Graetz-Nusselt Problem

It has been demonstrated that kg can be estimated by analogy with the Graetz-Nusselt problem governing heat transfer to a fiuid in a duct with constant wall temperature (SH= Nut) [30] and that the axial concentration profiles of NO and of N H 3 provided by the 1D model are equivalent and almost superimposed with those of a rigorous multidimensional model of the SCR monolith reactor in the case of square channels and of ER kinetics, which must be introduced to comply with industrial conditions for steady-state applications characterized by substoichiometric NH3 NO feed ratio, that is, a[Pg.401]

Wissler, E. H. and Schechter, R. S. 1959. The Graetz-Nusselt problem with extension for a Bingham plastic. Chem. Eng. Prog. Symp. Sen 29-34. 

This problem was first dealt with by Graetz (1850-1891) in 1883 [3.27], later in 1910 by Nusselt (1882-1957) [3.28] and by many other authors. It is also known as a Graetz or Graetz-Nusselt problem. It is described by the energy equation (3.232), in which, according to the suppositions made, the radial velocity component disappears, wr = 0, and the axial velocity is that of a Hagen-Poiseuille flow (3.223). With that the energy equation becomes... 

Heat transfer on Walls With Uniform Wall Temperature. Heat transfer in a duct with uniform wall temperature is known as the Graetz or Graetz-Nusselt problem. In this case, a fluid with a fully developed velocity distribution (Eq. 5.15) and a uniform temperature flows into the entrance, and the fluid axial conduction, viscous dissipation flow work, and energy resources are negligible. Graetz [14] and Nusselt [15] solved this problem as follows ... 

In many situations, solution for the Graetz-Nusselt problem is closely approximated by the "film model". In this model, radial temperature and concentration profiles are assumed to be uniform in the bulk fluid, whereas heat and mass transfer resistances are assumed to be close to the solid surface. Surface and bulk fluid properties being different, the specific radial mass and heat fluxes are given respectively by... 

The Graetz problem, also known as the Graetz-Nusselt problem, was originally formulated for the corresponding heat transfer case, which is represented by the PDE... 

The closed-form solution for the advection-dispersion equation expressed in dimensionless group that is referred to as the Graetz-Nusselt problem solution is given as... 

As a consequence of the calibration of the Graetz-Nusselt problem to give exactly the same value of the initial Sherwood number, both models behave in A similar way. There is a slight difference in the dynamics of the dissolution as the spherical blobs model by Powers et al. (1994) gives stronger TCE flux than the Graetz-Nusselt model. However, the complete dissolution time of the residual TCE content is the same for both models as it is shown in Figure 15.10. 

In this section we develop two such theories. The first is for mass transfer out of a solution in laminar flow in a short tube. This Graetz-Nusselt problem finds some application for blood oxygenators and artificial kidneys. The second example is the... 

Fig. 9.4-1. The Graetz-Nusselt problem. In this case, a pure solvent flowing laminarly in a cylindrical tube suddenly enters a section where the tube s walls are dissolving. The problem is to calculate the wall s dissolution rate and hence the mass transfer coefficient. The problem s solutions, based on analogies with heat transfer, are useful for designing artificial kidneys and blood oxygenators.

When will the Leveque limit of the Graetz-Nusselt problem (Equation 9.4-15) be valid ... 

The Nusselt problem is similar to the Graetz problem studied in Example 10.3, except the velocity profile is plug shaped. Starting with the transport equation... 

However, in the specific case of honeycomb catalysts with square channels, which is most frequent in SCR applications, the latter dependence is practically negligible, and an excellent estimate of the local Sherwood number, Sh, is provided by the Nusselt number from solution of the Graetz-Nusselt (thermal) problem with constant wall temperature, Nut, which is available in the heat transfer literature (113). The following correlation was proposed, accounting also for development of the laminar velocity profile ... 

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... 

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... 

Assuming an initially flat velocity profile, calculate and plot the Nusselt number as a function of the inverse Graetz number. Compare the soultions for a range of Prandtl numbers. Explain why the Graetz number may not be an appropriate scaling for the combined entry-length problem. 

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. 

Starting from the classical work by Graetz and Nusselt [157, 319], many authors considered the problem about the temperature distribution in a fluid moving in a tube under various assumptions on the type of flow, the tube shape, the form of boundary conditions, the value of the Peclet number, and some other simplifications (e.g., see [31, 70, 80, 108, 185,406]). In this section, we outline the most important results obtained in this field. 

In Fig. 3, the variation of local Nusselt number along the constant wall temperature tube is presented as a function of Peclet number, representing axial conduction in the fluid. For Pe = 50, which represents a case with negligible axial conduction, the solution of the classical Graetz problem, Nu = 3.66, is reached [44], while for Pe = 1, Nu = 4.03 [45] is obtained as the fully developed values of Nu. The temperature gradient at the wall decreases at low Pe values, thus the local and fully developed Nu values increase with decreasing Pe. 

H. J. Hickman, An Asymptotic Study of the Nusselt-Graetz Problem, Part 1 Large x Behavior, J. Heat Transfer, (96) 354-358,1974. 

The above problem is known as Graetz problem and can be solved as an eigenvalue problem. A similar problem is also seen for thermally developing heat transfer. The solution can be obtained for local Nusselt number (Nu(x)). 

---