mathematical modelling and system analysis of inorganic carbon in the aquatic environment

Ecological Modelling 152 (2002) 129–143

Mathematical modelling and system analysis of inorganiccarbon in the aquatic environment

B. Mukherjee a,*, P.N. Pandey a, S.N. Singh b

a P.G. Department of Zoology, Section of En�ironmental Biology, Ranchi Uni�ersity, Ranchi, Indiab P.G. Department of Physics, Ranchi Uni�ersity, Ranchi, India

Received 4 January 2001; received in revised form 27 February 2001; accepted 7 November 2001

Abstract

A mathematical model is designed, framed in difference equations which define the changes in inorganic carbon dueto photosynthesis, respiration, and deposition of calcium carbonate in aquatic systems. The main input for the modelare pH and alkalinity, and the output provides the entire carbon speciation at a given point in time. The model wasthen converted into a computer program (the MATCAT program), and simulated for precision analysis and generality.The results obtained from the model were verified from the overall carbon balance calculated from field data. Thesensitivity of the model was increased by adjusting the pH value. Fitted with appropriate regression equations, themodel accounts for changes in parameters such as oxygen, or the input of detergents into the water body. Thus themodel can be used for the rapid assessment of productivity and respiration in ecosystems, as well as the impact ofpollutants (such as detergents) entering into the water body. © 2002 Published by Elsevier Science B.V.

Keywords: Mathematical model; System analysis; Carbon budget; Carbon speciation; Photosynthesis; Respiration; Calciumdeposition

www.elsevier.com/locate/ecolmodel

1. Introduction

A common approach to ecosystem modellinghas been to define such systems in terms of aseries of compartments: producers, consumers,and decomposers linked by equations describingthe energy or material cycling (Golley, 1960;Deevey, 1970; Nisbet and Gurney, 1976, 1982;Kremer, 1983; Kremer and Nixon, 1978; Odum,1982; Rajar and Cetina, 1997; Hoch et al., 1998;

Dilao and Domingos, 2000). This took the formof energy flow diagrams and nutrient cyclingmodels using reservoirs (nutrient pools) and trans-port pathways along which material and energywere transferred from one compartment to theother (Fig. 1), and in natural conditions, main-tained a dynamic equilibrium.

Since photosynthesis and the resulting primaryproductivity brings in a dynamism to the ecosys-tem, starting the energy flow and material cyclingalong the food chain, so one of the key approachin the modelling of the ecosystem has been thequantification of the two opposing processes ofphotosynthesis and respiration (Inverznizzi and

* Corresponding author. Present address: Flat No. 14, Ka-lyani Apartments, Navin Mitra Road, Burdwan Compound,Ranchi 834 001, India. Tel.: +91-6515-60080.

E-mail address: bm–[email protected] (B. Mukherjee).

0304-3800/02/$ - see front matter © 2002 Published by Elsevier Science B.V.

PII: S0304-3800(01)00457-4

mailto:[email protected]

B. Mukherjee et al. / Ecological Modelling 152 (2002) 129–143130

Terpin, 1997), especially the oxygen balance(Burr, 1941; Tressler et al., 1940; Odum andOdum, 1955; Odum, 1956, 1957; Verduin, 1957;Beyers, 1963, 1965; Franzisket, 1969; Muscatineet al., 1981; Mukherjee et al., 1992, 1993; Wirtz,2000; Kayomoto et al., 2000).

Subsequently, inorganic carbon was found tobe a more suitable parameter (Beyers and Odum,1959; Smith and Jokiel, 1978) for modelling pro-ductivity and respiration. Only two parameters,pH and carbonate alkalinity can be used toanalyse the entire carbon speciation, and the inputand output of inorganic carbon to and from thevarious compartments. Work on the modelling ofinorganic carbon has been done by King (1970),Shapiro (1973), Young and King (1973), Kingand Novak (1970), Novak and Brune (1985), Poh-land and Suidan (1978), Leihr et al. (1988), Gol-ubyatnikov et al. (1998), Heymans and Baird(2000), De Xing and Covghenor (2000). However,these models are quite complicated, relate to spe-cific conditions, and do not take into account forthe addition or the deposition of carbonates.

Our approach has been to formulate a simplemodel based on hydrogen ion activity and rateconstants of polyprotic acids, differentiating basic

chemical equations in different frames, not onlyto account for photosynthesis and respiration butalso for factors involving addition and depositionof carbonates. The model can evaluate all theprocesses simultaneously.

2. Materials and methods

Chemical analysis of water samples was doneusing standard techniques (Greenberg et al.,1992). The method of Culberson et al. (1970),Strickland and Parsons (1972) was used to calcu-late alkalinity by the pH method.

