An example for a logistic equation with 'length-slices' instead of 'time-slices

Derive a logistic equation for a simple reaction of 2nd order running in a stationary ideal TFR, using 'length-slices' dz instead of time slices dt.

r = k c²
0 = - u_z dc/dz - k c^{2

dc/dz ~ ( c_j+1 -
c_j)/delta(z)}

rename delta(z) with dz:

c_j+1 = c_j (1-(dz*k/u_z)*c_j)

in terms of unisim:

X = Xv(1- (dt*n/m)*Xv)
with n = second order rate constant k and m linear flow velocity u_z (m/sec). You can take dt as length-slice in (m), then you get a simulation plot for the concentration versus tube length. Where have you reached 'full conversion' ? see a snapshot

And now let us control the simulation and it's results here:

The simulation was carried out with the following parameters:

X0 = initial concentration of educt = 1 (mol/l)
n = rate constant k = 1 (l /mol sec)

m = linear flow rate u_z= 0.5 (m/sec)

dt = length-slice dz = 0.1 (m)

As 100% conversion takes infinite time, we take a concentration decay down to 0.025 (mol/l). If we take the number of length steps in our simulation plot for an end concentration of 0.025 (mol/l), we get about 191 steps, - that means multiplied by 0.1 (slice-length) we get a reactor length of 19.1 m. The conversion is U = (1 - 0.025/1) = 0.975.

For 97.5 % conversion we need a tube with about 19 m length!

The reaction time (space time) for a second order reaction in the not back-mixed reactor type is:
t_R = (1/(k*C_A,0))(U_A/(1-U_A))

If you put the given values into the formula, you get t_R = 39 (sec) and with t_R = l/u you get:

l = t_R * u = 39 * 0.5 = 19.5 m

You see that the simulation exhibits sufficient results.

Additions: Compute the conversion U in the formula window and 'connect' ( by defining) it to one of the outputs U, V, W (simply by a formula U = ..., or V =... or W = ...) (by editing the script file unisim.vee). See what tube lengths are necessary for given conversions.

And: add a 'free running' (that means not coupled) reaction of first order with using the second variable Y of unisim (you can also add a conversion variable here!!). You can compare the two reactions.

*And last not least something trivial but nice to see:

the differential equation for the batch is:

-dc/dt = k* C²

and the logistic equation:

C_j+1 = C_j(1 - (k * dt *C_j))

if you write: u = dz/dt (linear velocity), you get:

dz = u*dt

if you take the logistic equation for the TFR (in dz slices):

C_j+1 = C_j(1- (dz*k/u)*C_j)

and input dz = u*dt

you get exactly the same formula for the TFR and the STR, and that is clear!!!!

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