Real flow coefficient

For the ideal case of frictionless flow, every term in the brackets is zero, resulting in a coefficient of unity. The same result may be produced in a real flow if the sum of all the terms in the brackets is zero. The friction hf is always positive, but in converging flow the sum of the other two terms can be negative. 

An alternative model for real flows is the dispersion model with the model parameters Bodenstein number (Bo) and mean residence time t, The Bodenstein number which is defined as Bo = uL/D characterises the degree of backmixing during flow. The parameter D is called the axial dispersion coefficient, u is a velocity and L a length. The RTD of the dispersed plug flow model ranges from PFR at one extreme (Bo = °) to PSR at the other (Bo = 0). The transfer function for the dispersion model with closed-closed boundaries is [10] ... 

Mass-transfer coefficients of solute species through the membrane may be calculated, using Eqs (4) and (5). Ds- efFective diffusion coefficients of solute species in the feed, carrier, and strip solutions are evaluated by extrapolating the plots of ti =f(U) to U Q. The magnitude ofDs is far from the real diffusion coefficient of solute complexes in liquids, because of some assumptions, mentioned above. Equation (20) at U 0 becomes undefined, however, for calculation mass-transfer coefficients of solutes at visible flow rates, this parameter (coefficient) is quite applicable. The reduction in the area for diffusion by the impregnable sections of the microporous membrane is accounted for by... 

The above relations describe frictionless flow processes. However, the outflow from safety valves and bursting discs is accompanied by friction losses. These are accounted for by the discharge coefiicient K. This coefficient is determined experimentally in the context of the certification process. It represents the ratio between the ideal and real flow velocities. 

Calculation of Flow Coefficient Accounting for Real Gas Effects... 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

It is shown in Section 9.9.5 that, with the existence of various bypass and leakage streams in practical heat exchangers, the flow patterns of the shell-side fluid, as shown in Figure 9.79, are complex in the extreme and far removed from the idealised cross-flow situation discussed in Section 9.4.4. One simple way of using the equations for cross-flow presented in Section 9.4.4, however, is to multiply the shell-side coefficient obtained from these equations by the factor 0.6 in order to obtain at least an estimate of the shell-side coefficient in a practical situation. The pioneering work of Kern(28) and DoNOHUE(lll who used correlations based on the total stream flow and empirical methods to allow for the performance of real exchangers compared with that for cross-flow over ideal tube banks, went much further and. 

The complex flow pattern on the shell-side, and the great number of variables involved, make it difficult to predict the shell-side coefficient and pressure drop with complete assurance. In methods used for the design of exchangers prior to about 1960 no attempt was made to account for the leakage and bypass streams. Correlations were based on the total stream flow, and empirical methods were used to account for the performance of real exchangers compared with that for cross flow over ideal tube banks. Typical of these bulk-flow methods are those of Kern (1950) and Donohue (1955). Reliable predictions can only be achieved by comprehensive analysis of the contribution to heat transfer and pressure drop made by the individual streams shown in Figure 12.26. Tinker (1951, 1958) published the first detailed stream-analysis method for predicting shell-side heat-transfer coefficients and pressure drop, and the methods subsequently developed... 

Important issues in groundwater model validation are the estimation of the aquifer physical properties, the estimation of the pollutant diffusion and decay coefficient. The aquifer properties are obtained via flow model calibration (i.e., parameter estimation see Bear, 20), and by employing various mathematical techniques such as kriging. The other parameters are obtained by comparing model output (i.e., predicted concentrations) to field measurements a quite difficult task, because clear contaminant plume shapes do not always exist in real life. 

Both contributions to the current obey the Butler-Volmer law. The current flowing through the conduction band has a vanishing anodic transfer coefficient, ac = 0, and a cathodic coefficient of unity, /3C — 1. Conversely, the current through the valence band has av — 1 and j3v = 0. Real systems do not always show this perfect behavior. There can be various reasons for this we list a few of the more common ones ... 

The net area of this intimate contact is called the real area of contact Areai. It is assumed that plastic flow occurs at most microscopic points of contact, so that the normal, local pressures correspond to the hardness aj, of the softer of the two materials that are in contact. The (maximum) shear pressure is given by the yield strength cry of the same material. The net load L and the net shear force Fs follow by integrating aj, and cry over the real area of contact Areai. That is, L = cs, Arca and Fs = ayAreai. Hence, the plastic deformation scenario results in the following (static) friction coefficient ... 

Lactic acid fermentation was the topic of a paper by Vaccari et al.35 In this work, lactic acid, glucose, and biomass were determined over the course of the reaction. The measurements were made in real time, using a bypass pump and flow-through cell for the NIR measurements. Instead of using normal chemomet-ric statistics, the authors used correlation coefficients, mean of differences, standard deviation, student s t-test, and the student test parameter of significant difference to evaluate the results. Under these restrictions, the results appeared fairly good, with the biomass having the best set of statistics. 

