OXYGEN TRANSFER AT TURBULENT MICROSCALE

Transcription

1 OXYGEN TRANSFER AT TURBULENT MICROSCALE Bimlesh Kumar* and T. Thiyam Devi Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati (INDIA) * bimk@iitg.ernet.in Received October 15, 2011 Accepted January 25, 2012 ABSTRACT An experimental and numerical investigation has been carried out to analyze the existence of local isotropy concept in geometrically similar surface aeration systems. Commercial software has been used in determining the small scale parameters such as microscale of turbulence and energy dissipation. Experimental observations have been carried out for oxygen transfer measurements in geometrically similar surface aeration systems. Two-film theory has been adopted in the measurement of oxygen transfer rate. It is found that microscale of turbulence and oxygen transfer are inversely related in geometrically similar surface aeration systems. Present work establishes the existence of local isotropy concept in higher turbulent flow conditions. Based on the local isotropy concept, the relationship of microscale of turbulence and rotational speed of rotor should exhibit the constancy, has been found in the present work. Empirical equation to simulate oxygen transfer rate in geometrical similar systems at turbulent microscale has been developed in the present work. Key Words : Energy dissipation, Geometrical similarity, Local isotropy, Microscale of turbulence, Turbulent flow INTRODUCTION In many industrial and biotechnological processes, stirring is achieved by rotating an impeller in a vessel containing a fluid. The vessel is usually a cylindrical tank equipped with an axial or radial impeller. When impellers are placed at the surface of fluid media, vessel is called as surface aerator. Surface aerators are a popular choice of aeration system because of their inherent simplicity and their competitive rate of oxygen transfer per unit effective input power under actual aeration conditions. The large majority of bioreactors used in the area of biotechnology are stirred-tank bioreactors, because stirring helps to distribute effectively gas and nutrients to the growing organism. Surface aeration systems are used in aerobic biological reactors to supply microorganisms with oxygen and to mix the fermentation broth 1. Surface aerators are very effective in supporting the growth of cells in suspension and on micro-carriers 2. Adequate oxygen mass transfer is critical characteristic to maintain when scaling up suspension cultures of mammalian cells 3-4. *Author for correspondence There are two types of stirring, laminar and turbulent in any aeration systems. Although laminar stirring has its difficulties and has been studied in the past 5-8 by many authors, in most industrial applications where large scale stirring vessels are used, turbulence is predominant. Turbulent flows are far more complicated and a challenging task to predict due to their chaotic nature The statistical theory of turbulence proposed by Kolmogorov 11 is now-a-days successfully applied to model the turbulent flow Based on statistical theory of turbulence or local isotropy concept, present work does an attempt to find the effect of turbulent microscale on oxygen transfer process in surface aeration systems. MATERIAL AND METHODS In turbulent flow, hydrodynamics of surface aerator is strongly dependent on the molecular mixing. It is known that the microscale processes are decisive for the dissipation of turbulence energy. According to Kolmogorov's theory 11, the turbulent flow field can be understood as a superposition of turbulent eddies of different orders of magnitude. This view is based on the 569

