5:2:1)

When applying Eq. 5.2.1, we have to consider all forms of energy and account for

all types of work. We usually derive simpliﬁed and useful relations by imposing

certain assumptions and incorporating other thermodynamic relations (equation

of state, Maxwell’s equations, etc.) into the energy balance equation. Therefore,

it is essential to identify all the assumptions made and examine whether they

are valid.

Let E denote the total amount of energy, in all its forms, contained in a system,

and let e denote the speciﬁc energy (energy per unit mass). In most chemical pro-

cesses, electric, magnetic, and nuclear energies are negligible. Therefore, we

restrict the treatment of the ﬁrst law of thermodynamics to systems where energy

is present in three forms: internal energy (U ), kinetic energy (KE), and potential

(gravitational) energy (PE); hence,

E ¼ U þ KE þ PE (5:2:2)

The speciﬁc energy (energy per unit mass) is

e ¼ u þ

þ gz (5:2:3)

where u is the speciﬁc internal energy, v is the linear velocity, and z is the vertical

elevation.

Below, we apply the energy balances for macroscopic systems. First, we derive

the energy balance equation for closed systems (batch reactors) and then for open

systems (ﬂow reactors). Microscopic energy balances, used to describe point-to-

point temperature variations inside a chemical reactor, are outside the scope of

this book.

5.2 ENERGY BALANCES 135

5.2.1 Batch Reactors

For batch reactors (closed systems), the energy balance is written in the common

form of the ﬁrst law of thermodynamics:

DE(t) ¼ Q(t)  W(t)(5:2:4)

where DE(t) ¼ E(t) 2 E(0) is the change in the energy of the system, Q(t) is the

heat added to the system, and W(t) is the work done by the system on the surround-

ings during operating time t. For a stationary system, the only energy changed is

internal energy; hence, DE(t) ¼ DU(t), and Eq. 5.2.4 reduces to

DU( t) ¼ Q(t)  W(t)(5:2:5)

For batch reactors, the work term consists of two components: shaft work (work

done by a mechanical device such as a stirrer) and work done by expanding the

boundaries of the system against the surroundings:

W ¼ W

PdV (5:2:6)

Expressing the internal energy in terms of the enthalpy (recall, H ; U þ PV and

dH ¼ dU þ pdVþ Vdp)

DU ¼ DH 

PdV

VdP

and, substituting into Eq. 5.2.6, the energy balance becomes

DH(t) ¼ Q(t)  W

(t) þ

VdP (5:2:7)

Most chemical reactors operate at isobaric or near isobaric conditions. Also, in gen-

eral, the enthalpy is a weak function of the pressure. Hence, the last term is rela-

tively small, and the energy balance equation reduces to

DH(t) ¼ Q(t) W

(t)(5:2:8)

In many instances, the reacting ﬂuid is not viscous, and the shaft work is small in

comparison with the heat added to the system; hence

DH(t)  Q(t)(5:2:9)

It is important to note that Eq. 5.2.9 is applicable only when all the assumptions

made in its derivation are valid: negligible kinetic, potential and electric energies,

negligible effect of pressure, and negligible viscous work.

For closed systems with chemical reactions, the enthalpy varies due to changes

in both composition and temperature. We usually select a reference state at some

136 ENERGY BALANCES

convenient composition and temperature T

. Assuming no phase change, the

change in the enthalpy of the reactor in operating time t is

DH(t) ¼

(t) þ

T(t)

)

dT 

T(0)

)

dT (5:2:10)

where

)

and

)

indicate the heat capacity of the reacting ﬂuid

initially and at time t, respectively. The ﬁrst term on the right-hand side of Eq.

5.2.10 represents the change in enthalpy due to composition changes (by chemical

reactions) at the reference temperature, T

, whereas the other two terms indicate the

change in the “sensible heat” (enthalpy change due to variation in temperature).

Note that since the enthalpy is a state quantity, the calculation of the enthalpy

difference is shown schematically in Figure 5.2. Also note that the summation in

the ﬁrst term on the right is over the independent reactions.

