The Graetz Problem

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, 

The thermal energy equation for the parallel-flow in a duct reduces to 

The fluid temperature entering the heated section is To and the heated duct-wall temperature is fixed at Tw. It is a bit unusual to use different scale factors for the two spatial coordinates r and z. The reason for doing so here is to facilitate a boundary-layer argument 

Assuming constant properties, the nondimensional energy equation is 

In this equation the Reynolds and Prandtl numbers are defined in the usual way as 

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

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,... 

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

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. 

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. 

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. 

Mikhailov M. D. and R. M. Cotta, 1997, Eigenvalues for the Graetz Problem in Slip-Flow, Int. Comm. Heat... 

For the uniform temperature boundary eondition in a cylindrical charmel, the fully developed Nusselt number decreases as Kn increases. For the no-slip condition Nu =3.6751, while it drops down to 2.3667 for Kn = 0.12, which is a decrease of 35.6 %. This decrease is due to the fact that the temperature jump reduces heat transfer. As Kn increases, the temperature jump also increases. Therefore, the denominator of Eq. (5.14) takes larger values. Similar results were found by [18]. They report approximately a 32 % decrease. However, [20] extended the Graetz problem to slip flow, where they find an increase in the Nusselt number for the same conditions without considering the temperature jump. We can see the same trend in the other two cases of constant wall heat flux for cylindrical and rectangular geometries. 

The separation of variables is a common technique used to solve linear PDEs. This technique will be discussed in detail in chapter 7. This technique yields ordinary differential equations for the eigenfunctions. In this section, we will present two numerical techniques for the Graetz problem. 

The Graetz problem (heat or mass transfer) in cylindrical coordinates with parabolic velocity profile is solved here. The governing equation for the eigenfunction is [15] [8]... 

In section 5.1.4, the Graetz problem was solved using the semianalytical technique. The solution obtained is numerical in x and analytical in z. The solution is obtained as a function of the Peclet number. The solution obtained compares well with the analytical solution reported in the literature. Our technique avoids calculation of special functions and at the same time provides solutions explicit in the Peclet number. In section 5.1.5, the semianalytical technique developed earlier was extended to the case when the initial condition is a function of x. 

Consider the Graetz problem discussed in example 5.6. The same problem in planar geometry is ... 

Consider the Graetz problem discussed in problem 11 of chapter 5.1. Solve this problem using numerical method of lines for Pe = 1,10 and 20. 

As an aside, it is worth mentioning, that the technique described earlier can also be used for solving partial differential equations in cylindrical coordinates. For example, consider the Graetz problem, [1]... 

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