ADVANCED MATHEMATICS

Date: 11th April 2014

Present by:

LINEAR SECOND ORDER PDE’s

Research Introduction:

DEVELOPMENT OF A PREDICTIVE CONTROL SYSTEM BASED IN MODEL OF A PROCESS OF FED-BATCH REACTION TO INCREASE THE PERFORMANCE OF PHB (POLYHIDROXYBUTIRATO) PRODUCTION USING VIRTUAL SENSORS AS CONTROL STRATEGY.

For developing a scale up process of production of PHA´s is necessary to know the metabolic pathway and the environmental conditions (pH, temperature, substrate and Oxygen concentration, etc.) that must be controlled in order to get a high yield of production. Optimization plays an important role here, because it let get high productivity reducing costs, due this, is important develop an advanced control system based in the model of this particular process. There are many research groups working in the optimal way to have a better yield in the PHA obtention, one of them, is the biotransformation group of the Universidad de Antioquia, which is working in the obtention of PHA by a microorganism called Ralstonia Eutropa. The main problem is related to establishing the right way of an advanced control system in PHA´s production and trying to estimate the variability of the system and avoiding the perturbations in it to

finally develop a high yield PHB production process.

Physical Problem Description:

This studied is related to the airlift bioreactor described by ( Nikakhtari & Hill, EnhancedOxygen Mass Transfer in an External LoopAirlift Bioreactor Using a Packed Bed, 2005). This includes the main equations and suppositions made by the authors.

For the past 3 decades, airlift bioreactors have been used both at research and industrial scales for aqueous fermentation and bioremediation purposes. Because oxygen has low aqueous solubility and is in high demand by exponentially growing microorganisms, the oxygen mass transfer rate is an important feature for aerobic fermentation and bioremediation processes. The oxygen mass transfer coefficient, KLa, is directly proportional to the rate at which oxygen can be transferred from the air phase to the aqueous medium ( Meng, Hill, & Dalai,2002).

Although considering the external loop airlift bioreactor (ELAB) as a completely stirred reactor is frequently used to predict oxygen mass transfer coefficients, it is not accurate for a larger ELAB, especially when there is a low liquid circulation rate. Also, in some studies the variation of gas-phase concentration has been neglected throughout the vessel. This can be a reasonable assumption for oxygen mass transfer but not for the mass transfer of volatile organic hydrocarbons that may drop from high inlet concentrations to near zero at the outlet of the ELAB ( Nikakhtari & Hill, Hydrodynamicand oxygen mass transfer in an external loopairlift bioreactor with a packed bed, 2005).

Oxygen KLa values fall below 100 h-1 in well-mixed bioreactors when pure water is used as the aqueous phase, but mixing solutes in the water can increase this value up to 1000 h-1. By using high aeration rates, KLa values for oxygen in bubble columns and loop bioreactors can reach similar values as those reported for well-mixed tanks. However, at aeration rates similar to those used in well-mixed tanks, KLa values tend to be an order of magnitude smaller in columns compared to well-mixed bioreactors due to poor mass transfer. Although several methods have been reported to enhance oxygen mass transfer rates in column reactors at low aeration rates, in this case it is shown that simply using a small amount of nylon packing significantly increases the oxygen mass transfer coefficient in an ELAB. A dynamic model was used to determine mass transfer coefficients and predict the dynamic oxygen profiles throughout the vessel both with and without the packed bed ( Nikakhtari& Hill, Enhanced Oxygen Mass Transfer inan External Loop Airlift Bioreactor Using aPacked Bed, 2005).

Experimental Apparatus and Procedures:

Fig.1 Schematic of the external loop airlift bioreactor.

Figure 1 shows a schematic diagram of the ELAB with packed bed. Specifications for the experimental column are listed in Table 1.

Tap water is used as the continuous phase and is deaerating using nitrogen gas. Then air, as the dispersed phase, should be instantaneously connected to the sparger and entering the ELAB through the sparger. This procedure is performed both with and without packing installed in the riser section of the ELAB.