Values for total inorganic carbon, free carbondioxide, carbonates and bicarbonates were calcu-lated using the basic dissociation equation forpolyprotic acids (Christian, 1986; Leihr et al.,1988).

Field studies were done on aquatic systems(ponds with a depth of 0.5 to 1 m, and streamswith an average depth of 0.2 m). Laboratorysimulation studies were conducted in an aquarium(dimensions: 0.5×0.5×1.5 m), to calculate theeffect of photosynthesis, respiration, and carbon-ate addition on the carbon speciation, and com-

Fig. 1. The basic ecosystem model showing nutrient pools, compartments (boxes), and transport pathways (arrows) in the flow ofmatter and energy.

B. Mukherjee et al. / Ecological Modelling 152 (2002) 129–143 131

pared with field data. Both pretreated water andsamples brought from the field were used to mon-itor the effect of detergents. Known concentra-tions of detergents were introduced into thesystem, and the change in alkalinity, pH, andphosphates were noted. Correlation equationswere derived from the data obtained, and werethen used to calculate the daily detergent input infield conditions (Mukherjee et al., 1994; Mukher-jee and Pankajakshi, 1995).

3. System modelling

For the construction of the system model wehave used the basic carbon cycle of aquatic sys-tems. Here the major reservoir of inorganic car-bon is the free and bound carbon present in theaquatic system (Fig. 1).

The major uptake from the reservoir is due tothe fixation of carbon dioxide to organic carbonthrough the process of photosynthesis by the phy-toplankton and macrophytes present in the waterbody. This organic carbon then moves throughthe food chain via the consumers and then to thedecomposers. The other components have beenleft out to simplify the process. Another route bywhich carbon dioxide may be utilised from thereservoir is the deposition of calcium carbonatethrough chemical processes linked withphotosynthesis.

The major return pathway to the reservoir isvia the respiratory process taking place in theproducers and consumers, and decomposition oforganic matter by the decomposers.

We have used such a system involving basicchemical equations relating the carbon dioxideequilibrium under different conditions to con-struct a model that can define the rates of thesethree processes (photosynthesis, respiration, andcalcium addition and precipitation), given appro-priate variable inputs.

3.1. System design

Carbon dioxide reacts with water to form aweak acid known as carbonic acid which dissoci-ates as follows:

CO2+H2O� H2CO3Carbonic acid

�H+

+ HCO3−

bicarbonate ion�H++ CO3

2−

carbonate ion(1)

The amount of CO2 present as simple solutionplus that in the form of carbonic acid is termed asfree CO2, while the amount present in the form ofcarbonates and bicarbonates is termed as boundcarbon dioxide. If a strong acid is added thebound carbon dioxide is converted into the freeform.

CaCO3+H2SO4 �CaSO4+H2O+CO2 (2)

The amount of acid required to accomplish this isa measure of the alkalinity.

Carbonic acid is a polyprotic that is, it hasmore than one ionisable proton, and the distribu-tion of the different species is a function of thepH. It is thus possible to calculate the fraction ofthe total inorganic carbon (� Ct) that exists in agiven form under a specific pH.

Processes which utilise carbon dioxide (such asphotosynthesis) shift the equilibrium to the rightincreasing the pH. In contrast, processes releasingcarbon dioxide (such as respiration) shifts theequilibrium to the left, decreasing the pH. Thecarbon balance and the pH may be altered by twoother processes: addition of carbonates or bicar-bonates or the deposition of calcium carbonate.The above four processes (photosynthesis, respira-tion, addition, and deposition) change the carbonbudget which can be represented as follows:

� Ct �H2CO3+HCO3−+CO3

2− (3)

Then the fractions of the various forms of carbonare as follows:

�0=H2CO3

� Ct

; �1=HCO3

−

� Ct

; �2=CO3

2−

� Ct

where �0+�1+�2=1 and �0, fraction of freecarbon dioxide (Table 1); �1, fraction of bicarbon-ates; �2, fraction of carbonates. Considering thetwo stage ionisation of carbonic acid we have:

H2CO3=H++HCO3− (4)

and the equilibrium constant:


Table 1Symbols and definitions

Total inorganic carbon� Ct

�� Ct Change in total inorganic carbon

Fraction of total inorganic carbon as free carbon�0

dioxideFraction of total inorganic carbon as bicarbonates�1

Fraction of total inorganic carbon as carbonates�2

K1 First dissociation constant of carbonic acidSecond dissociation constant of carbonic acidK2

