International Journal of Modern Mathematical Sciences, 2014, 11(1): 13-23
International Journal of Modern Mathematical Sciences ISSN:2166-286X
Florida, USA
Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx
Article
Mathematical Modelling of Depletion of Different Storage Vessels
of Drinking Water by the Formation of Heavy Toxic Metals in the
Drinking Water
Ogunmola, Oyewole Adeniyi and Ogbaji Eka Oche*
Department of Mathematics and Statistics, Federal University Wukari
* Author to whom correspondence should be addressed. E-mail: [email protected];
[email protected]
Article history: Received 1 April 2014, Received in revised form 10 July 2014, Accepted14 July 2014,
Published 20 July 2014.
Abstract: Deterministic model was used to determine the depletion of water storage vessels
dissolving in the stored water of pH value of 6.6. Stability analysis was carried out and it
showed that there is instability in the depletion of the vessels. The numerical analysis showed
that the depletion is in the form of negative exponential curve. Metal Vessel depletes the
most at all levels of concentration of the input of toxic metal values, followed by earthen
vessel, then Cement Reservoir and the least is Plastic vessel. Negative exponential curve well
fits the generated data for each level of concentration of the input toxic metal values for each
vessel. The estimated rates of depletion of vessel at each level for each vessel are also
significant. The estimated rate of depletion of metal vessel is the highest, followed by the
estimated rate of depletion of earthen vessel. The least estimated rate of depletion among the
four vessels is plastic.
Keywords: negative exponential, instability, water storage vessels and toxic metals
1. Introduction
Metals are often characterised and distinguished from non-metals by their physical properties, the
ability to conduct heat, and an electrical resistance that is directly proportional to temperature,
Copyright © 2014 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci. 2014, 11(1): 13-23
14
malleability, ductility and even lustre. These properties especially that of a temperature dependent
conductivity, at least allow us to define what a metal is in contrast to non-metals and metalloids. However
as mentioned above, all of these physical properties are lost after the metal has been chemically
transformed into a chemical compound that can be taken up by plants [1].
Heavy metals" are chemical elements with a specific gravity that is at least 5 times the specific
gravity of water. The specific gravity of water is 1 at 4°C (39°F). Simply stated, specific gravity is a
measure of density of a given amount of a solid substance when it is compared to an equal amount of
water. Some well-known toxic metallic elements with a specific gravity that is 5 or more times that of
water are arsenic, 5.7; cadmium, 8.65; iron, 7.9; lead, 11.34; and mercury, 13.546 [3]. In small quantities,
certain heavy metals are nutritionally essential for a healthy life. Some of these are referred to as the trace
elements (e.g., iron, copper, manganese, and zinc). These elements, or some form of them, are commonly
found naturally in foodstuffs, in fruits and vegetables, and in commercially available multivitamin
products [3]. Diagnostic medical applications include direct injection of gallium during radiological
procedures, dosing with chromium in parenteral nutrition mixtures, and the use of lead as a radiation
shield around x-ray equipment [3]. Heavy metals are also common in industrial applications such as in
the manufacture of pesticides, batteries, alloys, electroplated metal parts, textile dyes, steel, and so forth
[3]. Many of these products are in our homes and actually add to our quality of life when properly used.
Heavy metals become toxic when they are not metabolized by the body and accumulate in the soft tissues.
Heavy metals may enter the human body through food, water, air, or absorption through the skin when
they come in contact with humans in agriculture and in manufacturing, pharmaceutical, industrial, or
residential settings. Industrial exposure accounts for a common route of exposure for adults. Ingestion is
