Mass convection concentration

A more detailed picture of the 3D temperature and concentration distribution can be obtained by an appropriate numerical model. Besides the diffusion equations for heat and mass, convection caused by both thermal and solutal expansion needs to be taken into account. 

Like heal convection, mass convection is also complicated because of the complications associated with fluid flow such as the surface geomeiiy, flow regime, flow velocity, and the variation of the fluid properties and composition. Therefore, wc have to rely on experimental relations to determine mass transfer. Also, mass convection is usually analyzed on a mass basis rather than on a molar basis. Therefore, sve will present formulations in terms of mass concentration (density p or mass fraction iv) instead of molar concentration (molar density C or mole, fraction y). But the formulations on a molar basis can be obtained using the relation C piM where M is the molar mass. Also, for simplicity, wc will restrict our attention to convection in fluids that are (or can be treated as) binaiy mixtures. 

Consider the flow of air over the free surface of a water body such as a lake under isothermal conditions. If the air is not saturated, the concentration of water vapor will vtsry from a maximum at the water surface where the air is always saturated to the free steam value far from the surface. In heat convection, we defined the region in which temperature gradients exist as the thermal boundary layer. Similarly, in mass convection, we define the region of the fluid in which concentration gradients exist as the conceniration boundary layer, as shown in Figure 14 -38. In external flow, the thickness of the concentration boundary layer S,. for a. species A at a. specified location on the surface is defined as the normal distance y from the surface at which... 

ShenvQod number relations in mass convection for specified concentration at the surface corresponding to the Nusselt number relations in heat convection for specified surface temperature ... 

Figure 14a, compared with the stable equilibrium bubble shape (Figure 14b) or the stable droplet shape during the evaporation stage in (Figure 4a). Thus the initial flux of CO2 into ethanol at P < Pm is much faster than the reverse process of ethanol evaporation, which can be explained by the higher equilibrium concentration of CO2 in the ethanol-rich phase and also by the larger coefficient of internal mass transfer in comparison to the vapor phase. Clearly, fluid dynamics plays a very important role for both internal and external mass transfer, as illustrated by the very strong gravity convection (concentration plumes) clearly visible in Figures 4 and 14a.

The capacitive elements and fields in the model represent equilibrium thermodynamics part of the model. As the simulation proceeds, the matter inside the control volume represented by these elements changes reversibly from one equilibrium state to the next, i.e. the process is assumed to be quasi-static. The R-fields represent the non-equilibrium parts of the model, and they introduce the irreversibilities into the system. The R-field elements represented by MR in Fig. 10.4 introduce the irreversibility due to mass convection into the system (refer to Section 10.2.4). The R-field element represented by RS in Fig. 10.4 introduces the irreversibility due to the over-voltage phenomena (ohmic, concentration and activation losses). The other R-field elements introduce the irreversibilities due to the heat transfer phenomena. 

To understand the peculiarities of voltammograms, some regularities regarding the partial processes of Cu(II) and Sn(II) reduction are expedient to invoke. As noted earlier, most polyethers do not show a noticeable inhibition activity on copper substrate in the absence of halides. Such behavior is ako characteristic of other surfactants in mixed Cu(II) and Sn(II) solutions. However, when the Sn Sn equilibrium potential is approached, the specific voltammetric minimum develops. The effect of surfactants becomes detectable at rather low concentrations (some mgdm ). Its depth depends both on their molecular mass and concentration and on the intensity of forced convection. This minimum deepens and the effect of forced convection weakens when the molecular mass of surfactant grows. 

Concentration gradients for the analyte in the absence of convection, showing the time-dependent change in diffusion as a method of mass transport. 

Concentration gradient for the analyte showing the effects of diffusion and convection as methods of mass transport. 

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. 

Equations (12.40) to (12.45) describe the velocities u, v, w, the temperature distribution T, the concentration distribution c (mass of gas per unit ma.ss of mixture, particles per volume, droplet number density, etc.) and pressure distribution p. These variables can also be used for the calculation of air volume flow, convective air movement, and contaminant transport. 

The description of mass transfer requires a separation of the contributions of convection and mutual diffusion. While convection means macroscopic motion of complete volume elements, mutual diffusion denotes the macroscopically perceptible relative motion of the individual particles due to concentration gradients. Hence, when measuring mutual diffusion coefficients, one has to avoid convection in the system or, at least has to take it into consideration. 

Sundararajan et al. [131] in 1999 calculated the slurry film thickness and hydrodynamic pressure in CMP by solving the Re5molds equation. The abrasive particles undergo rotational and linear motion in the shear flow. This motion of the abrasive particles enhances the dissolution rate of the surface by facilitating the liquid phase convective mass transfer of the dissolved copper species away from the wafer surface. It is proposed that the enhancement in the polish rate is directly proportional to the product of abrasive concentration and the shear stress on the wafer surface. Hence, the ratio of the polish rate with abrasive to the polish rate without abrasive can be written as... 

Under realistic conditions a balance is secured during current flow because of additional mechanisms of mass transport in the electrolyte diffusion and convection. The initial inbalance between the rates of migration and reaction brings about a change in component concentrations next to the electrode surfaces, and thus gives rise to concentration gradients. As a result, a diffusion flux develops for each component. Moreover, in liquid electrolytes, hydrodynamic flows bringing about convective fluxes Ji j of the dissolved reaction components will almost always arise. 

The convective mass flows in and out are obtained by multiplying the respective concentrations by the volumetric flow rate, which is equal to Ag v. The diffusive mass flows are calculated from the inlet and outlet concentration gradients using the multiplying factor of Ag D. 

Both pH and the availability of nutrient ions in soil play important roles in rhizo-sphere dynamics and are often dependent on one another. Nutrient ions move in soil toward plant roots either by mass flow with the soil water or by diffusion. Mass flow is the result of bulk convective movements of the soil solution toward roots, whereas diffusion occurs in response to a concentration gradient for a particular ion, which results from its absorption by the root and depletion from the... 

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