FUNDAMENTALS OF MICROALGAE BIOTECHNOLOGY - D. S. DVORETSKY - 2015

4. MATHEMATICAL MODELING OF PROCESSES FOR OBTAINING PRODUCTS FROM MICROALGAE

4.1. Mathematical Modeling of the microalgae cultivation process

Since experimental studies of microalgae cultivation are both resource- and time-intensive, developing a mathematical model capable of describing the accumulation of microalgae Cell biomass, substrate depletion, and lipid accumulation is an urgent task.

The microalgae cultivation process can be represented by the following schematic diagram (Fig. 22).

Class="center">Fig. 22. Structural diagram of the biomass cultivation process: T - Temperature; S - vector of substrate component concentrations; I - insolation intensity; X - microalgae biomass concentration; K, μ, E - kinetic coefficients of the Biosynthesis process

The course of the intensive biomass cultivation stage is determined by the input parameters T, S, I, variations in which affect The amount of accumulated Lipids X, and internal parameters К_s, К_i, К_l, μ_max, Е, which depend on the Physical Properties of the medium, the substance, etc.

When developing a mathematical model for The kinetics of microalgae biomass growth and intracellular lipid accumulation, the following assumptions were made:

A1. The cultivation process of Chlorella biomass consists of two periods: biomass accumulation, during which cell lipid formation and accumulation do not occur; and the induction of stress conditions, under which lipid formation and accumulation take place.

A2. The hydrodynamic regime in the working volume of the photobioreactor is close to complete mixing, since the cultivation is carried out in a batch mode with intensive aeration using an air-gas mixture. The delivery of gas-mixture bubbles to the Cells is unhindered.

A3. Oxygen is uniformly distributed throughout the working volume of the photobioreactor, and its amount is sufficient for cellular METABOLISM/26.html">Energy Metabolism.

A4. Illumination within the working volume of the photobioreactor is uniform.

A5. Cell age is not taken into account when modeling the lipid formation process.

A6. Nitrogen-containing salts serve as the primary limiting substrate.

A7. Under stress conditions, a low concentration of nitrogen-containing salts is maintained in the nutrient medium.

A8. The cultivation process is carried out within a pH range optimal for biomass accumulation.

A9. Processes of Nutrition, Photosynthesis, reproduction, and others occur simultaneously.

A10. The effects of illumination and the concentration of nitrogen-containing salts on the growth rate are independent.

The mathematical model for the cultivation kinetics of the microalgae biomass Chlorella vulgaris includes the following equations.

1. Experimental studies on microalgae biomass accumulation (Fig. 23) have shown that the curve profile corresponds to the Verhulst logistic equation for restricted population growth [33]:

(1)

where X is The Cell biomass concentration, million cells/mL; μ is the specific growth rate, day^-1; K is the carrying capacity of the population, million cells/mL.

Fig. 23. Kinetics of microalgae biomass growth

2. The substrate depletion process during batch cultivation of microalgae is described by the equation [34]

(2)

where Y_ХS is the yield coefficient representing the amount of biomass formed ∆Х per amount of substrate consumed ∆S over a time interval ∆t, calculated from experimental data ((million cells • L)/(mL • g)); Y_РS is the yield coefficient representing the amount of lipids formed ∆Р per amount of substrate consumed ∆S over a time interval ∆t, calculated from experimental data; Р is the lipid content, %; ms is the maintenance coefficient.

In accordance with assumption D7, the consumption of the nitrogen-containing substrate does not occur during the accumulation of metabolic products; therefore, the second term is excluded from equation (2). For non-energy substrates such as nitrogen, ms = 0 [34]. Thus, equation (2) transforms into form (3)

(3)

3. The amount of lipids formed P is proportional to the accumulated biomass X of the microalgae. Lipid formation requires stress conditions—specifically, a reduced concentration of the substrate S (nitrogen-containing salt)—meaning that the amount of P must be proportional to 1/ Sr, where r is the exponent. The microalgae biomass concentration X reaches a maximum value K, after which further growth ceases. It can be assumed that the increase in the amount of lipids formed will cease in a similar manner. The maximum possible amount of lipid accumulation will then be determined by P_mах. Taking the above into account, the equation for calculating the amount of lipids formed can be written as

(4)

where q_P is the parameter characterizing the maximum lipid production rate per unit time, day^-1; P_mах is the maximum accumulated lipid content, %.

