Mass driving forces

In a vacuum (a) and under the effect of a potential difference of V volts between two electrodes (A,B), an ion (mass m and charge ze) will travel in a straight line and reach a velocity v governed by the equation, mv = 2zeV. At atmospheric pressure (b), the motion of the ion is chaotic as it suffers many collisions. There is still a driving force of V volts, but the ions cannot attain the full velocity gained in a vacuum. Instead, the movement (drift) of the ion between the electrodes is described by a new term, the mobility. At low pressures, the ion has a long mean free path between collisions, and these may be sufficient to deflect the ion from its initial trajectory so that it does not reach the electrode B. 

Mass transfer rates may also be expressed in terms of an overall gas-phase driving force by defining a hypothetical equiHbrium mole fractionjy as the concentration which would be in equiHbrium with the bulk Hquid concentration = rax ) ... 

Expressions similar to equations 6 and 7 may be derived in terms of an overall Hquid-phase driving force. Equation 7 represents an addition of the resistances to mass transfer in the gas and Hquid films. The analogy of this process to the flow of electrical current through two resistances in series has been analyzed (25). 

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the MaxweU-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair directiy based on the MaxweU-Stefan diffusivities. 

This rate equation must satisfy the boundary conditions imposed by the equiUbrium isotherm and it must be thermodynamically consistent so that the mass transfer rate falls to 2ero at equiUbrium. It maybe a linear driving force expression of the form... 

The Driving Force for Mass Transfer. The rate of mass transfer increases as the driving force, (7 — (7, is increased. can be enhanced as follows. From Dalton s law of partial pressures... 

Oxygen transfer rate (OTR) is estimated by the foUowiag standard procedure (12), where the rate of mass transfer per unit volume of Hquid is taken to be directly proportional to the driving force of the system... 

In considering the effect of mass transfer on the boiling of a multicomponent mixture, both the boiling mechanism and the driving force for transport must be examined (17—20). Moreover, the process is strongly influenced by the effects of convective flow on the boundary layer. In Reference 20 both effects have been taken into consideration to obtain a general correlation based on mechanistic reasoning that fits all available data within 15%. 

In the macroscopic heat-transfer term of equation 9, the first group in brackets represents the usual Dittus-Boelter equation for heat-transfer coefficients. The second bracket is the ratio of frictional pressure drop per unit length for two-phase flow to that for Hquid phase alone. The Prandd-number function is an empirical correction term. The final bracket is the ratio of the binary macroscopic heat-transfer coefficient to the heat-transfer coefficient that would be calculated for a pure fluid with properties identical to those of the fluid mixture. This term is built on the postulate that mass transfer does not affect the boiling mechanism itself but does affect the driving force. 

Likewise, the microscopic heat-transfer term takes accepted empirical correlations for pure-component pool boiling and adds corrections for mass-transfer and convection effects on the driving forces present in pool boiling. In addition to dependence on the usual physical properties, the extent of superheat, the saturation pressure change related to the superheat, and a suppression factor relating mixture behavior to equivalent pure-component heat-transfer coefficients are correlating functions. 

The use of molal humidity as the mass-transfer driving force is conventional and convenient because of the development of humidity data for, especially, the air—water system. The mass-transfer coefficient must be expressed in consistent units. 

Although equation 35 is a simple expression, it tends to be confusing. In this equation the enthalpy difference appears as driving force in a mass-transfer expression. Enthalpy is not a potential, but rather an extensive thermodynamic function. In equation 35, it is used as enthalpy pet mole and is a kind of shorthand for a combination of temperature and mass concentration terms. 

An analogy exists between mass transfer by diffusion and heat transfer by conduction. Each involves coHisions between molecules and a gradient as the driving force which causes flow. Eor diffusion, this is a concentration gradient for conduction, the driving force is an energy gradient. Eourier s... 

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... 

According to this analysis one can see that for gas-absorption problems, which often exhibit unidirectional diffusion, the most appropriate driving-force expression is of the form y — y tyBM,. ud the most appropriate mass-transfer coefficient is therefore kc- This concept is to he found in all the key equations for the design of mass-transfer equipment. 

It is important to understand that when chemical reactions are involved, this definition of Cl is based ou the driving force defined as the difference between the couceutratiou of un reacted solute gas at the interface and in the bulk of the liquid. A coefficient based ou the total of both uureacted and reached gas could have values. smaller than the physical-absorption mass-transfer coefficient /c . 

Equations (13-111) to (13-114), (13-118) and (13-119), contain terms, Njj, for rates of mass transfer of components from the vapor phase to the liquid phase (rates are negative if transfer is from the liquid phase to the vapor phase). These rates are estimated from diffusive and bulk-flow contributions, where the former are based on interfacial area, average mole-fraction driving forces, and mass-... 

Tray Efficiencies in Plate Absorbers and Strippers Compn-tations of the nnmber of theoretical plates N assnme that the hqnia on each plate is completely mixed and that the vapor leaving the plate is in eqnihbrinm with the liqnid. In actnal practice a condition of complete eqnihbrinm cannot exist since interphase mass transfer reqnires a finite driving-force difference. This leads to the definition of an overall plate efficiency... 

Although the right-hand side of Eq. (14-60) remains valid even when chemical reactions are extremely slow, the mass-transfer driving force may become increasingly small, until finally c — Cj. For extremely slow first-order irreversible reactions, the following rate expression can be derived from Eq. (14-60) ... 

Effect on mean mass-transfer driving force... 

External mass tran.sfer between the external surfaces of the adsorbent particles and the surrounding fluid phase. The driving force is the concentration difference across the boundary layer that surrounds each particle, and the latter is affected by the hydrodynamic conditions outside the particles. 

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