I
M. R. HOARE Department of Chemistry, University of Washington, Seattle
5, Wash.
Determining True Residence Times in Flow System Reactions The old problem of residence time in a flow system can be solved for any reaction given “almost plug flow” and a complete analysis of theoutlet stream-no knowledge of rate equations or reaction mechanism needed
XE
PROBLEX of describing the behavior of a reaction occurring in a gas-phase flolz-system a t constant pressure is essentially that of determining the true residence time of a mixture that may be expanding or contracting as a result of chemical change while it passes through the reaction zone. Several direct approaches to this problem ( I ? 2. 5: 6) have shown that, provided the kinetic order and stoichiometry of the reaction are completely known, it is possible, in simple cases, to derive the true residence time as an exact integral. However, these methods apart from being limited to the simplest types of system, are of no use whatever where the reaction concerned is not clear-cut and ishen it may not be possible to xvrite a meaningful rate equation. The purpose of this article is to show that, in an isothermal system without diffusion, the true residence time can always be calculated, in principle, irrespective of the detailed chemical changes that occur, provided only that there is sufficient knowledge of the composition of inlet and outlet streams and of the way in which the latter changes as a function of the input flow rate. Moreover, as will be shown, the procedure described may give a good approximation to the true residence time even where considerable eddy diffusion occurs in the s)-stem. Consider any reaction occurring in a reactor of uniform cross section under conditions of plug flow. Let x measure the fractional volume traversed through the reaction zone, so that reaction is initiated a t x = 0 and ceases a t x = 1. Under steady state conditions, the “degree of reaction,” f , which may be defined simpl>- as the fractional disappearance of a chosen principal reactant, will have a definite profile, [ ( x ) , along the reactor, rising to a measurable value, f 1 , a t the outlet. f will also be expressible as a function of the time of reaction, [ ( t ) , corresponding to a rate equation lshich may or may not be known. Let Fo and F1 be the volume
flow rates a t inlet and outlet, respectively, both measured a t the temperature of reaction. Fo and F1 tvill only be equal if: the over-all degree of reaction is zero ( f = O), or the reaction results in no net change in the number of molecules present. In general, the volume flowrate will have an unknown profile, F ( x ) , along the reactor, which cannot be determined by single measurements of Fo and F1 alone. This situation leads to tMo types of problem: Given either direct or indirect measurements of the flow rates Fo and F1, what is the true rate equation, [(t)? Given the rate equation f ( t ) and the input flow rate, F0,what is the over-all degree of reaction El? The first of these cases is of crucial importance in deriving kinetic data from laboratory flow-system experiments ; the second in predicting technical flow-system behavior from known rate equations
For generality, the inlet stream is assumed to be composed of “reactants” with initial volume floiv, f r , and “carrier-gas’’ with constant volume flow. fc, rshich takes no part in the reaction. The total volume-flow a t any point, x. in the reactor is then given by: F(x)
=
fc
+fJ1 +
(1 1
Y(X)[(X)I
where, in general, the functions ~ ( x ) and f ( x ) are unknown. K o w if the total volume of the reactor is V>then the residence time of the gas in a n elementary volume dx is: dt
=
Vdx/F(x)
= l’ dx/(.fc
(2)
+ frI1
(3)
dx)E(x)l)
By integrating the true residence time is obtained :
or : Direct and Indirect Measurements of Flow Ratio
In treating the first problem, it will be assumed that experimental results are available in the form of chemical analyses of the outlet stream rather than as direct measurements of the outlet flow rate. The extent of the volume change accompanying any system of reactions may be expressed by a parameter, Y , which will be defined as the ratio: total change in number of molecules in the system due to reactiont’total molecules of reactants consumed. In a simple reaction Y is either a positive integer or fraction or a negative fraction and its value may be determined by inspection of the stoichiometric equation. I n complex systems where competitive and secondary reactions occur, there is no characteristic value of Y and it must be thought of as varying with time and having a profile ~ ( x )along the reactor. But for every value of E obtained b) analysis of the outlet stream, a corresponding value of Y can be obtained, provided the analysis is sufficiently complete to permit the drawing up of a mass balance for the reaction.
