Graetz

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... 

GCp/kl the Graetz group Gz. and h/Cppu the Stanton group, St — Nu/(RePr)... 

Convective heat transfer to fluid inside circular tubes depends on three dimensionless groups the Reynolds number. Re = pdtu/ii, the Prandtl number, Pr = Cpiilk where k is the thermal conductivity, and the length-to-diameter ratio, L/D. These groups can be combined into the Graetz number, Gz = RePr4/L. The most commonly used correlations for the inside heat transfer coefficient are... 

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... 

Hanselman TA, Graetz DA, Wilkie AC (2003) Manure-borne estrogens as potential environmental contaminants a review. Environ Sci Technol 37 5471-5478... 

J. Graetz, Y. Lee, J.J. Reilly, S. Park, T. Vogt, T. Structures and thermodynamics of the mixed alkali alanates. Physical Review B Condensed Matter and Materials Physics, 71(18), 184115/1-184115/7 (2005). 

G. Sandrock, J. Reilly, J. Graetz, W.-M. Zhou, J. Johnson, J. Wegrzyn, Accelerated thermal decomposition of AlH for hydrogen-fueled vehicles, Appl. Phys. A 80 (2005) 687-690. 

Graetz et al. [188-190] studied the decomposition of two other polymorphs P-and y-AlHj in comparison to the a-polymorph, aU of which were freshly made by the organometaUic method developed by Brower et al. [180], Table 2.21 compiles thermodynamic and kinetic data extracted from their papers. It was found that at temperatures >1(X)°C, decomposition of P- and y-Aftf occurs with the initial exothermic phase transformation of P- and y-AUf — a-Aftf, which subsequently decomposes in the... 

Finally, it must be mentioned that Graetz and Reilly [188] clearly pointed out that the conventional organometalhc synthesis of AIH3 is a costly procedure. This factor, together with its on-board irreversibility, calls for novel, very cost-effective, and energetically efficient techniques to regenerate AIH3 from spent Al powder. 

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