Investigating Waste Water Treatment in a Closed Environment
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This study investigates the claim that water can be completely purified by maintaining a population of at least 10 microorganisms per cubic centimeter in a treatment tank for at least 8 days. It explores the growth patterns of various microorganisms and evaluates their effectiveness in wastewater treatment through a mathematical model and numerical simulations. The findings indicate that certain microorganisms, particularly 'multiplicanda numerosa,' offer the most efficient treatment capabilities while maintaining ecological balance.
![International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2451-2455 ISSN: 2249-6645
www.ijmer.com 2451 | Page
T. S. Nurudeen1
, R. A. Kareem2
, S. O. Salawu3
1, 2
Department of Mathematics, Lagos State Polytechnic, Ikorodu, Nigeria
3
Department of Mathematics, University of Ilorin, Ilorin, Nigeria
Abstract: This paper focuses on the investigation of the claim of ‘The Sound and Environmental Water and Effluent Re-
cycling company’ that water can be completely purified if a population of at least 10 organisms per cubic centimeter can be
maintained in its cleaning tank for a period of at least 8 days. We investigated the growth patterns of each of these
organisms and make recommendations about which one will provide the most efficient water treatment and how many
organisms should be introduced initially.
Keyword: Environment, Waste, Micro-organisms and Water Treatment,
I. INTRODUCTION
The importance of these kinds of investigations as well as the complimentary experimental works, say in large-scale
water treatment and redistribution networks, thus makes them of great relevance [1-6]. It should however be remarked that to
date, the literature on the waste water treatment is still quite sparse. This then forms the crux of the current investigation [8].
Untreated water supply causes a serious socio-ecological hazard. If the problem is not carefully controlled and monitored,
large communities can be exposed to extensive health risks. It then requires taking immediate corrective measure to redress
and mitigate against its impact in the society [9].
The use of micro organisms to clean waste water and make it safe for re-use is good idea for water treatment. The
micro-organisms feed on harmful bacteria, thus removing them from the water. The efficiency of this method depends upon
choosing the correct species of micro-organism and ensuring that it is able to thrive in the tank of waste water successfully
for long enough to consume all the bacteria [7]. However, it is important that the population declines after completing its
function, since the ecological balance could be upset by discharge number of micro-organism into the environment.
II. MODEL EQUATION
The growth of micro-organism in a closed environment (such as the cleaning tank) is determined by three principle factors.
The reproduction rate
The effect of increased crowding
The build-up of toxins produced as waste products by the organisms.
This can be modeled by the equation [9]
t
ducubuau
dt
du
0
2
1
Where tu the number of organisms per cubic centimeter of water is, a is the reproduction ratio, b is the crowding
coefficient, and c is a measure of the sensitivity of the organisms to toxins. The integral term is necessary since the toxins in
the environment are built up gradually over a period of time at a rate proportional to the number of organisms present.
The model equation in 1 which is in Integra-differential equation is reduced to differential equations of the form
Let
t
dutx
0
t
tudu
dt
d
dt
dx
0
tu
dt
dx
ux 2
Then 1 becomes
cuxbuau
dt
du
2
cuxbuauu 2
3
Investigating Waste Water Treatment in a Closed Environment](https://image.slidesharecdn.com/dx3424512455-130912044630-phpapp01/85/Investigating-Waste-Water-Treatment-in-a-Closed-Environment-1-320.jpg)
![International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2451-2455 ISSN: 2249-6645
www.ijmer.com 2455 | Page
Figure 8a
Figure 8b
V. CONCLUSION
Cases 1, 4 and 7 provide more efficient water treatment as evident in the observation compared to cases 2, 3, 5 and
6. From the figures, it is obvious that case 1 (i.e. Multiplicanda numerosa) with the largest amplitude curves brings the
logistic curve with slow decay will provide the most efficient water treatment because it can successfully thrive in the tank of
waste water to consume all the bacteria in fewer days and has the maximum permissible population below 0.05 per cubic
centimeter (see figure 8b) within water discharge into the environment. This is evident in figure 8a which shows the
comparison between 1, 4 and 7. So from figure 8b, 10 organisms should be introduced initially and this confirms the claim of
the SEWER scientists.
REFERENCE
[1] C. Zheng and G. D. Bennett Applied Contaminant Transport Modeling: Theory and Practices, Van Nostrand Rienhold, 1995.
[2] P. Dulal and S. Khan, “A time dependence mathematical model for dispersion of air pollutant from point sources,” International
Journal of Environmental Studies, vol. 35, pp. 197–208, 1990.
[3] M. Summerfield and W. Krebs, “Particle dispersion in a swirling confined jet flow source,” Particle Characterization, vol. 7, pp.
16–24, 1990.
[4] J. F. Sini, S. Anquetin, and P. G. Mestayer, “Pollutant dispersion and thermal effects in urban street canyon,” Atmospheric
Environment, Urban Atmospheric, vol. 30, pp. 2659–2677, 1996.
