A Two-Stage System for the Large-Scale Cultivation of Biomass: a Design and Operation Analysis Based on a Simple Steady-State Model Tuned on Laboratory Measurements

Abstract

The optimal design and operation at large scale of a continuous fermentation process including a biological reactor/photobioreactor and a gravity settler with partial recycle and purge of the biomass are addressed. The proposed method is developed with reference to microalgae (Scenedesmus obliquus) cultivation, but it can be applied to any fermentation process as well as to activated sludge wastewater treatment. A procedure is developed to predict the effect of process variables, mainly the recycle ratio (R), the solid retention time (θ _c ), the reactor residence time (θ), and the ratio between feed and purge flow rates (F _I /F _W ). It includes a simple steady-state model of the two units coupled in the process and the experimental measurement of basic kinetic data, in both the bioreactor and the settler, for the tuning of model parameters. The bioreactor is assumed as perfectly mixed, and a rigorous gravity-flux approach is used for the settler. The process model is solved in terms of dimensionless variables, and plots are given to allow sensitivity analyses and optimization of operating conditions. A discussion about washout is presented, and a simple method is outlined for the calculation of the minimum values of residence time (θ _min ) and recycle ratio (R _min ) and of the maximum allowed recycle ratio (R _{max,operating} ) and biomass purge rate (F _Wmax ). In particular, it is shown that the system is sensitive to the concentration of biomass lost from the top of the settler (C _X ^S). The proposed method can be useful for the design and analysis of large-scale processes of this type.

Appendix. Gravity Solid Flux Theory

The total flux of solids in a gravity settler (G _T ) is given by the following:

$$ {G}_T={G}_v+{G}_u $$

(34)

$$ {G}_u={C}_Xu $$

(35)

$$ u=\frac{RF_I+{F}_W}{A} $$

(36)

$$ {G}_v={C}_Xv $$

(37)

$$ v={v}_0{e}^{-\alpha {C}_X} $$

(38)

where

G _u :

Convective solid flux (kg m⁻² day⁻¹)

G _v :

Gravitational solid flux (kg m⁻² day⁻¹)

C :

Solid concentration (in this case, biomass concentration) (kg/m³)

u :

Convective settling velocity

A :

settler surface area

v :

Settling velocity by gravity

In the settler, the value of C _X increases from C _X ^U to C _X ^R. The solid fluxes are a function of the solid concentration according to Fig. S2 (Supplementary File), where the occurrence of a minimum value is evidenced.

The solid flux applied to the settler is given by the following:

$$ Gapp=\frac{C_X^U\left(1+R\right){F}_I}{A} $$

(39)

For a correct settler operation, it is necessary that G _app  < G _T , whatever the value of C _X within the settler (between C _X ^U and C _X ^R). The dependence of G _T on C _X , for a pre-established u value, is given by the following:

$$ {G}_T={C}_X{v}_0{e}^{-\alpha {C}_X}+{C}_Xu $$

(40)

Applying the first and second derivatives analysis to find the minimum point, we obtain the following:

$$ \frac{\partial {G}_T}{\partial {C}_L}=u+{v}_0{e}^{-\alpha {C}_L}-{v}_0{C}_L{e}^{-\alpha {C}_L}=0 $$

(41)

$$ \frac{\partial^2{G}_T}{\partial {C_L}^2}={v}_0{\alpha}^2{C}_L{e}^{-\alpha {C}_L}-2{v}_0{\alpha C}_L{e}^{-\alpha {C}_L}>0 $$

(42)

where C _L indicates the critical biomass concentration corresponding to the maximum, or limiting, solid flux (G _L ) ensured by the settler.

Equation (41) cannot be resolved analytically; however, using Eq. (42), a relationship easier to work can be found:

$$ {C}_L>\frac{2}{\alpha}\kern4.25em \left(\mathrm{condition}\ \mathrm{I}\right) $$

(43)

By replacing Eq. (43) in Eq. (41), we have the following:

$$ u<{v}_0{e}^{-2}\kern4.25em \left(\mathrm{condition}\ \mathrm{II}\right) $$

(44)

The value of the limiting flow of solids (G _L ) can be calculated from Eq. 40 with u given by Eq. 36:

$$ {G}_L={C}_L{v}_0{e}^{-\alpha {C}_L}+{C}_Lu $$

(45)

According to the literature [2], if it is assumed that v = 0 at the bottom of the settler, Eq. (45) in combination with Eq. (34) yields the following:

$$ {C}_X^R={C}_L+\frac{C_L{v}_0{e}^{-\alpha {C}_L}}{u} $$

(46)

