Transfer Coefficients in Laminar Flow

In principle, at least, we do not need mass-transfer coefficients for laminar flow, since molecular diffusion prevails, and the relationships of Chap. 2 can be used to compute mass-transfer rates. A uniform method of dealing with both laminar and turbulent flow is nevertheless desirable. 

Mass-transfer coefficients for laminar flow should be capable of computation. To the extent that the flow conditions are capable of description and the mathematics remains tractable, this is so. These are, however, severe requirements, and frequently the simplification required to permit mathematical manipulation is such that the results fall somewhat short of reality. It is not our purpose to develop these methods in detail, since they are dealt with extensively elsewhere [6, 7]. We shall choose one relatively simple situation to illustrate the general technique and to provide some basis for considering turbulent flow. 

Mass Transfer from a Gas into a Failing Liquid Film 

The problem is solved by simultaneous solution of the equation of continuity (2.17) for component A with the equations describing the liquid motion, the Navier-Stokes equations. The simultaneous solution of this formidable set of partial differential equations becomes possible only when several simplifying assumptions are made. For present purposes, assume the following  

The rate of absorption of gas is very small. This means that in Eq. (2.17) due to diffusion of A is essentially zero. 

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The RHSE has the same limitation as the rotating disk that it cannot be used to study very fast electrochemical reactions. Since the evaluation of kinetic data with a RHSE requires a potential sweep to gradually change the reaction rate from the state of charge-transfer control to the state of mass transport control, the reaction rate constant thus determined can never exceed the rate of mass transfer to the electrode surface. An upper limit can be estimated by using Eq. (44). If one uses a typical Schmidt number of Sc 1000, a diffusivity D 10 5 cm/s, a nominal hemisphere radius a 0.3 cm, and a practically achievable rotational speed of 10000 rpm (Re 104), the mass transfer coefficient in laminar flow may be estimated to be ... 

Mass Transfer Coefficients in Laminar Flow Extraction from the PDE Model... 

Annuli Approximate heat-transfer coefficients for laminar flow in annuh may be predicted by the equation of Chen, Hawkins, and Sol-berg [Tron.s. Am. Soc. Mech. Eng., 68, 99 (1946)] ... 

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. 

By analogy, the mass-transfer coefficient for laminar flow in spiral or coiled channels should vary as N rather than as... 

For laminar and turbulent flows, we need appropriate correlation equations for the friction coefficient, heat transfer coefficient, and mass transfer coefficient. For laminar flow in the ranges of 5 X 106 > Re > iO3, and Pr and Sc > 0.5, we have the following relations for the coefficients ... 

Knudsen and Katz [54] have shown that it is valid for Re Pr- djl >10. Equation 19.21 cannot be used for long tubes, since it would yield zero heat transfer coefficient. Sarti et al. have employed a different Equation 19.16 to estimate the heat transfer coefficient for laminar flow in circular tubes (shown in Table 19.1). 

Recently, Phattaranawik et al. [48] have used several equations to estimate the heat transfer coefficient in laminar and turbulent flow regimes. They found that Equation 19.22 is the most suitable for laminar flow, while the Dittus-Boelter equation was most suitable for turbulent conditions. 

The pioneering conclusion drawn by Tuckerman and Pease in 1982 [1] that the heat transfer coefficient for laminar flow through microchannels may be greater than that for turbulent flow, accelerated research in this area. Many experimental [2-6], numerical [7-10], and analytical [11-14] studies have been performed, with some focusing on the effects of roughness [15-21] and temperature-variable thermophysical properties of the fluid [22-26]. 

As seen in the previous section, flow is considered to be laminar when Re < 2300 and turbulent when Re > 104. Transition flow occurs in the range of 2300 < Re < 104. Few correlations or formulas for computing the friction factor and heat transfer coefficient in transition flow are available. In this section, the formula developed by Bhatti and Shah [45] is presented to compute the friction factor. It follows ... 

We explore here mass transfer coefficients in terms of the stream function. Write an expression for the surface-averaged dimensionless mass transfer coefficient for laminar flow of an incompressible Newtonian fluid around a stationary gas bubble in terms of the appropriate dimensionless numbers and the dimensionless stream function h. Use the approach velocity of the fluid and the radius of the bubble to construct a characteristic volumetric flow rate to make dimensionless. 

Correlations are available for predicting pressiffe drops and convective heat transfer coefficients for laminar flow inside and outside of ducts, tubes, and pipes for pipes with longitudinal and peripheral fins for condensation and boiling and for several different geometries used in compact heat exchangers. No attempt is made to discuss or summarize these correlations here. They are presented by Hewitt (1992). 

Sherwood et al. (19Z5) report that transition from laminar to turbulent occurs in the Reynolds number range from 250 to 500. Although they do not report any correlations for the liquid mass-transfer coefficient in turbulent flow, Treybal (1980) reports the following correlation for a liquid film with constant surface concentration at a somewhat higher range of Reynolds numbers. 

Here Dg is the diffusivity of the salt, and the Reynolds number is defined by Eq. tl7-35bl with d = height of feed channel. This equation is similar to Eq. tl7-35aT but predicts a higher mass transfer coefficient. For laminar flow in a tube of length L and radius R with a bulk velocity U], the average mass transfer... 

In theory it is not necessary to have experimental mass-transfer coefficients for laminar flow, since the equations for momentum transfer and for diffusion can be solved. However, in many actual cases it is difficult to describe mathematically the laminar flow for geometries, such as flow past a cylinder or in a packed bed. Hence, experimental mass-transfer coefficients are often obtained and correlated. A simplified theoretical derivation will be given for two cases in laminar flow. 

Laminar Flow Although heat-transfer coefficients for laminar flow are considerably smaller than for turbulent flow, it is sometimes necessary to accept lower heat transfer in order to reduce pumping costs. The heat-flow mechanism in purely laminar flow is conduction. The rate of heat flow between the walls of the conduit and the fluid flowing in it can be obtained analytically. But to obtain a solution it is necessary to know or assume the velocity distribution in the conduit. In fully developed laminar flow without heat transfer, the velocity distribution at any cross section has the shape of a parabola. The velocity profile in laminar flow usually becomes fully established much more rapidly than the temperature profile. Heat-transfer equations based on the assumption of a parabolic velocity distribution will therefore not introduce serious errors for viscous fluids flowing in long ducts, if they are modified to account for effects caused by the variation of the viscosity due to the temperature gradient. The equation below can be used to predict heat transfer in laminar flow. 

The recommended procedure is to calculate the heat transfer coefficient using both mechanisms and select the higher value as the effective heat transfer coefficient (h). For baffled condensers, the vapor shear effects vary for each typical baffle section. The condenser should be calculated in increments with the average vapor velocity (Vy) for each increment used to calculate vapor shear heat transfer coefficients. When the heat transfer coefficients for laminar flow and for vapor shear are nearly equal, the effective heat transfer coefficient (h) is increased above the higher of the two values. The table below permits the increase to be approximated ... 

Buller and Kilburn (1981) performed experiments determining the heat transfer coefficients for laminar flow forced air cooling for integrated circuit packages mounted on printed wiring boards (thus for conditions differing from that of a flat plate), and correlated he with the air speed through use of the Colburn J factor, a dimensionless number, in the form of... 

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