Mass transfer diluted systems

Kcfl Overall volumetric gas-pbase mass-transfer dilute systems kmoP[(s-m )(mole-fraction solute in gas)] (lbmol)/[(b-ft (mole-fraction solute in gas)]... 

Kia Overall volumetric gas-phase mass-transfer dilute systems (lbmol)/[(hft3)(lbf/in2 solute partial pressure)]... 

Cost of electrical power Equipment Cost (expressed as power cost) Efficiency of oxygen mass transfer Dilute system, lOg// ... 

Equations 27—32 are appHcable only to dilute, immiscible systems. If the amount of mass transfer is significant ia comparison to the total dow rates, more compHcated treatments of differential contactors are required (5,28). 

Mass-Transfer Principles Dilute Systems When material is transferred from one phase to another across an interface that separates the two, the resistance to mass transfer in each phase causes a concentration gradient in each, as shown in Fig. 5-26 for a gas-hquid interface. The concentrations of the diffusing material in the two phases immediately adjacent to the interface generally are unequal, even if expressed in the same units, but usually are assumed to be related to each other by the laws of thermodynamic equihbrium. Thus, it is assumed that the thermodynamic equilibrium is reached at the gas-liquid interface almost immediately when a gas and a hquid are brought into contact. 

For systems in which the solute concentrations in the gas and hquid phases are dilute, the rate of transfer may be expressed by equations which predic t that the rate of mass transfer is proportional to the difference between the bulk concentration and the concentration at the gas-liquid interface. Thus... 

Tbe mass-transfer coefficients k c and /cf by definition are equal to tbe ratios of tbe molal mass flux Na to tbe concentration driving forces p — Pi) and (Ci — c) respectively. An alternative expression for tbe rate of transfer in dilute systems is given by... 

When two or more gases are absorbed in systems involving chemical reac tions, the situation is much more complex. This topic is discussed later in the subsection Absorption with Chemical Reac tion. Graphical Design Method for Dilute Systems The following notation for multicomponent absorption calculations has been adapted from Sherwood, Pigford, and Wilke (Mass Transfer, McGraw-Hill, New York 1975, p. 415) ... 

This is the gas-phase mass-transfer limited condition, which can be substituted into Eq. (14-71) to obtain the following equation for calculating the height of packing for a dilute system ... 

For an isothermal absorber involving a dilute system in which a liquid-phase mass-transfer limited first-order irreversible chemic reaction is occurring, the packed-tower design equation is derived as... 

For a dilute system in which the liquid-phase mass-transfer limited condition is valid, in which a veiy fast second-order reaction is involved, and for which Nna E veiy large, the equation... 

The treatment here is restricted to the Langmuir or constant separation factor isotherm, single-component adsorption, dilute systems, isothermal behavior, and mass-transfer resistances acting alone. References to extensions are given below. Different isotherms have been considered, and the theory is well understood for general isotherms. 

Equation (3.5) can be used to establish a one-to-one correspondence among all composition scales for which mass exchange is feasible. Since most environmental applications involve dilute systems, one can assume that these systems behave ideally. Hence, the transfer of the pollutant is indifferent to the existence of other species in the waste stream. In other words, even if two waste streams contain species that are not identical, but share the same composition of a particular pollutant, the equilibrium composition of the pollutant in an MSA will be the same for both waste streams. Hence, a single composition scale, y, can be used to represent the concentration of the pollutant in any waste stream. Next, (3.5) can be employed to generate Ns scales for the MSAs. For a given set of corresponding composition scales y,x, X2,..., xj,..., it is thermodynamically and practically feasible to transfer the pollutant from any waste stream to any MSA. In addition, it is also feasible to transfer the pollutant from any waste stream of a composition y/ to any MSA which has a composition less than the xy obtained from (3.5b). 

This chapter has introduced Fick s law for dilute solutions and has shown how this law can be combined with mass balances to calculate concentrations and fluxes. The mass balances are made on a small rectangular box. When the box becomes very small, the mass balances become the differential equations used to solve various pharmaceutical transport problems. Thus, this chapter discussed many mathematical equations. Although these mathematical equations are tedious, they are essential to each specific application of mass transfer in pharmaceutical systems to be discussed in the rest of this book. 

Toor, H. L. (1962). Mass transfer in dilute turbulent and non-turbulent systems with rapid irreversible reactions and equal diffusivities. AIChE Journal 8, 70-78. 

In this section we will develop the mathematical tools to describe mass transfer at diffusive boundaries. Again, it is our intention to demonstrate that diffusive boundaries have common properties, although the physics controlling them may be different. We will then apply the mathematical tools to the process of dilution of a pollutant cloud in an aquatic system (ocean, lake, river). Here the boundary is produced by the localized (continuous or event-like) input of a chemical that first leads to a confined concentration patch. The patch is then mixed into its environment by diffusion or dispersion. Note that in this case the physical characteristics on both sides of the... 

The HETP of a column, valid for either distillation or dilute-gas absorption and stripping systems in which constant molar overflow can be assumed, and in which no chemical reactions occur, is related to the height of one overall gas-phase mass-transfer unit, HQG, by the equation ... 

Since mass transfer in packed or spray towers occurs differentially rather than stagewise, their performance should be expressed in terms of the number of transfer units (NTU) rather than the number of theoretical stages (NTS). For dilute systems, the number of transfer units is given in terms of the terminal concentrations and the equilibrium relation by... 

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