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JOURNAL TITLE:
International journal of energy technology and policy
USER JOURNAL TITLE: International journal of energy technology and policy
ARTICLE TITLE:
A mathematical modelling of the intensity of contaminants (CO 2 ) on occupancy level of a space
in continuous use
ARTICLE AUTHOR:
Oke, Ademola
VOLUME:
6
ISSUE:
5-6
MONTH:
YEAR:
2008
PAGES:
502-514
ISSN:
-
OCLC #:
PATRON:
Oyedokun, Oluwafemi
PATRON NOTES:
primo.exlibrisgroup.com:primo3-Article-s
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502
Int. J. Energy Technology and Policy, Vol. 6, Nos. 5/6, 2008
A mathematical modelling of the intensity of
contaminants (CO2) on occupancy level of a space
in continuous use
S.A. Oke*, O. Damisa and O.I. Oyedokun
Department of Mechanical Engineering,
University of Lagos, Nigeria
E-mail:-E-mail:-E-mail:-*Corresponding author
O.G. Akanbi and A.D. Adeyeye
Department of Industrial and Production Engineering,
University of Ibadan, Nigeria
E-mail:-E-mail:-Abstract: This study investigates the Indoor Air Quality (IAQ) of a room
apartment and focused on the number of occupants that a space or an apartment
can accommodate as a result of increase in the concentration level of the CO2
contaminant present within the space. The law of mass action was applied to
generate a relation between the concentration of CO2 produced and the
concentration of the reactants (i.e., O2 and CO). A decay equation was also
used to relate the variation in the number of occupants with the level of the
concentration of CO2 within the space only at steady state conditions.
Keywords: contaminants; air quality; indoor; room ventilation; CO2;
concentration level.
Reference to this paper should be made as follows: Oke, S.A., Damisa, O.,
Oyedokun, O.I., Akanbi, O.G. and Adeyeye, A.D. (2008) ‘A mathematical
modelling of the intensity of contaminants (CO2) on occupancy level of a space
in continuous use’, Int. J. Energy Technology and Policy, Vol. 6, Nos. 5/6,
pp.502–514.
Biographical notes: S.A. Oke graduated in Industrial Engineering from the
University of Ibadan, Nigeria with a Bachelor and Master’s Degrees in 1989
and 1992, respectively. He worked for the IDM Services Limited as a
consultant. He lectures in the Department of Mechanical Engineering,
University of Lagos. He has reviewed papers for several international journals.
O. Damisa is a Senior Lecturer at the Department of Mechanical Engineering,
University of Lagos, Nigeria. He has BSc in Mechanical Engineering with
First Class Honours in 1973. He also has DIC and PhD at Imperial College,
London in 1981. His research interests also include vibration and dynamics
analysis of systems. He has published several papers in choice journals.
I.O. Oyedokun, is a BSc (Hons) First Class graduate of the Department of
Mechanical Engineering, University of Lagos.
Copyright © 2008 Inderscience Enterprises Ltd.
A mathematical modelling of the intensity of contaminants (CO2)
503
O.G. Akanbi, PhD received his BS from Ohio State University, USA, his MSc
and an MBA from Oklahoma State University, and his PhD from the
University of Ibadan. He is a member of Nigerian Institute of Industrial
Engineering, Member of the Nigerian Society of Engineers, and a Registered
Engineer with the Council of Registered Engineers of Nigeria. He was the
General Manager of Conpole Nigeria, Limited. He has presented several
international and local papers and reviewed journals. Currently, he lectures at
the University of Ibadan, Nigeria.
A.D. Adeyeye has a BSc (Ife) and MSc (Ibadan). He is currently a PhD student
of the Department of Industrial Engineering, University of Ibadan. He teaches
in the same department.
