Correlation function, turbulence parameters

Gouesbet, G., Berlemont, A. and Picart, A. (1984), Dispersion of discrete particles by continuous turbulent motions. Extensive discussion of the Tchen s theory, using a two-parameter family of lagrangian correlation functions, Phys. Fluids, 27(4), 827. 

The mass-transfer coefficients depend on complex functions of diffii-sivity, viscosity, density, interfacial tension, and turbulence. Similarly, the mass-transfer area of the droplets depends on complex functions of viscosity, interfacial tension, density difference, extractor geometry, agitation intensity, agitator design, flow rates, and interfacial rag deposits. Only limited success has been achieved in correlating extractor performance with these basic principles. The lumped parameter deals directly with the ultimate design criterion, which is the height of an extraction tower. 

When both phases are in turbulent flow, or when one phase is discontinuous as in bubble flow, it is not presently possible to formulate the proper boundary conditions and to solve the equations of motion. Therefore, numerous experimental studies have been conducted where the holdups and/or the pressure drop were measured and then correlated as a function of the operating conditions and system parameters. One of the most widely used correlations is that of Lockhart and Martinelli (L12), who assumed that the pressure drop in each phase could be calculated from the equations... 

All models for turbulent flows are semiempirical in nature, so it is necessary to rely upon empirical observations (e.g., data) for a quantitative description of friction loss in such flows. For Newtonian fluids in long tubes, we have shown from dimensional analysis that the friction factor should be a unique function of the Reynolds number and the relative roughness of the tube wall. This result has been used to correlate a wide range of measurements for a range of tube sizes, with a variety of fluids, and for a wide range of flow rates in terms of a generalized plot of/ versus /VRe- with e/D as a parameter. This correlation, shown in Fig. 6-4, is called a Moody diagram. 

This correlation (C3) is intended to apply for turbulent-turbulent horizontal flow in pipes, and was developed to give better pressure-drop prediction for higher pressures and larger-diameter pipes. On an entirely empirical basis, the quantity APtp/aPl is given as a function of liquid volume-fraction of the feed, with a quantity >Pq pl/ l po as a parameter. For this correlation aP l is evaluated as the pressure-drop based on the total mass-flow using the liquid-phase properties. The parameter po ph/ Lpo is defined as... 

Qfis may ask the reason for the functional form of Eq. (6-4). Physical reasoning, based on the experience gained with the analyses of Chap. 5, would certainly indicate a dependence of the heat-transfer process on the flow field, and hence on the Reynolds number. The relative rates of diffusion of heat and momentum are related by the Prandtl number, so that the Prandtl nunfber is expected to be a significant parameter in the final solution. We can be rather confident of the dependence of the heat transfer on the Reynolds and Prandtl numbers. But the question arises as to the correct functional form of the relation i.e., would one necessarily expect a product of two exponential functions of the Reynolds and Prandtl numbers The answer is that one might expect this functional form since it appears in the flat-plate analytical solutions of Chap. 5, as well as the Reynolds analogy for turbulent flow. In addition, this type of functional relation is convenient to use in correlating experimental data, as described below. 

In particular, the loose packing of particles promotes an open powder structure that is less adhesive and flows and disperses more readily. There is a strong correlation between the interaction parameters derived by IGC and the in vitro data that play an important role in the prediction of aerosol performance of dry powder inhalation formulations. Enhanced dispersibility is particularly important for DPI devices, where performance strongly depends on powder deaggregation at relatively low dispersion forces. Clearly, high turbulence is preferable for dispersion, but it inevitably leads to high pressure differentials, which may prevent many devices from functioning correctly. In addition, low dispersion forces for supercritically produced... 

Davis, Ouwerkerk and Venkatesh developed a mathematical model to predict the conversion and temperature distribution in the reactor as a function of the gas and liquid flow rates, physical properties, the feed composition of the reactive gas and carrier gas and other parameters of the system. Transverse and axial temperature profiles are calculated for the laminar flow of the liquid phase with co-current flow of a turbulent gas to establish the peak temperatures in the reactor as a function of the numerous parameters of the system. Also in this model, the reaction rate in the liquid film is considered to be controlled by the rate of transport of reactive gas from the turbulent gas mixture to the gas - liquid interface. The predicted reactor characteristics are shown to agree with large-scale reactor performance. For the calculations of the mass transfer coefficient in the gas phase, kg, Davis et al. used the same correlation as Johnson and Crynes, but multiplied the calculated values arbitrarily by a factor 2 to include the effect of ripples on the organic liquid film caused by the high SOj/air velocities in the core of the reactor. 

Fig. 14. The transition from stripes to chemical turbulence as a function of malonic acid concentration, as characterized by the following quantities L, the total length of the calculated grain boundaries v, the average speed of pattern motion correlation length and A, wavelength of the pattern. Other control parameters were held fixed at the values given for Figure 11. (From [10])...

Nu ist referred to the Nusselt number as a dimensionless parameter characterizu both the physical properties of the coolant and dynamic characteristics of the flow. For the turbulant flow in the cooling channels of the core, different empirical correlations for Nu are available. All correlations are expressed as a function of Reynolds number, specifying hydraulic condition and prandtl number Pr, for physical properties of the coolant. 

A simphfied way is to decouple the two approaches first, the stabihty condition serves as the close law for the simphfied algebraic conservation equations, as described by the EMMS model for gas—hquid and gas—solid systems. The nonlinear optimization problem can therefore be solved to obtain the global or local structure parameters which are then be used to derive the closure law or correlations for the drag, bubble-induced turbulence and even the correction factors for the kernel functions of bubble coalescence and breakup for PBEs. 

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