Graetz equation

The Graetz equation arises in the analysis of heat transfer to fluids in laminar flow... 

Circular Tubes For horizontal tubes and constant wall temperature, several relationships are available, depending on the Graetz number. For 0.1 < Ngz < 10 Hausens [A//g. Waermetech., 9, 75 (1959)], the following equation is recommended. 

For flow inside helical coils, Reynolds number less than 10,000, substitute the term DJDjY for (L/Di) where the latter appears in the apphcable equation for straight tubes (frequently as part of the Graetz number). 

The calculation of heat transfer film coefficients in an air-lift bioreactor is more complex, as small reactors may operate under laminar flow conditions whereas large-scale vessels operate under turbulent flow conditions. It has been found that under laminar flow conditions, the fermentation broths show non-Newtonian behaviour, so the heat transfer coefficient can be evaluated with a modified form of the equation known as the Graetz-Leveque equation 9... 

Gas, cells, 464, 477, 511 characteristic equation, 131, 239 constant, 133, 134 density, 133 entropy, 149 equilibrium, 324, 353, 355, 497 free energy, 151 ideal, 135, 139, 145 inert, 326 kinetic theory 515 mixtures, 263, 325 molecular weight, 157 potential, 151 temperature, 140 velocity of sound in, 146 Generalised co-ordinates, 107 Gibbs s adsorption formula, 436 criteria of equilibrium and stability, 93, 101 dissociation formula, 340, 499 Helmholtz equation, 456, 460, 476 Kono-walow rule, 384, 416 model, 240 paradox, 274 phase rule, 169, 388 theorem, 220. Graetz vapour-pressure equation, 191... 

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... 

This is a linear equation whose solution can be determined by the method of separation of variables. Indeed, this is what Graetz did, and the details can be found in several fluid-mechanics texts. Here we will use a relatively simple implicit finite-difference technique to determine the solution in a spreadsheet. 

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 ... 

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). 

For known values of the parameters in the kinetic equation for a specific reactive mix, it is easy to calculate the dimensionless factors y and v. Then the flow pattern in the mold filling process is completely determined by the dimensionless Da and Gz Numbers and the boundary conditions. The Damkohler Number characterizes the ratio of the rates of chemical reaction and convective heat transfer and the Graetz Number is a measure of the ratio of the convective heat flux due to a moving liquid to the heat flux due to the conductivity of the liquid. 

By assuming van der Waals s equation for the vapour, Graetz found ... 

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 ... 

FIGURE 23.12 Variation of Sherwood number with Graetz number. Results are given for Newtonian and non-Newtonian blood analogue fluids and bovine blood for flat sheet BOs. The solid line depicts Equation 23.6. The dashed line gives the predictions of Equation 23.9. (From Wickramasinghe, S.R., Han, B., Garcia, J.D., and Specht, R., AIChE J., 51, 656, 2005. With permission.)... 

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... 

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]... 

Consider a variant of Graetz problem discussed in examples 3.2.14 and 3.2.15 (Villadsen and Michelsen, 1978). The governing equation and boundary conditions are ... 

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]... 

Consider the Graetz problem in planar geometry (chapter 5.1, exercise problem 11). The governing equations and boundary/initial conditions are ... 

Consider the Graetz problem with axial conduction.[8] [7] The governing equation is ... 

The equation (3-202) with conditions (3-203), (3-204), and (3-212) is known as the Graetz problem. The condition (3-212) does not make sense in the present context. Nevertheless, this equation is just the well-known heat equation with a variable coefficient, and there is a long history of working on an exact solution of the Graetz problem. 

Thus, application of the lumping procedure to the original formulation leads to the extended Graetz problem described by the equation and boundary conditions below ... 

---