Biochemical Engineering Fundamentals, Part 1 - Bailey J., Ollis D. 1989

Transport phenomena in biotechnological systems
Mass transfer involving freely rising and falling bodies
Determination of interfacial area and gas holdup of a dispersed system

Having determined k_l using one of the formulas above, we must then find the specific interfacial area a' per unit volume. The value of a' can be estimated by calculation based on the known diameter of the sparger orifices and other Reactor design data, as well as by photographing the process or other Methods. If the bubble residence time in the reactor is t_b, the volumetric flow rate per sparger orifice is F₀, and the total number of (identical) orifices is n, then the interfacial area per unit volume a' (neglecting coalescence and changes in D due to hydrostatic pressure changes or absorption) is determined by the equation

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Below we will consider the parameters on the right-hand side of Equation (8.39); in particular, we will study in detail the physical factors determining bubble size and, on this basis, evaluate the feasibility of calculating bubble size as a function of reactor operating parameters, impeller design, and liquid properties. Many of the concepts presented here for systems with rising single bubbles or bubble swarms are also applicable to The Study of transport processes in mechanically agitated reactors.

Bubble size in bioreactors is determined by three key interrelated factors; similar factors also play a major role in The formation of dispersions of a second, sparingly soluble liquid phase. These factors include The Mechanism of bubble formation, bubble breakup into smaller bubbles, and coalescence (bubble merging). Bubble formation is caused by the instability of the gas stream entering the liquid phase, which leads to the breakup of a single jet into individual bubbles. Bubble breakup depends on the competition between surface tension, which stabilizes the bubble, and local hydromechanical forces tending to disrupt it.

The probability of bubble coalescence is determined by The properties of the gas-liquid interface. In aqueous systems typical of biochemical engineering processes, coalescence is primarily influenced by substances dissolved in the liquid phase, such as Fatty acids, polyols, electrolytes, and ketones. The addition of these substances suppresses coalescence. In the subsequent Structure/133.html">Discussion, we will consider two extreme cases: coalescing dispersed systems (e.g., air-pure Water) and non-coalescing dispersed systems (e.g., air-aqueous electrolyte solution).

We begin our analysis by studying The process of gas bubble formation as gas enters at a volumetric flow rate F₀ through a single orifice of diameter d. We will denote the initial diameter of the forming bubble by D₀. In this situation, two regimes can be distinguished. At low gas flow rates, bubbles form sequentially at the orifice, one after another. The simplest approach to analyzing this regime is based on the balance of forces acting on the bubble leaving the orifice. This occurs when the buoyant force (пD₀³∆pg)/6 is balanced by the retaining force пσd:

A more detailed theory of this issue is presented in [21].

At a certain critical gas flow rate F₀*, a transition from individual gas bubbles to a continuous jet is observed at the sparger orifice. Methods for the precise calculation of F₀* have not yet been developed; however, it is known that the critical gas flow rate should lie in the range

If the gas flow rate is higher than F₀*, gas bubbles are formed by the breakup of the gas jet due to interfacial instability. The stability theory developed by Rayleigh predicts that the diameter of bubbles formed by the breakup of a gas jet in a laminar liquid flow should be approximately equal to

D₀ = d_l(12п/0,485)^1/3 = 4,27d      (8.42)

provided that [21]

In biochemical engineering processes, this condition is typically satisfied.

When a gas stream is introduced into a viscous culture medium, the primary resistance to the formation of new bubbles is exerted not so much by the surface tension of the bubbles as by the viscosity of the liquid medium. When bubbles are formed, The ratio of the bubble diameter to the sparger orifice diameter (D/d) is determined by the equation

Here, the Reynolds and Froude numbers related to the sparger orifice are given by Equations (8.45) and (8.46) respectively:

Sometimes, in a gas-liquid dispersed system, the gas bubbles have a smaller diameter than at the outlet of the gas distribution device. This phenomenon may be caused by bubble instability under the action of forces exerted on them by the moving continuous phase. For a dispersed liquid or gas phase in a continuous liquid phase, the maximum size (diameter) of bubbles or droplets of the dispersed phase that rise (sink) freely or are transported by agitation is determined by the balance of opposing forces:

1. Dynamic pressure т (the sum of the differences in shear stress and normal stress), which tends to elongate the droplets and eventually break them up into smaller droplets; this process is opposed by the two forces listed below.

