Average mass transfer coefficient

If the velocity of the flowing stream is uniform over a veiy deep region of hquid (total thickness, 6r V ), the time-averaged mass-transfer coefficient according to penetration theoiy is given oy... 

An accurate nasal model must also account for the airflow rate and the concentration of the inspired gas. Aharonson et al, conclusively demonstrated that the uptake coefficient," or average mass-transfer coefficient, over the entire nose for acetone, ozone, sulfur dioxide, and ether increased with increasing airflow rate. 

Now, the average mass transfer coefficient corresponding to the total gas-liquid interface of the heterogeneous part (it is equal to R tt) can be obtained after integrating (R ) T over the total interface (R = R /R) and dividing it by the total interface ... 

Knowing the average mass transfer coefficient, the absorption rate referring to the total gas-liquid interface can be given for zero- and first-order reactions, respectively as follows (see also Eq. 7,9,12,14) ... 

For the average mass transfer coefficient over the time interval A, one thus obtains... 

Compared with the Higbie penetration theory, the main difference consists in the fact that both the spatial and temporal dependence of the concentration are taken into account. In other words, the elements of liquid are characterized not only by their time of contact A with the interface, but also by their dimension x0 along the main flow direction. The average mass transfer coefficient should be, therefore, defined as... 

Denoting, for the simplicity of writing, the multiple average mass transfer coefficient also by k, dimensional considerations lead to... 

In 1951,Danckwerts [4] proposed the surface renewal model as an extension ofthe penetration model. Instead of assuming a fixed contact time for all fluid elements, Danckwerts assumed a wide distribution of contact time, from zero to infinity, and supposed that the chance of an element ofthe surface being replaced with fresh liquid was independent of the length of time for which it has been exposed. Then, it was shown, theoretically, that the averaged mass transfer coefficient at the interface is given as... 

It should be noted that the local mass transfer coefficient can only be obtained experimentally and is case specific. An analytical relationship for the local mass transfer rate coefficient can be obtained if a mathematical expression describing the gradient of the dissolved concentration at the NAPL-water interface is known. Unfortunately, the local mass transfer coefficient usually is not an easy parameter to determine with precision. Thus, in mathematical modeling of contaminant transport originating from NAPL pool dissolution, k(t, x,y) is often replaced by the average mass transfer coefficient, k(t), applicable to the entire pool, expressed as [41]... 

The effects of aquifer anisotropy and heterogeneity on NAPL pool dissolution and associated average mass transfer coefficient have been examined by Vogler and Chrysikopoulos [44]. A two-dimensional numerical model was developed to determine the effect of aquifer anisotropy on the average mass transfer coefficient of a 1,1,2-trichloroethane (1,1,2-TCA) DNAPL pool formed on bedrock in a statistically anisotropic confined aquifer. Statistical anisotropy in the aquifer was introduced by representing the spatially variable hydraulic conductivity as a log-normally distributed random field described by an anisotropic exponential covariance function. 

Numerically evaluated average mass transfer coefficients, k, based on the average of 200 different realizations of the log-normal hydraulic conductivity, as a function of C,JC,X for several variances of Y=lnK (a =0.1,0.2,0.3,0.4, and 0.5) for a hydraulic gradient of dhldx=0.01 and mean log-transformed hydraulic conductivity of 7=0.8 are shown in Fig. 7a. The results indicate that for increasing C,JC,X there is a significant increase in k. Low values of the anisotropy... 

Fig. 7a, b Average mass transfer coefficient as a function of a aquifer anisotropy ratio for several variances of the log-transformed hydraulic conductivity distribution b variance of the log-transformed hydraulic conductivity distribution where open circles represent numerically generated data and solid lines represent linear fits. All model parameter values are identical with those used in Fig. 6... 

Model simulations were performed in order to examine the effect of humic substances on the effective average mass transfer coefficient. The behavior of ke, defined in Eq. (47), as a function of time in the presence and absence of humic substances is illustrated in Fig.9a. Note that the parameter Lhis set equal to zero when dissolved humic substances are not present. Clearly, the results indicate that the presence of humic substances increases the mass transfer from the PCE pool into the aqueous phase. An increase in the mass transfer coefficient leads to faster dissolution of the PCE pool. 

The time invariant, average mass transfer coefficient of a single component rectangular NAPL pool can be expressed by the following functional relationship ... 

On the other hand, mass-transfer inside tbe viscous polymer particles will occur by molecular diffusion. According to an analytical solution Co the diffusion equation inside tbe particles, average mass-transfer coefficient between time 0 is given as follows (Newman, 1931) ... 

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