Analysing Batch Reactor Data

* One of the purposes for which batch reactors are used is rate law determination.* Concentration (or any other convenient variable) measured as a function of time is the data typically available* several methods are available for this analysis - differential, integral, initial rate, half life, etc.

Differential Method* data is available as CA vs. t (where A is a reactant)* for constant volume batch reactor with nth order reaction we can writedCAdt��� kCn

A orln� � dCA

dt � � lnk � nlnCA

* Thus if we can plot ln� � dCA

dt � vs. lnCA from the data, we will get a straight line whose slope gives the order of thereaction and whose intercept gives the logarithm of the rate constant.

* But how to get dCAdt given data of CAvs. time?

* several methods exist for this as well - numerical differentiation, graphical method, and polynomial fit method

Numerical Differentiation* in this method, the derivative is evaluated using finite difference formulas -dCAdt � i � CA � i � CA � i 1

ti � ti 1, where i refers to the number of the data point.

* such formulas can be forward or backward difference* a three-point differentiation formula in general will give more accurate results than the above:dCAdt � 0 � � 3CA0 4CA1 � CA2

2∆tdCAdt � 1 � CA2 � CA0

2∆tdCAdt � 2 � CA0 � 4CA1 3CA2

2∆t* the data should have been collected in equal time intervals

Graphical Method* in this method, a smooth curve is drawn through the experimental data points on a CAvs. t graph.* at each time instant of interest, tangents are drawn to this curve, the slope of the tangent line is the derivative value at

that time instant* this method is attractive since we have a lot of control over the quality of results, and no assumptions regarding

uniformity of sampling time intervals, etc. are required.* but it can be tedious, specially for large data sets or many experiments.

Polynomial Fit Method* a polynomial of suitable order has to be fitted to the data* the derivative can be then evaluated by differentiating the polynomial expression* extreme care is necessary to make sure the fit is sensible - in general the best lower order polynomial that fits the data

reasonably should be chosen rather than a very high order polynomial that goes through all the data points* a plot of the data and the fitted polynomial curve can be an important visual tool to ascertain that we doing the sensible

thing

Differential Method, non-constant volume batch reactor* similar analysis is possible for non-constant volume batch reactor* lets say that V (total volume) is the measured variable* balance equation is written as dNA

dt��� kCn

AV* this can be written in terms of volume as NA0

VδyA0V0

dVdt� kCn

A

1

* since NA� NA0

�1 � XA � � NA0

�1 � V � V0

δyA0V0 �* furthermore,

CA� NA0

V�1 � XA � � NA0

� �δyA0 � 1 � V0

� V �δyA0V0V

� CA0

� �δyA0 � 1 � V0

� V �δyA0V

* thusCA0

δyA0

dlnVdt

� k�

CA0

� �δyA0 � 1 � V0

� V �δyA0V � n

* or

ln� dlnV

dt � � lnk � ln� Cn � 1

A0�δyA0 � n � 1 � � nln

� �δyA0 � 1 � V0

V� 1 �

* and thus a straight line plot can be obtained to get the order or reaction from V vs. t data

Integral Method* an order of reaction has to be assumed* then the batch reactor equation is integrated with the rate law of assumed order* the quantities are bunched together in such a manner as to get a linear equation* plot the data for this line* the assumed order is correct if the data actually fall on this line, if not, another trial is required* the slope and intercept of this straight line should give the rate constant value

Example:* It can be shown that for first order reactions (Constant Volume Batch Reactor),ln�CA � � ln

�CA0 � � kt so, ln

�CA � has to be plotted vs. time

* For second order reactions,1

CA

� 1CA0� kt so 1

CAhas to be plotted vs. time, to get k

* n-th order reaction (n �� 1): C1 � nA

� C1 � nA0

� k�n � 1 � t so plot C1 � n

A vs. t to get k

* rate laws of even more complex nature can be used in this method* however the assumption of order that has to be made a-priori makes integral method inconvenient at times* judging whether the data actually fall on the prescribed line also tends to be an art, specially if the available data points

are not very numerous

Method of Initial Rates* the reactor is operated using different initial concentrations.* from the initial slope of the concentration vs. time data, the rates corresponding to these initial concentrations are

obtained.* plotting ln

�rate � vs. ln

�CA0 � will give the order and the rate constant, just like in the differential method

* the big disadvantage of this method is that lot of experiments are required

Method of Excess* this method is for rate laws having dependence on concentration of more than 1 species* the concentration of all but 1 species is kept constant* the order of reaction wrt the 1 species is determined by usual methods (perhaps the method of initial rates?)* in some sense method of excess is a sub-method - a variation of the differential or integral or other methods, for a

special situation

Method of half-lives

2

* half-life is the time taken for the concentration of a reactant to fall to one-half its original value* it is just the same half-life we have studied when we dealt with radioactivity* the half-life is the measured variable in this method, typically as a function of the initial concentration

Example:* consider C1 � n

A� C1 � n

A0� k�n � 1 � t (nth order reaction, n is not 1)

* half-life is the time for the concentration to fall to half it original value, or time at which CA� CA0 � 2 �

* thus t1 � 2 � C1 nA0

k � n � 1 � � � 0 � 5 � 1 � n � 1 �* if we have a set of data of half-life vs. the initial concentration, to get k and n, we have to plot ln

�t1 � 2 � vs. ln

�CA0 � , the

slope will be 1 � n. k can be obtained from the intercept

* note that for first order reactions the half-life is independent of the initial concentration, so one experiment at anyarbitrary initial concentraction can give the rate constant

* many experiments are necessary here also, and waiting to get to half-life may be expensive* a ’fractional life’ method may easily be used, performing all calculations and experiments to the time when the con-

centration is something like 10% of the initial concentration

3