2. Surface tension forces σ/D of the particle, which tend to restore the droplet to a spherical shape corresponding to the minimum surface energy.

3. Resistance of the dispersed phase to deformation due to its viscosity; this resistance is proportional to (here, the subscript d denotes that the

parameter belongs to the dispersed phase).

In gas-liquid systems, force 3 is negligible compared to force 2. In liquid-liquid dispersed systems, force 3 should be relatively large, but, as we will see later in Example 9.2, even in this case, surface tension forces play the dominant role.

The Effect of the last two factors decreases proportionally to D^-ß, where ß is a positive number. It follows that at a certain critical diameter D_c, the dynamic pressure will overcome the two opposing resistances, and the bubbles or droplets will break up into smaller particles. The critical diameter must obviously satisfy the following equality:

where m₁ and m₂ are constants.

If surface tension forces far exceed the forces due to the viscosity of the dispersed phase, then at the critical bubble diameter

According to equations (8.48), the maximum bubble size is equal to the product of the dimensionless constant m₁ and the surface tension divided by the dynamic pressure. Consequently, an increase in dynamic pressure will lead to the breakup of bubbles into increasingly smaller diameters.

In theoretical or empirical equations describing the relationship between the maximum diameter of a stable bubble (or droplet) and the Properties of the liquid medium and flow, a similarity criterion based on equations (8.48) is commonly used. Thus, the Weber number We is defined as

The critical Weber number We_c is nothing other than the value of We at D = D_c; according to equation (8.48), We_c is a characteristic constant. Experimental work and theoretical calculations have shown that for a pure air–water system, We_c is approximately equal to unity (more precisely, We_c = 1.05).

To calculate the maximum diameter of stable bubbles using the concepts and equations presented, we need to determine the dynamic pressure values т corresponding to different flow regimes. For freely rising bubbles, т is determined by the equation

Here, u_t is the terminal velocity of the bubbles, which in the case of spherical bubbles can be determined from equation (8.32). In complex turbulent flow, determining the dynamic pressure is difficult. Turbulence is to be expected, for example, in bubble columns near the aerating nozzle. Having first encountered this problem here, we will proceed to discuss some General Concepts related to turbulence that are also useful when considering several subsequent topics.

In the statistical theory of turbulence developed by A. N. Kolmogorov and other researchers, the turbulent flow field is viewed as the result of a superposition of eddies or velocity fluctuations, characterized by a length scale or frequency of fluctuations and magnitude. The largest-scale eddy elements, or primary eddies, have the scale of the main flow. These primary eddies are unstable and break down into smaller ones, which, due to their instability, break down into even smaller-scale eddies, and so on. This cascade of transformations from large-scale to small-scale eddies is accompanied by a transfer of kinetic energy until it is eventually dissipated as heat. The kinetic energy flux through the eddy cascade causes the directional Nature of the primary eddies (which depends on reactor geometry, nozzle design, impellers, etc.) to gradually decay. Kolmogorov's theory states that small-scale eddies are statistically independent of the primary eddies and are locally isotropic (spatially uniform). The length scale λ0 of the smallest-scale eddies that dissipate turbulence energy is determined by the equation

where P/V_l is the power input per unit volume.

In studying the effects of turbulence, time-averaged parameters are typically used. Thus, the ROOT-mean-square velocity u_rms ≡ <u²(t)>^1/2 (the brackets denote the time average of instantaneous velocity fluctuations) reflects the average magnitude of local velocity variations. For length scales l much smaller than the scale of the primary eddies and much larger than λ₀, the dependence of the root-mean-square eddy velocity on the characteristic scale I is given by the expression

where а is a constant.

Returning to The problem of finding an expression for the dynamic pressure, which could then be used to determine the Weber number, we note that we can use the equation

This expression is a measure of the turbulent shear stress acting on a bubble of diameter D_c. From equations (8.49), (8.52), and (8.53), we obtain

Here, а' is a constant. Thus, an increase in the power input per unit volume leads to a decrease in the maximum size of stable bubbles.