10−pHH+

Carbonate alkalinityAChange in AdAChange in H+dH+

Gross primary productionGppNpp Net primary production

RespirationR

� Ct= [H2CO3]+K1[H2CO3]

[H+]+

K1K2[H2CO3][H+]2

(11)

� Ct

[H2CO3]=

[H2CO3][H2CO3]

+K1[H2CO3]

[H+][H2CO3]

+K1K2[H2CO3][H+]2[H2CO3]

Dividing both sides with [H2CO3] we have:

1�0

=1+K1

[H+]+

K1K2

[H+]2(12)

or

1�0

=[H+]2+K1[H+]+K1K2

[H+]2(13)

or

�0=[H+]2

[H+]2+K1[H+]+K1K2

(14)

Similarly �1 and �2 can be expressed in terms ofK1, K2 and [H+] as follows:

�1=K1[H+]

[H+]2+K1[H+]+K1K2

(15)

and

�2=K1K2

[H+]2+K1[H+]+K1K2

(16)

The total inorganic carbon can be expressed asa function of [H+] and alkalinity so long asalkalinity is due to inorganic carbon and hydroxylions only. The basic equation for alkalinity is:

A= [HCO3−]+2[CO3

2−]+ [OH−]− [H+] (17)

The total alkalinity in freshwaters generally re-mains constant, apart from seasonal changes dueto evaporation and precipitation, and this has abearing on salinity. However, it changes due toaddition of carbonates and bicarbonates or due todeposition of calcium carbonate. The change ininorganic carbon due to addition or deposition isequal to half the change in total alkalinity (Smithand Key, 1975).

The relation between total alkalinity and inor-ganic carbon can be derived by rearranging Eq.(17).

K1=[H+][HCO3

−][H2CO3]

=4.3×10−7 (5)

again:

HCO3−=H++CO3

2− (6)

and the equilibrium constant for this reaction is:

K2=[H+][CO3

2−][HCO3

−]=4.8×1011 (7)

Therefore,

HCO3−=

K1[H2CO3][H+]

(8)

and

CO32− =

K2[HCO3−]

[H+](9)

If we also express CO32− in terms of [H2CO3] and

[H+] then:

CO32− =

K2[HCO3−]

[H+]or

CO32− =

K2×K1[HCO3]/[H+][H+]

so

CO32− =

K2×K1[H2CO3][H+]2

(10)

so Eq. (3) can also be expressed as:


A=�1� Ct+2�2� Ct+ [OH−]− [H+] (18)

or

A=� Ct(�1+2�2)+ [OH−]− [H+] (19)

or

� Ct(�1+2�2)+A− [OH−]+ [H+] (20)

or

� Ct=A− [OH−]+ [H+]

(�1+2�2)(21)

Since [H+] and [OH−] are negligible compared toA, we can write Eq. (21) as:

� Ct=A

�1+2�2

(22)

So we can calculate all the inorganic carbon spe-cies in the aquatic environment just by measuringthe pH and total alkalinity. Taking the values of�1 and �2 from Eqs. (15) and (16), we can derive� Ctas:

� Ct

=A

(K1[H+]/([H+]2+K1[H+]+K1K2))+2(K1K2/([H+]2+K1[H+]+K1K2)) (23)

or

� Ct

=A

(K1[H+]+2K1K2)/([H+]2+K1[H+]+K1K2)(24)

or

� Ct=A([H+]2+K1[H+]+K1K2)

K1[H+]+2K1K2

(25)

Similarly free carbon dioxide, bicarbonates, andcarbonates can be evaluated as follows:

CO2=A [H+]

K1[H+]+2K1K2

(26)

HCO3−=

AK1[H+]K1[H+]+2K1K2

(27)

and

CO32− =

AK1K2

K1[H+]+2K1K2

(28)

4. Mathematical model

We have already seen that total inorganic car-bon is a function of total alkalinity and pH. Ascarbon dioxide is utilised in photosynthesis thereis a decrease in the total inorganic carbon, and aconsequent increase in pH. The rate of utilisationof total inorganic carbon can be assessed by mea-suring the total alkalinity of the water body, andthe changes in pH with time. Similarly, thechanges in the above parameters with communityrespiration during the night can be measured bythe same method.

