Turbulent flow mass transfer coefficients

An exclusively analytical treatment of heat and mass transfer in turbulent flow in pipes fails because to date the turbulent shear stress Tl j = —Qw w p heat flux q = —Qcpw, T and also the turbulent diffusional flux j Ai = —gwcannot be investigated in a purely theoretical manner. Rather, we have to rely on experiments. In contrast to laminar flow, turbulent flow in pipes is both hydrodynamically and thermally fully developed after only a short distance x/d > 10 to 60, due to the intensive momentum exchange. This simplifies the representation of the heat and mass transfer coefficients by equations. Simple correlations, which are sufficiently accurate for the description of fully developed turbulent flow, can be found by... 

This result suggests diet the mass transfer coefficient varies as Djg, a result intermediate to those of the film theory and tha panetration theory. In fact, the j power dependence of the Sherwood number on the Schmidt number is observed in a number of correlations for mass transfer in turbulent flow in conduits aed for flow about submerged objects. (See Section 2.4-3.)... 

Fouling phenomena diminish as concentration polarisation decreases. Concentration polarisation can be reduced by increasing the mass transfer coefficient (high flow-velocities) and using low(er) flux membranes. Also the use of various kinds of turbulence promoters w ill reduce fouling, although fluidised bed systems and rotary module systems seem not veiy feasible from an economical point of view for large scale applications but they may attractive for small scale applications. 

Both heat and mass transfer in turbulent flow are generally not amenable to analytical treatment. It is customary in these cases to resort to what is termed dimensional analysis. This device consists of grouping the pertinent physical parameters of the system into a number of dimensionless groups, thus reducing the number of variables that must be dealt with, and evaluating the undetermined coefficients experimentally. It is a powerful tool for arriving at a first qualitative description of complex systems and eases enormously the experimental work required to quantify the relationship. 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Mass-Transfer Coefficient Denoted by /c, K, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theoiy, the surface renewal theoiy, and the penetration the-oiy, all of which pertain to ideahzed cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flowthrough banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. 

The mass-transfer coefficients depend on complex functions of diffii-sivity, viscosity, density, interfacial tension, and turbulence. Similarly, the mass-transfer area of the droplets depends on complex functions of viscosity, interfacial tension, density difference, extractor geometry, agitation intensity, agitator design, flow rates, and interfacial rag deposits. Only limited success has been achieved in correlating extractor performance with these basic principles. The lumped parameter deals directly with the ultimate design criterion, which is the height of an extraction tower. 

Tlie power for laminar flow is proportional to agitation rate, N2, and if the flow is turbulent the power is proportional to N3Dt2. Let us assume the mass transfer coefficients remain constant (Kha unchanged) ... 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

For flow in a smooth pipe, the friction factor for turbulent flow is given approximately by the Blasius equation and is proportional to the Reynolds number (and hence the velocity) raised to a power of -2. From equations 12.102 and 12.103, therefore, the heat and mass transfer coefficients are both proportional to w 75. 

Obtain the Taylor-Prandtl modification of the Reynolds Analogy between momentum transfer and mass transfer (equimolecular counterdiffusion) for the turbulent flow of a fluid over a surface. Write down the corresponding analogy for heat transfer. State clearly the assumptions which are made. For turbulent flow over a surface, the film heat transfer coefficient for the fluid is found to be 4 kW/m2 K. What would the corresponding value of the mass transfer coefficient be. given the following physical properties ... 

When two or more phases are present, it is rarely possible to design a reactor on a strictly first-principles basis. Rather than starting with the mass, energy, and momentum transport equations, as was done for the laminar flow systems in Chapter 8, we tend to use simplified flow models with empirical correlations for mass transfer coefficients and interfacial areas. The approach is conceptually similar to that used for friction factors and heat transfer coefficients in turbulent flow systems. It usually provides an adequate basis for design and scaleup, although extra care must be taken that the correlations are appropriate. 

