Mass-transfer coefficients in laminar flow

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... 

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 ... 

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. 

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. 

In practical situations like these, it has become customary to describe the mass-transfer flux in terms of mass-transfer coefficients. The relationships of this chapter are then rarely used directly to determine mass-transfer rates, but they are particularly useful in establishing the form of the mass-transfer coefficient-rate equations and in computing the mass-transfer coefficients for 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. 

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. 

These equations are identical to the equations describing mass transfer in monolithic reactors. For monolithic reactors it was shown [14] that when the reaction rate is very fast compared to the mass transfer rate in the fluid domain, the boundary condition of Eq. (1 5) becomes identical to the standard heat transfer boundary condition of constant wall temperature when the reaction rate is very slow compared to the mass transfer rate in the fluid domain, the boundary condition of Eq. (IS) becomes identical to the standard heat transfer boundary condition of constant heat flux. The influence of the boundary conditions on the mass transfer coefficient in case of laminar flow is discussed in the following section. 

Davis, Ouwerkerk and Venkatesh developed a mathematical model to predict the conversion and temperature distribution in the reactor as a function of the gas and liquid flow rates, physical properties, the feed composition of the reactive gas and carrier gas and other parameters of the system. Transverse and axial temperature profiles are calculated for the laminar flow of the liquid phase with co-current flow of a turbulent gas to establish the peak temperatures in the reactor as a function of the numerous parameters of the system. Also in this model, the reaction rate in the liquid film is considered to be controlled by the rate of transport of reactive gas from the turbulent gas mixture to the gas - liquid interface. The predicted reactor characteristics are shown to agree with large-scale reactor performance. For the calculations of the mass transfer coefficient in the gas phase, kg, Davis et al. used the same correlation as Johnson and Crynes, but multiplied the calculated values arbitrarily by a factor 2 to include the effect of ripples on the organic liquid film caused by the high SOj/air velocities in the core of the reactor. 

Mass Transfer Coefficients in Ducts of Various Geometries for Laminar Flow... 

Particularly when fluids flow past immersed objects, the local mass-transfer coefficient varies with position on the object, due especially to the separation of the boundary layer from the downstream surfaces to form a wake. This phenomenon has been studied in great detail for some shapes, e.g., cylinders [8]. The average mass-transfer coefficient in these cases can sometimes best be correlated by adding the contributions of the laminar boundary layer and the wake. This is true for the second entry of item 5, Table 3.3, for example, where these contributions correspond respectively to the two Reynolds-number terms. 

It is possible to predict theoretically the mass transfer rate (or flux AT,) across any surface located in a fluid having laminar flow in many situations by solving the differential equation (or equations) for mass balance (Bird et al, 1960, 2002 SkeUand, 1974 Sherwood et al, 1975). Our capacity to predict the mass transfer rates a priori in turbulent flow from first principles is, however, virtually nil. In practice, we follow the form of the integrated flux expressions in molecular diffusion. Thus, the flux of species i is expressed as the product of a mass-transfer coefficient in phase y and a concentration difference in the forms shown below ... 

In the case when pure CO2 gas is used as the feed gas, the mass transfer resistance in the membrane phase can be neglected. In this case, according to the Pick s law, which is usually used to describe the mass transfer resistance in the membrane phase, the membrane thickness and tortuosity has little effect on mass transfer [12]. Leveque s correlation is a common equation used to describe the tube side mass transfer. Because the liquid absorbent flow inside the hollow fiber is typically laminar, the mass transfer coefficient in the tube can be estimated from the Leveque s correlation [13] ... 

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. 

Obtain the Taylor-Prandtl modification of the Reynolds analogy between momentum and heat transfer and write down the corresponding analogy for mass transfer. For a particular system, a mass transfer coefficient of 8,71 x 10 8 m/s and a heat transfer coefficient of 2730 W/m2 K were measured for similar flow conditions. Calculate the ratio of the velocity in the fluid where the laminar sub layer terminates, to the stream velocity. 

The modeling of mass transport from the bulk fluid to the interface in capillary flow typically applies an empirical mass transfer coefficient approach. The mass transfer coefficient is defined in terms of the flux and driving force J = kc(cbuik-c). For non-reactive steady state laminar flow in a square conduit with constant molecular diffusion D, the mass balance in the fluid takes the form... 

---