This led us to formulate a mathematical modelthat could simultaneously calculate the changes intotal inorganic carbon and chemical speciationdue to the four processes: photosynthesis, respira-tion, addition and deposition. The basis of ourmathematical model is Eq. (22), and we have usedthe differential equation as a rate measurer asfollows:

d� Ct

dH+

=�1+2�2/(dA/dH+)−A(d/dH+)(�1+2�2)

(�1+2�2)2

(29)

or

d� Ct

dH+ =1

(�1+2�2)×

dAdH+

Part 1

−A

(�1+2�2)2

Part 2

×� d

dH+�1+d

dH+2�2�

Part 3

(30)

Putting the values of �1 and �2, differentiating,and then combining the three parts we have:


d� Ct

dH+ =[H+]2+K1[H+]+K1K2

K1[H+]+2K1K2

×dA

d[H+]+

AK1([H+]+4K2[H+]+K1K2)(K1[H+]+2K1K2)2 (31)

or

d� Ct

d[H+]=

AK1([H+]2+4K2[H+]+K1K2)(K1[H+]+2K1K2)2 +Photosynthesis and respiration

[H+]2+K1[H+]+K1K2)(K1[H+]+2K1K2)2 ×

dAd[H+]

Addition and deposition of carbonates

(32)

The first part of the model accounts for changesin total inorganic carbon due to photosynthesisand respiration, and there is no change in alkalin-ity. While the second part of the model accountsfor the changes in total inorganic carbon due toaddition of carbonates and bicarbonates, andthere is a change in alkalinity.

5. Results

5.1. Precision analysis

The model was then subjected to precision anal-ysis. This consisted initially of testing both theparts of the model separately. The first part forphotosynthesis and respiration, while the secondpart for addition and deposition of carbonates.

The method consisted of the derivation of theamounts of inorganic carbon through a stepwiseprocess involving hourly calculations from fielddata, and then assessing the output from themodel.

The model was then subjected to optimisationby changing the parameters and testing the sensi-tivity followed by appropriate tuning of the modelto control the accuracy of the output. It was thenconverted into a computer program and the out-put was tested for different aquatic environmentsto assess its generality by varying the functionsand conditions. The model was then fitted withappropriate regression equations to define thechanges in other parameters. Once programmedin this way the model was able to provide a fairlyaccurate prediction (as confirmed by t-test involv-ing results obtained from both the processes, they

show no statistical difference at the 95% confi-dence limit) of the pattern of changes for a givenset of variables, not only for natural conditions,but also for aquatic systems receiving pollutantscontaining carbonates or bicarbonates (such asdetergents).

5.1.1. Analysis of the changes in inorganic carbonbudget due to photosynthesis and respiration

For analysis of the changes in inorganic carbonand for the formulation of the carbon budget, weselected an unpolluted pond ecosystem in Tiril atRanchi. The initial carbon dioxide concentrationwas 10 mg l−1, the alkalinity was 1.73 milliequiv-alents per litre (meq l−1), and the initial pH was7.30 (Table 2) at the start of the photosyntheticprocess at 06:00 h.

Since the concentration of carbon dioxide in thewater body was relatively low, its continuousuptake through the photosynthetic process de-creased the concentration to zero at 10:00 h. Oncefree carbon dioxide was used up, the producershad to depend on the reserve inorganic carbonpresent in the form of bound carbon (carbonatesand bicarbonates). Thus inorganic carbon wasborrowed from the system reserve.

Let us analyse Table 3 which has been derivedfrom Table 2. The table shows the changes inconcentration of the different species of inorganiccarbon at a given point in time due to the twoprocesses photosynthesis and respiration duringthe day light hours. Photosynthesis continuedfrom 06:00 to 13:00 h as indicated by a continu-ous increase in pH. From 13:00 h the rate ofphotosynthesis decreased and community respira-tion became dominant being indicated by a de-crease in pH. The various carbon species werecalculated using Eqs. (25)– (28). Table 3 gives usthe following stepwise output.

The total change in inorganic carbon due toboth photosynthesis and respiration during theperiod 06:00–13:00 h was −0.316382838 gC.This can be differentiated as follows:


(a) Change in free carbon dioxide =−0.199163343(b) Carbon borrowed from the carbon reserve as bicarbonates =−0.234438972(c) This leads to an increase in carbonates =+0.117219487

=−0.316382838Addition of a, b, and c

Table 2Data on the diel flux of physical and chemical parameters in a freshwater pond at Ranchi (Tiril)

CO2 (mg l−1) O2 (mg l−1)Time (h) Temperature (°C) pH Alkalinity (meq l−1)

Water Air

24.0 7.30 1.73 10 8.0006:00 20.024.5 25.0 8.4307:00 1.73 10 8.40

9.20101.738.6108:00 26.526.510.80509:00 1.7327.0 8.6326.5

29.010:00 8.7830.0 1.73 0 13.2011:00 30.0 8.79 1.73 0 13.4030.0

31.031.512:00 9.08 1.73 0 13.801.73 0 14.409.2231.513:00 31.5

9.01 1.73 014:00 13.0032.0 31.030.5 8.97 1.7315:00 032.0 12.80

32.016:00 29.0 8.97 1.73 0 12.8028.5 8.84 1.73 017:00 12.2031.0

18:00 28.0 8.76 1.73 0 11.2030.0

Table 3Analysis of the photosynthetic activity based on the changes in concentration of different carbon species (Alkalinity 1.73)