the most common route of exposure in children [3]. Children may develop toxic levels from the normal
hand-to-mouth activity of small children who come in contact with contaminated soil or by actually
eating objects that are not food (dirt or paint chips) [3].
This model and its solutions could aid in decision supporting the usage of drinking water stored
in different storage vessels. The model used to illustrate the effect of toxic heavy metals emitted in a
drinking water. The particular the area whose water quality was the motivation for the study is the zone
C senatorial of Benue state. Rain water is used as the source of water. It is assumed that, there is no acid
rain in the area. We have presented a deterministic mathematical model for toxic heavy metals of drinking
water in different storage vessels. Stability analysis of the model was also carried out.The primary
objective of the present study is to use deterministic model to investigate the rate at which toxic heavy
metals of drinking water emitted from different storage vessels.
The basic assumptions is: the there is no acid rain in the area.
1.1. Model Equation
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Int. J. Modern Math. Sci. 2014, 11(1): 13-23
15
dP
 PH        
dt
(1.1)
dM
 MH  a  b  c  d
dt
(1.2)
dE
 EH  i  j  k  l
dt
(1.3)
dR
 RH  v  x  y  z
dt
(1.4)
2. Numerical Results
The values of the parameters in equations in section 1.1 are obtained from the results of the work
of reference [4]), the experiment upon which the mathematical models in this research work dwells on.
The values on the minimum, average and maximum concentrations of the toxic metals are showed in
tables 1-4. The Toxic metals referred to here are Cadmium (Cd), Chromium (Cr), Manganese (Mn) and
Nickel (Ni). These values are given below:
Table 1: The minimum, average and maximum concentrations of the toxic metals in Metal vessel
Level of
Concentrations
Minimum
Average
Maximum
Cadmium (Cd)
Chromium (Cr) Manganese (Mn)
Nickel (Ni)
0.0280
0.0536
0.0760
0.0350
0.0660
0.0920
0.0540
0.0737
0. 0.0950
0.6660
0.7860
0.9220
Table 2: The minimum, average and maximum concentrations of the toxic metals in Earthen vessel
Level of
Concentrations
Minimum
Average
Maximum
Cadmium (Cd)
Chromium (Cr) Manganese (Mn)
Nickel (Ni)
0.0011
0.0021
0.0042
0.0110
0.0210
0.0290
0.0210
0.0277
0.0350
0.0780
0.1845
0.7650
Table 3: The minimum, average and maximum concentrations of the toxic metals inPlastic vessel
Level of
Concentrations
Minimum
Average
Maximum
Cadmium (Cd)
Chromium (Cr) Manganese (Mn)
Nickel (Ni)
0.0010
0.0013
0.0020
0.0010
0.0019
0.0030
0.0010
0.0013
0.0020
0.0890
0.1572
0.3210
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Int. J. Modern Math. Sci. 2014, 11(1): 13-23
16
Table 4: The minimum, average and maximum concentrations of the toxic metals in Cement Reservoir
vessel
Level of
Concentrations
Minimum
Average
Maximum
Cadmium (Cd)
Chromium (Cr) Manganese (Mn)
Nickel (Ni)
0.0013
0.0021
0.0031
0.0020
0.0033
0.0060
0.0010
0.0013
0.0020
0.0890
0.1305
0.1750
2.1. Stability Analysis of the Toxic Heavy Metal State in the Model
Definition:
If f(t, x)=A(t)x + g(t),where A(t) is an n*n matrix function of t and g(t) is an n*1 vector function then,
X’= A(t)x + g(t)
(2.1)
Is called linear system of ordinary differential equations. Then the following theorem taken from
Grinshaw,1990, p.87 will be important in our work.
Theorem 2.1
The following theorem gives conditions for the stability of a linear system with constant coefficient.
Suppose that A is a constant matrix with eigenvalues 1 , 2 ,..., n .Then the stability of a solution of the
system (2.5) is determined according to the following criteria:
(a) If Re( i )<0 for all i=1,2,…,n, then there is uniform and asymptotic stability (UAS) .
(b) If Re( i )≤0 for all i=1,2,…,n, and the algebraic multiplicity equals the geometric multiplicity whenever
Re( i )=0 for any i, then there is uniform stability.
(c) If Re( i )˃0 for at least on i, or the algebraic multiplicity is greater than the geometric multiplicity, then
there is instability.
To find stability analysis of the model, we have find the Jacobian matrix.
 f 1 f 1 f 1 f 1 
 E P M R 


