Kinetics and Ideal Reactor Models
2.2 BATCH REACTOR (BR) .1 General Features
A batch reactor (BR) is sometimes used for investigation of the kinetics of a chem- ical reaction in the laboratory, and also for larger-scale (commercial) operations in which a number of different products are made by different reactions on an intermittent basis.
A batch reactor, shown schematically in Figure 2.1, has the following characteristics:
(1) Each batch is a closed system.
(2) The total mass of each batch is fixed.
(3) The volume or density of each batch may vary (as reaction proceeds).
(4) The energy of each batch may vary (as reaction proceeds); for example, a heat exchanger may be provided to control temperature, as indicated in Figure 2.1.
(5) The reaction (residence) time t for all elements of fluid is the same.
(6) The operation of the reactor is inherently unsteady-state; for example, batch composition changes with respect to time.
(7) Point (6) notwithstanding, it is assumed that, at any time, the batch is uniform (e.g., in composition, temperature, etc.), because of efficient stirring.
As an elaboration of point (3), if a batch reactor is used for a liquid-phase reaction, as indicated in Figure 2.1, we may usually assume that the volume per unit mass of material is constant (i.e., constant density), but if it is used for a gas-phase reaction, this may not be the case.
Liquid surface Stirrer
Liquid contents (volume = V) Vessel (tank)
Heat exchanger (if needed)
Figure 2.1 Batch reactor (schematic, liquid-phase reaction)
2.2 Batch Reactor (BR) 27
2.2.2 Material Balance; Interpretation of ri
Consider a reaction represented by A + . . . -+ products taking place in a batch reactor, and focus on reactant A. The general balance equation, 1.51, may then be written as a material balance for A with reference to a specified control volume (in Figure 2.1, this is the volume of the liquid).
For a batch reactor, the only possible input and output terms are by reaction, since there is no flow in or out. For the reactant A in this case, there is output but not input.
Equation 1.5-1 then reduces to
rate offormation of A by reaction = rate of accumulation of A or, in mol s-l, sayr,
(-rA)V = -dnAldt,
(2.2-1)
where V is the volume of the reacting system (not necessarily constant), and nA is the number of moles of A at time t. Hence the interpretation of r, for a batch reactor in terms of amount nA is
(-rA) = -(l/V)(dnAldt) (2.2-2)
Equation 2.2-2 may appear in various forms, if nA is related to other quantities (by normalization), as follows:
(1)
If A is the limiting reactant, it may be convenient to normalize nA in terms of fA, the fractional conversion of A, defined byfA = @A0 - nA)inAo WV (2.2-3) j
where n&, is the initial amount of A; fA may vary between 0 and 1. Then equation 2.2-2 becomes
cerA) = (nA,lV)(dfAldt) (2.2-4) ~
(2) Whether A is the limiting reactant or not, it may be convenient to normalize by means of the extent of reaction, 5, defined for any species involved in the reac- tion by
d[ = dnilvi; i = 1,2, . . . , N (2.2-5) 1
‘Note that the rate of formation of A is rA, as defined in section 1.4; for a reactant, this is a negative quantity. The rate of disappearance of A is (-r.& a positive quantity. It is this quantity that is used subsequently in balance equations and rate laws for a reactant. For a product, the rate of formation, a positive quantity, is used. The symbol rA may be used generically in the text to stand for “rate of reaction of A” where the sign is irrelevant and correspondingly for any other substance, whether reactant or product.
Then equation 2.2-2 becomes, for i = A,
(-rA) = -(v,lV)(dSldt)
(2.2-6)
/(3) Normalization may be by means of the system volume V . This converts nA into a volumetric molar concentration (molarity) of A, CA, defined by
If we replace nA in equation 2.2-2 by cAV and allow V to vary, then we have
(-).A) = 2!2$ - ?$ (2.2-8)
Since (-?-A) is now related to two quantities, CA and V, we require additional information connecting CA (or nA) and V. This is provided by an equation of state of the general form
v = v(nA, T, P)
(3a) A special case of equation 2.2-8 results if the reacting system has constant vol- ume (i.e., is of constant density). Then dVldt = 0, and
(-,-A) = -dc,/dt (constant density)
(2.2-10)
Thus, for a constant-density reaction in a BR, r, may be interpreted as the slope of the CA-t relation. This is illustrated in Figure 2.2, which also shows that rA itself depends on t , usually decreasing in magnitude as the reaction proceeds, with increasing t .
rAl = slope at cA1, tl
rA2 = slope at cA2, tp
Figure 2.2 Interpretation of rA for an isothermal, constant-density batch system
2.3 Continuous Stirred-Tank Reactor (CSTR) 29
For a reaction represented by A + products, in which the rate, ( -rA), is proportional to CA, with a prOpOrtiOnality Constant kA, show that the time (t) required to achieve a specified fractional conversion of A (fA) is independent of the initial concentration of reactant cAO.
Assume reaction occurs in a constant-volume batch reactor.
SOLUTION
The rate law is of the form
(-rA) = kACA
If we combine this with the material-balance equation 2.2-10 for a constant-density reac- tion,
-dc,ldt = kACA From this, on integration between CA0 at t = 0 and CA at t,
t = (IlkA) ln(CAo/CA) = (l/k,) h[l/(l - fA)]
from equation 2.2-3. Thus, the time t required to achieve any specified value of fA under these circumstances is independent of cAO. This is a characteristic of a reaction with this form of rate law, but is not a general result for other forms.