Algebraic solutions transfer

To illustrate the system behavior, the ternary mixture 1 = iso-propanol, 2 = water, and 3 = air is considered here. In order to obtain an algebraic solution, both the dif-fusivities of iso-propanol in air and iso-propanol in water vapor were assumed to be approximately the same, which is not far from reality. The liquid phase mass transfer resistance was negligibly small, as will be shown below. The phase equilibrium constants K/,c and Kjrs were calculated with activity coefficients from van Laar s equation. Water vapor diffuses 2.7-fold faster in the inert gas air than iso-propanol. The ratio of the respective mass transfer coefficients kj3 equals the ratio of the respective diffusivities to the power of 2/3rd according to standard convective mass transfer equations Sh =J Re, Sc). 

The variations in At and the heat-transfer coefficients must be taken into account when Eqs. (16) and (17) are integrated. Under some conditions, a graphical or stepwise integration may be necessary, but algebraic solutions are possible for many of the situations commonly encountered with heat-transfer equipment. 

An algebraic solution of the SMB can be derived for the ideal linear model, i.e., for the model assuming a linear isotherm and instantaneous equilibrium everywhere in the coluiim, and neglecting axial dispersion and the resistances to mass transfer. Such analytical solutions were derived by Rhee et al. [22,23] and by Zhong et al. [24]. 

The purpose of this chapter is to provide a comprehensive discussion of some simple approaches that can be employed to obtain information on the rate of heat and mass transfer for both laminar and turbulent motion. One approach is based on dimensional scaling and hence ignores the transport equations. Another, while based on the transport equations, does not solve them in the conventional way. Instead, it replaces them by some algebraic expressions, which are obtained by what could be called physical scaling. The constants involved in these expressions are determined by comparison with exact asymptotic solutions. Finally, the turbulent motion is represented as a succession of simple laminar motions. The characteristic length and velocity scales of these laminar motions are determined by dimensional scaling. It is instructive to begin the presentation with an outline of the basic ideas. 

In what follows, the preceding evaluation procedure is employed in a somewhat different mode, the main objective now being to obtain expressions for the heat or mass transfer coefficient in complex situations on the basis of information available for some simpler asymptotic cases. The order-of-magnitude procedure replaces the convective diffusion equation by an algebraic equation whose coefficients are determined from exact solutions available in simpler limiting cases [13,14]. Various cases involving free convection, forced convection, mixed convection, diffusion with reaction, convective diffusion with reaction, turbulent mass transfer with chemical reaction, and unsteady heat transfer are examined to demonstrate the usefulness of this simple approach. There are, of course, cases, such as the one treated earlier, in which the constants cannot be obtained because exact solutions are not available even for simpler limiting cases. In such cases, the procedure is still useful to correlate experimental data if the constants are determined on the basis of those data. 

The general solution of the algebraic equation system obtained by means of the internal [(14.11a) and (14.11b)] conditions withm= 1,2,..., Mandexternal boundary conditions [(14.3a) and (14.3b)] are given in Appendix B. The mass-transfer rate on the upstream side of the membrane can be given, for that case, as follows ... 

Due to the complexity of mass transfer between gas-liquid-solid phases, it is difficult to evaluate the average value of mass transfer coefficient ki from the literature. A realistic way to evaluate ki is to use the algebraic expression of solution and by regression to obtain the experimental data rather than by regression with solving the set of non-linear differential equations. 

To illustrate the use of Equation 2.18 in interpreting osmotic data, we will consider osmotic responses of pea chloroplasts suspended in external solutions of various osmotic pressures. It is customary to plot the volume V versus the reciprocal of the external osmotic pressure, l/n°, so certain algebraic manipulations are needed to express Equation 2.18 in a more convenient form. After transferring r1 — P1 to the left-hand side of Equation 2.18 and then multiplying both sides by VwrCwf II0 — r1 + / ), can be shown to equal RT -n -/(Jl° — r1 +/>1). The measured chloroplast volume V can be... 

The systematic approach described above for solving radiation heal transfer problems is very suitable for use with today s popular equation solvers such as lili.V, Mathcad, and Matlab, especially when there are a large number of surfaces, and is known as the direct melhod (formerly, the matrix method, since it resulted in matrices and the solution required a knowledge of linear algebra). The second method described below, called the network method, is based on Ihe electrical network analogy. 

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