Mass solution flow velocity profile

The rotating disc electrode (RDE) is the classical hydrodynamic electroanalytical technique used to limit the diffusion layer thickness. However, readers should also consider alternative controlled flow methods including the channel flow cell (38), the wall pipe and wall jet configurations (39). Forced convection has several advantages which include (1) the rapid establishment of a high rate of steady-state mass transport and (2) easily and reproducibly controlled convection over a wide range of mass transfer coefficients. There are also drawbacks (1) in many instances, the construction of electrodes and cells is not easy and (2) the theoretical treatment requires the determination of the solution flow velocity profiles (as functions of rotation rate, viscosities and densities) and of the electrochemical problem very few cases yield exact solutions. 

The above-presented formulas give a solution of the conjugation boundary-value problem (3.75), (3.76) for constant coefficients A,f> = const in the final form. Some typical 1 ( /-distributions have been shown in Figure 3.18. Dotted line 0 denotes the air velocity distribution if there were no motion of EPR elements, S = 0. The dotted line 3 corresponds to that if there were no EPR in the duct at all, A = 0 or 6 = 0. Carrying flow velocity profiles 1 (for ft = 10) and 2 (for ft = 40) lie between them. It can be verified analytically from (3.79) that the case /J -> oo corresponds to no EPR at all in accordance with (3.77) (the droplet mass tends to zero, m —> 0) and with (3.78). 

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

In turbulent flow, properties such as the pressure and velocity fluctuate rapidly at each location, as do the temperature and solute concentration in flows with heat and mass transfer. By tracking patches of dye distributed across the diameter of the tube, it is possible to demonstrate that the liquid s velocity (the time-averaged value in the case of turbulent flow) varies across the diameter of the tube. In both laminar and turbulent flow the velocity is zero at the wall and has a maximum value at the centre-line. For laminar flow the velocity profile is a parabola but for turbulent flow the profile is much flatter over most of the diameter. 

The mass transfer coefficient is usually obtained from correlations for flow in non-porous ducts. One case is that of laminar flow in channels of circular cross-section where the parabolic velocity profile is assumed to be developed at the channel entrance. Here the solution of LfivfiQUE(7), discussed by Blatt et al.(H>, is most widely used. This takes the form ... 

The distribution of the solute between the mobile and the stationary phases is continuous. A differential equation that describes the travel of a zone along the column is composed. Then the band profile is calculated by the integration of the differential mass balance equation under proper initial and boundary conditions. Throughout this chapter, we assume that both the chemistry and the packing density of the stationary phase are radially homogeneous. Thus, the mobile and stationary phase concentrations as well as the flow velocities are radially uniform, and a one-dimensional mass balance equation can be considered. 

As stated in Eq. 4.156, it might appear that a the solution (i.e., u(r)) would exist for any value of the parameter Re/. However, the velocity profile must be constrained to require that the net mass flow rate is consistent with m = pU Ac, where U is the mean velocity used in the Reynolds number definition. Based on the integral-constraint relationship,... 

Solution of the system, of course, requires boundary conditions. These are written as in the earlier cases as no-slip on the channel walls (i.e., / = 0), and a mass-flow-rate constraint on the velocity profiles. 

The simulator used was a DISMOL, described previously by Batistella and Maciel (2). All explanations of the equations used, the solution methods, and the routine of solution are described in Batistella and Maciel (5). DISMOL is a simulator that permits changes in feed composition, feed temperaturethe evaporation rate, as well as feed flow rate. The effective rate of surface evaporation is obtained from the kinetic theory of gases. The liquid film thickness is obtained by mass balance and geometry of the evaporator. The temperature in the liquid obeys the Fourier-Kirchhoff equation. The solution of the velocity profile requires knowledge of the viscosity and the liquid film thickness over the evaporator. The solution for the temperature and the concentration profiles requires knowledge of the velocity profiles, which determine the convective heat and mass fluxes. 

The boundary conditions are zero velocity at the walls and zero slope at any planes of symmetry. Analytical solutions for the velocity profile in square and rectangular ducts are available but cumbersome, and a numerical solution is usually preferred. This is the reason for the transient term in Equation 16.7. A flat velocity profile is usually assumed as the initial condition. As in Chapter 8, is assumed to vary slowly, if at all, in the axial direction. For single-phase flows, u can vary in the axial direction due to changes in mass density and possibly to changes in cross-sectional area. The continuity equation is just AcUp = constant because the cross-channel velocity components are ignored. 

Many numerical and series solutions for the laminar boundary layer model of mass transfer are available for situations such as flow in coeduits under conditions of fully developed or developing concentration or velocity profiles. Skellaed31 provides a particularly good summary of these results. The laminar boundary layer model has been extended to predict tha effects of high mass transfer flux on the mass transfer coefficient from a flat plate. The results of this work ate shown in Fig. 2.4-2 and. in com rest to the other theories, iedicate a Schmith number dependence of Ihe correction factor. 

