With reversible exothermic reactions, the combination of thermodynamic and kinetic parameters means that increasing temperatures give rise to decreasing net forward reaction rates. This may seem a little counter- intuitive. Let us go back to Eq. 4.13:
a1 a2 r
E E ' H . (Eq. 4.13) For exothermic reactions, “'HR < 0” means that Ea2 is greater than Ea1. Thus, as the temperature rises, the term “exp{-Ea2/RT}” increases in magnitude faster that the term “exp{-Ea1/RT}”. This is why the ratio k1/k2
(=Keq) diminishes with rising temperature. Plotting xA vs T, we find that the equilibrium conversion decreases with increasing temperature.
Figure 4.2 Qualitative sketch of the equilibrium conversion xA,eq vs. T for a reversible exothermic reaction As already discussed, when working with irreversible or reversible-endothermic reactions, high reaction rates can be achieved by raising the temperature to as high a level as is safe and practical. But when working with reversible-exothermic reactions, the choosing operating conditions is less straightforward. In designing a reactor, it is necessary to take account of two competing effects. First, both the forward and reverse rates of reaction increase with rising temperature, so the kinetics is favoured by increasing the reaction temperature.
However, equilibrium limitations are more restrictive at higher temperatures. In other words, the maximum attainable conversion, xA,eq takes on smaller values with increasing temperature. This is because, with increasing temperature, the reverse reaction picks up speed faster than the forward reaction so equilibrium, where the net rate of reaction is zero, is approached sooner. Let us look at the xA vs. T diagram for a reversible exothermic reaction in greater detail.
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T xA
xA,eq
Constant xA line CONVERSION
TEMPERATURE
Constant reaction rate lines increasing rA
Figure 4.3 Several features of the xA vs. T diagram
At constant reaction rate (Figure 4.3), the conversion initially rises rapidly with increasing temperature and then begins to fall – due to the faster rising rate of the reverse reaction. The characteristics of the reversible- exothermic reaction can be better understood by following a horizontal (constant conversion) line in Figure 4.3. At low temperatures, the reaction rate would be low but would initially increase with rising temperature as we move towards the right in Figure 4.3, i.e. towards higher temperatures. At the other extreme, the intersection of the constant xA-line (horizontal line) with the equilibrium line xA,eq represents the point where the net reaction rate rAis zero. Moving from left to right, between these two extremes, the reaction rate passes through a maximum.
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Thus, for reversible exothermic reactions, there exists an ‘optimum’ operating temperature (corresponding to the maximum reaction rate) for a fixed conversion. The line through these maxima gives a trajectory of maximum reaction rates as a function of the conversion, which we will call “the locus of maximum reaction rates” and discuss in the next section. It is important to remember that the present discussion is independent of the design of any particular type of reaction vessel.
4.3.1 The Locus of Maximum Reaction Rates
We will next derive an equation for the locus of maximum reaction rates for a simple reversible-exothermic reaction
1 , 2
k k
A mo B
Assuming CB0 = 0, the rate equation for the netconsumption of the reactant “A” may be written as
A 1 A0 A 2 B A0 1 1 2 A
r k C ( 1 x ) k C C { k ( k k )x } (Eq. 4.19) where k1 k e10 'E / RT1 , k2 k e20 'E / RT2 and ' E2 > ' E1. The maximum reaction rate as a function of temperature can now be found by differentiating Eq. 4.19 with respect to the temperature, at constant conversion:
A
A 1 1 2
A0 A0
x
dr dk dk dk
0 C C x
dT dT dT dT
Đ ã Đ ã
ă á ă á
â ạ â ạ A (Eq. 4.20) Rearranging
1 1 2
A
dk dk dk
x 0
dT dT dT
Đ ã
ăâ áạ (Eq. 4.21) and solving for xA
2
1
1
A,opt 1 2 E 2
2 RT
2 20 1 eq
E
1 RT
2 10
dk
1 1
x dT
dk dk E 1
E 1 1
dT dT 1 R T k e E K ( T )
E 1
R T k e
' '
' '
' '
Đ ã
Đ ã
ăâ áăạâ áạ
Đ ã
Đ ã
ă áă á
â ạâ ạ
(Eq. 4.22)
Having derived an expression for xA,opt( T ) along the locus of maximum reaction rates, the corresponding maximum reaction rates can be derived as a function of temperature, by substituting Eq. 4.22 into
A A0 1 1 2 A,opt
r C { k ( k k )x }. (Eq. 4.23)
1 2 eq
A,max A0 1 1 A0
2 2
1 eq 1 eq
1 1 ( k k ) K
r C k k C 1
E 1 E 1
1 1
E K E K
' '
' '
ê º ê
ô ằ ô
ô ằ ô
ô Đ ã ằ ô Đ ã
ô ă á ằ ô ă á
ô â ạ ằ ô â ạ
ơ ẳ ơ
ºằ
ằằ
ằằẳ
(Eq. 4.24)
For a given conversion, the reactor volume would therefore be a minimum if the temperature of operation is selected to be that corresponding to the maximum reaction rate.
Both the constant rate lines in Figure 4.3 and the locus of maximum reaction rates are independent of the type of reactor used.
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