H+ Species of inorganic carbon (moles m−3)pHTime (h)

� Ct H2CO3 HCO3− CO32−

5.01187234×10−8 1.929600754 0.2012544406:00 1.726692617.30 0.0016536938.4307:00 3.71535229×10−9 1.722783766 0.014571308 1.686424915 0.021787542

2.45470892×10−9 1.706942769 0.00950661708:00 1.6648719878.61 0.03256384809:00 0.0340295951.6619408090.0090603941.7050307992.34422882×10−98.63

1.6353991590.006311831.689011409 0.0473004211.65958691×10−98.7810:0011:00 8.97 1.07151931×10−9 1.662740529 0.003961872 1.587557314 0.071221343

9.08 0.831763771×10−9 1.64349458812:00 0.00300131 1.550988914 0.08950554213:00 0.602559586×10−9 0.1188731981.4922536380.0020910099.22 1.613217916

+0.117219487−0.234438972−0.199163343−0.316382838Total

� Ct=H2CO3+HCO3−+CO32−=−0.199163343−0.234438972+0.117219487=−0.316382838.


Table 4Analysis of the respiratory activity based on the changes in concentration of different carbon species (Alkalinity 1.73)

Time (h) H+pH Species of inorganic carbon (moles m−3)


13:00 9.22 0.602559586×10−9 1.613217916 0.002091097 1.492253638 0.11887318Photosynthesis stops here as indicated by the decrease in pH

0.977237221×10−9 1.656206603 0.003579999.01 1.5725322414:00 0.0773733888.9715:00 1.071519305×10−9 1.66283136 0.003956521 1.587749676 0.0711251628.9716:00 1.071519305×10−9 1.66283136 0.003956521 1.587749676 0.071125162

1.445439771×10−9 1.681581483 0.0054531958.84 1.6225657517:00 0.05387171218:00 1.737800829×10−98.76 1.713067307 0.006710707 1.660491956 0.045864642

+0.099849391 +0.004619610Total +0.168238318 −0.073008538

� Ct=H2CO3+HCO3−+CO32−=0.004619610+0.168238318−0.073008538=0.099849391.

Thus the carbon budget is balanced so far as thephotosynthetic period is concerned.

We can now ask the question whether the bi-carbonates or carbonates are used during theprocess, how do they provide carbon dioxide, andhow do the carbonates increase? This can be seenfrom Eq. (33):

2HCO3− �CO2+CO3

2− +H2O (33)

Thus two moles of bicarbonates provide one moleof carbon dioxide and one mole of carbonate ion.Taking values from Table 3, we see that this isexactly the case.

2HCO3−

0.234438972 moles m−3�

CO2+CO32− +H2O

0.117219486 moles m−3 (0.234438972/2)(34)

The total period of photosynthesis was 7 h(06:00–13:00 h) during which 0.234438972 molesof inorganic carbon was borrowed from the re-serve bound carbon or carbonate alkalinity (spe-cifically bicarbonate ions). The total amount ofinorganic carbon fixed during the daytime isknown as gross primary production, since respira-tion also takes place during the day which releasesinorganic carbon, so that the total change ininorganic carbon shown in Table 3 represents netprimary production. So,

Npp=0.31638238 moles m−3 per day

=0.316382838×12 g

=3.796594056 gC m−3 per day

Photosynthesis stops at 13:00 h as indicated bya decrease in pH and an increase in inorganiccarbon concentration (Table 4). However, therewas no significant decrease in the amount of light.Studies on various green algae have shown thathigh or low temperature generally decreases pho-tosynthesis by various algal species (Novak andBrune, 1985) and so does low carbon dioxideconcentration (Burr, 1941; Oswald, 1960; Wrightand Mills, 1967; King, 1970). At an alkalinitybelow 2 equiv. m−3 (equivalents per metre cube)the algae become carbon limited when the concen-tration of carbon dioxide decrease to about 0.003moles m−3. In out study, the concentration ofcarbon dioxide decreases to about 0.002 molesm−3 (Table 4), limiting photosynthesis, and sodoes the high water temperature (31.5 °C) at13:00 h (Table 2).