2 interpretation of the temporal course of turbulent fluctuating velocities at a point in the flow field, which can be explained as the superposition of different frequencies of different amplitudes (amount of fluctuating velocities). The largest eddies are produced by the stirrer head. They give their kinetic energy up cascade-like to ever smaller eddy. This energy transport is not prevented by the viscosity forces, as long as eddies and their Reynolds numbers are sufficiently large. The viscosity forces only dominate in the case of small eddies and ensure that the energy of flow is converted into (dissipated as) heat. In this range of eddy sizes local isotropy prevails, although the main flow is anisotropic. In other words, the small eddy elements characterized by high wave numbers are completely statistically independent of the main flow. Kolmogorov 11 postulated two similarity hypotheses. The laws of statistical distribution for locally-isotropic turbulence are clearly determined by the kinematic viscosity v and the power per unit mass =P/ V. Dimensional analysis gives the following relationship for the linear dimension of a turbulence element (1) where is the dimension of the smallest turbulence element, whose energy due to the viscosity is directly converted into heat. has become known as the Kolmogorov's microscale of turbulence. Energy transfer from larger to smaller turbulence elements is independent on viscosity for all turbulence elements in between with dimensions >. Microscale has been defined as a length scale on which eddy gives Reynolds number = 1. As the Reynolds number is E measure of the inertial forces to the viscous forces, this implies that at these small eddies are dominated by viscous forces. All processes in mixing tank are related to length scales. The various length scales (macro mixing and micromixing) involved depend on the reactor geometry, the flow field and the fluid properties. Some theoretical investigations have been done in light of macromixing and micromixing The performance of stirred tank depends not only on the kinetic rate processes but also on the interfacial mass transfer, which is influenced significantly by the length of the mixing scales. Measurements Experimental and numerical scheme has been adopted to calculate oxygen transfer rate and shear rates respectively in circular surface aerator. Experimentation Stirring experiments are generally carried out in small laboratory devices (d<1 m), in which the microscale turbulence is predominant 20. Oxygen transfer rate at different speed in different sized tanks has been determined experimentally. The cross- sectional areas of the circular tanks tested H DC Motor h b N l Flange are A=1m 2 and 0.52m 2. A schematic diagram of the aerator is shown in the Fig.1. Conditions of geometric similarity i.e., A/d = 2.88, H/d = 1.0, l/d=0.3, b/d = 0.24 and h/h = 0.94 as suggested by Rao and Kumar21 were maintained in all the surface aerators. According to two-film theory 22, the oxygen transfer coefficient at T o C, K L a T may be expressed as follows K L a T = [ln(c s - C 0 ) - ln(c s - C t )]/t (2) D Shaft Rotor blades Fig. 1: Schematic diagram of a surface aerator d 570

3 The value of K L a T can be obtained as slope of the linear plot between ln (C s - C t ) and time t. The value of K L a T can be corrected for a temperature other than the standard temperature of 20 o C as K L a 20, using the Vant- Hoff Arrhenius equation K L a T = K L a 20 (T - 20) (3) Where is the temperature coefficient equal to 1.02 for pure water. Once the rotor starts rotating, DO meter reading is noted at regular interval up to the point when the DO values reaches 80% of the saturation value or above. Thus the known values of DO measurements in terms of C t at regular intervals of time t (including the known value of C O at t = 0) a line is fitted, by linear regression analysis of Equation (2), between the logarithm of (C s - C t ) and t, by assuming different but appropriate values of C s such that the regression that gives the minimum standard error of estimate is taken and thus the values of K L a T and C s were obtained simultaneously. Oxygen transfer modeling by two-film theory assumes that a single and constant value of C s is adequately representative of the equilibrium DO for the liquid phase oxygen mass transfer for the entire aeration systems and the transfer process is predominately liquid phase mass transfer controlled and the gas phase resistance to transfer can be ignored. Now by fixing the value of C s, the value of K L a T has been determined by the best fit straight line, semi-logarithmic plot of (C s - C t ) and t. It may be noted that the value of C s used in the log-deficit approach can be based on field measurement, published value, or simple assumption. It is a common practice to fix the value of C s around the maximum DO value with some increment in it, as long as it gives the best fit. The slope of such a straight line is equal to - K L a T. The values K L a 20 are computed using Equation 3 with = 1.02 as per the standards for pure water 23. Thus, the values of K L a 20 were determined for different rotor speeds N of the rotor in all of the geometrically similar tanks. Numerical scheme In the present work, microscale of turbulence and corresponding energy dissipation of surface aeration systems have been calculated by using commercial software Visimix. The Visimix program can be helpful in analyzing the mixing parameters in a stirred tanks It can be useful in investigating different scenarios for scaleup and changes in agitator and vessel configuration. According to literature 25,26, the most intensive turbulence is created in eddies behind the agitator blades, and it is completely dissipated in a turbulent jet formed around the agitator by the discharge flow. In the case of agitators with flat radial blades, the jet is roughly symmetrical with respect to the agitator plane, and its height is about 1.5 of the blade height. The mean value of the kinetic energy of turbulence, E at the radius r is defined as where v' is the mean square root velocity of turbulent pulsations corresponding to the largest local linear scale of turbulence. Steady-state transport of the turbulent component of kinetic energy can be described as de q dr where q is the circulation flow rate through the rotor, v e is the eddy viscosity and h j is the local linear scale of turbulence which is approximately equal to 1.5b. Equation 5 is solved for v' = 0 at r =. The value of on the other boundary (r =d/2 ) is calculated using an estimated value of the maximum dissipation rate in the flow past the blades 8 d dr E = 3v' 2 where v o is the axial velocity of the rotor. The dimensions of the m zone (length, height and width) are l, b and b/2, respectively. The mean value of dissipation is estimated as The unit of turbulent dissipation rate, is W/kg. Equation 7 is solved numerically to get the value of at different rotational speed. /2 de 2rhj e 2rhj 0 dr 3 2rN v Sin l m o / m Nb / 6 r (4) (5) (6) (7) 571