[5] O. D. Makinde, R. J.Moitsheki, and B. A. Tau, “Similarity reductions of equations for river pollution,” Applied Mathematics and
Computation, vol. 188, no. 2, pp. 1267–1273, 2007.
[6] R. J. Moitsheki and O. D. Makinde, “Symmetry reductions and solutions for pollutant diffusion in a cylindrical system,” Nonlinear
Analysis: Real World Applications, vol. 10, no. 6, pp. 3420–3427, 2009.
[7] SEWER, „‟ The uses of Micro-organisms to clean waste water and make it safe for re-use‟‟ 2009
[8] T. chinyoka and O.D. Makinde, „‟ Analysis of Nonlinear Dispersion of a poolutant ejected by an External Source into a Channel
Flow: Hindaawi Publication Corporation Mathematical Problems in Engineering. p17. 2010
[9] M. S. Kalamkin, „‟ Mathematical Modelling‟‟: Classroom notes in Applied Mathematics. 2010.
[10] T. I eldho, „‟Management of Waste Water‟‟: Department of Civil Enginering, IIT Bombay, 2011.](https://image.slidesharecdn.com/dx3424512455-130912044630-phpapp01/85/Investigating-Waste-Water-Treatment-in-a-Closed-Environment-5-320.jpg)
Investigating Waste Water Treatment in a Closed Environment
- 1. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2451-2455 ISSN: 2249-6645 www.ijmer.com 2451 | Page T. S. Nurudeen1 , R. A. Kareem2 , S. O. Salawu3 1, 2 Department of Mathematics, Lagos State Polytechnic, Ikorodu, Nigeria 3 Department of Mathematics, University of Ilorin, Ilorin, Nigeria Abstract: This paper focuses on the investigation of the claim of ‘The Sound and Environmental Water and Effluent Re- cycling company’ that water can be completely purified if a population of at least 10 organisms per cubic centimeter can be maintained in its cleaning tank for a period of at least 8 days. We investigated the growth patterns of each of these organisms and make recommendations about which one will provide the most efficient water treatment and how many organisms should be introduced initially. Keyword: Environment, Waste, Micro-organisms and Water Treatment, I. INTRODUCTION The importance of these kinds of investigations as well as the complimentary experimental works, say in large-scale water treatment and redistribution networks, thus makes them of great relevance [1-6]. It should however be remarked that to date, the literature on the waste water treatment is still quite sparse. This then forms the crux of the current investigation [8]. Untreated water supply causes a serious socio-ecological hazard. If the problem is not carefully controlled and monitored, large communities can be exposed to extensive health risks. It then requires taking immediate corrective measure to redress and mitigate against its impact in the society [9]. The use of micro organisms to clean waste water and make it safe for re-use is good idea for water treatment. The micro-organisms feed on harmful bacteria, thus removing them from the water. The efficiency of this method depends upon choosing the correct species of micro-organism and ensuring that it is able to thrive in the tank of waste water successfully for long enough to consume all the bacteria [7]. However, it is important that the population declines after completing its function, since the ecological balance could be upset by discharge number of micro-organism into the environment. II. MODEL EQUATION The growth of micro-organism in a closed environment (such as the cleaning tank) is determined by three principle factors. The reproduction rate The effect of increased crowding The build-up of toxins produced as waste products by the organisms. This can be modeled by the equation [9] t ducubuau dt du 0 2 1 Where tu the number of organisms per cubic centimeter of water is, a is the reproduction ratio, b is the crowding coefficient, and c is a measure of the sensitivity of the organisms to toxins. The integral term is necessary since the toxins in the environment are built up gradually over a period of time at a rate proportional to the number of organisms present. The model equation in 1 which is in Integra-differential equation is reduced to differential equations of the form Let t dutx 0 t tudu dt d dt dx 0 tu dt dx ux 2 Then 1 becomes cuxbuau dt du 2 cuxbuauu 2 3 Investigating Waste Water Treatment in a Closed Environment
- 2. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2451-2455 ISSN: 2249-6645 www.ijmer.com 2452 | Page III. NUMERICAL SOLUTION The claim of SEWER scientists that water can be completely purified if a population of at least 10 organisms per cubic centimeter can be maintained in its cleaning tank for a period of at least 8 days. We investigated by a MATLAB code for equations 2 and 3 that give the plots below on which the investigation is made. The maximum permissible population within water discharged into the environment is 0.05 per cubic centimeter. We considered the micro-organism (multiplicanda numerosa, solo anti-socials and reproductio sensitova) whose growth is determined by the three principle factors a, b and c defined above. Alternatively, they could use the combination of these three. These can be categorized into 7 cases for the organism coefficients. Case 1 Multiplicanda numerosa 003.0,01.0,20 cba Case 2 Solo anti-socialis 01.0,5.0,5 cba Case 3 Reproductio sensitova 08.0,001.0,5.0 cba Case 4 Multiplicanda numerosa and Solo anti-socialis 013.0,51.0,25 cba Case 5 Multiplicanda numerosa and Reproductio sensitova 083.0,011.0,5.20 cba Case 6 Solo anti-socialis and Reproductio sensitova 09.0,501.0,5.5 cba Case 7 Multiplicanda numerosa, Solo anti-socialis and Reproductio sensitova 093.0,511.0,5.25 cba In summary the following table shows the different cases. Case Organism Reproduction rate Crowding effect Toxin Build-up 1 mn 20 0.01 0.03 2 sa 5.0 0.5 0.01 3 rs 0.5 0.001 0.08 4 mn + sa 25 0.51 0.013 5 mn + rs 20.5 0.011 0.083 6 sa+ rs 5.5 0.501 0.09 7 mn + sa+ rs 25.5 0.511 0.093 Where mn = Multiplicanda numerosa sa= Solo anti-socialis rs= Reproductio sensitova Plot for cases 1 -7 are given in figures 1 – 7 below for 50 days in which the analysis of the results are based. The number of organisms per cubic centimetre chosen by experiments as initial values is 10, 20 and 30. The plots are given together for comparison from which the following inferences are drawn.
