Complex Systems
5.2 MEASURES OF REACTION EXTENT AND SELECTIVITY .1 Reaction Stoichiometry and Its Significance
For a complex system, determination of the stoichiometry of a reacting system in the form of the maximum number (R) of linearly independent chemical equations is de- scribed in Examples 1-3 and 1-4. This can be a useful preliminary step in a kinetics study once all the reactants and products are known. It tells us the minimum number (usu- ally) of species to be analyzed for, and enables us to obtain corresponding information about the remaining species. We can thus use it to construct a stoichiometric table cor- responding to that for a simple system in Example 2-4. Since the set of equations is not unique, the individual chemical equations do not necessarily represent reactions, and the stoichiometric model does not provide a reaction network without further informa- tion obtained from kinetics.
Spencer and Pereira (1987) studied the kinetics of the gas-phase partial oxidation of CH, over a Moo,-SiO, catalyst in a differential PFR. The products were HCHO (formalde- hyde), CO, C02, and H,O.
(a)
Obtain a set of R linearly independent chemical equations to represent the stoi- chiometry of the reacting system.(b) What is the minimum number of species whose concentrations must be measured experimentally for a kinetics analysis?
SOLUTION
(a) The system may be represented by
{(CH,, O,, H,O, CO, CO,, HCHO), (C, 0, H))
Using manipulations by hand or Mathematics as described in Example 1-3, we obtain the following set of 3 (R) equations in canonical form with CH,, O,, and HZ0 as components, and CO, CO,, and HCHO as noncomponents:
CH, + ;02 = 2H,O + CO (1)
CH, + 20, = 2H,O + CO, (2)
CH, + 0, = H,O + HCHO (3)
These chemical equations may be combined indefinitely to form other equivalent sets of three equations. They do not necessarily represent chemical reactions in a reaction net- work. The network deduced from kinetics results by Spencer and Pereira (see Example 5-8) involved (3), (l)-(3), and (2) as three reaction steps.
(b) The minimum number of species is R = 3, the same as the number of equations or noncomponents. Spencer and Pereira reported results in terms of CO, CO,, and HCHO, but also analyzed for O2 and CH, by gas chromatography. Measurements above the min- imum number allow for independent checks on element balances, and also more data for statistical determination of rate parameters.
5.2 Measures of Reaction Extent and Selectivity 91
5.2.2 Fractional Conversion of a Reactant
Fractional conversion of a reactant, fA for reactant A, say, is the ratio of the amount of A reacted at some point (time or position) to the amount introduced into the system, and is a measure of consumption of the reactant. It is defined in equation 2.2-3 for a batch system, and in equation 2.3-5 for a flow system. The definition is the same whether the system is simple or complex.
In complex systems, fA is not a unique parameter for following the course of a re- action, unlike in simple systems. For both kinetics and reactor considerations (Chap- ter 18) this means that rate laws and design equations cannot be uniquely expressed in terms of fA, and are usually written in terms of molar concentrations, or molar flow rates or extents of reaction. Nevertheless, fA may still be used to characterize the over- all reaction extent with respect to reactant A.
5.2.3 Yield of a Product
The yield of a product is a measure of the reaction extent at some point (time or po- sition) in terms of a specified product and reactant. The most direct way of calculating the yield of a product in a complex system from experimental data is by means of a stoichiometric model in canonical form, with the product as a noncomponent. This is because that product appears only once in the set of equations, as illustrated for each of CO, CO,, and HCHO in Example 5-1.
Consider reactant A and (noncomponent) product D in the following set of stoichio- metric equations:
IV&A + . . . = vnD + . . . +other equations not involving D The yield of D with respect to A, YDiA, is
moles A reacted to form D YD/A =
mole A initially
(5.2-la)
moles A reacted to form D
= x moles D formed
mole D formed mole A initially
_ bAiD nD - llDo
(BR, constant or variable p)
(5.2-lb)
, vy,s, FD~‘~DO _
FAO
(flow reactor, constant or variable p) (5.2-1~)
_ iuy,s, cD - cDo (BR or flow reactor, constant p)
(5.2-ld)
VD CA0
where IvAID is the absolute value of vA in the equation involving D, and nDo, FD,, cDo
refer to product D initially (each may be zero).
The sum of the yields of all the noncomponents is equal to the fractional conversion of A:
N
kz+, “IA = kg+1 Ty -
bAik nk - llko _ nAo - “ A = fA nA0
(5.2-2)
where k is a noncomponent index, C is the number of components, and N is the number of species.
For a simple system with only one noncomponent, say D,
YDIA = fA (simple system) (5.2-2a)
As defined above, YDIA is normalized so that
0 5 Y,,, 5 1 (5.2-3)
5.2.4 Overall and Instantaneous Fractional Yield
The fractional yield of a product is a measure of how selective a particular reactant is in forming a particular product, and hence is sometimes referred to as se1ectivity.l Two ways of representing selectivity are (1) the overall fractional yield (from inlet to a particular point such as the outlet); and (2) the instantaneous fractional yield (at a point). We consider each of these in turn.
For the stoichiometric scheme in Section 5.2.3, the overall fractional yield of D with respect to A, S,,,, is,.
iD/A = moles A reacted to form D mole A reacted
_ bAlD nD - lZDo (BR, constant or variable p)
VD nAo - IzA
(5.2-4a) (5.2-413)
_ I"AI~ F~ -F~o
(flow reactor, constant or variable p) (5.2-4~)
VD FAo -FA _ IVAIDCD - coo
(BR, or flow reactor, constant p) (5.2-4d)
VD cAo - cA
n
From the definitions of fA, Yo,A, and SD,,, it follows that ,.
YDIA = ~ASDIA (5.2-5)
The sum of the overall fractional yields of the noncomponents is unity:
bAlk *k - nko _ nAo - nA = 1 (5.2-6)
llAo - nA
AS in the cases of fA and Yn/A, SD/A is normalized in the definitions so that
1
‘Other definitions and notation may be used for selectivity by various authors.
5.2 Measures of Reaction Extent and Selectivity 93 The instantaneous fractional yield of D with respect to A is
rate of formation of D