4 RESULTS AND DISCUSSION The microscale of turbulence has been calculated by using the Equation 1. Based on the dimensional argument, the dissipation rate is given: = A v' 3 / L (8) Where, A is a constant and L is a length-scale characteristic. The length scale, L, can be taken as proportional to the impeller diameter D 27,28. Now, in Equation 8 v' can be represented as ND, where N is the rotation of impeller (ND) 3 /D =N 3 D 2 (9) By substituting the value of in Equation 1 and assuming that same fluid has been used in the experimental and numerical investigations (kinematic viscosity is almost same of in other words same temperature has been maintained), Equation 1 for a constant rotor diameter tank can be modified as (10) Fig. 2 shows the representation of the Equation 10 for both the tanks. It can be seen from the Fig. 2 that N 3/4 is almost constant (variation may be due to difference in temperature in real experimentations). For medium tank, mean (µ) is N 3 / N 3 4 N Const. A = 1 m 2 [= & = 1.89e-5] A = 0.52 m 2 [= & = 2.21e-5] No. of of Runs Runs Fig. 2 : Relationship of turbulent microscale with rotational speed and standard deviation ( σ ) is 2.21e-5, whereas for big tanks, µ and σ is and 1.89e-5 respectively. One interesting observation can be made from the (Fig. 2) that big tank is having lower constant term than the medium tank. It may be due to that bigger tank dissipate the energy more and faster than the medium tank. For geometrical similar systems, bigger tanks require bigger rotor diameter and thus dissipates more energy. It has demonstrated that oxygen transfer rate in mixing tank depends on the turbulence intensity, expressed as a function of (m) 1 E -3 1 E -4 A = 0.52 m 2 A = 1 m 2 Equation 11 1 E -5 1 E K L a (1/minute) Fig. 3 : Correlation between turbulent microscale with oxygen transfer the dissipated energy rate 29,30. Energy dissipation rate is governed by the smallest scale of turbulence in stirred tanks. The relationship of microscale of turbulence with corresponding oxygen transfer coefficient of geometrically similar surface aerators has been shown in the Fig. 3. It can be seen from the (Fig. 3), all the experimental and numerical observations fall on a unique curve. Equation representing such unique curve is given below ln K ln La e KLa (11) 10 e Oxygen transfer is inversely related to the microscale of turbulence. Low values of microscale (mean higher dissipation) give high values of oxygen transfer coefficient. It is also seen from the (Fig. 3) that scale up can be achieved through microscale or energy dissipation criteria in geometrically similar systems. 572

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