 f 2 f 2 f 2 f 2 
 E P M R 
J= 

 f 3 f 3 f 3 f 3 
 E P M R 
 f f f f 
 4 4 4 4
 E P M R 
Copyright © 2014 by Modern Scientific Press Company, Florida, USA
(2.2)
Int. J. Modern Math. Sci. 2014, 11(1): 13-23
17
Let
f1  PH        
f 2  MH  a  b  c  d
f 3  EH  i  j  k  l
f 4  RH  v  x  y  z
f1
E
f 2
E
f 3
E
f 4
E



J= 



f1
f
f
 H , 1  0, 1  0
P
M
R
f
f
f
 0, 2  0 2  H , 2  0
P
M
R
f
f
f
 H , 3  0, 3  0, 3  0
P
M
R
f
f
f
 0, 4  0, 4  0, 4  H
P
M
R
 0,
0
H
0
0
0
H
0
0
0
0
H
0
0
0
0
H







The corresponding characteristics equation is
J  I =0
Where J is jacobian matrix, I is identity matrix and  is eigenvalue
I=
 1 0 0 0 
 010 0 


 0 0 1 0 ,


 0 0 0 1





I = 



0 H 0 0  
 0 0 H 0 

 
J  I  H 0 0 0   

 
0 0 0 H  

 
-

H
0 -
0 0
10 0 0
0 1 0 0 
0 010

0 0 0 1

0 0 0 
0  0 0 
0 0  0  =0,

0 0 0 

0
0
H
0
-(H)
H-
H
0
-  0 =0
0 H-
-
0
H
-
H
0
0
0
0
H
0
0
- 0
0 H-
0
Copyright © 2014 by Modern Scientific Press Company, Florida, USA
=0
Int. J. Modern Math. Sci. 2014, 11(1): 13-23
 (H   )
- H
-(H)(H   )
0 -