1
Introduction
From various research findings, it seems that a great number of people are exposed to an
indoor air environment longer than they are to an outdoor environment (ASHRAE, 1996,
1989; Carpenter, 1996; Emmerich and Persily, 1997; Levine et al., 1994; Vaculik and
Plett, 1993). The obvious reason is that many people spend most of their time for daily
activities indoors. Apart from outright display and sales of goods in the open air, even
marketing activities that are field-focused involve moving from one office to another or
from one household to another. Thus, many activities such as manufacturing and
maintenance operations, which are factory-based, are indoor activities. People engaged in
indoor activities seem to be in the majority since the outdoor air environment
may be uncomfortable to the delicate body of human being (Leephakpreeda et al.,
2001). Although CO2 is not the only indoor air pollutant, the focus of this paper seems
to be on CO2 since it can be used as an index of Indoor Air Quality (IAQ).
As Leephakpreeda et al. (2001) noted, this may be because CO2 is a human-waste fluid
and if CO2 concentration is controlled at the desired level, then other pollutants will be
controlled at acceptably low levels as well.
Previous studies on modelling of the intensity of contaminants have been conducted
on a worldwide scale in Mexico (Cynthia et al., 2008) and the UK (Kolokotroni and
Katsoulas, 2002), among other locations, while no documentation seems to exist for the
Nigerian case. Cynthia et al. (2008) evaluated the impact of an improved wood burning
stove (Patsari) in reducing personal exposures and indoor concentrations of particulate
matter (PM2.5) and carbon monoxide (CO) in 60 homes in a rural community in Mexico.
Kolkotroni et al. (2002) investigated the performance of thermal passive ventilation
stacks in the classrooms of a school in the UK during the summer. Measurements of air
temperature, carbon dioxide (CO2) and air velocity were carried out, from which
ventilation rates in the classrooms and stacks were calculated. Other studies relevant to
the current work are as follows. Leephakpreeda et al. (2001) reported on the theoretical
and experimental studies of occupancy-based control of the ventilation to demonstrate the
practical usefulness of the occupancy-based scheme for real time air ventilation control
and energy saving. In Carpenter (1996), it was reported that CO2-based Demand Control
Ventilation (DCV) was applicable and it saved energy when compared with constant
ventilation. Ke and Mumma (1997) determined the changeable occupancy for the
ventilation control based on the CO2 concentration in the outdoor air and the return air.
504
S.A. Oke et al.
Vaculik and Plett (1993) investigated the control strategies of the CO2-based DCV by
simulations for a typical occupancy profile during a working day. Kagi et al. (2007) noted
that there are various emission sources of chemical contaminants. These emissions cause
air pollution. Air pollution occurs as a result of the combustion of fossil fuels (i.e.,
petroleum, gasoline and natural gas) and the discharge of gaseous chemicals from
industry and transport into the surrounding atmosphere.
Kagi et al. (2007) reported that in heavily industrialised areas, there is a high
incidence of lung cancer and chronic bronchitis due to the inhaling of dust and smoke
particles. In coal-mining areas, ‘black lung’ is a common occurrence in miners due to the
inhalation of coal dust. In the plastic industry, asbestos dust, a carcinogen, causes
pulmonary cancer. Various hydrocarbons that are released into the atmosphere from car
exhausts include carbon monoxide. Carbon monoxide, a poisonous gas produced because
of incomplete combustion, combines with haemoglobin in red blood corpuscles to form
carboxyhaemoglobin. This substance impedes the transport of oxygen in the blood.
Carbon dioxide (CO2) is another contaminant present in the ambient air at a level of
0.03% and is generated by the respiration of human beings and animals. An individual is
able to generate 4.72 × 10–3 litres per second of CO2 during respiration. The effects
of carbon dioxide contamination by volume (or concentration) become severe as the
concentration increases. 1–2% CO2 continuous exposure leads to headaches and
dysproea, 3% CO2 can lead to severe headaches, 5% CO2 exposure can lead to mental
depression, 6% CO2 exposure can lead to visual impairment and a 10% concentration
of CO2 with respect to air volume can lead to unconsciousness.
The structure of the paper is as follows: introduction, methodology, case study,
discussion and conclusion. The introduction provides the motivation for the study and
a justification for the choice of topic. In Section 2, methodology discuses the approach
utilised to solve the problem. Section 3 is a case description of practical instances that
verify the workability of the model. Section 4 presents a discussion of important points.
Section 5 presents the concluding remarks.
