Question

The 2-day and 4-day BOD values of a sewage
sample are 100 mg/L and 155 mg/L,respectively. The value of BOD rate constant(expressed in per day) i

Answer 1

The value of the BOD rate constant (expressed per day) is 0.299.

Given:

The 2-day and 4-day BOD values of a sewage

the sample is 100 mg/L and 155 mg/L, respectively.

To Find:

The value of the BOD rate is constant.

Solution:

To find the value of the BOD rate constant we will follow the following steps:

As we know,

BOD is biochemical oxygen demand which is the measure of oxygen required by the aerobic water organism to decompose organic matter.

Also,

$BOD \: concentration (c)= lo(1 - {e}^{kt} )$

Here,

k is the BOD rate constant.

t is time in days.

According to the question:

In case 1:

BOD concentration = 100 mg/L

time = 2 day

In case 2:

BOD concentration = 155 mg/L

time = 4 day

Now,

By making equations of both cases by putting values in the above formula we get,

$100 \: = lo(1 - {e}^{2k} ) \: eq - 1$

$155\: = lo(1 - {e}^{4k} ) \: e q - 2$

Now,

Taking the ratio of both equations we get,

$\frac{150}{100} = \frac{1 - {e}^{4k}}{1 - {e}^{2k}}$

$1.55 = \frac{1 - {e}^{4k}}{1 - {e}^{2k}}$

$1.55 - 1.55{e}^{2k} = (1 - {e}^{4k} )$

Let e-2k = x

Then,

x² - 1.55x + 0.55 = 0

Now,

Solving for the values of x we get,

x = 1 or x = 0.55

x = 1 is not possible because any negative power of e cannot be greater than 1.

So, e –2K = 0.55

Taking log on both sides s we get,

-2K = In 0.55

K = 0.299

Henceforth, the value of BOD rate constant (expressed in per day) is 0.299.

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