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Box models

Box models are sometimes used in environmental modelling. For example, to simulate the fate of an organic pesticide in a lake one could consider two boxes: the lake water and the fish

The box model would be:

Such a system consists of one component (the pesticide) and two boxes. It may be described with two equations (in general one equation for each component in each box) such as:

This is a set of ordinary differential equations (ODE) and it may be solved given some initial conditions. An artificial example is given below showing how box models may be solved with SMath Studio. Many more examples of this kind of problems may be found in the book: J.B.Snape, I.J.Dunn, J.Ingham, J.E.Prenosil Dynamics of Environmental Bioprocesses, VCH, Weinheim, 1995. The example described below is a simple one-box model with two inflows and one outflow. It could be a reactor tank in an industry, for example. The microbial reactions are simulated according to a relatively complex model taken from the literature, to show the capabilities of SMath. The results of the simulation are presented graphically, and compared with some data. Adding more compartments simply requires adding more equations to solve. Diffusion between for example lake water and bottom sediments may also be simulated, as described in Snape et al (1995) p.429, etc.

Stirred Tank Reactor

A one-box model with five components: 1- Microbes (biomass) 2- Acetate (CH3COO-) 3- Sulphate (SO4-2) 4- Sulphide (HS-) 5- Bicarbonate (HCO3-)

Two pipes bring two aqueous solutions into the reactor tank: one pipe brings acetate, and the other brings sulphate. In the tank the two solutions are mixed and a redox reaction takes place where microbes reduce sulphate to sulphide, and at the same time they oxidize acetate to bicarbonate.

The sulphate reduction reaction may be written as: CH3COO- + SO4-2 --> 2HCO3- + HS- Acetate is also used by microbes to grow by creating biomass (C5H7O2N): 5/2 CH3COO- + NH4+ + 3/2 H+ --> C5H7O2N + 3 H2O(l) Many simplifications are made here: enough ammonium is added through the aqueous solutions going into the reactor tank. Also acid-base reactions are not considered in this example: H2S(aq), H2S(g) and CO3-2 are excluded. The system may be described with some initial conditions and five ordinary differential equations (ODE), one for each component in the single box.

Vol 1 m 3^* :

Definition of the tank reator:

F.in_acetate 0.002 m 3^* hr 1 -^* :

F.in_SO4 0.002 m 3^* hr 1 -^* :

F.out F.in_acetate F.in_SO4 + :

(the volume of the tank is fixed)

Vol F.out 1 -^* hr 250

Residence time:

Ac.in 0.001 mol L / * :

SO4.in 0.010 mol L / * :

The concentrations of the influents: (acetate and sulphate) Note that sulfate is in excess.

The "microbial" model developped by Jin and Bethke will be used, see: - Jin, Q., & Bethke, C. M. (2005). Predicting the rate of microbial respiration in geochemical environments. Geochimica et Cosmochimica Acta, Vol.69, p.1133–1143. - Jin, Q., & Bethke, C. M. (2007). The thermodynamics and kinetics of microbial metabolism. American Journal of Science, Vol.307, p.643–677.

Define the thermodynamic potential factor (F_T)

ΔG0 2 586.845 - (* 12.243 (+ 744.004 - (369.41 - (+ (- (kJ mol / * :

ΔG0 kJ mol / 48 -

ΔG.C 145 * kJ * mol 1 -^* :

χ 6 :

Note, these two values taken from: Jin, Roden and Giska, Geomicrobiology Journal 30 (2013) 173-185

function used below to avoid "log" of zero or negative numbers:

x mn 1050 -^x 1050 -^≤ (* (x x 1050 -^> (* (+ :

In the integration of the ODEs, a vector "y" will contain the variables (concentrations) of the components: X (biomass, mg/L), acetate, sulphate, sulphide, and bicarbonate; y = (X, ac, so4, hs, hco3)

T 298.15 K * :

y ΔG.A ΔG0 - R.m T * y 4 el mn ln 2 y 5 el mn ln * + y 3 el mn ln - y 2 el mn ln - (* - :

y F.T 1 y ΔG.A - ΔG.C + χ R.m * T * / exp - :

The rate equations: the microbial process is calculated with a dual Monod-equation. Parameters taken from: Jin, Roden and Giska, Geomicrobiology Journal, Vol.30 (2013) p.173-185

K.max 1.3 106 -^* mol g s * / * :

K.ac 5106 -^* mol L / * :

K.SO4 68106 -^* mol L / * :

D 108 -^s 1 -^* :

Y 3.8 103^* mg mol / * :

mol biomass carbon generated per mol SO4 reduced:

biomass generated per mol SO4 reduced

5 Y * 113.115 g mol / * (1 -^* 0.17

functions used to avoid negative concentrations which might be generated by numerical oscillations, etc:

x c x x 0 > (* mol * L 1 -^* :

x cx x x 0 > (* mg * L 1 -^* :