From 14:00 h the inorganic carbon increasesdue to respiration, and a decrease in photosynthe-sis. Analysis of the data in Table 4 shows thatfrom 13:00 to 18:00 h (5 h), the total inorganiccarbon increases by 0.099849301 moles m−3, andwe can make the following budgetary analysis:


=+0.004619610(a) Increase in free carbondioxide moles m−3

=+0.168238318(b) Increase in bicarbonatesmoles m−3

=−0.073008538(c) Decrease in carbonatesmoles m−3

Addition of 1, 2, and 3 =0.099849391moles m−3

Thus the carbon budget is so far again balanced.Now we can calculate the rate of gross primaryproduction, net primary production, andrespiration.

=0.99849391 moles m−3 in 5 hThe respiratoryrate

=0.99849391/5 moles m−3 h−1

=0.019969878 moles m−3 h−1

=0.239638536 gC m−3 h−1

=0.019969878×17 moles m−3

(for the remaining 17 h whenonly respiration occurred=0.339487026 moles m−3)=0.339487026×12 gC m−3

=4.073855112 gC m−3

The hourly gross primary production can thenbe evaluated as:

Gpp=Npp+R=0.316382838+0.239638536

=0.556021374 gC m−3 h−1

and the total respiration R24=5.751324864 gCm−3 day−1.

Once respiration becomes dominant with thedecrease of photosynthesis, the loan of inorganiccarbon from the reserve is returned back torecharge the carbon dioxide system for utilisationthe next day. From Tables 3 and 4 we can inferthat:

loan repayment to bicarbonate species

=0.234438972 moles m−3

loan repayment per hour=0.168238318/5

=0.033647636 moles m−3 h−1

number of hours required

=0.234438072/0.033647636=7 h (13:00–20:00)

Simultaneously, the carbonate ions then reactwith the excess H+ ions to form bicarbonate ionsand in turn decrease in concentration:

H++CO32− =HCO3

− (35)

loan repayment to carbonate species

=0.117219487 moles m−3

loan repayment per hour=0.073008538/5

=0.014601707 moles m−3 h−1

number of hours required

=0.117219487/0.014601707

=8 h (up to 21:00 h)

After repayment of the loan, any increase ininorganic carbon will appear as free carbon diox-ide. Since the loan is repaid upto 21:00 h (from13:00 to 21:00 h), the amount of carbon dioxideevolved through respiration from 21:00 to 06:00 hthe following day when photosynthesis starts in-volves 9 h. The total amount of inorganic carbonreleased during this period is:

0.019969878×9=0.179728902 moles m−3

=0.179728902×12 g

=2.156746824 gC m−3

so the amount of carbon dioxide measured nextday will amount to:

moles � Ct×mol. wt. of CO2

mol. wt. of C

=2.156746824×4412

=7.908071688 mg l−1

It should be noted that photosynthesis and respi-ration does not change the alkalinity.

5.1.2. Testing the model output for photosynthesisand respiration

Now we can test the first part of the modelwhich defines photosynthesis and respiration. Inthis case, the second part of the model is equal tozero because there is no change in alkalinity.


Table 5Comparison between the stepwise calculation and model out-put

Process Change in inorganic carbon (moles m−3)

Model outputStepwise Differencecalculation

Photosynthesis −0.31682838 −0.33052362 +0.014140782

Respiration +0.099849391 +0.117730175 −0.017880784

period=8.26; average H+ for the respiratory pe-riod=8.99. Table 5 shows the comparison be-tween the stepwise calculation and the modeloutput. The model output is somewhat higher, ofthe order of 0.014–0.018 moles in m−3. Thedifference is predictable since we are taking aver-age pH. However, the sensitivity of the model canbe increased by taking pH average �0.01 (Table6).

5.1.3. Analysis of the carbon budget due toaddition of carbonates

Having tested the first part of the model wenow move on to test the second part of the modelwhich defines the changes in inorganic carbon dueto addition or deposition of carbonates. Table 7provides an analysis of the carbon budget due tothe addition of carbonates, as well as the changesin concentration of the different species of inor-

d� Ct

dH+ =AK1([H+]2+4K2[H+]+K1K2)

(K1[H+]+2K1K2)2

Using the model with the following consider-ations: A=1.73; K1=4.3×10−7; K2=4.8×10−11; average H+ for the photosynthetic

Table 6Results obtained after increasing the sensitivity of the model by using pH average �0.01

Process Change in inorganic carbon (moles m−3)

Stepwise calculation Model output After increasing sensitivity

Photosynthesis −0.33052362−0.31682838 −0.32485894+0.099849391Respiration +0.117730175 +0.113149825