0 H
H -
18
=0

2
  ( H   ) ( 2 )- H ( H   ) H =0


 3 ( H   )  H 3( H   ) =0
(  3  H 3 )( H   )=0
 (  3  H 3 )=0 or ( H   )=0
1  H , 2  H , 3 
 H  iH 3
 H  iH 3
, 4 
2
2
From theorem 2.1c the rate at which the toxic heavy metals emitted from different storage vessels are
unstable as 1 , 2  0 .
2.2. Effects of the Minimum Concentrations of Toxic Metals
Figure 1 shows the effect of the minimum concentration of the toxic metals on the four vessels
(Plastic vessel, Metal vessel, Earthen vessel, and Cement Reservoir). The four vessels show depletions
in the form of negative exponential curves; whereby, they each decreases with approximate constant rate
as time increases. Metal vessel depletes the most, having the highest rate of depletion. It depletes about
twice the rate of depletion of Earthen vessel. Earthen vessel depletes about twice the rate of depletion of
Cement Reservoir. Cement Reservoir depletes a little more than Plastic vessel. The Plastic vessel shows
the smallest and slowest depletion, only a little of it depletes at the end of time.
12
Vessel (mcg)
10
8
P(min)
6
M(min)
4
E(min)
R(min)
2
0
0
2
4
6
8
10
12
Time
Fig. 1: Changes in the four vessels when concentrations of toxic metals are at minimum values and pH
of 6.6 water
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Int. J. Modern Math. Sci. 2014, 11(1): 13-23
19
2.3. Effects of the Average Concentrations of Toxic Metals
Figure 2 shows the effect of the average concentrations of the toxic metals on the four vessels.
The four vessels show more depletion in the form of negative exponential curves than those in Figure 1.
Metal vessel depletes the most; it depletes about 1.9 times the rate of depletion of Earthen vessel. The
Earthen vessel depletes at about 1.5 times the rate of depletion of Cement Reservoir. Cement Reservoir
depletes a little higher rate than that of Plastic vessel.
Each of the vessels depletes more than when toxic metals are at minimum concentration levels.
2.4. Effects of the Maximum Concentrations of Toxic Metals
Figure 3 shows the effect of the maximum concentrations of the toxic metals on the four vessels
(Plastic vessel, Metal vessel, Earthen vessel, and Cement Reservoir). The four vessels show more
depletion in the form of negative exponential curves than those in Figure 2. Metal vessel depletes the
most, depleting about 1.5 times the rate of depletion of Earthen vessel; Earthen vessel depletes 1.2 times
the rate of depletion of Cement Reservoir; Cement Reservoir depletes 1.5 times the rate of depletion of
Plastic vessel.
Each of the vessels depletes more than when toxic metals are at average concentration levels.
12
Vessel(mcg)
10
8
P(avg)
6
M(avg)
4
E(avg)
R(avg)
2
0
0
2
4
6
8
10
12
Time
Fig. 2: Changes in the four vessels when concentrations of toxic metals are at average values and pH of
water is 6.6x10-6
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Int. J. Modern Math. Sci. 2014, 11(1): 13-23
20
12
Vessel (mcg)
10
8
P(max)
6
M(max)
4
E(max)
R(max)
2
0
0
2
4
6
8
10
12
Time
Fig. 3: Changes in the four vessels when concentrations of toxic metals are at maximum values and pH
6.6 water
3. Fitness of Negative Exponential Curve to the Generated Data
To determine if negative exponential curve fits the generated data for each vessel, regression
analysis is carried out. By this also, we can determine the rate of depletion of the weight of the vessels
where there is fitness and significant parameter estimates.
A statistical negative exponential curve is given as 𝑦𝑖𝑡 = 𝛽0 𝑒 −𝛽1𝑡 + 𝜀𝑡 , 𝑖 = 1,2,3
where,
𝑦𝑖𝑡 = weight of vessel at level 𝑖 in mcg unit at time 𝑡
𝛽0= the initial weight of vessel at time 𝑡 = 0
𝛽1= the rate of depletion of the vessel as time progresses
𝜀𝑡 = random error associated with the weight of vessel at time 𝑡.
𝑖 = 1(𝑚𝑖𝑛𝑖𝑚𝑢𝑚), 2(𝑎𝑣𝑒𝑟𝑎𝑔𝑒), 3(𝑚𝑎𝑥𝑖𝑚𝑢𝑚) level of concentrations of input toxic metals
values.
̂
The estimated negative exponential curve fit is given by 𝑦̂𝑡 = 𝛽̂0 𝑒 𝛽1𝑡 . Using least square method to obtain
∑ 𝑦𝑡−∑ 𝑦 ∑ 𝑡
the 𝛽̂0 and 𝛽̂1 , for a linear transformation of the negative exponential curve, we have, 𝛽̂1 = ∑ 2
2
𝑡 −(∑ 𝑡)
and 𝛽̂0 = 𝑦̅ − 𝑏𝑡.̅ Then the linearly transformed estimated negative exponential curve will be 𝑙𝑛𝑦̂𝑡 =
𝑙𝑛𝛽̂0 + 𝛽̂1 𝑡.
SPSS is employed to carry out the analysis on obtaining the fitness of the curve and the estimated
parameter values in the curve for each of the vessels. Below is the result from SPSS output:
Copyright © 2014 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci. 2014, 11(1): 13-23
a. For the Plastic vessel
Level
Adj. 𝑅 2
DW
F-Value
21
Coefficients
t-value
Minimum
1.000
1.516
1.015𝐸10∗
𝛽0 = 2.303
𝛽1 = −0.015
𝛽0 = 2663822.164∗
𝛽1 = −100769.178∗
Average
1.000
2.479
2.502𝐸10∗
𝛽0 = 2.303
