CALCULATION OF MASS TRANSFER COEFFICIENT AND POWER NUMBER CORRELATION

AIM:

To calculate the volumetric mass transfer coefficient (K

_{L}a ) using power number correlation.INTRODUCTION:

The traditional method to determine mass transfer coefficient (K

_{L}a ) for large antibiotic fermentor has been based upon log mean relationship between the inlet and exit equilibrium dissolved oxygen concentration and the actual dissolved oxygen in the fermentor. The resulting K_{L}a is then related to power unit volume, agitator speed and superficial gas velocity. This method has a number of disadvantages. The dissolved oxygen in the fermentor varies from top to bottom of the fermentor. The K_{L}a value can be determined on-line from O_{2}analysis and a single dissolved oxygen measurement at any stage of the fermentor. An aerobic cell will be able to utilize a richer carbon source fully only if oxygen can also be maintained at a higher concentration in the direct vicinity of the cell. The overall mass transfer coefficient K

_{L}a is a measure of the ability of fermentor to supply oxygen to cells. The solubility of oxygen in aqueous solutions under 1 atm of air and near ambient temperature is of the order of 10ppm. Oxygen is sparingly soluble in aqueous solution and hence gas liquid mass transfer should be increased. Agitation enhances gas liquid mass transfer. K

_{L}a depends on power input and power consumption and quantity of air supplied. K_{L}a is highly dependent on the culture conditions, with age, cell concentration and growth rate having a major effect. Consistent K_{L}a values can be obtained at the same point in each run. In low viscosity liquids, K_{L}a is a relatively strong function of gassing rate, with K_{L}a and hold up increasing monotonically with power input and similar values are obtained regardless of agitator type. Increasing viscosity reduces K_{L}a and at the highest viscosities. K_{L}a is independent of the gassing rate, as is the power input. Experiment is conducted in water so that K_{L}a is highly dependant on the physiochemical nature of the solution in low viscosity systems, as shown by the results correlated by Van’t reit. For pure water he proposed:

K

_{L}a = 0.026 (P/V)^{0.4}Vs^{0.5}And for strongly ionic solutions:

K

_{L}a = 0.002 (P/V)^{0.7}Vs^{0.2}These correlations show how K

_{L}a varies and also how the dependence of K_{L}a on the power input and gassing rate varies with a change in ionic strength. This is considered to be due to the change in bubble coalescing properties. In pure water, bubbles coalesce easily forming bubbles of low surface area to volume ratio when ascend stabilized by reducing hold up and interfacial area. In ionic solutions, the interface is stabilized by ions and coalescence is reduced. This maintains smaller bubble sizes and leads to a higher hold up and therefore higher interfacial areas. The power consumption in a bioreactor is reduced by aeration. Power number is used to calculate the power required to rotate a given impeller at a given speed. The power number depends on dimensions of tank and impeller, the distance of impeller from the tank floor, the liquid depth and the dimensions of the baffles, if these are used. The number and arrangement of the baffles and the number of blades in the impeller must also be fixed. The variables that enter the analysis are the important measurements of tank and impeller, the viscosity µ and density ρ of the liquid, the speed n, and because the Newton ’s law applies, the dimensional constant g.

The various linear measurements can all be converted to dimensionless ratios, called “shape factors”, by dividing each of them by one of their number, which is arbitrarily chosen as basis. The diameter of the impeller is taken as D

_{a}and of the tank as D_{t}and the shape factors are calculated by dividing each of the remaining measurements by the magnitude of D_{a}or D_{t}. Let the shape factors, so defined, be denoted by S_{1}, S_{2}, S_{3,………..}S_{n.}When the shape factors are temporarily ignored and the liquid is assumed “Newtonian”, the power P is a function of the remaining variables, or

P = ψ (n, D

_{a}, g_{c}, µ, g, ρ)Applying law of homogeneity, we get

Pg

_{c}/N^{3}D_{a}^{5}ρ = ψ (nD_{a}^{5}ρ/µ, n^{2}D_{a}/g) where Pg_{c}/N^{3}D_{a}^{5}ρ power no._{a}

^{2}

N

^{2}D_{a}/g Froude no.In use turbulent regime, the power input is independent of Reynolds no.