Table 7Analysis of the changes in inorganic carbon due to addition of carbonates

Alkalinity (eq m−3) Species of inorganic carbon (moles m−3)H+pH

HCO3−� Ct H2CO3 CO32−

1.348962883×10−7 1.691315162 0.403773142 1.287084039 0.000457981.288 6.870.08811239 1.2838238441.288 7.53 2.951209227×10−8 0.000457981.374024312

0.0462728681.656 8.79 1.621810097×10−9 1.5634542641.615623936 0.058968041.7104907090.0029488471.8241942027.413102413×10−109.131.932 0.110754645

2.300 5.248074602×10−109.28 0.1778329142.124540112 1.9443341710.0010311759.45 0.2742775672.576 2.027444860.001672943.548133892×10−10 2.303395372

2.318157046 0.001818434 2.056672222.576 0.2596613889.42 3.801893963×10−10

2.398832919×10−10 2.761123084 0.0012829183.220 2.2996803319.62 0.460159834

Total +1.069807922 −0.402490224 +1.012596292 +0.459701854

� Ct=H2CO3+HCO3−+CO32−=−0.402490224+1.012596292+0.459701854=+1.069807922.


Table 8Results obtained after increasing the sensitivity of the model by using pH average �0.01

Change in inorganic carbon (moles m−3)Process

Stepwise calculation Model output After increasing sensitivity

−1.119327672Addition of carbonates −1.07816436−1.069807922

Differences +0.04951975 +0.03800851

ganic carbon. The total change in inorganic car-bon is +1.069807922 moles m−3. Free carbondioxide decreases due to the following reaction:

CO2+CaCO3+H2O=Ca(HCO3)2 (36)

while the concentration of carbonate and bicar-bonate ion increases. The total change can berepresented as follows:

� Ct1.069807922

= H2CO3−0.402490224

+ HCO3−

1.012596292+ CO3

2−

0.459701854

(37)

Thus the budget of inorganic carbon is againbalanced.

5.1.4. Testing the model output for addition ofcarbonates and deposition of calcium carbonate

Now we can test the second part of the modelwith the database presented in Table 7, and withthe following considerations:

Change in alkalinity (dA)=3.220−1.288=1.932

Change in inorganic carbon due to addition ordeposition of carbonates is equal to half thechange in alkalinity (Smith and Key, 1975), andH+ is equal to average H+.

d� Ct

dH+ =[H+]+K1[H+]+K1K2

K1[H+]+2K1K2

×0.5 dAdH+ (38)

Table 8 shows the comparison between the step-wise calculation and the model output. The sensi-tivity can be increased by modulating the pH. Thefirst part of the model is more sensitive to pHthan the second (Figs. 2 and 3).

Fig. 2. Sensitivity test through regression analysis of the firstpart of the model.

Fig. 3. Sensitivity test through regression analysis of thesecond part of the model.


Table 9Analysis of the changes in inorganic carbon due to photosynthesis, respiration, and addition and deposition of carbonates

Alkalinity (eq m−3) H+pH Species of inorganic carbon (moles m−3)


9.10 8.35 4.466835922×10−9 9.096811892 0.092541822 8.908540124 0.0957299383.548133892×10−9 10.24708113 0.08275122710.30 10.028659818.45 0.1356700973.548133892×10−9 7.700233776 0.0621839318.45 7.5360996987.74 0.10195015

8.397.31 4.073802778×10−9 7.293512355 0.06766022 7.141704271 0.0841478643.630780548×10−9 8.301189152 0.0686062648.34 8.1251657788.44 0.1074171113.89045145×10−9 8.572384523 0.0759352658.41 8.3928985198.60 0.10355074

8.147.22 7.22435960×10−9 7.304271325 0.1120047026 7.125574108 0.0472129457.677351413×10−9 7.504522258 0.079940967 7.3491625837.50 0.0754187088.33

−1.59228963 −0.012600855 −1.559377541Total −0.02031123

Table 10Comparison between the stepwise calculation and model output of the rates of utilisation of inorganic carbon for various processes

Study site d� Ct (moles m−3)Processes

Stepwise method Model output Differences

0.792289Basundhara pond 0.807020Photosynthesis and respiration −0.015Addition and deposition 0.800000 0.799915 +0.00008

3.52715 3.599680Ranchi lake −0.070Photosynthesis and respiration

0.813196Dasong stream 0.812984Photosynthesis +0.0002

5.2. Generality analysis

Having tested the model for precision we nowgo on to test the generality of the model based onthe sensitivity test. The mathematical model cannow be expressed as:

d� Ct

dH+ =AK1([H+]2+4K2[H+]+K1K2)

(K1[H+]+2K1K2)2

(pH average �0.01)

+[H+]2+K1H++K1K2

K1[H+]+2K1K2(H+ final+H+ initial/2�0.052)

×0.5 dAdH+

The model was then tested for changes in inor-ganic carbon for the various processes (photosyn-thesis, respiration, addition and deposition)taking place simultaneously (Table 9), while Table10 gives a comparison between stepwise calculateddata, and model output of changes in four differ-ent systems. The differences between the two are

of the order of 0.0008–0.070 moles and thusprovides a precise idea of the rate of metabolicprocesses, and the chemical speciation of the inor-ganic carbon in fresh waters in general.