Significantly, According to the theory of isotropic turbulence, the power input per unit volume is the key parameter determining both the scale of the resulting eddies and the intensity of turbulent velocity fluctuations comparable in length scale to the sizes of bubbles and droplets. Local isotropic turbulence is a theoretical phenomenon that is by no means always observed in practice; nevertheless, It is important to keep in mind the physical interpretation of the energy transfer mechanism (energy from compressed gas or mechanical agitation) to gas bubbles, droplets, flocs, or mycelial pellets. Furthermore, equations (8.52) and (8.54) suggest that the value of P/V_l will have a significant effect on Mass transfer coefficients in bubble columns and stirred tanks. Another crucial aspect of this theory must be emphasized: the primary role is played by The amount of energy per unit volume, rather than the method of its transmission. For instance, it is of little consequence whether the energy is supplied by a compressed gas stream or an impeller, nor is the design of the latter critical, and so on. This generalization is again theoretical, but in many cases, theoretical and corresponding experimental data show good agreement.

Having independently considered the processes of coalescence, formation, and breakup of bubbles, we can now proceed to study the various possible types of interaction between these processes in a bubble Column. First, let us assume that bubble coalescence occurs slowly in the two-phase dispersed system under consideration. If the initial bubble diameter D₀ is smaller than the maximum stable bubble diameter D_c characteristic of the conditions of highest dynamic pressure (typically in the bubble formation region of a bubble column), then the characteristic bubble size will be equal to D₀ (Table 8.3). If, however, D₀ is larger than D_c, the bubbles can be expected to break up into smaller ones with a characteristic diameter of D_c.

Table 8.3. Dependence of the characteristic bubble diameter D on their coalescence tendency and the ratio between the bubble diameter at formation (D₀) and the maximum diameter of stable bubbles (D_c)

On the other hand, if coalescence occurs rapidly, the resulting bubbles will merge and grow in size until their diameter exceeds D_c. Then, the reverse process of bubble breakup will begin. In this case, the initial bubble size D₀ can only be maintained in the vicinity of the sparger and has virtually no effect on the actual bubble size in the reactor. Given that velocity fluctuations in the turbulent regime, and consequently D_c, are generally different at various points in the reactor, each point in a coalescing system is characterized by a tendency toward equilibrium between bubble coalescence and breakup, so that ultimately bubbles with diameter D_c are formed.

These considerations must be taken into account when designing the relevant equipment. In a non-coalescing system, the dissipation of energy spent on gas dispersion is most effective in the zones of bubble formation and dispersion. However, if coalescence processes play a major role, the dissipation of energy spent on gas dispersion occurs more or less uniformly throughout the entire volume of the liquid phase. As we will see later (Ch. 9), these theoretical Conclusions served as the basis for the development, study, and use of stirring and mixing devices for bioreactors of various designs; for this reason, they and some other qualitative conclusions are currently perhaps the most practically valuable results of the theoretical work discussed above. Indeed, reactor characteristics such as spatial heterogeneity of flow structures, features of turbulent flows, and volume fractions of gas and liquid phases are in most cases so complex that their quantitative estimation by calculation is extremely difficult, and the necessary numerical data can usually only be obtained using empirical equations. When choosing the latter, however, we must not forget the physical basis of the phenomena just discussed. Thus, an equation describing a coalescing pure air–water system is highly likely to be inapplicable to a process involving a relatively non-coalescing two-phase system.