MODEL:

An airlift loop reactor consists of three segments, namely the riser, down-comer and gas–liquid separator. In order to simplify the model, the gas–liquid separator is first not included to focus on the riser and down-comer. When the axial dispersion model is

applied to the riser and down-comer respectively, the differential equations governing the component balances of a substance flowing through a column reactor by oxygen concentrations in the liquid and gas phases

Fig.2 Schematic parts of the external loop airlift bioreactor.

Dispersion Model: According to Fick´s Law, the rate of diffusion of a substance A is proportional to the negative of the concentration gradient of A.

J=−D ∂ c∂ x

Where for the diffusion phenomena in one dimension is described by:

1D

∂ c∂ x

= ∂2 c∂ x2

The diffusion phenomena in two dimensions taking in account the dynamic stage:

∂2 c∂ z2 + 1

z∂c∂ z

= 1D

∂ c∂ t

Deriving the applicable differential equation for describing the concentration of some component A in fluid phase i as a function of location and time for a packed or fluidized

bed , in both phases for a gas –sparged liquid, and for all phases in other similar systems, may be done simply by considering a material balance on a differential volume of the system using the well-known Shell-Balance technique referring to an element of fluid phase. In such a treatment, each fluid phase may be considered separately. The equation for fluid phase i is, in word form,

(Rate of A in by dispersion)+(Rate of A in by convection)=(Rate of A out by dispersion)+(Rate of A out by Convection)+(Rate of accumulation of A within the volume element)-(Rate of mass transfer of A into fluid phase i from another phase within the volume element)-(Rate of introduction of A into fluid phase i from a source within the volume element).

The dispersion term represent the combined effects of diffusion and dispersion due to convective stirring caused by the relative flows of fluid phase i and the packing or other phase or by velocity gradients.

The sparger orifice diameter does not significantly affect hydrodynamic parameters, such as gas hold up and circulation time. The equations needed to predict the hydrodynamics of this ELAB Gas holdup relationships are:

Without Packing: θGR=1.06 J GR0.701 (1)

With Packing:

θGR=(−2.75+0.272 hp+4.03∅ S ) J GR0.701

(2)

These equations are obtained over a range of packing heights of 0.05-0.8 m, porosities of 0.90-0.99, and gas superficial velocities of 0.003-0.016 m/s. To determine the liquid velocity in the riser section

U LR=CF E (3)

Where E is the gas holdup driving force for liquid circulation, given by

E=( θGR

(1−θGR )−2(AR

A D)

2 )0.92

(4)

And CF is the friction resistance for liquid flow, given by

Without PackingC F=19.1 (5)

With Packing CF=−54.3−7.53hp+71.4∅ S (6)

The axial mixing in the ELAB is evaluated by the Bodenstein number:

B0=U LR L/ D (7)

The Bodenstein number in an ELAB without packing is 47 in and 42.6 for a porosity value of 0.963.

Considering oxygen mass transfer between the air and liquid phases, two partial differential equations can be written to predict oxygen concentrations over time and position in these phases:

∂ c∂ t

=D ∂2 c∂ z2−ULR

∂ c∂ z

+K Lα (c¿−c )(8)

∂ y∂ t

=−ULR∂ y∂ z

+ KLα

1−θGR

θGR(c¿−c )(9)

ASSUMPTIONS:

There are some assumptions in writing these equations. Operating conditions such as gas flow rate and liquid volume are constant; therefore, gas holdup, gas and liquid velocities, and liquid dispersion remain constant and can be determined by the hydrodynamic eqs 1-7.

Flow and dispersion in the radial and angular directions are assumed to be negligible and the gas phase flows in a plug flow pattern. Also the variation of gas velocity as a result of oxygen mass transfer and hydrostatic pressure has been ignored, which is reasonable in a relatively small ELAB for low soluble oxygen. For oxygen transfer from air to water, the liquid-phase limits the mass transfer rate. The oxygen concentration in the liquid phase at the air interface (c*) is related to the bulk air phase oxygen concentration according to Henry’s law:

y=Hc ¿ (10)

Equations 8 and 9 are linear partial differential equations and can be solved simultaneously by numerical finite differencing. Because, at low aeration rates, there is no air bubbles in the downcomer, mass transfer only occurs in the riser section. The riser can be divided into N finite difference elements, and the downcomer is assumed to be a plug flow column for liquid, as shown in Figure 2.