2
Mathematical modelling
2.1 Definition of terms
y:
The concentration of carbon (IV) oxide produced; it is represented by
percentage of air volume in that environment
f:
The velocity constant, which is similar to the frequency of the chemical reaction,
the unit is cycles/sec
Aa:
The initial concentration of oxygen in the air; it is also represented by
percentage of air volume
Bb:
The initial concentration of carbon (II) oxide, it is expressed as percentage of air
volume
a and b: Combining moles of oxygen and carbon (II) oxide, respectively
yc:
Complementary function
yp:
Particular solution
ψ:
Constant
A mathematical modelling of the intensity of contaminants (CO2)
505
C:
Q:
Q c:
Ci:
Co:
Increase in contamination concentration
Volumetric flow per second per person of the incoming air into the space
Volume of pollutant (Carbon (IV) oxide) produced per person within the space
Initial concentration of contaminant in the space at time zero
Concentration of the contaminant in the incoming air to the space
V:
n:
vn:
Volume of the space ≠ IV per person
Number of air changes per hour for the whole space
Volume at the instant of n.
2.2 Problem analysis
The concentration of carbon (IV) oxide produced by the chemical reaction between
carbon (II) oxide and oxygen can be determined by using the Law of Mass Action, which
states that under constant temperature, the velocity of a chemical reaction is proportional
to the product of the concentrations of the reacting substances.
Considering the chemical reaction: 2CO(g) + O2(g)
2CO2(g)
The combining ratio of CO to O2 is 2 : 1. Mathematically, the Law of Mass Action is:
dy
ay
by
= f Aa −
Bb −
.
dt
a +b
a +b
(1)
As carbon (II) oxide is released to the atmosphere, it reacts almost immediately with the
oxygen in the air, depending on the temperature of the air, to yield carbon (IV) oxide.
This paper assumes that the concentration of CO2 produced as a result of this reaction is
the same of the volume of CO produced was stored and made to accumulate for the same
period of time it took to release the CO to the atmosphere, and the gas reacted with
oxygen. Then, the gas was released, with an initial percentage by volume of air, Bb and
consequently reacted with oxygen, having an initial percentage by volume of air, A.
By solving the ordinary differential equation in equation (1), we have:
fab 2
Ab + aB
dy
+ f
y + fAB.
y=
2
dt
a+b
( a + b)
(2)
The solution to the equation above can be classified into complementary and particular
equations, i.e., y = yc + yp.
Now, solving for the particular function:
ab
Ab + aB
dy
+ f
y= f
2
dt
a+b
( a + b)
2
y .
ab
Ab + aB
.
Let R =
f and P = f
2
( a + b)
a+b
(3)
506
S.A. Oke et al.
Then equation (3) becomes:
dy
+ Ry = Py 2 .
dt
(4)
Equation (4) is a first order linear and non-homogeneous differential equation. Hence,
using Bernoulli’s equation to solve it,
y −2
dy
+ Ry −1 = P.
dt
(5)
Let
Z = y −1 .
(6)
Therefore
dz
dy
= − y −2
.
dt
dt
Multiply through equation (5) by –1:
dy
− Ry −1 = − P.
dt
− y −2
(7)
(8)
Substituting equations (6) and (7) into (8) yields:
dz
− Rz = − P.
dt
(9)
Solving equation (9) by the use of Integrating Factor (IF),
IF = exp ∫ − Rdt = e − Rt
(10)
Z . IF = ∫ − P . IF dt
(11)
Ze − Rt = ∫ − Pe− Rt dt =
But
Z=
P − Rt
e +ψ.
R
P
P + ψ Re Rt
+ ψ e − Rt =
.
R
R
(12)
(13)
Also, Z = 1/y for equation (6), which implies that y = 1/z.
Therefore,
R
.
P + ψ Re Rt
This is the particular solution.
Now, solving for the complementary function,
yp =
(14)
yc = constant = φ
(15)
A mathematical modelling of the intensity of contaminants (CO2)
dyc
= 0.
dt
Putting equations (15) and (16) into the differential equation:
507
(16)
φ R = fAB
(17)
fAB
.
R
(18)
φ=
Hence,
yc =
Thus,
y=
fAB
.