Substituting Eq. 5.2.10 into Eq. 5.2.8, the general energy balance equation of

closed systems becomes

Q(t) ¼

(t) þ

T(t )

)

dT 

T(0)

)

þ W

(t)(5:2:11)

Figure 5.2 Determination of enthalpy change for reacting systems.

5.2 ENERGY BALANCES 137

It is convenient to select the initial temperature as the reference temperature, T

T(0), and Eq. 5.2.11 reduces to

Q(t) ¼

(t) þ

T(t )

) dT þ W

(t)(5:2:12)

Equation 5.2.12 is the integral form of the energy balance equation for batch reac-

tors, relating the reactor temperature, T(t), to the extents of the independent reac-

tions, X

(t), the heat added to the reactor, Q(t), and the mechanical work done

by the system, W

(t), during operating time t.

The differential form of the energy balance equation is obtained by differentiat-

ing Eq. 5.2.12 with respect to time:

Q(t) ¼

)

(t)(5:2:13)

where Q

(t) is the rate heat is added to the reactor, and W

(t) is the rate of mech-

anical work done by the reactor at time t. For an ideal batch reactor, the temperature

is uniform throughout the reactor, and the rate of heat transfer is expressed by

Q(t) ¼ US(t)[T

 T(t)] (5:2:14)

where U is the overall heat transfer coefﬁcient, S(t) is the heat transfer area at time t,

and T

is the temperature of the external heating (or cooling) medium. The heat

transfer area S(t) relates to the reactor volume by

S(t) ¼



(t)(5:2:15)

where (S/V) is the heat transfer area per unit volume, a system characteristic that

depends on the geometry of the reactor and the heat exchange area (wall, coil,

etc.). Substituting Eqs. 5.2.14 and 5.2.15 into Eq. 5.2.13 and rearranging,

(t)

)



 T(t)] 

)



(t)

)

(5:2:16)

Equation 5.2.16 expresses the changes in the reactor temperature during the reactor

operation. The ﬁrst term on the right-hand side represents the rate heat transferred to

the reactor at time t divided by the heat capacity of the reacting ﬂuid:

Q(t)

)

(t)

)



 T(t)] (5:2:17)

138 ENERGY BALANCES

To reduce Eq. 5.2.16 to a dimensionless form, we use the dimensionless temp-

erature, deﬁned by Eq. 3.3.4, u ¼ T/T

, the dimensionless extent, deﬁned by

Eq. 2.7.1, Z

¼ X

/(N

tot

)

, and the dimensionless operating time, deﬁned by

Eq. 4.4.3, t ¼ t/t

. Dividing both sides of Eq. 5.2.16 by T

and (N

tot

)

and multi-

plying both sides by the characteristic reaction time, t

, we obtain

)



 u) 

tot

)



)



)

(5:2:18)

Equation 5.2.18 is the dimensionless, differential energy balance equation of ideal

batch reactors, relating the reactor dimensionless temperature, u(t), to the dimen-

sionless extents of the independent reactions, Z

(t), at dimensionless operating

time t. Note that individual dZ

/dt’s are expressed by the reaction-based design

equations derived in Chapter 4.

Equation 5.2.18 is not conveniently used because the heat capacity of the react-

ing ﬂuid,

), is a function of the temperature and reaction extents, and con-

sequently, it varies during the operation. To simplify the equation and obtain

dimensionless quantities for heat transfer, we deﬁne a heat capacity of the reference

state and relate the heat capacity of the reacting ﬂuid at any instant to it by

Heat capacity

of the reacting

fluid

;

Heat capacity

of the

reference state

Correction

factor



Mathematically, it is written as

) ; (N

tot

)

CF(Z

, u)(5:2:19)

where

is the speciﬁc molar heat capacity of the reference state and CF(Z

, u)isa

correction factor that adjusts the value of the heat capacity of the reacting ﬂuid as

and u vary. We discuss the determination of

in detail below. Substituting

Eq. 5.2.19 into Eq. 5.2.18 and noting that (N

tot

)

¼ C

CF(Z

, u)



 u)





CF(Z

, u)

)

tot

)



()

(5:2:20)

5.2 ENERGY BALANCES 139