First order kinetics in CMBR

Notes on course in Environmenal Engineering

Problem statement

Lets consider 3 continously mixed batch reactors (CMBR) with first-order decay kintetics (\(k_1\)) for a substance with initial concentration \(C_0\).

The parameters for each reactor are listed below:

• CMBR 1: C\(_0\) = 10 mg/L , \(k_1\) = 0.1 s\(^{-1}\)
• CMBR 2: C\(_0\) = 10 mg/L , \(k_1\) = 0.2 s\(^{-1}\)
• CMBR 3: C\(_0\) = 10 mg/L , \(k_1\) = 0.5 s\(^{-1}\)

1. Plot the concentration of each reactor as a function of time.
2. ¿What is the concentration of the substance in each reactor after 10 sec?

Solution

The concentration of a susbstance in a CMBR that decays with first order kinetics can be modeled as follows:

\[ C(t) = C_o e^{-k_1t} \tag{1}\]

Lets implement this equation numerically through a function:

from numpy import linspace, exp
import matplotlib.pyplot as plt

def conc(C0, k1, time):
    '''Concentration in a CMBR with first-order decay
    '''
    return C0 * exp(-k1 * time)

Note that our function receives the initial concentration C0, the first-order decay rate k1, and the time. Therefore, we can use the same function to model the 3 CMBRs.

Plot of concentration

time    = linspace(0, 20) # time in sec
k1      = [0.1, 0.2, 0.3] # fist-order decay kinetics (1/s)
C0      = [ 10,  10,  10] # initial concentrations (mg/L)

for i, k in enumerate(k1):
    Ct = conc(C0[i], k, time)
    plt.plot(time, Ct, label="CMBR %i: $k_1 = %s s^{-1}$" % (i+1, k) )

plt.xlabel("time [s]")
plt.ylabel("concentration [mg/L]")
plt.legend()
plt.grid()
plt.show()

Concentration of substance after 10 sec

We can use our function considering 10 sec as the input time to ompute the concentration of the substance.

t_eval = 10 # sec

for i, k in enumerate(k1):
    conc_t = conc(C0[i], k, t_eval)
    print ("CMBR %i - C(t=%ss): %1.3f [mg/L]" % (i+1, t_eval, conc_t) )

CMBR 1 - C(t=10s): 3.679 [mg/L]
CMBR 2 - C(t=10s): 1.353 [mg/L]
CMBR 3 - C(t=10s): 0.498 [mg/L]

Follow up questions

1. How much time is required for the concentration to be zero in each reactor?
2. Calculate the time it takes for each reactor to reach half of the initial concentration (\(t_{1/2})\).