R
R
fAB
+
.
P + ψ Re Rt
R
(19)
(20)
Equation (20) gives the value of the concentration of carbon (IV) oxide produced due to
the chemical reaction between carbon (II) oxide and oxygen. Carbon (IV) oxide can be
regarded as a pollutant for man but an important gas for the photosynthesis in plants.
This paper focuses on its effect on man; hence, we can conclude against this background
that it is a contaminant. In a time increment dt, the contamination in a space increases due
to the influx of the contaminated outside air into the space and that generated by the
occupants.
Therefore,
QCo
CI =
+ Qc dt per person in the space.
10000
(21)
Contamination levels are usually referred to as parts per 10,000. Now, let us assume that
there is no build up of pressure in the space and that natural ventilation occurs,
so that no air storage term exists in the continuity of air flow equation. Contamination
leaves with the effluent air. The concentration that leaves = Co. For natural ventilation,
the extract air flow is equal to the inflow air rate; only, the level of contamination carried
by this flow differs.
QC
Co =
dt per person in the space.
10000
(22)
The net change in the contamination level in the space over the time increment dt is
CI – Co,
QCo
Qc
dc =
+ Qc −
dt per person in the space.
10000
10000
(23)
Expressing dc as a concentration in parts per 10,000 of air/volume of room space,
we have:
508
S.A. Oke et al.
dc
1 QCo
Qc
=
+ Qc −
dt .
10000 v 10000
10000
(24)
Re-arranging
dc QC QCo + 10000 Qc
.
+
=
dt
v
v
(25)
Equation (25) can be solved by the use of an IF
IF = exp ∫
Qt
Q
dt = e v .
v
Therefore
Qt
1
C . IF∫ (QCo + 10000Qc ) . e v dt
v
(26)
Qt
Qt
v 1
C e v = . (QCo + 10000Qc ) e v + β
Q v
(27)
Qt
C ev =
Qt
1
(QCo + 10000Qc ) e v + β .
Q
(28)
To determine the value of the constant β, let us consider the boundary condition at time
t = 0, c = Ci.
Ci = Co + 10000
Qc
+ β.
Q
(29)
Therefore
β = Ci − Co + 10000
Qc
Q
.
(30)
Hence, putting equation (30) into (28)
Qt
10000Qc
Ce v = Co +
Q
Qtv
Qc
e + Ci − Co + 10000
Q
.
(31)
Dividing through by eQt/v, we have:
10000Qc
C = Co +
Q
Qt
−
v
C
e
+
i
Qc
Co + 10000
Q
− Qtv
e .
(32)
Re-arranging and factorising, we have:
10000Qc
C = Co +
Q
Qt
Qt
−
−
v
v
1
e
C
e
.
−
+
i
(33)
A mathematical modelling of the intensity of contaminants (CO2)
509
Equation (33) is the general expression for the contamination within a space at any time t.
Consider the fact that,
Q=
vn
3600
(34)
where n is the number of air changes per hour for the whole space. Putting equation (34)
into equation (33) yields:
10000Qc
− nt
− nt
C = Co +
= (1 − e ) + Ci e .
v
n
3600
(
)
(35)
Note that n is different from vn since n refers to number of air changes per hour for the
whole space, while vn is the volume at that instant. We further state that
C − Ci e − nt
3.6 × 107 Qc
− Co =
− nt
(1 − e )
nV
(36)
C − Ci e − nt − Co (1 − e − nt ) 3.6 × 107 Qc
=
.
(1 − e − nt )
nV
(37)
By taking the inverse of both sides of equation (37), we have:
(C − C e
i
(1 − e − nt )
− nt
− Co (1 − e
− nt
))
=
nV
.
3.6 × 107 Qc
(38)
From equation (38),
V=
3.6 × 107 Qc (1 − e − nt )
(C − C e
i
− nt
− Co (1 − e− nt ) )
(39)
.
Now, let the volume of the space be represented by Vs while the expected number of
occupants be N. This implies that
N=
Vs
.
N
(40)
Therefore, putting equation (40) into (39) yields:
3.6 × 107 Qc (1 − e − nt )
Vs
=
.
