HUNGARIAN JOURNAL 
OF INDUSTRIAL CHEMISTRY 

VESZPREM 
Vol. 30. pp. 191 -192 (2002) 

MATHEMATICAL EQUIVALENCE OF INFINITE MIXED FLOW REACTORS 
IN SERIES AND PLUG FLOW REACTOR 

T. RENGANATHAN and K. KRISHNAIAH 

(Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600 036, INDIA) 

Received: December 30, 2001; Revised: July 8, 2002 

~is paper. pro~es the ~athem~tical equivalenc~ of infinite mixed flow reactors in series and a plug flow reactor. in the 
?me d~mam usmg an Impulse mput. The proof IS mathematically less complicated, compared to the previous statements 
m the literature. 

Keywords: chemical reactors, mixed flow reactor, plug flow reactor, residence time distribution, convergence 

Introduction 

The basic concept, that infinite mixed flow reactors 
(MFR) in series give the same performance as a plug 
flow reactor (PFR) is well known in chemical 
engineering for many years. But, only a few 
mathematical proofs of this concept are available in 
literature. Villermaux [1] proved the convergence in the 
Laplace domain. Molin and Gervais [2] showed the 
convergence in time domain with step input using 
limiting values of incomplete gamma function. Chen [3] 
used asymptotic equation for treatment of incomplete 
gamma function and proved the convergence in time 
domain with step input. In this work, the mathematical 
equivalence of a plug flow reactor and infinite mixed 
flow reactors in series is shown in time domain using an 
impulse input (Dirac delta function). The mathematical 
complexity is less, compared to the previous works. 

Proof of Equivalence 

A series of mixed flow reactors (MFRs) with delta input 
are shown in Fig.l. The dimensionless exit age 
distribution function, E( 8), for N tanks in series can be 
derived easily [4] as 

N(N8)N -le -NO 
E(B) = (N -1)! (l) 

where 8 = th: is the dimensionless time, t is the time 
variable and -r the average residence time of the entire 
system. The response curves, Eq.( 1 ), for different 

number of MFRs in series (N = 1, 2, 5, 20, 50, 100, 200, 
500 and oo) are shown in Fig.2. As the number of tanks 
approaches infinity, the response approaches an impulse 
output with a time lag of average residence time r ( 8 = 
1) of the entire system. This response is characteristic of 
a piug flow reactor with delta input. 

By definition of plug flow, each cross-section should 
correspond to an ideal mixer. Therefore intuitively a 
plug flow reactor can be considered as infinite mixed 
flow reactors in series. This is validated here 
mathematically, by proving that the function E((}) tend~ 
towards a Dirac delta function with point of impulse at 
e = 1, as N tends towards oo, which is the response of a 
plug flow reactor to an impulse input. 

Using Stirling's approximation, 

N! =NN e-N .J21rN 
and rearranging, Eq.( 1) can be written as 

e{-N(-In9-l-t6)-ln9+(112)1nN) 

E(fJ)= .fbi 

(2) 

(3) 

To show that Eq.(3) approaches a Dirac delta 
function when N tends towards infinity, the following 
have to be proved [5]: 

such that 

E(fl) = oo 
=0 

... 

8=1 
O:t=l 

J E(8)dfJ = l 

(4a) 

(4b) 



192 

2 

Fig.l Schematic ofMFRs in series 

At 8= 1, fromEq.(3), 

E(8)= JN 
En 

N 

Therefore as N approaches=, E((J) tends towards oo. 

(5) 

10-

9 

8 

7 

6 

& 200 

~ 
5 -

4 

For values of e in the intervals [0,1) and (1, oo], -ln8 
-1 +8 is always positive. Therefore, the leading term in 
the exponent of Eq.(3) i.e. -N(-ln8 -1+8) tends towards-
·oo as N approaches oo. So in these intervals, E((J) tends 0 -~:=.::~~~!ZlL_:C~ss~~=~ 
towards 0 as N tends to =. 

Now, since negative values ofe are inadmissible, 

J E(O)d8 = J E(8)d8 = ~ J 8N-te-Ne d8 (6) 
_.,.. o (N -1)! o 

Using the definition of Gamma function [5], 

J= 8 N-1e-Ne dB = r(N) = (N -1)! NN NN 
0 

(7) 

Therefore, from Eq.(6), 

JE(O)dO = 1 N?.1 (8) 
0 

Thus for infinite MFRs in series, the function E( 8) 
converges to a Dirac delta function characteristic of a 
plug flow reactor. 

Conclusion 

Using an impulse input, the equivalence of a series of 
infinite mixed flow reactors to a plug flow reactor is 
proved mathematically in the time domain. The proof is 
less complicated compared to the previous works in the 
literature. 

SYMBOLS 

E( 8) dimensionless exit age distribution function 
N number of MFRs in series 
t time variable, s 

0 0.5 1 

8 

1.5 

Fig.2 RTD curves for MFRs in series 

Greek Letters 

e dimensionless time 
1: average residence time, s 
r(N) Gamma function 

REFERENCES 

2 

1. VILLERMAUX J.: Genie de la Reaction Chimique, 
Tee et Doc, Lavoisier, Paris, 1993 

2. MOLIN P. and GERVAIS P.: AIChE J., 1995, 41, 
1346-1348 

3. CHENW. Y.: Hung. J. Ind. Chern., 1995,23,21-24 
4. LEVENSPIEL 0.: Chemical Reaction Engineering, 3rd 

Edn., John Wiley & Sons, New York, pp. 321-323, 
1999 

5. WYLIE C. R.: Advanced Engineering Mathematics, 
3rd Edn., McGraw-Hill, New York, 1966 


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