By expressing u from Eq. (41) and introducing it in Eq. (46), it can be obtained that:

$$ {C}_X^R=\frac{\alpha {C}_L^2}{\alpha {C}_L-1} $$

(47)

Using condition I above (Eq. 43), it results to the following:

$$ {C}_X^R>\frac{4}{\alpha}\kern3.25em \left(\mathrm{condition}\ \mathrm{III}\right) $$

(48)

Equation (47) is a second-degree equation in C _L :

$$ {C}_L^2-{C}_X^R{C}_L+\frac{C_X^R}{\alpha }=0 $$

(49)

whose roots are as follows:

$$ {C}_L^{\prime }=\frac{C_X^R}{2}+\sqrt{\left({\frac{C_X^R}{4}}^2-\frac{C_X^R}{\alpha}\right)} $$

(50)

$$ {C}_L^{\prime \prime }=\frac{C_X^R}{2}-\sqrt{\left({\frac{C_X^R}{4}}^2-\frac{C_X^R}{\alpha}\right)} $$

(51)

Only C _L ′ is the useful root, as C _L ″ does not respect conditions I and III. A correct operation of the settler requires that:

$$ {G}_{app}={G}_L $$

(52)

according to which, it can be finally obtained that:

$$ \frac{F_I}{A}=\kern0.5em \frac{v_0\alpha {C}_L^2{e}^{-\alpha {C}_L}}{\left(1+R\right){C}_X^U} $$

(53)

where C _L can be expressed by Eq. (50) as a function of C _X ^R, which is in turn a function of C _X ^U, R, θ, and θ _c according to Eq. (15). In summary, if F _I , A, C _X ^U, θ, and θ _c are known, a single value of R can be calculated from Eq. (53).

The recycle ratio is indeed the key variable: if R value is too high, the convective velocity will be dominant in the settler and will not allow to thicken the solid enough for the sedimentation. A limiting situation will be reached when C _X ^U = C _X ^R (rupture of the reactor operation). To calculate this value, which is called critical recycle ratio (R _C ), it is sufficient to express ${C}_X^R$ from a mass balance around the settler:

$$ {F}_IR{C}_X^R+{F}_W{C}_X^R+{F}_S{C}_X^S={F}_I\left(1+R\right){C}_X^U $$

(54)

and to apply Eq. (48) (condition III).

Two cases can be considered to calculate the R _c value:

• 1. When C _X ^S = 0, it results to the following:

$$ {R}_C=\left(\frac{C_X^U-\frac{4}{\alpha}\frac{F_W}{F_I}}{\frac{4}{\alpha }-{C}_X^U\ }\ \right) $$

(55)

• 2. When C _X ^S ≠ 0, we have the following:

$$ {R}_C=\left(\frac{1+\left(\frac{F_W}{F_I}-1\right)\frac{C_X^S}{C_X^U}-\frac{4}{\alpha}\frac{F_W}{F_I}\frac{1}{C_X^U}}{\frac{C_X^S}{C_X^U}+\frac{4}{\alpha {C}_X^U}-1\ }\ \right) $$

(56)

In any case, the system must be operated with the following:

$$ R\le {R}_C\kern0.5em \left(\mathrm{condition}\ \mathrm{IV}\right) $$

(57)

and Eq. (53) holds under this condition only.

The value of C _X ^R _{min,operating} , i.e., the minimum biomass concentration to correctly operate the settler, is given by Eq. (48) (condition III), then from Eq. (15) R _{max,operating} can be calculated:

$$ {R}_{\mathit{\max}, operating}=\kern0.5em \frac{C_X^U\left(1-\frac{\theta }{\theta_c}\right)\ }{\left(\frac{4}{\alpha }-{C}_X^U\right)\ } $$

(58)

It is noteworthy that the procedure outlined above assumes v = 0 at the bottom of the settler. If this condition is not verified, Eq. (46) does not hold any more. Instead, by combining Eq. (45) with Eq. (36), where the value of u is expressed by Eq. (41), it can be obtained that:

$$ {C}_X^R{v}_0{e}^{\left(-\alpha {C}_X^R\right)}-{C}_X^R{v}_0{e}^{\left(-\alpha {C}_L\right)}\left(1-\alpha {C}_L\right)-\alpha {C}_L^2{v}_0{e}^{\left(-\alpha {C}_L\right)}=0 $$

(59)

This equation can be solved numerically for C _L , because v₀, α, and C _X ^R(θ _c ) are fixed values, depending on C _L , only. In this case, the value of C _X ^R _{min,operating} cannot be expressed analytically as for Eq. (47). These concepts are qualitatively shown in Fig. S3 (Supplementary File).