Analogies, heat/mass transfer

Heat Transfer Heat-transfer rates are gener ly large despite severe axial dispersion, with Ua. frequently observed in the range 18.6 to 74.5 and even to 130 kW/(m K) [1000 to 4000 and even to 7000 Btu/(h fF °F)][see Bauerle and Ahlert, Ind. Eng. Chem. Process Des. Dev., 4, 225 (1965) and Greskovich et al.. Am. Tn.st. Chem. Eng. J., 13,1160 (1967) Sideman, in Drewet al. (eds.). Advances in Chemical Engineering, vol. 6, Academic, New York, 1966, p. 207, reviewed earlier work]. In the absence of specific heat-transfer correlations, it is suggested that rates be estimated from mass-transfer correlations via the heat-mass-transfer analogy. 

Obtain the Taylor-Prandtl modification of the Reynolds analogy between momentum and heat transfer and write down the corresponding analogy for mass transfer. For a particular system, a mass transfer coefficient of 8,71 x 10 8 m/s and a heat transfer coefficient of 2730 W/m2 K were measured for similar flow conditions. Calculate the ratio of the velocity in the fluid where the laminar sub layer terminates, to the stream velocity. 

One other measurement technique that has been used to measure Kl over a shorter time period, and is thus more responsive to changes in wind velocity, is the controlled flux technique (Haupecker et al., 1995). This technique uses radiated energy that is turned into heat within a few microns under the water surface as a proxy tracer. The rate at which this heat diffuses into the water column is related to the liquid film coefficient for heat, and, through the Prandtl-Schmidt number analogy, for mass as well. One problem is that a theory for heat/mass transfer is required, and Danckwert s surface renewal theory may not apply to the low Prandtl numbers of heat transfer (Atmane et al., 2004). The controlled flux technique is close to being viable for short-period field measurements of the liquid film coefficient. 

The analogy between mass transfer and heat transfer is summarized in Kq. The mass transfer coefficient s relation to Ap in mass transfer is analogous to the role that the overall heat transfer coefficient is to AT in heat transfer. 

In the case of pore formation in systems with random fluxes, one has to take into account the influence of pore formation on the heat/mass transfer. In most cases, this factor can be considered as the little perturbation of the situation without pore formation, analogically to the solution given in [5]. The pore formation influences the values of U, Um n, C/max- The value of energy of a subsystem is approximately given by ... 

Analogously to mass transfer, the following equation is used for external heat transfer ... 

For a given wet-bulb temperature, both X and are fixed. The relation between and T then depends on the ratio hyjky. The close analogy between mass transfer and heat transfer provides considerable information on the magnitude of this ratio and the factors that affect it. 

M. K. El-Riedy, Analogy Between Heat and Mass Transfer by Natural Convection From Air to Horizontal Tubes, Int. J. Heat Mass Transfer (24) 365-369,1981. 

Heat-transfer coefficients can be estimated, if not available otherwise, through the heat-mass-transfer analogies discussed in Chapter 2. For the gas phase, the heat-transfer coefficient must be corrected for mass transfer (Treybal, 1980) ... 

Analogously to mass transfer, for heat transfer there is a necessity to calculate the heat conductivity A in the pores, which appears in the Fourier s law relating the heat flux through a surface with the temperature difference along the path of heat flow... 

Because of the analogy between mass transfer by diffusion and heat transfer by conduction in a boundary layer, correlations for mass transfer and heat transfer to particles are similar. For mass transfer to a single isolated sphere,... 

Application of the same analogy between mass transfer and heat transfer to channel shapes other than the square introduces errors in the estimation of the interphase mass transfer coefficients, which range approximately within 20%, depending on the reaction kinetics and on the channel geometry. On the other... 

Quite frequently heat transfer problems have their analogs in mass transfer. This can be fortuitous if recognized, as time and labor can be saved when solving a given problem. One approach that can be very helpful in exposing the similarities between two problems requires the use of dimensionless... 

Jaturonglumlert and Kiatsiriroat (2010) considered heat and mass transfer for the combined convective and FIR drying of fruit leather, and found that the ratio between the heat and mass transfer coefficients for the combination technique could not be obtained from the heat-mass transfer analogy. Hence, modified correlations for predicting the ratio of heat and mass transfer coefficients in term of the heat transfer Nusselt number were developed. 