To determine the coefficients included in equations (1), (3), and (4), a series of experimental studies was conducted.

The maximum carrying capacity of the population K = 31 million cells/mL was determined from the graph presented in Fig. 23 and corresponded to the maximum cell biomass concentration achieved.

It is known [7, 32 - 34] that the factors exerting the strongest influence on the specific growth rate μ are temperature T, light intensity I, and the concentration of nitrogen-containing salts S, i.e.,

μ = μ(Т, 1, 5). (5)

The Effect of each factor was examined while holding The values of the other factors constant at their optimum levels.

Experimental studies on the cultivation of the microalga Chlorella vulgaris in nutrient media were carried out at various potassium nitrate concentrations (Fig. 24).

Analysis of the graph shown in Fig. 24 leads to the Conclusion that the inhibition of biomass growth by elevated substrate concentrations S is described by the Andrews equation [4]

(6)

where μ_max = 1.1 day^-1 is the maximum specific growth rate of the microorganisms; S is the Substrate Concentration (nitrogen-containing salts), g/L; K_s = 1.06 g/L is the half-saturation constant; K_ing = 8.4 is the inhibition constant, g/L, I = const = 5600 lx. The values of μ_max, K_s, and K_ing were determined from the graph in Fig. 24.

Fig. 24. Specific growth rate μ as a function of substrate concentration S

The effect of light intensity I on the microalgae growth kinetics is most frequently described by the Michaelis-Menten Equation [35]

(7)

where К₁ = 15,000 lx is the light half-saturation constant, determined by Processing experimental data at a nitrogen-containing salt concentration of S = 5 g/L (Fig. 25).

Fig. 25. Specific growth rate μ as a function of light intensity I

During microalgal cultivation, the maximum specific growth rate μm_ах—which is part of the formulas for calculating μ(S)_1=const and μ (l)_S=const—depends on the temperature T. The constants K_s and K_ing exhibit a weaker dependence on temperature T [34]; therefore, the equation for calculating the maximum specific growth rate μm_ах was formulated as a power function of temperature T

μ_mах(Т ) = А₀ + А₁Т + А₂Т ^2, (8)

where А₀ = -1,3 • 10³; А₁ = 8,6 ; А₂ = -0,01 are coefficients obtained by solving the system consisting of three equations (9) for temperatures of 300, 303, and 308 K.

In accordance with assumption D10, we adopt the relationship [34]

μ = 0,5(μ(S)_l=const + μ(l)_S=const).

The value Y_xs is determined as the mean value obtained by processing the experimental curves of microalgae biomass accumulation X(t) and substrate amount S(t) using the formula [7]

(10)

where 0 and n are the first and last days of the experiment, respectively, Y_xs = 6,05 (million cells • L)/(mL • g).

The specific rate of product biosynthesis q_p was calculated analogously to the formula for the specific Cell Growth Rate [7] using the equation

(11)

The values q_p = 1,5; P_mах = 32% and r = 2 were determined by processing experimental data on lipid accumulation kinetics.

The system of ordinary differential equations (1), (3), (4), and (5) was solved using the 4th-order Runge-Kutta method [35] in the Matlab environment [36]. The initial conditions at t = 0 are:

Х(0) = 2, S(0) = 5, Р(0) = 4. (12)

The results of the solution are presented in Figs. 26 - 28.

Fig. 26. Biomass cell count, million cells / mL

Fig. 27. Substrate amount, g/L

Fig. 28. Lipid content, wt. %

Analysis of the obtained results leads to the conclusion that the maximum microalgae biomass growth and intracellular lipid accumulation are achieved within a period of up to 15 days, after which it is advisable to terminate the biosynthesis stage.

4.2. Mathematical Modeling of Cell Disruption

Cell disruption is commonly described by a first-order kinetic model, as, for instance, applied by Hetherington et al. [37] for Yeast cells disrupted to extract Proteins. A similar first-order kinetic model is used to describe the disruption of microalgae cells via ultrasound or high-pressure homogenization (HPH):

(13)

where C_d is the number of cells disrupted per unit volume in time t (cells/mm³); C₀ is the number of intact cells initially present in the culture per unit volume, or the initial culture cell concentration (cells/mm³); t is the duration of the disruption experiment; K is the disruption rate constant. For ultrasonic disruption, t is the duration of ultrasound exposure (s); K is the ultrasonic disruption rate constant (1/s). For HPH, t is the number of passes through the HPH valve, and K is the HPH disruption rate constant (1/pass).