Numerous turbulent mass-transfer relationships are given in Eqs. (39)-(50), Table VII. Although the most important ones in practical applications are those for channels and tubes, several other configurations also have been investigated because of their hydrodynamic interest. Generally, it is not possible to predict mass-transfer rates quantitatively by recourse to turbulent flow theory. An exception to this is for the region of developing mass transfer, where a Leveque-type correlation between the mass-transfer coefficient and friction coefficient/can be established ... 

Considerable interest has been generated in turbulence promoters for both RO and UF. Equations 4 and 5 show considerable improvements in the mass-transfer coefficient when operating UF in turbulent flow. Of course the penalty in pressure drop incurred in a turbulent flow system is much higher than in laminar flow. Another way to increase the mass-transfer is by introducing turbulence promoters in laminar flow. This procedure is practiced extensively in enhanced heat-exchanger design and is now exploited in membrane hardware design. 

Since the term local in the case of turbulent flow refers to a path of length xa, the local turbulent mass transfer coefficient should be defined as the average... 

For the turbulent motion in a tube, the mass transfer coefficient k is proportional to the diffusion coefficient at the power of 2/3. It is easy to realize by inspection that this value of the exponent is a result of the linear dependence of the tangential velocity component on the distance y from the wall. For the turbulent motion in a tube, the shear stress t r0 = const near the wall, whereas for turbulent separated flows, the shear stress is small at the wall near the separation point (becoming zero at this point) and depends on the distance to the wall. Thus, the tangential velocity component has, in the latter case, no longer a linear dependence on y and a different exponent for the diffusion coefficient is expected. For separated flows, it is possible to write under certain conditions that [90]... 

It is of interest to note that the exponent of the diffusion coefficient is indeed 3/4 (hence different from that in a tube) and that Richardson [91] has obtained a value of 2/3 for the exponent of the velocity while studying experimentally the heat and mass transfer in turbulent separated flows. 

The critical rotational speed of the impeller for complete suspension is approximately 541 rpm (9 tps) (eq. (3.114)). This rotational speed results in a Reynolds value of about 2.54 X 105, and tlius the flow is turbulent. Here, it has to be noted that the mass transfer coefficient strongly depends on the rotational speed below the critical rotational speed needed for complete suspension, and weakly depends on the rotational speed above its critical value at which there ate no particles remaining in rest for longer than 1-2 s in a fixed position. 

Hollow-fiber (capillary)-type membrane oxygenators are the most widely used today, and comprise two main types (i) those where blood flow occurs inside the capillaries and (ii) those where there is a cross-flow of blood outside the capillaries. Although in the first type the blood flow is always laminar, the second type has been used more extensively in recent times, as the mass transfer coefficients are higher due to blood turbulence outside capillaries and hence the membrane area can be smaller. Figure 15.3 shows an example of the cross-flow type membrane oxygenator, with a built-in heat exchanger for controlling the blood temperature. 

There are several reports in the literature of the critical Reynolds number at which turbulent film flow commences. These values of NRe t are usually determined from the breaks which appear in the curves of film thickness, surface velocity of the film, heat or mass transfer coefficients in the film, etc., when plotted against NHe. Some of the numerical values proposed by various investigators are listed in Table I. 

When two phases are mixed together (gas-liquid, immiscible liquid-liquid), a fine dispersion of bubbles or drops and a high specific interfacial area are produced because of the intensive turbulence and shear. For this reason, resistance to interphase mass transfer is considerably smaller than in conventional equipment. In addition, a wide range of gas-liquid flow ratios can be handled, whereas in stirred tanks the gas-flow rate is often limited by the onset of flooding. Mass transfer coefficients (kLa) can be 10-100 times higher than in a stirred tank. 

An important exception to laminar flow is the calculation of velocity profiles in industrial electrochemical reactors. These often function in the turbulent regime in order to maximize the yield of the process, given that the mass transfer coefficients are higher. 

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