On Effective viscosity models for gas-stirred ladle systems
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DIPAK M A Z U M D A R , Assistant Professor, is with the Department of Metallurgical Engineering, Indian Institute of Technology, Kanpur, UP, 208 016, India. Manuscript submitted April 20, 1988.
[3]
It is readily apparent that this spatially dependent expression for effective viscosity can be reduced to an appropriate bulk average form, once the two fundamental quantities k and e are represented in terms of their respective equivalent average values. Thus, starting with the definition of specific turbulence kinetic energy, k [=0.5(E 2 + 172 + rPz)] and assuming energy-containing eddies to be primarily isotropic, the following expression is obtained: k = -3 E2 2
[41
Further, taking the average fluctuating velocity component, t7, to be proportional to the mean speed of liquid recirculation, U (e.g., E = C1/_7),[10~ a simple expression for k results in terms of the macroscopic variable, U, as 3 k = - c ~ tY 2
2
[51
The turbulence energy dissipation rate, e, can also be deduced once the specific rate of energy input to the gasstirred system is known. Considering an efficiency factor r/tm governing the generation and dissipation of turbulence kinetic energy, e can be readily estimated from the following relationship:
[2]
In Eq. [2], C represents a proportionality constant, which was deduced through comparison of values of Ixe from the above equation and average spatial values of Ixe obtained from an elaborate numerical computation embodying the k - e turbulence model. More recent work [4'9] on the hydrodynamics of gas-stirred melts, however, appears to indicate that the k - e model overestimates the values of various turbulence parameters within such systems. In view of this, it is likely that the numerical value assigned to the empirical constant C (=5.5 • 10-3) 181 is somewhat uncertain and, therefore, needs further consideration. Since estimates of C based solely on theoretical considerations t8j appear to be unreliable and, moreover, since Ixe, the effective viscosity, is not a measurable quantity, starting with Eq. [2] and following the previously taken approach, it is not possible to derive an appropriate value
METALLURGICAL TRANSACTIONS B
Ixe ~- IX, = C ~ p k 2 / e
Ill
In reviewing the applicability of the above equation to typical gas-stirred ladle systems, Sahai and Guthrie rS~ subsequently pointed out several difficulties and ambiguities. Through a detailed consideration of gas-liquid coupling phenomena during gas-injection operation, these authors, ~8~ on the basis of dimensional arguments, proposed a physically more plausible prescription of effective viscosity, e.g., Ixe = CpLL[Q( 1 - a ) g / D ] ~/3
for the constant C. Toward this, it is therefore evident that a fresh approach has be to adopted. Alternative to the procedures adopted I81 in deducing the above relationship (viz., Eq. [2]), one can also proceed to develop an explicit expression starting from a fundamental definition of "effective" or "turbulent" viscosity. Thus, considering k ~/2 as the characteristic velocity scale a n d k3/2/e
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