Fig.3 Schematic ELAB showing the Finite Difference Sections.

Using backward differencing for both time and space dimensions and substitution of c*

from eq 10, gives the following algebraic equations:

cnt =( AL+BL ) cn−1

t−1 +(1−2 AL−EL) cnt−1+( AL−BL ) cn+1

t−1+EL yn

t−1

H(11)

ynt =2 BG yn−1

t−1 +(1−2 BG−V G

H ) y nt−1+V G cn

t−1

(12)

Where;

AL=D ∆ t( ∆ z )2

(13)

BL=U LR ∆ t(2∆ z )

(14 )

BG=UGR ∆ t

∆ z(15)

EL=KLα ∆ t (16)

V G=KLα ∆ t1−θGR

θGR(17)

These equations are applied from n > 1 to N over space and from t > 0 to tfinal over time. Two boundary conditions for the liquid phase, one for the gas phase, and one initial condition for each phase are required.

BOOUNDARY CONDITIONS:

1. The boundary and initial conditions for the gas phase are simply:

y0t = y¿ (18 )

yn0= y¿ (19 )

2. The initial condition for the liquid phase is the dissolved oxygen concentration after dearation:

cn0=cmin(20)

3. The first boundary condition for the liquid occurs at the inlet of the riser, where it is mixed with the downcomer liquid. The concentration of oxygen in the downcomer is the same as the riser outlet after a time lag given by the residence time in the downcomer:

c1t =( AL+BL) c¿

t−1+( 1−2 AL−EL) c1t−1+( AL−BL) c2

t−1+EL y1

t−1

H(21)

Where cIN is the oxygen concentration in the outlet liquid from the downcomer:

c¿t =cmin t <tdelay (22)

c¿t =c¿

t−t delay t >t delay(23)

Where;

t delay=H D

J LD(24 )

J LD=J LR AR

AD(25)

4. The second boundary condition for liquid phase is at the top of the riser. Here there is no change in the oxygen concentration, as the liquid exits the riser:

c Nt =( AL+BL ) cN−1

t−1 +(1−AL−EL−BL ) cNt−1+

EL yNt−1

H(26)

Equation 8 is a parabolic equation and for stability purposes, all coefficients in eq 11 needs to be equal to or greater than zero, which results in the following limit:

D ∆ t(∆ z )2

≤ 0.5(27)

CONCLUSIONS

A mathematical model considering an ELAB as a distributed column with respect to both the liquid and gas phases should be developing to predict mass transfer of oxygen with respect to both time and space. The model must be compare experimental oxygen transfer data and determinate the KLa for ELABs with low liquid circulation rates, as compared to a completely stirred reactor. The model should correctly predicted wavy oxygen concentrations in the liquid phase and small oxygen losses in the air phase.

The Model must be adjusted to reality or changes in the dimensions of the ELAB. It is expected get good approximations in the partials differential equations that allow get the most realistic behavior of the Oxygen Diffusion in order to get the major distribution of oxygen in the medium and due this get a high yield of the PHB production.

Are pretty important analyses the stability of the parabolic differential equation, because due to this, we can get bad data or not a good model results.

REFERENCES

Meng, A. X., Hill, G. A., & Dalai, A. K. (2002). Hydrodynamic Characteristics in an External Loop Airlift Bioreactor Containing a Spinning Sparger and a Packed Bed. Ind. Eng. Chem., 41, 2124-2128.

Nikakhtari, H., & Hill, G. A. (2005). Hydrodynamic and oxygen mass transfer in an external loop airlift bioreactor with a packed bed. Biochemical Engineering Journal, 27, 138–145.

Nikakhtari, H., & Hill, G. A. (2005). Enhanced Oxygen Mass Transfer in an External Loop Airlift Bioreactor Using a Packed Bed. Ind. Eng. Chem., 44, 1067-1072.