5.3. Computer program (MATCAT)

The model was then converted into a computerprogram, the matrix calculator program (MAT-

CAT) for the rapid assessment of the rates of thevarious processes in different environments (Table10).

K=K1×K2; VXj=values of H+;

VYj=values of A

Then

dchlj=VYj×K1[(VXj)2+4K2(VXj)+K ]

[K1(VXj)+2K ]2(39)

dch2j=[(VXj)2+K1(VXj)+K ]

K1(VXj)+2K(40)


and

dchj=dch1j+dch2j (41)

5.4. Extension of the model

The model was then expanded to calculatesimultaneously the changes in other parameters inthe aquatic environment. This was done by fittingappropriate regression equations to the model.We can find out for instance, the amount ofoxygen liberated due to photosynthesis or theamount of nitrates or phosphorus taken up dur-ing the process.

5.5. Changes in oxygen

We used the data presented in Tables 2–4 tocalculate the changes in the concentration of inor-ganic carbon and defined by the following regres-sion equation:

O2 moles m−3= −0.6643� Ct+1.5129

(r= −0.9230; P�0.001) (42)

Fig. 4 shows a plot of the equation. The oxygenconcentration was calculated with the model andcompared with observed values at given points intime. Table 11 gives a comparison of the ob-served, and model output values.

The model can also be utilised to calculate theentry of pollutants such as detergents into theaquatic body since it changes the carbonspeciation.

5.6. Estimation of detergent input

The detergent input can be estimated from themodel by fitting appropriate regression equations.Since, the detergents contain a large amount offillers in the form of polyphosphates and sodiumcarbonate, the entry of detergent increases theconcentration of carbonates and phosphates(Mukherjee et al., 1993, 1994) leading to a changein the entire carbon speciation, pH, and alkalin-ity. We measured the changes in the parametersby introducing known amounts of detergents inlaboratory simulation studies as well as in fieldsituations where the aquatic body was receivingdaily detergent inputs. The entry was defined bythe following regression equation:

DI mg 1−1=18.83� Ct(mg 1−1)+118.63

(43)

(r=0.9052; P�0.001), DI=detergent input

Thus the model can be fitted with appropriateregression equations to provide a detailed analysisof the various parameters of a water body in aparticular time frame (Fig. 5).

6. Conclusion

System analysis and mathematical modellingform powerful tools in the assessment of thestructure and function of ecological and environ-mental systems, and the mode of developmentand oscillation in the presence of a stress. Steps in

Fig. 4. Regression curve of oxygen versus inorganic carbon inthe aquatic systems studied.

Table 11Comparison of the amounts of oxygen (mg 1−1) betweenobserved and model generated values

Time (h)

13:00 18:00

11.2014.40Observed valuesModel output 11.9914.11


Fig. 5. Regression curve of detergent input and change ininorganic carbon that can be fitted into the model to accountfor the entry of detergents.

carbon dioxide by the phytoplankton from boundcarbon thus testing various hypothesis forwardedduring the years.

The model was then subjected to precision anal-ysis, the first part of the model was found to besensitive, while the second part of the model wasrelatively less sensitive to pH changes. The modelwas optimised and converted into a computerprogram. Data collected from various aquatic sys-tems were then tested, and the model was foundto have a high degree of precision and generalityin predicting the changes in inorganic carbon.

Since the chemical rhythm in the aquatic envi-ronment resonates in tune with the photosyntheticand respiratory activities, it has been shown thatthe model can be fitted with appropriate regres-sion equations to account for changes in oxygen,phosphorus, and other elements linked with pro-ductivity. It can also account for the exogenousentry of carbon in the inorganic or organic form,as well as the influx of detergents.

The uniqueness of the model is its simplicityand output, exemplified by the fact that the entirecarbon speciation with time, and the rate of inputand output of carbon from the inorganic pool canbe modelled using only two parameters, pH andalkalinity. The model can be used for the rapidassessment of the water quality and the manage-ment of the aquatic ecosystem in order to main-tain its steady state or to maximise the output ofa particular resource.

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