Heat transfer model solution procedure

Solution of the required column height is achieved by integrating the two component balance equations and the heat balance equation, down the column from the known conditions Xi , yout and TLin, until the condition that either Y is greater than or X is greater than Xqui is achieved. In this solution approach, variations in the overall mass transfer capacity coefficient both with respect to temperature and to concentration, if known, can also be included in the model as required. The solution procedure is illustrated by the simulation example AMMON AB. 

The design of two-phase contactors with heat transfer requires a firm understanding of two-phase hydrodynamics in order to model effectively the heat- and mass-transfer processes. In this chapter we have pointed out areas where further theoretical and experimental research is critically needed. It is hoped that design engineers will be motivated to test the procedures presented, in combination with their use of the details from the original references, in the solution of pragmatic problems. 

The basic discretization of the two-fluid model equations is similar to the approximations of the corresponding transport equations for single phase flow. A minor difference is that the two-fluid model equations contain the novel phase fraction variables that have to be approximated in an appropriate manner. More important, to design robust, stable and accurate solution procedures with appropriate convergence properties for the two-fluid model equations, emphasis must be placed on the treatment of the interface transfer terms in the phasic momentum, heat and mass transport equations. Because of these extra terms, the coupling between the different equations is even more severe for multiphase flows than for single phase flows. 

The hybrid solution procedure described in the previous section is computationally more demanding than one that does not rely on the CFD package to predict the heat transfer from the exhaust gas. In fact, this simpler approach was adopted in die early stages of the project, the heat transfer process was modelled using a mean heat transfer coefficient estimated from correlations for convective heat transfer in annuli. However, it was soon realized that this method has a high degree of uncertainty when the heat transfer process takes place under unsteady-state conditions and when the thermal entry length spreads over an appreciable extent of the domain. These conditions are always met in the application under study. 

The transport equations for laminar motion can be formulated, in general, easily and difficulties may lie only in their solution. On the other hand, for turbulent motion the formulation of the basic equations for the time-averaged local quantities constitutes a major physical difficulty. In recent developments, one considers that turbulence (chaos) is predictable from the time-dependent transport equations. However, this point of view is beyond the scope of the present treatment. For the present, some simple procedures based on physical models and scaling will be employed to obtain useful results concerning turbulent heat or mass transfer. 

In this section the classical heat and mass transfer theories are examined. The singular surface jump conditions for the primitive quantities, as derived in the framework of the standard averaging procedures, are approximated by the classical chemical engineering stagnant film theory normally used in chemical reactor models. The relevant transport phenomena solutions and the classical theories on heat and mass transfer considering both low- and high mass transfer rates are summarized in the subsequent subsections. 

Model Studies. General Procedure. The theoretical amount of sodium or potassium hydroxide (corrected for the amount of water present), an equivalent amount of 4-methylphenol, 0.05 equivalent of the phase transfer catalyst, biphenyl (internal standard, 0.25 equivalents) and methylene chloride or a 3 to 1 mixture of chloro-benzene/methylene chloride were heated at reflux. The amount of solvent used was chosen to give a solution which would be 20% in final product (g of formal/ml of solvent X 100 = 20%). Aliquots were removed at timed intervals and were worked up with methylene chloride and 1.2N HCl. These samples were analyzed by vpc and the yields of the product were calculated, based on an internal standard... 

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