The initial condition can be written as

Сd (0) = 0. (14)

Solving equation (13) with the initial condition (14), we obtain С_d = С₀(1 - е^-Кt). (15)

Next, we convert to dimensionless variables С_d = С_о — С (16)

where C is the number of intact (viable) cells present in the culture per unit volume at time t (cells/mm³).

Substituting (16) into (15), we obtain С₀ - С = С₀(1 - е^-Кt). (17)

In terms of dimensionless variables, equation (17) can be written as D = (1 - е^-Кθ), (18)

In equation (18), D is the dimensionless cell concentration of the microalgal culture; K is the dimensionless rate constant of cell disruption;

θ is the dimensionless time; t_max is the maximum duration of the experiment during which disruption was carried out (25 min for ultrasonic Treatment, 5 passes for HVD).

The value D characterizes the fraction of intact culture cells subjected to the disruption process over time t and can be interpreted as the degree of disruption. The dimensionless disruption rate constant K characterizes the efficiency of microalgal cell disruption. A higher value of K indicates more rapid cell breakage.

4.3. Mathematical models of the Extraction Process

The extraction process from plant material is illustrated in Fig. 29 [22]. The extraction process can be divided into several stages.

Fig. 29. Scheme of the lipid extraction process from a cell

Stage 1. A mixture of non-polar and polar organic Solvents penetrates through The cell membrane into the Cytoplasm.

Stage 2. The mixture interacts with the lipid complex. During this interaction, the non-polar organic solvent surrounds the lipid complex and forms Structure/103.html">Van der Waals interactions with the neutral lipids within the complex, while the polar organic solvent similarly surrounds the lipid complex and forms Hydrogen Bonds with the polar lipids. These Hydrogen bonds are strong enough to replace the lipid binding in lipid-protein associations of the cell membrane.

Stage 3. An organic solvent-lipid complex is formed and separates from the cell membrane.

Stage 4. The organic solvent-lipid complex diffuses through the cell membrane.

Stage 5. Lipids diffuse through the static organic solvent film surrounding the cell into the bulk organic solvent.

Furthermore, lipid recovery during organic solvent extraction from microalgal biomass is described by a first-order kinetic equation [22] m_е = m₀(1- e^-kt), (19)

where m_e is the amount of lipids extracted by the organic solvent in time t (g of lipids/g of dry microalgal biomass); m₀ is the initial amount of lipids present in the cells (g of lipids/g of dry microalgal biomass); k is the mass transfer coefficient of lipids from microalgal cells into the organic solvent (1/s); t is the extraction time (s).

The similarity between equations (19) and (15) indicates a uniform approach to describing cell disruption and lipid extraction processes.

According to [38], lipid extraction from cells can be described using the fundamental mass transfer equation

(20)

where K is the mass transfer coefficient representing the amount of substance transferred through a unit surface area per unit time at a driving force (difference between operating and equilibrium concentrations) equal to unity, is the amount of substance transferred from one phase to another per unit time, mol/s; F is the interfacial contact area, m²; ∆с is the driving force of the mass exchange process, defined as the difference between operating and equilibrium concentrations in corresponding units, m³/m³.

In general, the extraction process can consist of three components:

- “internal” diffusion, which characterizes the mass transfer of a substance inside the cell from the core to the surface (quantitatively assessed by the internal diffusion coefficient D_вн);

- mass transfer within the immediate diffusion boundary layer surrounding the cell, moving from its surface to the boundary of the boundary layer (quantitatively assessed by the coefficient D);

- mass transfer by the moving extractant As a result of convective diffusion from the boundary layer into the solution (quantitatively assessed by the mass transfer coefficient β).

Therefore, the overall mass transfer coefficient K can be calculated as the reciprocal of the total mass transfer resistance from the core of one phase (the intracellular space) to the core of the other phase (the solution), i.e.,

(21)

where 2r is the cell diameter; n is a coefficient; D_вн is the internal diffusion coefficient; D is the molecular diffusion coefficient; δ is the thickness of the diffusion boundary layer; β is the mass transfer coefficient.

