Transfer coefficient analysis

The preceding/following electron-number coefficients (y) used here are slightly different and less general than those employed by B R (namely, Yp and y ) in their transfer coefficient analysis D3q. (1)]. They are, however, related and for a preceding dissociation, the expressions that link them are... 

The main conclusion to be drawn from these studies is that for most practical purposes the linear rate model provides an adequate approximation and the use of the more cumbersome and computationally time consuming diffusing models is generally not necessary. The Glueckauf approximation provides the required estimate of the effective mass transfer coefficient for a diffusion controlled system. More detailed analysis shows that when more than one mass transfer resistance is significant the overall rate coefficient may be estimated simply from the sum of the resistances (7) ... 

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. 

Gal-Or and Hoelscher (G5) have recently developed a fast and simple transient-response method for the measurement of concentration and volumetric mass-transfer coefficients in gas-liquid dispersions. The method involves the use of a transient response to a step change in the composition of the feed gas. The resulting change in the composition of the liquid phase of the dispersion is measured by means of a Clark electrode, which permits the rapid and accurate analysis of oxygen or carbon dioxide concentrations in a gas, in blood, or in any liquid mixture. 

Pavlushenko et al. (P4) in their dimensional analysis considered Ks, the volumetric mass transfer coefficient, to be a function of pc, pc, L, Dr, N, Vs, and g. They determined the following relationship for the dimensionless groupings ... 

In their analysis, however, they neglected the surface tension and the diffusivity. As has already been pointed out, the volumetric mass-transfer coefficient is a function of the interfacial area, which will be strongly affected by the surface tension. The mass-transfer coefficient per unit area will be a function of the diffusivity. The omission of these two important factors, surface tension and diffusivity, even though they were held constant in Pavlu-shenko s work, can result in changes in the values of the exponents in Eq. (48). For example, the omission of the surface tension would eliminate the Weber number, and the omission of the diffusivity eliminates the Schmidt number. Since these numbers include variables that already appear in Eq. (48), the groups in this equation that also contain these same variables could end up with different values for the exponents. 

Most studies on heat- and mass-transfer to or from bubbles in continuous media have primarily been limited to the transfer mechanism for a single moving bubble. Transfer to or from swarms of bubbles moving in an arbitrary fluid field is complex and has only been analyzed theoretically for certain simple cases. To achieve a useful analysis, the assumption is commonly made that the bubbles are of uniform size. This permits calculation of the total interfacial area of the dispersion, the contact time of the bubble, and the transfer coefficient based on the average size. However, it is well known that the bubble-size distribution is not uniform, and the assumption of uniformity may lead to error. Of particular importance is the effect of the coalescence and breakup of bubbles and the effect of these phenomena on the bubble-size distribution. In addition, the interaction between adjacent bubbles in the dispersion should be taken into account in the estimation of the transfer rates... 

Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection can be neglected in comparison with those of forced convection. 

By dimensional analysis, derive a relationship for the heat transfer coefficient h for natural convection between a surface and a fluid on the assumption that the coefficient is a function of the following variables ... 

The calculations for the experimental reaction rates are based on an unsteady state heat transfer analysis. We computed the overall heat transfer coefficient of the system and estimated the experimental rates as follows dT... 

GP 9] [R 16[ The extent of external transport limits was made in an approximate manner as for the internal transport limits (see above), as literature data on heat and mass transfer coefficients at low Peclet numbers are lacking [78]. Using a Pick s law analysis, negligible concentration differences from the bulk to the catalyst sur-... 

Dang, N.D.P., Karrer, D.A. and Dunn, I.J. (1977). Oxygen Transfer Coefficients by Dynamic Model Moment Analysis, Biotechnol. Bioeng. 19, 853. 

The theoretical approach by Samec based on the ion-free compact layer model established that the true apparent transfer coefficient is obtained after correction for concentration polarization effect [1] [see Eq. (14)]. Subsequent studies by Samec and coworkers on the ferricyanide-Fc system provided values of a smaller than the expected 0.5. Preliminary attempts to rationalize this behavior were based on defining effective interfacial charges and separation distance between reactants [79]. The inconclusive trends reported in these studies were ascribed to complications arising from ion pairing of the ferro/ferricyanide ions. Later analysis of the same system appeared to show that k i is... 

The complex flow pattern on the shell-side, and the great number of variables involved, make it difficult to predict the shell-side coefficient and pressure drop with complete assurance. In methods used for the design of exchangers prior to about 1960 no attempt was made to account for the leakage and bypass streams. Correlations were based on the total stream flow, and empirical methods were used to account for the performance of real exchangers compared with that for cross flow over ideal tube banks. Typical of these bulk-flow methods are those of Kern (1950) and Donohue (1955). Reliable predictions can only be achieved by comprehensive analysis of the contribution to heat transfer and pressure drop made by the individual streams shown in Figure 12.26. Tinker (1951, 1958) published the first detailed stream-analysis method for predicting shell-side heat-transfer coefficients and pressure drop, and the methods subsequently developed... 

The analysis could be improved by dividing the tube length into sections, calculating the heat transfer coefficient and pressure drop over each section, and totalling. 

From that same section, we know that the steady state gain and the time constant are dependent on the values of flow rate, liquid density, heat capacity, heat transfer coefficient, and so on. For the sake of illustration, we are skipping the heat transfer analysis. Let s presume that we have done our homework, substituted in numerical values, and we found Kp = 0.85 °C/°C, and xp = 20 min. 

The analysis was later modified to include some of the factors neglected by Nusselt1213. One of these is the effect of buoyancy forces acting on the liquid film. This results in the pl term in Equation 15.80 being replaced by Pl(pl — Pv ) Such buoyancy forces are usually only important close to the critical point. In most cases, the two most important factors that cause a significant deviation from Equation 15.80 are the presence of vapor shear forces and noncondensable gases in the vapor. Vapor shear forces act to increase the heat transfer coefficient, whereas noncondensable gases act to decrease it. 

ORNL small-break LOCA tests Experimental investigation of heat transfer and reflood analysis was made under conditions similar to those expected in a small-break LOCA. These tests were performed in a large, high-pressure, electrically heated test loop of the ORNL Thermal Hydraulic Test Facility. The analysis utilized a heat transfer model that accounts for forced convection and thermal radiation to steam. The results consist of a high-pressure, high-temperature database of experimental heat transfer coefficients and local fluid conditions. 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

The proper analysis of liquid-liquid systems requires an understanding of the system. When the model equations are derived so that the heat-transfer coefficient does not include the lack of knowledge of the gross fluid mechanics, a correlation for the coefficient, formulated from experiment analysis, is most useful. The correlation can be employed in physical situations where the gross fluid mechanics is entirely different, as in a different-diameter pipe. 

Depending on the driving force we choose to employ in our analysis, there are several definitions of mass transfer coefficients that may be considered appropriate for use. If we consider an arbitrary interface between a fluid and the external surface of a catalyst particle, we might choose to define a mass transfer coefficient based on a concentration driving force (kc) as... 

Heat Transfer to the Containing Wall. Heat transfer between the container wall and the reactor contents enters into the design analysis as a boundary condition on the differential or difference equation describing energy conservation. If the heat flux through the reactor wall is designated as qw, the heat transfer coefficient at the wall is defined as... 

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