𝛽1 = −0.031
𝛽0 = 1923766.813∗
𝛽1 = −158181.156∗
Maximum
1.000
1.769
4.371𝐸10∗
𝛽0 = 2.303
𝛽1 = −0.040
𝛽0 = 2006462.440∗
𝛽1 = −209070.493∗
b. For the Metal vessel
Level
Adj. 𝑅 2
DW
F-Value
Minimum
1.000
1.792
1.893𝐸11∗
𝛽0 = 2.303
𝛽1 = −0.169
𝛽0 = 1019878.584∗
𝛽1 = −435105.999∗
Average
1.000
2.644
7.396𝐸10∗
𝛽0 = 2.303
𝛽1 = −0.249
𝛽0 = 409708.851∗
𝛽1 = −271959.767∗
Maximum
1.000
2.137
5.690𝐸10∗
𝛽0 = 2.303
𝛽1 = −0.328
𝛽0 = 279047.881∗
𝛽1 = −238534.905∗
F-Value
Coefficients
Coefficients
t-value
c. For the Earthen vessel
Level
Adj. 𝑅 2
DW
Minimum
1.000
1.553
3.927𝐸10∗
𝛽0 = 2.303
𝛽1 = −0.044
𝛽0 = 1781405.224∗
𝛽1 = −198178.676∗
Average
1.000
1.905
38.498𝐸10∗
𝛽0 = 2.303
𝛽1 = −0.066
𝛽0 = 1654286.589∗
𝛽1 = −291510.784∗
Maximum
1.000
1.492
1.485𝐸11∗
𝛽0 = 2.303
𝛽1 = −0.096
𝛽0 = 1537403.696∗
𝛽1 = −385307.877∗
t-value
d. For the Cement Reservoir vessel
Level
Adj. 𝑅 2
DW
F-Value
Minimum
1.000
2.447
7.977𝐸09∗
𝛽0 = 2.303
𝛽1 = −0.016
𝛽0 = 2175377.375∗
𝛽1 = −89426.455∗
Average
1.000
2.185
5.722𝐸10∗
𝛽0 = 2.303
𝛽1 = −0.037
𝛽0 = 2421677.808∗
𝛽1 = −239213.095∗
Maximum
1.000
2.652
1.339𝐸11∗
𝛽0 = 2.303
𝛽1 = −0.072
𝛽0 = 1948107.731∗
𝛽1 = −365916.185∗
Coefficients
t-value
Significant at 5% level of significance.
Copyright © 2014 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci. 2014, 11(1): 13-23
22
Since the F-values are significant at all the levels for each vessel, the negative exponential curves
at all the three levels of concentrations of the input toxic metals for each of the vessel fits the generated
data. The adjusted coefficient of determination (Adj. R2) values showed that variations in the weight of
vessels are completely determined by time. The Durbin Watson (DW) test for first order of serial
correlation shows absence of serial correlation in the residuals. The parameter estimates are all significant
for each level in each vessel. Therefore, the estimated vessel weight for at each level for each vessel is
given in the linear regression equations below:
For Plastic vessel
𝑙𝑛𝑦̂1𝑡 = 𝑙𝑛2.303 − 0.015𝑡
𝑙𝑛𝑦̂2𝑡 = 𝑙𝑛2.303 − 0.031𝑡
𝑙𝑛𝑦̂3𝑡 = 𝑙𝑛2.303 − 0.040𝑡
where 1,2,3 are minimum, average and maximum levels of concentration of input toxic metals
respectively. The estimated rates of depletion of Plastic vessel are 0.015, 0.031 and 0.040 at the minimum,
average and maximum levels respectively.
For Metal vessel
𝑙𝑛𝑦̂1𝑡 = 𝑙𝑛2.303 − 0.169𝑡
𝑙𝑛𝑦̂2𝑡 = 𝑙𝑛2.303 − 0.249𝑡
𝑙𝑛𝑦̂3𝑡 = 𝑙𝑛2.303 − 0.328𝑡
where 1,2,3 are minimum, average and maximum levels of concentration of input toxic metals
respectively. The estimated rates of depletion of Metal vessel are 0.169, 0.249 and 0.328 at the minimum,
average and maximum levels respectively.
For Earthen vessel
𝑙𝑛𝑦̂1𝑡 = 𝑙𝑛2.303 − 0.044𝑡
𝑙𝑛𝑦̂2𝑡 = 𝑙𝑛2.303 − 0.066𝑡
𝑙𝑛𝑦̂3𝑡 = 𝑙𝑛2.303 − 0.096𝑡
where 1,2,3 are minimum, average and maximum levels of concentration of input toxic metals
respectively. The estimated rates of depletion of Earthen vessel are 0.044, 0.066 and 0.096 at the
minimum, average and maximum levels respectively.
For Cement Reservoir vessel
𝑙𝑛𝑦̂1𝑡 = 𝑙𝑛2.303 − 0.016𝑡
𝑙𝑛𝑦̂2𝑡 = 𝑙𝑛2.303 − 0.037𝑡
𝑙𝑛𝑦̂3𝑡 = 𝑙𝑛2.303 − 0.072𝑡
Copyright © 2014 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci. 2014, 11(1): 13-23
23
where 1,2,3 are minimum, average and maximum levels of concentration of input toxic metals
respectively. The estimated rates of depletion of Cement Reservoir vessel are 0.016, 0.037 and 0.072 at
the minimum, average and maximum levels respectively.
4. Conclusion
We used deterministic model to determine the depletion of water storage vessels dissolving in the
stored water of pH value of 6.6. The stability analysis showed that there is instability in the depletion of
the vessels, meaning that the vessels will continue to deplete. The numerical analysis showed that the
depletion is in the form of negative exponential curve. Metal Vessel depletes the most at all levels of
concentration of input toxic metal values. This is followed by Earthen vessel, then Cement Reservoir and
the least is Plastic vessel.
To determine if the distribution of the depleting vessels followed negative exponential curve and
then to determine the estimated depleting rates of the vessels, statistical regression analysis is carried out.
The results showed that negative exponential curve fits the generated data for each level of concentration
of the input of toxic metal values for each vessel. The estimated rates of depletion of vessel at each level
for each vessel are also significant. The estimated rate of depletion of metal vessel is the highest, followed
by the estimated rate of depletion of Earthen vessel. The least estimated rate of depletion among the four
vessels is plastic.
References
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