Two situations involviag tmasport of a species lo a liquid in laminar flow can he mudeled by the system in Fig, 2.3-16. One might represent absorption (tom a gas stream into liquid Rowing over pieces of packing in a column, The other geometry could represent dissolution of a solid into a flowing liquid film or the transport from a membrane surface into a solution in laminar flow over the membrane. Both these simple systems illustrate die efleet of velocity profiles or shear rales on mass transfer to a fluid. 

Laminar Flow. The Graetz or Leveque solutions25 26 for convective heat transfer in laminar flow channels, suitably modified for mass transfer, may be used to evaluate the mass transfer coefficient where the laminar parabolic velocity profile is assumed to be established at the channel entrance but where the concentration profile is under development down the full length of the channel. For all thin-channel lengths of practical interest, this solution is valid. Leveque s solution26 gives ... 

The simplest treatments of convective systems are based on a diffusion layer approach. In this model, it is assumed that convection maintains the concentrations of all species uniform and equal to the bulk values beyond a certain distance from the electrode, 8. Within the layer 0 x < 5, no solution movement occurs, and mass transfer takes place by diffusion. Thus, the convection problem is converted to a diffusional one, in which the adjustable parameter 8 is introduced. This is basically the approach that was used in Chapter 1 to deal with the steady-state mass transport problem. However, it does not yield equations that show how currents are related to flow rates, rotation rates, solution viscosity, and electrode dimensions. Nor can it be employed for dual-electrode techniques or for predicting relative mass-transfer rates of different substances. A more rigorous approach begins with the convective-diffusion equation and the velocity profiles in the solution. They are solved either analytically or, more frequently, numerically. In most cases, only the steady-state solution is desired. 

It should be noted that all investigations of flow stability of polymerizing liquids are few in number and have been carried out up till now only for unidimensional problems. The problem of stability of steady rheokinetic two-dimensional flows to local hydrodynamic perturbations has not been discussed in the literature yet. Obviously the problem can be solved (the solution is difficult from the technical point of view), for example, by numerical methods solving the problem on unsteady development of the flow of polymerizing mass directly after a forced local change of the profile of the flow velocity. 

In most cases, proper conditions for p, T, v or their derivatives at inflow-and outflow boundaries can be set according to the physical setup of the problem. For example if the mass flow is known in advance, the velocity at the inlet can be determined as soon as a velocity profile is chosen - the pressure level will be fixed at the outlet. On the other hand, if the pressure at the inlet and at the outlet is prescribed, the velocity cannot be chosen since the mass flow rate is part of the solution. 

A power-law non-Newtonian solution of a polymer is to be heated from 288 K to 303 K in a concentric-tube heat exchanger. The solution will flow at a mass flow rate of 210 kg/h through the inner copper tube of 31.75 mm inside diameter. Saturated steam at a pressure of 0.46 bar and a temperature of 353 K is to be condensed in the armulus. If the heater is preceded by a sufficiently long unheated section for the velocity profile to be fully established prior to entering the heater, determine the required length of the heat exchanger. Physical properties of the solution at the mean temperature of 295.5 K are ... 

Schematic of biocatalyst immobilized onto the membrane surface. Qualitative substrate and product concentration profiles are reported in order to show external mass transfer resistances in the film of thickness 5, when solution flows with velocity U. 

The condition expressed by Equation 8.2 is usually fulfilled and channel flow occurs in the laminar range. This allows the problem to be treated analytically and the solutions presented in this chapter rely on this assumption. Accounting for the simultaneous development of the velocity profile compared with concentration/temperature fields requires numerical evaluation, and the importance of this effect is measured by Prandtl s number for heat transfer or by Schmidt s number for mass transfer ... 

The approximations given by Equations 8.35 are the solution to Leveque s problem given in Equation 8.30 with a linear wall reaction. Since the formulation of the problem leads to a linearized velocity profile in a planar boundary layer, laminar flows (parabolic velocity profiles) in curved channels are more susceptible to present higher deviations from these results. For a fully developed flow in a round tube, the error associated with Equation 8.35b is 1.4 and 0.13% for aPe ,lz equal to 100 and 1000, respectively. Lopes et al. [40] observed that these differences are visible mainly for Da — 00 and calculated corrections to account for these effects. It was shown that in the mass transfer-controlled limit. 

A polymer solution (n of 0.5 K at 90 F of 51 Ibm fT viscosity activation energy of 14,900 Btu/lb mole) is fed into a 1-in. i.d. stainless steel tube (10 ft long) at a mass flow rate of 750 Ibm/h and a temperature of 90 F. The velocity profile is fully developed before the solution enters the heated tube. Heat is supplied by steam condensing at 20 psia. 

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