The dual Monod-equation with the thermodynamic potential factor:

Note: y=(X,ac,so4,hs,hco3)

y r.redox K.max y 1 el cx * y 2 el c K.ac y 2 el c + / * y 3 el c K.SO4 y 3 el c + / * y F.T * y F.T 0 > (* :

for sulphate (flow in - flow out + reactions):

y dSdt F.in_SO4 SO4.in * (F.out y 3 el c * (- Vol / y r.redox (- :

biomass: production minus outflow minus decrease:

y dXdt Y y r.redox * (F.out y 1 el cx * Vol / (- D y 1 el cx * (- :

sulphide and bicarbonate: production minus outflow:

y dHSdt y r.redox (F.out y 4 el c * Vol / (- :

Note: y=(X,ac,so4,hs,hco3)

y dHCO3dt 2 y r.redox (* F.out y 5 el c * Vol / (- :

For acetate: flow in - flow out - consumption. The rate of acetate consumption is the sum of the sulphate reduction process CH3COO- + SO4-2 → 2HCO3- + HS- and the biomass growth with the reaction: 5/2 CH3COO- + NH4+ + 3/2 H+ => C5H7O2N + 3 H2O(l) The molecular mass of C5H7O2N is 113.115 g/mol.

y dAcdt F.in_acetate Ac.in * (F.out y 2 el c * (- Vol / 2 Y 52 / (* 113.115 g mol / * (1 -^* + (y r.redox * (- :

Initial conditions:

Cell_dry_mass 1012 -^:

At the start there are 5x10^5 cells/mL in the reactor tank

biomass 5104^* 1 mL / * Cell_dry_mass * g * :

biomass mg L / 0.05

Using C5H7O2N for biomass:

carbon biomass 5 * 113.115 g mol / * (1 -^* :

carbon mol L / 2.21 106 -^*

Initially the tank contains only water, except for some microbes:

Note: y=(X,ac,so4,hs,hco3)

y.initial 0.05 106 -^106 -^106 -^106 -^51 mat :

check for possible mistakes in the equations above by controlling that the units are correct:

y.initial ΔG.A kJ mol 1 -^* 82.3

y.initial F.T 0.9184

Note that the units for biomass concentration are "mg/L" (mass concentration) while for the other components the concentration is expressed as "mol/L" (amount concentration).

y.initial dSdt mol L 1 -^* s 1 -^* 5.55 109 -^*

y.initial dXdt mg L 1 -^* s 1 -^* 5.55 108 -^* -

y.initial dAcdt mol L 1 -^* s 1 -^* 5.54 1010 -^*

y.initial dHSdt mol L 1 -^* s 1 -^* 9.6691 1013 -^* -

Finally the system of ODEs may be solved to calculate the concentrations in the reaction tank. A "plugin" is used that requires a Jacobian matrix (the rates for each component) and the initial conditions. The "plugin" can not work with units, so the units must be removed:

t y dydt y dXdt hr L * mg / * y dAcdt hr L * mol / * y dSdt hr L * mol / * y dHSdt hr L * mol / * y dHCO3dt hr L * mol / * 51 mat :

The time here must be units-less, corresponding to the values in the Jacobian, in this example "hours"

t.start 0 :

t.end 1000 :

n.steps 50 :

Note: the "plugins" solving the ODEs require that the variables "t" (a scalar) and "y" (a vector) in the Jacobian may not be defined above: they must be symbolic.

The calculations are performed here with the "plugin" which implements an algorithm called Gear's BDF (Backward Differentiation Formulae). The results are stored in the matrix "res".

res y.initial t.start t.end n.steps t y dydt dn_GearsBDF :

The matrix "res" contains the time as the first column, and the other columns contain the component concentrations (X,ac,so4,hs,hco3) at the time given in the first column. The concentrations are in the same order as in the vectors "y.initial" and "dydt".

time res 1 col :

(the time in hours, as defined in "dydt" above)

end n.steps 1 + :

Cells X 106 -^* Cell_dry_mass / :

j 1 end range logCells j el Cells j el log10 : for

Carb X 5 * 113115 / :

X res 2 col :

Ac res 3 col :

SO4 res 4 col :

HS res 5 col :

HCO3 res 6 col :

time X augment time logCells augment time Ac augment time HCO3 augment

Example of comparing a simulation with some experimental data

data 0010014005900742 mat :

time SO4 augment time HS augment time HS 32070 * augment data 2 1 sys

Note that sulphate is in excess and reaches a steady state value, while the steady state concentration of acetate is close to zero. The biomass stabilises to a value corresponding to the substrate amounts entering the reactor tank.

Plot the thermodynamic potential factor in the reactor tank

j 1 end range F.Tcalc j el res j j 26 submatrix transpose F.T : ΔG.Acalc j el res j j 26 submatrix transpose ΔG.A mol kJ / * : 21 line for

time F.Tcalc augment time ΔG.Acalc augment