+
where, for ideal gases, B = f7/(fc f?) is the mole fraction of reactants in the input stream. If now the “nominal residence time,” t’, is defined as the time that would elaspe if there were no reaction, then :
+
t’ = V / ( f C
f7)
(6)
and
If ~ ( x )and ((x) could be measured directly, for example by the analysis of small samples extracted a t points along the reactor, then the value of t could be determined straightforwardly from this integral. Usually the most that can be done in laboratory practice is to analyze the outlet gases a t different input Row rates, thus obtaining Y and not as functions of t but of t’. Nevertheless, it is always possible to approximate the integral in the following manner. VOL. 53, NO. 3
MARCH 1961
197
An Analysis of isothermal Plug Flow System Outlet gases from an isothermal reactor are analyzed at different input flow rates to determine the fractional disappearance ( E ) of some chosen reactant and the “net change in the number of molecules” present per number of molecules reacting
(VI. A nominal residence time is calculated for each inlet velocity, where nominal residence time for the gases spent in the tube, i s as if no reaction took place. A plot i s made of fraction disappearance (.$) vs. nominal residence time (t’) for experimental data obtained. In addition, a plot can b e made of the net change in number of molecules ( v ) that resulted for each conversion of fractional disappearance (.$) that was observed. Pick some value of nominal residence time-e.g., fa’-and construct a graph of ( E ) vs. distance ( x ) for a distance between zero and one, b y assuming that this curve i s similar in shape to the (6)vs. ( f ’ ) curve. From points taken from the (6)vs. ( x ) curve, evaluate the integral:
for all values of .$(x)and for values of v ( x ) corresponding to the observed value of g. Evaluation of integral gives a ration residence time assuming the conversion curve vs. distance developed in the step above, or which when multiplied b y t,’ gives the residence time t, associated with the conversion (a a t the end of nominal residence time fa’. Calculations in the two preceding steps for different arbitrary values tb’, f c ’ , etc., to arrive a t residence times fa, tb, f,, etc., for values of .$n.b,c, etc., are repeated. From this new curve, another reacPlot a new reaction curve . $ o . b , c vs. tion curve can be calculated, until subsequent calculations d o not change the reaction curve. This procedure gives the “true” reaction curve.
Having plotted the “nominal reaction curve” t(t’) from experimental measurements of [, over a range of different input flow rates, pick a point t,’ and assume that the shape of the curve [(t‘) over the interval (0, tL’) is the same as that of the true reaction profile [ ( E ) over the region (0,1), under the conditions of input flow rate specified by t$’. If the integral is evaluated in this way for a set of t,’ values, an improved reaction curve is obtained which may be used in turn as a second approximation. By this iterative process, the true reaction curve [ ( t ) is eventually derived. That the iterations must converge to the true curve is clear from the following considerations: If the reaction leads to an expansion, then the first approximation to ( ( x ) and ~ ( x is ) an over-estimate, which leads to an under-estimate of t. This leads to a n over-estimate of t on the next iteration and so on. Since the successive approximations oscillate about the true curve, it follows that any limit obtained can only be the true curve. The same is true whether the reaction leads to expansion or contraction. In applications of the method convergence was found to be very rapid, differences converging to within experimental error in about five iterations for the highest degree of reaction and less for lower regions of the reaction curve. The procedure is tedious to carry out graphically, but is obviously well suited to computer programming. [A detailed description of the method and its application to some complex reaction systems have been given (3, 4 ) . ]
198
It may be remarked that the form of the integral in Equation 7 leads naturally to the following conditions. all of which are in agreement with expectation. (i) If Y is positive, t < t’; if Y is negative, t > t’. (ii) As ( +. 0 for all x, t t’ the reac(iii) If [ = 1 for all x-i.e., tion is instantaneous upon entry into the reactor-and v is a constant, then: t’ = t/(l B”). (iv) If B