- 3. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2451-2455 ISSN: 2249-6645 www.ijmer.com 2453 | Page Case 1 Figure 1: a = 20; b = 0:01; c = 0:03 Case 2 Figure 2: a = 5; b = 0:5; c = 0:01 Case 3 Figure 3: a = 0:5; b = 0:001; c = 0:08 Case 4 Figure 4: a = 25; b = 0:51; c = 0:013
- 4. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2451-2455 ISSN: 2249-6645 www.ijmer.com 2454 | Page Case 5 Figure 5: a = 20:5; b = 0:011; c = 0:083 Case 6 Figure 6: a = 5:5; b = 0:501; c = 0:09 Case 7 Figure 7: a = 25:5; b = 0:511; c = 0:093 IV. DISCUSSION OF RESULTS It is observed from figures 1, 4, and 7 that number of organisms is more than 10 so growth of organisms is limited due to crowding effects. Toxin build-up only dominating in the long run, hence the population is insensitive to toxins. They are all good methods of treatment. Also for figures 2, 3, 5 and 6, it is observed that the number of organisms is less than 10. Growth is immediate due to lack of crowding effects and checked by accumulation of toxins. The populations are sensitive to toxins. They are not good methods of waste water treatment.
- 5. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2451-2455 ISSN: 2249-6645 www.ijmer.com 2455 | Page Figure 8a Figure 8b V. CONCLUSION Cases 1, 4 and 7 provide more efficient water treatment as evident in the observation compared to cases 2, 3, 5 and 6. From the figures, it is obvious that case 1 (i.e. Multiplicanda numerosa) with the largest amplitude curves brings the logistic curve with slow decay will provide the most efficient water treatment because it can successfully thrive in the tank of waste water to consume all the bacteria in fewer days and has the maximum permissible population below 0.05 per cubic centimeter (see figure 8b) within water discharge into the environment. This is evident in figure 8a which shows the comparison between 1, 4 and 7. So from figure 8b, 10 organisms should be introduced initially and this confirms the claim of the SEWER scientists. REFERENCE [1] C. Zheng and G. D. Bennett Applied Contaminant Transport Modeling: Theory and Practices, Van Nostrand Rienhold, 1995. [2] P. Dulal and S. Khan, “A time dependence mathematical model for dispersion of air pollutant from point sources,” International Journal of Environmental Studies, vol. 35, pp. 197–208, 1990. [3] M. Summerfield and W. Krebs, “Particle dispersion in a swirling confined jet flow source,” Particle Characterization, vol. 7, pp. 16–24, 1990. [4] J. F. Sini, S. Anquetin, and P. G. Mestayer, “Pollutant dispersion and thermal effects in urban street canyon,” Atmospheric Environment, Urban Atmospheric, vol. 30, pp. 2659–2677, 1996. [5] O. D. Makinde, R. J.Moitsheki, and B. A. Tau, “Similarity reductions of equations for river pollution,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1267–1273, 2007. [6] R. J. Moitsheki and O. D. Makinde, “Symmetry reductions and solutions for pollutant diffusion in a cylindrical system,” Nonlinear Analysis: Real World Applications, vol. 10, no. 6, pp. 3420–3427, 2009. [7] SEWER, „‟ The uses of Micro-organisms to clean waste water and make it safe for re-use‟‟ 2009 [8] T. chinyoka and O.D. Makinde, „‟ Analysis of Nonlinear Dispersion of a poolutant ejected by an External Source into a Channel Flow: Hindaawi Publication Corporation Mathematical Problems in Engineering. p17. 2010 [9] M. S. Kalamkin, „‟ Mathematical Modelling‟‟: Classroom notes in Applied Mathematics. 2010. [10] T. I eldho, „‟Management of Waste Water‟‟: Department of Civil Enginering, IIT Bombay, 2011.