N ( C − Ci e− nt − Co (1 − e − nt ) )
(41)
From equation (41),
N=
V s {C − Ci e − nt − Co (1 − e − nt )}
3.6 × 107 Qc (1 − e − nt )
.
(42)
510
S.A. Oke et al.
Under normal atmospheric conditions, i.e., when there are no pollutants like carbon (II)
oxide or carbon (IV) oxide released into the air by any human activity, the volume
of carbon (IV) oxide is 0.03% air volume. This paper assumes that the concentration
of these gases should be represented by their percentage composition of air volume.
It implies that:
Co = 0.03 + y.
(43)
By putting equation (20) into (43) yields:
Co = 0.03 +
R
fAB
+
.
Rt
P + ψ Re
R
(44)
ab
Ab + aB
.
Recall that: R =
f , and P = f
2
a+b
( a + b)
Putting the corresponding value of P and R into equation (44) yields:
( Ab + aB ) f
Co = 0.03 +
Ab + aB
( Ab + aB) a + b
(a + b) f ab
+
ψ
e
2
a+b
( a + b)
+ AB(a + b) (45)
( Ab + aB )
t
But a = 1, b = 2, and
(2 A + B ) f
Co = 0.03 +
2 A+ B
2
3
ψ
f
A
B
(2
)
e
+
+
3
+ 3 AB .
2A + B
t
(46)
Therefore, putting equation (46) into (42), we have:
(2 A + B ) f
V s C − Ci e − nt − 0.03 +
2 / 3 f + ψ (2 A + B) e( 2 A + B 3 )
N=
{3.6 × 107 Qc (1 − e− nt )}
3
3 AB
− nt
+
−
(1
e
)
t 2 A + B
. (47)
Case study
3.1 Case study I
A classroom having a volume of 283 m3 undergoes 1–5 air changes per hour due to
natural ventilation. The school is located in an industrial estate where the environment
is seriously polluted by CO from the manufacturing companies within the estate.
Now, assuming that the concentration of CO2 a person can produce during respiration is
4.72 × 10–6 m3/s. The critical value of the concentration of CO2 under consideration
here is 1%.
A mathematical modelling of the intensity of contaminants (CO2)
S/N
511
A%
B%
C%
N
1
21.00
0.020
0.030
242.0
2
20.90
0.030
0.045
238.0
3
20.70
0.050
0.075
213.0
4
20.30
0.080
0.120
220.0
5
20.10
0.085
0.127
218.0
6
19.90
0.087
0.130
217.0
7
19.80
0.088
0.132
217.0
Specimen calculation
For case I
Given that A = 21%, B = 0.02% and b = 2, a = 1; Cc = 1% or 100 parts per 10,000
Co =
AB (a + b)
21 × 0.02 × 3
=
= 0.03% or 3 parts per 10,000.
Ab + aB
21 × 2 + 0.02
Therefore, N =
(Cc − Co ) nV (100 − 3) × 1.5 × 283
=
= 242.
3.6 × w7 × Qc
3.6 × w7 × 10 −6
3.2 Case study II
A media house planned on building a cinema hall that would operate for six hours a day
with the use of mechanical ventilating equipment having air changes per hour value of
four. The company has two pieces of land located at different places where patronage
rates are almost the same. The environmental conditions of the two areas are as stated
below:
Relative humidity
L
V
50%
51%
Ambient temperature
20°C
21°C
Atmospheric pressure
1.013 bars
1.013 bars
0.02%
0.08%
21%
20%
Percentage of CO released by other companies
Percentage of O2 available in air
The critical concentration of CO2 is assumed to be 1%. The initial concentration of CO2
present in the supposed hall is assumed to be equal to that of the ambient environment.
The company needs advice on where to build the cinema hall for maximum profit.
Assuming that f = 1 cycle per hour and ψ = 1.