P ∞ n

^{3}D_{a}^{5 }: N_{p}is a constant.Laminar flow relation is given by

P ∞ n

^{2}D_{a}^{3 }or N_{p}∞ 1/N_{Re}The proportionality constant in each case depends on impeller geometry.

The oxygen transfer rate per unit volume is given by

Qo

_{2=}K_{L}a (C^{*}- C), where a interfacial area per liquid volume,(C

^{*}- C) Overall concentration driving force.The ratio of power requirement in aerated versus non-aerated vessels P

_{g}/P versus the aeration no N_{a }can be correlated.N

_{a}= F_{g }/ N^{i}D_{j}^{3 }; F_{g}air flow rateN

_{a}= (F_{g}/D_{j}^{2})/ N_{j}D_{j }where N_{a}= superficial gas velocity / impeller tip velocity.The air / gas flow rate can be calculated from the ‘vvm’. The overall mass transfer coefficient can also be given by

K

_{L}a = k (P/V)^{K }V_{s}^{β}Where (P/V) is the power per unit volume

Vs is the superficial gas velocity.

K

_{L}a = 2 ×10^{-3}(P/V)^{0.7}(V_{s})^{0.2}S^{-1}.EQUIPMENT AND EXPERIMENT METHOD:

The studies that will be reported here were conducted on 5l (working volume 2.5lit) reactor. This reactor is equipped with flat four-blade Rushton turbine. Height 265mm, diameter of tank 146mm, diameter of impeller 58mm, inter impeller distance 93mm, distance between last impeller and tank 212mm. The tank has two baffles with a width and height of 16 and 25mm. The impeller is run with 96W(24Volt X4 Amp) motor. The power input to the motor was measured by monitoring voltage and ampere on a multimeter. Effective power was calculated by assuming the efficiency to be 90%. All studies were conducted with water.

Ungassed power number (N

_{p}):The effective power number was calculated from the Rushton power equation

N

_{p}= (P_{eff})_{g}/ N^{3}D_{a }^{5}ρGassed power number (P

_{g}):Ratio of P

_{g}/P as a function of aeration number was plotted. P_{g}/P increased with increase in liquid level below turbine.DATA USED:

Diameter of impeller = 58 mm

Density of water = 1000kgm

^{-3}Viscosity of water = 10

^{-3}PaFORMULAE USED:

Power, P = √3 VI cosф (watts),

where, V = Voltage, Volts

I = current, Amps

Effective power P

_{eff}= 0.9 P watts.Reynolds number N

_{re}= D^{2}Nρ/µWhere D = diameter of impeller

N = Revolution per second

ρ = Density of water

µ = Viscosity of water

Power number N

_{p}= P_{eff }/ N^{3}D^{5}ρRESULTS:

The K

_{L}a values were determined and were found be:For vvm = 0 ;K

_{L}a =For vvm = 0.5 ;K

_{L}a =For vvm = 1 ;K

_{L}a =For vvm = 1.5 ;K

_{L}a =Ratio of P

_{g}/P as a function of aeration number was plotted. It was found that both aeration number and P_{g}/ P ratio increase with impeller velocity.No significant variation was observed in the correlation between N

_{p}and N_{Re}for the gassed and ungassed cases.CONCLUSION:

In a multiturbine fermentor, it would be desirable to be able to estimate the power input of each turbine in the fermentor. Using this information, a modeling approach can be used to relate stage wise mass transfer coefficients to stagewise power inputs. It would then be possible to predict the axial dissolved oxygen distribution of the fermentor and how the distribution reacts to changes in such process variables as pressure, airflow rate, the agitator speed. It would also be possible to optimize the agitator configuration to obtain the most efficient use of power inputs.

The aeration number was plotted against the ratio of gassed to ungassed power absorption. The trend should show that the power drawn decreases to as much as 50% of ungassed power drop during aeration. The ungassed power drawn is important in selecting the appropriate operating speed of agitators, especially for fixed motors. For instance, in certain plants, the nutrient medium is sterilized in the fermentor and as it is necessary to turn the air off during this operation, the impeller would draw maximum power. Also compressor failure causing loss of aeration during operation would increase the power draw to the maximum load at the operating speed. In order to prevent over load of the agitator motors during these conditions, the operating speed must be specified with respect to the ungassed power for fixed speed motors.

**INFERENCE:**

Aeration of a liquid decreases the power consumption during agitation because an aerated liquid is less dense than an unaerated one.

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