Graetz problem

FIGURE 8.7 Numerical versus analytical solutions to the Graetz problem with ari/R = 0.4. 

Solution A transformation to dimensionless temperatures can be useful to generalize results when physical properties are constant, and particularly when the reaction term is missing. The problem at hand is the classic Graetz problem and lends itself perfectly to the use of a dimensionless temperature. Equation (8.52) becomes... 

A basic element of the thermal dynamics of the DPF is the heat transfer between the gas in the channel and the porous wall. In case of a porous wall having small wall thermal Peclet number PeT (as is always the case for a DPF as shown by Bissett and Shadman (1985)) the problem degenerates to the following modified Graetz problem ... 

The function A stands for the modification to the usual Nusselt number for non-porous wall (the so-called Graetz problem) due to the suction-distorted... 

The Graetz problem considers the thermal entry of an incompressible fluid in a circular pipe with a fixed velocity profile. The situation is illustrated in Fig. 4.16. The Graetz problem is a classic problem in fluid mechanics, and one that permits an analytic solution. After some hydrodynamic entry length, the velocity profile approaches a steady profile that is,... 

Fig. 4.16 Illustration of the Graetz problem. A fully developed parabolic velocity profile is established in a circular duct and remains unchanged over the length of the duct. There is a sudden jump in the wall temperature, and the fluid temperature is initially uniform at the upstream wall temperature. The thermal-entry problem is to determine the behavior of the temperature profile as it changes to be uniform at the downstream wall temperature. Because the flow is incompressible, the velocity distribution does not depend on the varying temperatures.

A numerical solution procedure is reasonably flexible in accommodating variations of problems. For example, the Graetz problem could be solved easily for velocity profiles other than the parabolic one. Also variable properties can be incorporated easily. Either of these alternatives could easily frustrate a purely analytical approach. The Graetz problem can also be worked for noncircular duct cross sections, as long as the velocity distribution can be determined as outlined in Section 4.4. 

For the purposes of this exercise, assume that Sc = 0.85. The mass fraction is already nondimensional. However, it will be important to create a normalized mass fraction based on AT, which is the difference between the inlet CO mass fraction and the surface value. Use an analogy with the nondimensional temperature in the Graetz problem. 

By analogy with the Graetz problem for the Nusselt number, determine quantitatively the behavior of the Sherwood number as a function of z, the axial distance from the start of the catalytic section... 

In the spirit of the Graetz problem (i.e., impose a parabolic velocity profile) develop a nondimensional form of the species-continuity equation. Use the following scale factors and dimensionless variables ... 

Develop and discuss a set of boundary conditions to solve the Graetz problem. Take particular care with the effects of surface reaction, balancing heterogeneous reaction with mass diffusion from the fluid. A second Damkohler number should emerge in the surface boundary condition,... 

This is a linear parabolic partial differential equation that can be readily solved as soon as boundary conditions are specified. There is a symmetry condition at the centerline, and it is presumed that the mass fraction Yk vanishes at the wall, Yk = 0. It is important to note that it has been implicitly assumed that the velocity profile has been fully developed, such that the similarity solution / is valid. This assumption is analogous to that used in the Graetz problem (Section 4.10). 

This expression states that the product of the mean mass fraction and the overall mass flow rate must equal the integral over the channel width of the local mass flow rate of species ft. An analogous definition for the energy flow was used to define a mean temperature in the Graetz problem (Section 4.10). In nondimensional terms,... 

This is the classic Graetz problem as discussed in Section 4.10. 

The situation considered in this section is traditionally termed the Graetz problem. 

Notter, R.H. and Sleicher, C.A., A Solution to the Turbulent Graetz Problem m. Fully Developed and Entry Region Heat Transfer Rates , Chem. ne Sci., Vol 27, pp. 2073-2093, 1972. 

Sellars, J. R., M. Tribus, and J. S. Klein Heat Transfer to Laminar Flows in a Round Tube or Flat Conduit The Graetz Problem Extended, Trans. ASME, vol. 78, p. 441, 1956. 

Example 7.4 Modified Graetz problem with coupled heat and mass flows The Graetz problem originally addressed heat transfer to a pure fluid without the axial conduction with various boundary conditions. However, later the Graetz problem was transformed to describe various heat and mass transfer problems, where mostly heat and mass flows are uncoupled. In drying processes, however, some researchers have considered the thermal diffusion flow of moisture caused by a temperature gradient. 

Transport of heat or mass to the wall of a single, circular channel under laminar flow conditions is known as the classical Graetz problem [10]. For heat transport only, the energy equation contains axial convection and radial conduction ... 

J.R. Sellars, M. Tribus and J.S. Klein, Heat transfer to laminar flow to a round tube or flat conduit— The Graetz problem extended, Trans. Am. Soc. Meek Eng. 75 441 (1956). L.P.B.M. Janssen and M.M.C.G. Warmoeskerken, Transport Phenomena Data Companion, Edward Arnold, London, 1987. 

A. Akyurtlu, J.F. Akyurtlu, K.S. Denison, and C.E. Hamrin, Jr., Application of the general purpose, collocation software, PDECOL to the Graetz problem. Comp, Chem. Eng. 70 213 (1986). 

Bar-Cohen, A., State of the Art and Trends in the Thermal Packaging of The Electronic Equipment, ASME Journal of Electronic Packaging, 1992, 114, 257-270. Barron, R.F, Wang, X. Ameel, T.A. and Warrington, R.O., The Graetz Problem Extended to Slip-Flow, Int. J. Heat Mass Transfer, 1997, 40(8), 1817-1823. 

Randall, F.B., Wang, X. and Ameel, T.A, The Graetz Problem Extended to slip flow, Int. J. Heat Mass Transfer, 1997, 40, 1817-1823. 

Heat transfer by forced convection inside micro tube, generally referred as the Graetz problem, has been extended by Barron et al. [11] and Larrode and al. [12] to include the velocity slip described by Maxwell in 1890 [13] and the temperature jump [14] on tube surface, which are important in micro scale at ordinary pressure and in rarefied gases at low-pressure. 

The problem given by eq.(59) subject to the conditions (60) is referred to as extended Graetz problem in honor of the pioneering work [5]. To solve this problem we need the eigenvalues m and the eigenfunctions y [R] of the eigenproblem ... 

The solution of the extended Graetz problem, eqns. (59, 60), is a special case from the solution given by Mikhailov and Ozisik in the book [20] ... 

Barron R. F, X. Wang, R. O. Warrington, and T. Ameel, 1996, Evaluation of the eigenvalues for the Graetz problem in slip-flow, Int. Comm. Heat Mass Transfer 23 (4), 1817-1823. 

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