Considering location L:
Qt
10, 000Qc N
v
−
C = Co +
1
e
Q
Qt
−
+
c
e
i v
512
S.A. Oke et al.
but Co = y =
R=
R
fAB
( Ab + aB ) f
,
+
and R =
p + ψ R e Rt
R
a + b
ab
p= f
2
( a + b)
21 × 2 + 0.02
2
= 14.0067, and p = 2 = 0.222
3
3
Co =
- ×-
+
=
+ 0.03 = 0.03%
14.0067× 6
0.222 + 14.0067e- × 1037
or 3 parts per 10,000
Now, determining the volume of persons per volume of room space.
10000 × 4.72 × 10−6 × N × 3600
−4t
−4t
100 = 3 +
(1 − e ) + 3e .
V ×4
Since t = 6 hrs, n = 4,
169.92 N
−24
−24
100 = 3 +
(1 − e ) + 3e .
4V
This is approximately equal to:
100 = 3 +
169.92 N
4V
97 × 4 N
= = 2.283 persons per 1m3 ;
169.92 V
V
= 0.438 m3 per person.
N
3.3 Case study III
A lecture theatre has a volume of 1500 cm3. The maximum level of carbon dioxide
at the end of the maximum usable time of the theatre is 0.1%, assuming an initial
concentration of 0.03%, equal to that in the outside air used for ventilation.
Natural ventilation can provide 1.5 air changes per hour. If each occupant generates CO2
at a rate of 5 × 10–6 m3/s and the maximum number of occupants that can use the space
for the maximum time is 113, we can determine the maximum allowable time as follows.
Qt
−
− Qtv
10000Qc N
v
C = Co +
1 − e + ci e
Q
10000 × 5 × 10−6 × 113 × 3600
−1.5t
−1.5t
10 = 3 +
(1 − e ) + 3e
×
1500
1.5
2.04
= e −1.5t
9.04
t = 1 hr.
A mathematical modelling of the intensity of contaminants (CO2)
513
Considering location V:
Co =
R
fAB
+
p + ψ R e Rt
R
R=
20 × 2 + 0.08
= 13.36
3
P=
2
= 0.222
32
Co =
13.36
20 × 0.08
+
= 0.12% or 12 parts per- + 13.36e13.36×6
13.36
Now determining the space per person:
100 = 12 +
169.92 N
4V
N 88 × 4
=
= 2.07 persons per m3 .
V 169.92
4
Discussion
The results obtained have been able to affirm the concept that the number of occupants
that can conveniently stay in an apartment or space reduces as the concentration of the
contaminant (especially CO2) increases. Consequently, the percentage by volume of O2
needed to respire comfortably drops. This has been verified by case study I. In case
study II, the adverse effect, of the indiscriminate release of CO into the atmosphere, has
been seen on profit maximisation. The increase in the amount of CO in location V
reduced the number of persons per m3 (volume). Consequently, for a specific hall
volume, the number of people that can conveniently stay in the hall in V, will be lower
than that in L, hence, the company will not make as much profit in V as in L. Therefore, it
will be advisable, based on the foundation of convenience on the part of the viewers and
profit on the part of the company, to build the cinema house in location L.
5
Conclusion
The decay equation introduced in this paper has been restricted to fully mixed
contamination, and is also applicable so long as there is no change in the contamination
generation rate during the time period covered by the integration. However, the
expressions are general in that they may be applied sequentially to time periods having
constant contamination generation rates or constant occupation levels. It may be applied
in parallel to any number of non-reacting contaminants that may be present together
within the space under study (Douglas). The paper has been able to validate the fact that
the occupancy rate is adversely affected by the concentration (volume in air) of the
contamination generated within the vicinity of the space utilised.
514
S.A. Oke et al.
References
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Air-conditioning Engineers, Inc., Atlanta.
Carpenter, S.C. (1996) ‘Energy and IAQ impacts of CO2-based demand-controlled ventilation’,
ASHRAE Trans, Vol. 102, pp.80–88.
Cynthia, A.A., Edwards, R.D., Johnson, M., Zuk, M., Rojas, L., Jimenez, R.D.,
Riojas-Rodriguez, H. and Masera, O. (2008) ‘Reduction in personal exposures to particulate
matter and carbon monoxide as a result of the installation of a Patsari improved cooking stove
in Michoacan Mexico’, Indoor Air, Vol. 18, No. 2, pp.93–105.
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