Now, returning to the parameters of equation (8.39), which defines the interfacial area per unit volume a', let us consider the bubble residence time t_b in the reactor. This parameter can be found by integrating the bubble rise velocity over the reactor height h_r:

Here, in the right-hand expression, it is approximately assumed that the bubble rise velocity is equal to its terminal velocity. For individual small bubbles at low Reynolds numbers, the terminal velocity can be determined using equation (8.32). If, however, we are dealing with large spherical or hemispherical bubbles of diameter D in a Newtonian fluid, the terminal velocity value determined by the following equation should be used in the corresponding calculations

Calculating the characteristic velocity of bubble movement in a flow is more complex because, first, some bubbles influence the movement of neighboring ones, and second, during movement, bubbles can coalesce or break up into smaller bubbles. Comparing equations (8.33) and (8.35), it can be seen that for identical Sc and Sh, to a first approximation

and, consequently,

Any real dispersed system is characterized by a certain bubble size distribution. In this regard, the problem of determining the characteristic or average bubble size arises. In equations (8.35) to (8.38), the bubble diameter D refers to the surface-averaged diameter or the Sauter mean diameter D_sm:

Here, m_j is the number of bubbles of diameter D_j.

The expression nF₀t_b in equation (8.39) represents the total volume of bubbles in the reactor. The ratio of the volume of all bubbles to the reactor volume (gas volume to reactor volume) is called the gas holdup H. If the value of H is known from other laboratory or plant data, or from literature sources (Example 8.1), it can be substituted into the equation

In calculations involving the material balance equation of the liquid phase only, for example in equations (8.14) and (8.15), it is more convenient to use the parameter a', which is the ratio of the interfacial area to the volume of the liquid phase, as well as the parameter a — the ratio of the interfacial area to the total volume of the reaction mixture (liquid + gas). These two parameters, which determine the size of the interfacial area, are related through the gas holdup H:

а'(1—H) = а      (8.61)

Be careful when using mass transfer equations published in the literature; check which parameter was used in the original work — k_la or k_ld'.

Example 8.1. Empirical Equations for Determining gas holdup.

Bubble column*:

H/(1 — H)⁴ = 0,20(Во)^1/8(Gа)¹/¹²F_r      (8П1.1)

where Bo is the Bond number (Bo = gd_t²p_c/σ); Ga is the Galileo number (Ga = gd_t³/u_c²); Fr is the Froude number

u_G— superficial gas velocity; d_t— column (reactor) diameter.

* Chakravarty М., Begum S., Singh H. D., Barrah J. N., Iyengar M. S., Biotech. Bioeng. Symp., 4, 373 (1973).

Laboratory airlift column*:

Gas holdup in the draft tube (with sparger):

Gas holdup in the annulus:

Gas holdup above the baffle;

Н₃ =7,5∙10^-3u^0,88      (8П1.4)

Total gas holdup of the column:

H = 0,003u0,⁸⁸

Here, μ_l is the liquid phase viscosity at the column Temperature, cP; μ_H2O is the viscosity of water at the column temperature, cP; σ is the gas-liquid interfacial tension, dyn/cm; u is the superficial gas velocity, cm/s; A_int is the cross-sectional area of the draft tube, cm²; A_ann is the cross-sectional area of the annulus, cm².

Laboratory airlift column**

In the draft tube:

where u_d is the superficial gas velocity in the draft tube;

Stirred tank reactor***

Provided that the reactor diameter is approximately equal to its height and Re_i^0,7 (NiD_i/u)^0,3 < 2∙10⁴,

At Re_i^0,7 (N_iD_i/u)^0,3 > 2∙10⁴

and

where a₀, a₁ are the interfacial area per unit volume of culture; Re_i is the impeller Reynolds number (Re_i = pN_iD_i²/μ_c); u is the superficial gas velocity (for an empty reactor); u_t is the bubble rise velocity. The Definitions of the symbols N_i, D_i, Р, р, σ are given in the text (see Section 8.4). Stirred tank reactors*

For the water–air system, the data can be expressed by the equation

where P is the power in hp; V is the volume of the liquid phase excluding gas, m³; u is the superficial velocity, m/h; and H is the gas phase volume fraction (at 0.02 < H < 0.2).

* Chakravarty M., Begum S., Singh H. D., Barrah J. N., Iyengar M. S., Biotech. Bioeng. Symp., 4, 373 (1973).

** Hatch R. T., Ph. D. Thesis in Food Science and Nutrition, p. 150, Massachusetts Institute of Technology, Cambridge, Mass., 1973.

*** Calderbank P. H., Trans. Inst. Chem. Eng., 36, 443 (1958).