This PPT is all about Mathematical modeling for waste water treatment. It contains introduction to waste water and waste water treatment. Aim for mathematical model. Then we took some assumptions for black box system. It contains system characterization. I mentioned variables and parameters that is used in mathematical model. causal relationship and formulation of mathematical model for single tank system and double tank system. To treat waste water we are using organisms like bacteria, protozoa etc. it contain picture of method of waste water treatment.

1. Submitted To:
Dr. Jai Prakash Tripathi
Assistant Professor
Department of Mathematics
Mathematical Model for
Waste Water Treatment
Submitted By:
Jagrati Mehra
2020IMSBMT015
Int. M.Sc. B.Ed. 4th semester
CENTRAL UNIVERSITY OF RAJASTHAN

2. Introduction
 Waste water have several undesirable components, the
organic and inorganic pollutants that are potentially
harmful to the environment and human health.
 The aim of waste water treatment is the removal of
contaminants from water.
 Water treatment carried out three stages:
1. Primary
2. Secondary
3. Tertiary

4. Aim
To analyze the dimension of tank and
efficiency of double tank is over a single
tank.

5. ● The rate of pollutant digestion by the organisms is assumed proportional to the
organism concentration.
● The rate at which organisms multiply is directly proportional to the pollutant
concentration.
● We have considered volume V of water having uniform concentrations of
pollutant and organisms.
● Polluted water flows into the tank at the rate Q and clean water flows out at the
same rate
● Usually the concentration of incoming polluted water varies but for simplicity
we consider it to be fixed.
Assumptions

6. Objects
Tank, Organisms and
Polluted water
Open/Closed
Closed System as no
external factors affect the
growth and the digestion of
pollutant by organism
Static vs Dynamic
Deterministic vs
Stochastic
As the change in
concentration of pollutant
and growth of bacteria
predicted with certainty .
Hence the system is
Deterministic.
System Characterization
As the variable are time
dependent so system is
dynamic.
White box/ Black Box
Black Box
02
01
03
04
05

7. Concentration of
pollutant in tank
Variables
Concentration of
organisms in tank
C(t) B(t)

8. MERCURY
It’s the closest planet to the Sun and the
smallest in the Solar System
VENUS
Venus has a beautiful name and is the second
planet from the Sun
ABOUT THE DISEASE
MARS
Despite being red, Mars is actually a cold place.
It’s full of iron oxide dust
Concentration of polluted water flowing into the tank
Parameters
C*
Q
D
R1
R2
V
Flow rate of polluted water into the tank and from
the tank
the rate at which organisms multiply is directly proportional to
the pollutant concentration with proportionality constant R2
The rate of pollutant digestion by the organisms is assumed
proportional to the organism concentration with proportionality
constant R1
Death Rate Constant (organisms)
Volume of Tank

9. C(t) C*(t)
D
B`(t)
C`(t)
V
B(t)
Causal Relationship

10. Formulation
• Formulation will be done by considering pollutant level and organisms
separately.
• Conservation of mass law for pollutant over time interval (t, ∆t).
• We will formulate two times :
1. For single tank system
2. For double tank system

11. For Single Tank System
Change in pollutant level in tank = pollutant arriving from inflow – Pollutant
leaving in outflow – pollutant digested by organisms.
𝒅𝑪
𝒅𝒕
=
[C∗ − C(t)] Q
V - R1 C(t) B(t) (1)
Change in organisms population in tank = organisms created in reproduction –
Organism lost via death rate – Organisms washed away in outflow
𝒅𝑩
𝒅𝒕
= [ R2 C(t) – D -
Q
V ] B(t) (2)

12. • We can extend this concept to formulate equations of Two Tank system with volume
V1 and V2.
• In First Tank, Pollutant Concentration will be C1 organism concentration will be B1.
• In Second Tank, Pollutant Concentration will be C2 organism concentration will be B2
• The Outflow from Tank 1 will form inflow to tank 2 Growth and Digestive behavior will
be same of organisms in both tanks.
• ODE for first tank in two tanks systems will be:
For Two Tank System
𝒅𝑪𝟏
𝒅𝒕
=
[C∗ − C1(t)] Q
V𝟏
- R1 C1(t) B1(t) (3)
𝒅𝑩𝟏
𝒅𝒕
= [ R2 C1(t) – D -
Q
V1
] B1(t) (4)

13. For Two Tank System :
Change in pollutant level in tank = pollutant arriving from inflow – Pollutant
leaving in outflow – pollutant digested by organisms.
𝒅𝑪𝟐
𝒅𝒕
=
[C1(t) − C2(t)] Q
V2
- R1 C2(t) B2(t) (5)
Change in organisms population in tank = organisms created in reproduction –
Organism lost via death rate + Organisms arriving in inflow from tank one –
Organisms washed away in outflow
𝒅𝑩𝟐
𝒅𝒕
= [ R2 C2(t) – D ] B2(t) +
Q
V2
[ B1(t) – B2(t) ] (6)
Change in polluted concentration in tank two in (t, ∆t) is given as

14. Thank You

15. Any Questions?