Formula (21) is applicable when the liquid phase moves at a low velocity (for example, during agitation in a stirred vessel); accordingly, all three components are present.

If the liquid phase is stationary, the substance will move exclusively by diffusion from the cell core to the surface and further from the surface into the solution. In this case, the “boundary layer” essentially occupies the entire volume of the Reactor. Formula (21) can then be rewritten as follows:

(22)

If the liquid phase moves at a high velocity (intensive agitation), the boundary layer around the cell will be absent, and external mass transfer will not affect the process, meaning that ideal mixing is effectively achieved. In this case, The rate of the process is determined solely by the internal diffusion of molecules within the cell:

(23)

To express the magnitude of the internal diffusion coefficient within cells, it is proposed to introduce a correction factor B into Einstein's equation for free diffusion, which accounts for all complexities of the process:

(24)

where R is the universal gas constant, J/(mol • K); T is the absolute temperature, K; N₀ is the Avogadro constant; ɳ is the viscosity in N/(s • m²); r is the cell radius, m.

The value of the diffusion coefficient D can be found using the Arrhenius equation

(25)

where D₀ is the pre-exponential factor, which is numerically equal to the diffusion coefficient at infinitely high temperature; E_a is the activation energy.

There are several Methods for carrying out extraction [7]: extraction with agitation, fixed-bed extraction, single- and multi-stage extraction, as well as co-current and counter-current extraction.

Extraction with agitation is the simplest extraction method. The solid phase (biomass after thickening or filtration) and the liquid phase (extractant) are charged into a stirred vessel. The process then proceeds under agitation.

The concentration of the extracted substance in the extractant varies over time in accordance with the equation

(26)

where V_ж is the volume of the liquid phase in the apparatus; C_ж is the concentration of the dissolved substance in the extract at time t; βV is the liquid-solid mass transfer coefficient based on the volume of the liquid phase.

The value Cpж in the formula denotes the concentration of the dissolved substance in the liquid that is in equilibrium with its content in the solid phase C_Т. It is assumed that the relationship between them is determined by Raoult's law

(27)

where γ_жт is the partition coefficient indicating how many times the equilibrium concentration of the solute in the liquid (when further dissolution ceases) exceeds its concentration in the solid phase. Typically, the extractant is chosen such that the solubility of the target solute in it is significantly higher than in the solid phase (i.e., γ_жт >> 1).

On the other hand, the concentration of the solute in the solid phase is given by the equation

or

(28)

where V_т is the volume of the solid phase in the apparatus.

Solving equations (27) and (28) allows us to determine how the solute concentrations change in both the liquid and solid phases.

Figure 30 shows that the concentration of the substance in the solid phase drops sharply from its initial value C_т0 to the equilibrium value C_тр. Meanwhile, the concentration in the liquid phase increases less dramatically—from C_ж0 = 0 to the equilibrium value C_жр. However, after a certain point in time, the concentration in the liquid C_ж begins to exceed that in the solid phase, i.e., C_ж > C_т. It might seem that the solute should therefore transfer back from the liquid into the solid phase, but in reality, extraction continues until the equilibrium values C_тр and C_жр are established. If we instead compare C_т and C_ж / γ_жт = C_т*, the plot takes a different form (Fig. 31).

Fig. 30. Variation of solute concentrations in the solid (1) and liquid (2) phases during extraction

Fig. 31. Variation of the driving force of the extraction process

This graph demonstrates that by the end of the process, the driving force of mass transfer approaches zero.

At equilibrium, the following equation must hold true in all cases

(29)

On the other hand, at equilibrium, the total amount of substance dissolved in both the Solid and liquid phases must equal the initial amount present at THE START OF the extraction process:

(30)

Solving these equations simultaneously yields

(31)

This relationship clearly shows that the solute concentration in the biomass after a single extraction stage is significantly lower than beforehand. This value decreases as the partition coefficient γ_жт and the liquid-to-solid volume ratio V_ж / V_т increase.

It is also evident that the final concentration is unaffected by the mass transfer coefficient βv; it merely accelerates or decelerates the extraction process while leaving The ultimate outcome unchanged.