Analogies Between Heat and Mass Transfer

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. 

The mass transfer coefficients, Kg and Ky, are overall coefficients analogous to an overall heat transfer coefficient, but the analogy between heat and mass transfer breaks down for mass transfer across a phase boundary. Temperature has a common measure, so that thermal equilibrium is reached when the two phases have the same temperature. Compositional equilibrium is achieved at different values for the phase compositions. The equilibrium concentrations are related, not by equality, as for temperature, but by proportionality through an equilibrium relationship. This proportionality constant can be the Henry s law constant Kh, but there is no guarantee that Henry s law will apply over the necessary concentration range. More generally, Kyy is a function of composition and temperature that serves as a (local) proportionality constant between the gas- and liquid-phase concentrations. 

The analogies between heat and mass transfer are reflected in the equations used to describe them. Thermal conduction is described by Fourier s law, which in one dimension is... 

Equation (1-47) is identical in form to the species continuity equation, Eq. (1 -38), and this leads to close analogies between heat and mass transfer as discussed in the next section. 

The formal analogy between heat and mass transfer under the conditions of no dissipation, low mass flux and constant properties can be completed as follows. Equations (1-38) and (1-39) and the boundary conditions considered in this book apply to heat transfer if one replaces c by pCfT, by p C T, ... 

Accuracy of an Analogy Between Heat and Mass Transfer and the Lewis Correlation for Conditioners and Cooling Towers... 

The Stefan convection, however, does not alter the analogy between heat and mass transfer because the laws governing the change in the Nusselt number with the Stefan flow are identical to those governing Shs. 

The analogy between heat and mass transfer makes it possible to obtain the solutions of many mass-transfer problems at low mass-transfer rates from the results of heat transfer in similar situations. 

Understand Ihe concenralion gradient and the physical mechanism of mass transfer, B Recognize the analogy between heat and mass transfer,... 

The analogy between heat and mass transfer also applies to cylindrical and spherical geometries. Repeating the approach outlined in ( hapter 3 for heat conduction, we obtain the following analogous relations for steady onedimensional mass transfer through nonrcacting cylindrical and spherical layers (Fig. 14—21)... 

Consider a circular pipe of inner diameter D = 0.015 m whose inner surface is I covered with a layer of liquid water as a result of condensation (Fig. 14-49). In I order to dry the pipe, air at 300 K and 1 atm is forced to flov/ through if with an average velocity of 1.2 m/s. Using the analogy between heat and mass transfer, determine the mass transfer coefficient inside the pipe for fully developed flov/. 

SOLUTION The liquid layer on the inner surface of a circular pipe is dried by blowing air through it, The mass transfer coefficient is to be determined. Assumptions 1 The low mass flux model and thus the analogy between heat and mass transfer is applicable since the mass fraction of vapor in the air is low (about 2 percent for saturated air at 300 K). 2 The flow is fully developed, Properties Because of lov/ mass flux conditions, v/e can use dry air properties for the mixture at the specified temperature of 300 K and l atm, for which v = 1.58 X 10 mVs (Table A-15). The mass diffusivity of water vapor in the air at 300 K is determined from Eq. 14-15 to be... 

Which is less than 2300 and thus the flow is laminar. Therefore, based on the analogy between heat and mass transfer, the Nusselt and the Sherv/ood numbers in this case are Nu = Sh = 3.66, Using the definition of Sherwood number, itie mass transfer coefficient is determined to be... 

EXAMPLE 14-11 Analogy between Heat and Mass Transfer... 

Assumptions 1 The low mass flux conditions exist so that the Chilton-Colburn analogy between heat and mass transfer is applicable (v/ill be verified). 2 Both air and naphthalene vapor are ideal gases. 

Using Ifie analogy between heat and mass transfer, the average heat transfer coefficient is determined from Eq. 14-89 to be... 

Wilhelm Nusselt (1882-1957) was nominated Professor of Theoretical Mechanical Engineering at the Technische Hochschule, Karlsruhe in 1920. Between 1925 and 1952 he taught at the Technische Hochschule, Munich. In 1915 he published his fundamental work The Fundamental Laws of Heat Transfer , in which he introduced dimensionless groups for the first time. Further important investigations included heat transfer in film condensation, cross current heat transfer and the analogy between heat and mass transfer in evaporation. 

The engineering approach to the mass-transfer suggests full analogy between heat and mass-transfer when making use of dimensionless hydrodynamic and transport criteria. For our problems, and with the numerical subscripts to indicate the appropriate standard components of our chemical systems, we have ... 

The ratio (h/M Ay)> termed the psychrometric ratio, lies between 0.96 and 1.005 for air-water vapor mixtures thus it is nearly equal to the value of humid heat c,. If the effect of humidity is neglected, the adiabatic saturation and wet-bulb temperatures and T, respectively) are almost equal for the air-water system. Note, however, that and are conceptually quite different. The adiabatic saturation temperature is a gas temperature and a thermodynamic entity while the wet-bulb temperature is a heat and mass transfer rate-based entity and refers to the temperature of the liquid phase. Under constant drying conditions, the surface of the drying material attains the wet-bulb temperature if the heat transfer is by pure convection. The wet-bulb temperature is independent of surface geometry as a result of the analogy between heat and mass transfer. 

This equation can also be used for gases from 79, = 1 to = lO, but it gives lower values of the Nusselt number than Fig. 12.6 at higher Reynolds numbers. Equation (12.69) is plotted in /-factor form in Fig. 21.4, in the section of Chap. 21 dealing with the analogies between heat and mass transfer. 

The various forms of the penetration theory can be classified as surface-renewal models, implying either formation of new surfaee at frequent intervals or replacement of fluid elements at the surface with fresh fluid from the bulk. The time or its reciprocal, the average rate of renewal, are functions of the fluid velocity, the fluid properties, the the geometry of the system and can be accurately predicted in only a few special cases. However, even if tj must be determined empirically, the surface-renewal models give a sound basis for correlation of mass-transfer data in many situations, particularly for transfer to drops and bubbles. The similarity between Eqs. (21.44) and (15,20) is an example of the close analogy between heat and mass transfer. It is often reasonable to assume that tj-is the same for both processes and thus to estimate rates of heat transfer from measured mass-transfer rates or vice versa. 

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. 

Rose [47, 48] has suggested an approximate solution for forced convection condensation along a flat plate in the presence of noncondensable gas using an analogy between heat and mass transfer. He points out that, in this situation, the diffusion problem for the vapor-gas... 

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