The heat- mass-transfer analogy wiU be used to estimate the mass-transfer coefficient (Jr,... 

Since the liquid is pure water, it has no mass-transfer coefficient. However, for such processes (see Chap, 7), we need convecbon heat-transfer coefficients for both gas and liquid. These can be estimated, in the absence of directly applicable data, through the heat- mass-transfer analogy. Thus, from Eq. (6.70), and assuming/ - jfi,... 

In order to use Eq. (7.26) for determination of Y, it is necessary to have at hand appropriate values of hQ/ky, thepsychrometric ratio.% Values of and ky can be estimated independently for the particular shape of the wetted surface by correlations like those of Table 3.3, using the heat- mass-transfer analogy if necessary. Alternatively, experimental values of the ratio can be employed. Henry and Epstein [10] have critically examined the data and methods of... 

Mass-transfer coefficients must be those associated with die individual liquid and gas phases overall coefficients wilt not serve. The correlations of Chap. 6 wiU provide these for Berl saddles and Raschig rings, and additional data are summarized in Ref. 14. The Ic-type coefficients are not suitable because in some cases the transfer is opposite the concentration gradient, but k can be converted into F s (Table 3.1). Similarly data in the form of /// can be converted into Fa [Eqs. (8.24) and (8.3l)j. Heat-transfer coefficients can be estimated, if not otherwise available, through the heat- mass-transfer analogy. For the gas, the correction for mass transfer [Eq. (3.70)] provides... 

Grbber [33] computed the approach to equilibrium temperature of spherical particles immersed in an agitated fluid of constant temperature. Since the concentration of the solution in a well-stirred vessel in continuous flow is essentially uniform throughout at the effluent value, Grober s result can be adapted to cocurrent adsorption through the heat- mass-transfer analogy, as in Fig. 11.27. Here the approach of the particles to equilibrium concentration with the effluent liquid is expressed as the Muiphree stage efficiency,... 

Another concept sometimes used as a basis for comparison and correlation of mass transfer data in columns is the Clulton-Colbum analogy (35). This semi-empirical relationship was developed for correlating mass- and heat-transfer data in pipes and is based on the turbulent boundary layer model... 

Below about 0.5 K, the interactions between He and He in the superfluid Hquid phase becomes very small, and in many ways the He component behaves as a mechanical vacuum to the diffusional motion of He atoms. If He is added to the normal phase or removed from the superfluid phase, equiHbrium is restored by the transfer of He from a concentrated phase to a dilute phase. The effective He density is thereby decreased producing a heat-absorbing expansion analogous to the evaporation of He. The He density in the superfluid phase, and hence its mass-transfer rate, is much greater than that in He vapor at these low temperatures. Thus, the pseudoevaporative cooling effect can be sustained at practical rates down to very low temperatures in heHum-dilution refrigerators (72). 

To model convection drying both the heat transfer to the coated web and the mass transfer (qv) from the coatiag must be considered. The heat-transfer coefficient can be taken as proportional to the 0.78 power of the air velocity or to the 0.39 power of the pressure difference between the air in the plenum and the ambient pressure at the coatiag. The improvement in heat-transfer coefficients in dryers since the 1900s is shown in Figure 20. The mass-transfer coefficient for solvent to the air stream is proportional to the heat-transfer coefficient and is related to it by the Clulton-Colbum analogy... 

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... 

The analogy between heat and mass transfer holds over wider ranges than the analogy between mass and momentum transfer. Good heat transfer data (without radiation) can often be used to predict mass-transfer coefficients. 

On occasion one will find that heat-transfer-rate data are available for a system in which mass-transfer-rate data are not readily available. The Chilton-Colburn analogy provides a procedure for developing estimates of the mass-transfer rates based on heat-transfer data. Extrapolation of experimental or Jh data obtained with gases to predict hquid systems (and vice versa) should be approached with caution, however. When pressure-drop or friction-factor data are available, one may be able to place an upper bound on the rates of heat and mass transfer, according to Eq. (5-308). 

For systems other than air-water vapor, the value of h /k c, may differ appreciably from unity, and the wet-bulb and adiabatic-saturation temperatures are no longer equal. For these systems the psychrometric ratio may be obtained by determining h /k from heat- and mass-transfer an ogies such as the Chilton-Colburn analogy [Ind. Eng. Chem., 26, 1183 (1934)]. For low humidities this analogy gives... 

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