The mathematical method of Aris assumes a doubly infinite pipe (as does Taylor), with both the velocity distribution and the diffusion coefficients constant in the direction of flow. Hence in any real pipe, the length would have to be long enough so that the buildup of the velocity profile at the entrance would not invalidate the doubly infinite pipe assumption. Thus there are some practical restrictions on the method used by Aris. 

The modeling of real immobilized-enzyme column reactors, mainly the fluidized-bed type, has been described (Emeiy and Cardoso, 1978 Allen, Charles and Coughlin, 1979 Kobayashi and Moo-Young, 1971) by mathematical models based on the dispersion concept (Levenspiel, 1972), by incorporation of an additional term to account for back-mixing in the ideal plug-flow reactor. This term describes the non-ideal effects in terms of a dispersion coefficient. 

The fact that the velocity of a fluid changes from layer to layer is evidence of a kind of friction between these layers. The layers are mathematical constructs, but the velocity gradient is real and a characteristic of the fluid. The property of a fluid that describes the internal friction or resistance to flow is the viscosity of the material. Chapter 4 is devoted to a discussion of the measurement and interpretation of viscosity. For now, it is enough for us to recall that this property is quantified by the coefficient of viscosity 77 of a material. The coefficient of viscosity has dimensions of mass length-1 time-1, kg m-ls-1 in SI units. In actual practice, the cgs unit of viscosity, the poise (P), is widely used. Note that pure water at 20°C has a viscosity of about 0.01 P = 10-3kgm-ls-1... 

Reference type of ww type of treatment system no, and type of ozone reactors (operating pressure) ozone production capacity nominal// real liquid flow-rate ozone yield coefficient Y(OJM) (M = COD) investment ozonation stage only specific costs (without annuity) remarks... 

Which experimental method should be used depends on the type of reactor and how it will be operated, and if clean or process water is to be used for the measurement. Nonsteady state methods are generally simpler and faster to perform if kLa is to be determined in clean water without reaction. For processes that are operated at steady state with a reaction, determination of kLa using steady state methods are preferred, since continuous-flow processes need not be interrupted and operating conditions similar to the normal process conditions can be used. This is especially important for systems with reactions because the reaction rate is usually dependent on the concentration of the reactants present. They are thus often applied for investigations of the mass transfer coefficient under real process conditions with chemical reactions kLa(02) or biological activity kLa(02), e. g. in waste water treatment systems. 

Real experiments for the determination of external mass transfer coefficients are used as an example for virtual experiments with CFD. Here experimental studies (Williamson et al., 1963 Wilson and Geankopolis, 1966) on the flow of two liquids, namely water and a propylene glycol-water mixture, through a packed bed of spherical particles made from solid benzoic acid are... 

It may be recalled that it was deduced from the similarity solution for flow over a flat plate that Six) = 5/jRex. The difference between the value of the coefficient in this equation, i.e., 5, and the value in Eq. (3.136), i.e., 4.64, has no real significance since, in deriving the similarity solution result, it was arbitrarily assumed that the boundary layer thickness was the distance from the wall at which u became equal to 0.99 m. 

The microanalytical methods of differential thermal analysis, differential scanning calorimetry, accelerating rate calorimetry, and thermomechanical analysis provide important information about chemical kinetics and thermodynamics but do not provide information about large-scale effects. Although a number of techniques are available for kinetics and heat-of-reaction analysis, a major advantage to heat flow calorimetry is that it better simulates the effects of real process conditions, such as degree of mixing or heat transfer coefficients. 

In the middle of cells and in faces that are perpendicular to the flowing direction, the borders are branched, which means that the effective number of borders, equivalent to that in a real system, is different from five. The number of independent borders with constant by height radius and length L can be determined by the electro-hydrodynamic analogy between current intensity and liquid flow rate through borders, both being directly proportional to the cross-sectional areas [6,35]. This analogy indicates that the proportionality coefficients (structural coefficients B = 3) in the dependences border hydroconductivity vs. foam expansion ratio and foam electrical conductivity vs. foam expansion ratio, are identical [10]. From the electrical conductivity data about foam expansion ratio it follows... 

For different systems, we have different signs of the real and imaginary part of Landau coefficient /. Here, we will keep our attention focused to flow past a circular cylinder, that works as a prototypical model for bluff-body flow instability. This instability begins as a linear temporal instability and its first appearance with respect to the Reynolds number is referred to as Hopf bifurcation. Thus, the Reynolds number at which the first bifurcation occurs is given by Rccr- Thus, above Rccr the value of <7 > 0 signifies linear instability. One of the most important aspect of this linear instability is the subsequent non-linear saturation that can be adequately explained by the Landau s equation, if only R is positive. We will focus upon this type of flow only in the next. 

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