Friction correlations

The most often used friction correlation for open channel flows is due to Manning (Trans. Inst. Civ. Engrs. Ireland, 20, 161 [1891]) and is equivalent to... 

Kim, N.-H, Yun, f.-H., and Webb, R. L. Heat transfer and friction correlations for wavy plate fin-and-tube heat exchangers. Journal of Heat Transfer 119 (1997) August, pp. 560-567,... 

The friction correlations in the turbulent regime are compared against conventional correlations in Fig. 2.20b. Predictions for nitrogen flow by Choi et al. (1991) agree very well with conventional results the Wu and Little (1984) correlation is similar to those two in its trend, but the predicted values are much higher in mag-... 

Fig. 2.19 Laminar flow and friction correlations. Reprinted from Sobhan and Garimella (2001) with permission...

Fig. 2.20 (a) Dependence of the friction factor on Reynolds number for tube of diameter 705 pm. Reprinted from Maynes and Webb (2002) with permission, (b) Turbulent flow friction correlations. Reprinted from Sobhan and Garimella (2001) with permission... 

Additional data are included in the friction correlation of Specchia and Baldi [Chem. Eng. Sci. 32, 515-523 (1977)], which is represented by ... 

R. L. Webb, E. R. G. Eckert, and R. J. Goldstein, Generalized Heat Transfer and Friction Correlations for Tubes With Repeated-Rib Roughness, Int. J. Heat Mass Transfer (15) 180-184,1972. 

HYNES - The force that enters the friction correlation function is the complete force arising from the solvent molecules interacting with the Sj 2 reaction system held fixed at the transition state (zero asymmetric stretch). It includes all contributions from the rotational and translational (as well as intramolecular vibrational and librational) motions of the H2O solvent molecules. 

Currently the standard TRACE code heat transfer (Dittus-Boelter) and fluid pressure drop (Churchill and Moody) correlations are applied to the gas cooler. Use of the Churchill correlation and Moody curves, and mathematical representations of the curves, for calculation of the single-phase friction factor in a variety of flow-channel geometries is a common engineering practice. Information on the TRACE default correlations is available in the TRACE theory manual (Reference 12-9). A surface roughness of 2E-6 m is used with the TRACE single phase friction correlations. In order to match the HB24 pressure drop prediction, additional frictional flow factors are included in the hydraulic model. The TRACE model also includes plenums to provide a location to specify form loss factors for the gas cooler. The heat transfer and pressure drop correlations would have been updated as the cooler design was determined and as test data was collected. 

However, it is possible that friction events from rubbing between fractured surfaces can be generated at low load levels also during the loading part of the cycle. This is depicted in the two correlation plots of Figure 5. In the plot at the bottom, these events are marked with a rectangle. It was decided that in addition to the previous filter, another filter based in load level should be added. Acoustic emission events were thus accepted only if they occurred at a load higher than 85% of the maximum load level of the test. 

In this report problem of information processing in MIA pulse flaw detectors by means of cross correlation function is considered. Such processing promises to increase the sensitivity and to reduce the noises, mainly the frictional ones. 

The lubricating properties of tears are an important feature in normal blinking. Kalachandra and Shah measured the coefficient of friction of ophthalmic solutions (artificial tears) on polymer surfaces and found no correlation with viscosity, surface tension or contact angle [58]. The coefficient of friction appears to depend on the structure of the polymer surfaces and decreases with increasing load and sliding speed. 

Predicting the solvent or density dependence of rate constants by equation (A3.6.29) or equation (A3.6.31) requires the same ingredients as the calculation of TST rate constants plus an estimate of and a suitable model for the friction coefficient y and its density dependence. While in the framework of molecular dynamics simulations it may be worthwhile to numerically calculate friction coefficients from the average of the relevant time correlation fiinctions, for practical purposes in the analysis of kinetic data it is much more convenient and instructive to use experimentally detemiined macroscopic solvent parameters. 

The issues of the correlation of adliesion and of viscoelastic relaxation with friction are currently being investigated using AFM and LFM. Although friction does not correlate with the adliesion energy between two... 

VER occurs as a result of fluctuating forces exerted by the bath on the system at the system s oscillation frequency O [5]. Fluctuating dynamical forces are characterized by a force-force correlation function. The Fourier transfonn of this force correlation function at Q, denoted n(n), characterizes the quantum mechanical frequency-dependent friction exerted on the system by the bath [5, 8]. 

Equation (C3.5.2 ) is a function of batli coordinates only. The VER rate constant is proportional to tire Fourier transfonn, at tire oscillator frequency Q, of tire batli force-correlation function. This Fourier transfonn is proportional as well to tire frequency-dependent friction q(n) mentioned previously. For example, tire rate constant for VER of tire Emdamental (v = 1) to tire ground (v = 0) state of an oscillator witli frequency D is [54]... 

Chemical reaction dynamics is an attempt to understand chemical reactions at tire level of individual quantum states. Much work has been done on isolated molecules in molecular beams, but it is unlikely tliat tliis infonnation can be used to understand condensed phase chemistry at tire same level [8]. In a batli, tire reacting solute s potential energy surface is altered by botli dynamic and static effects. The static effect is characterized by a potential of mean force. The dynamical effects are characterized by tire force-correlation fimction or tire frequency-dependent friction [8]. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

The Boltzmann constant is ks and T the absolute temperature. — is the Dirac delta function. Below we assume for convenience (equation (5)) that the delta function is narrow, but not infinitely narrow. The random force has a zero mean and no correlation in time. For simplicity we further set the friction to be a scalar which is independent of time or coordinates. 

As discussed above the errors in the trajectory are correlated with the missing rapid motions. In contrast to the friction approach of estimating the variance, which may affect long time phenomena, we identify our errors as the missing ( filtered ) high frequency modes. We therefore attempt to account approximately for the fast motions by choosing the trajectory variance accordingly. 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

Pressure Drop. The prediction of pressure drop in fixed beds of adsorbent particles is important. When the pressure loss is too high, cosdy compression may be increased, adsorbent may be fluidized and subject to attrition, or the excessive force may cmsh the particles. As discussed previously, RPSA rehes on pressure drop for separation. Because of the cychc nature of adsorption processes, pressure drop must be calculated for each of the steps of the cycle. The most commonly used pressure drop equations for fixed beds of adsorbent are those of Ergun (143), Leva (144), and Brownell and co-workers (145). Each of these correlations uses a particle Reynolds number (Re = G///) and friction factor (f) to calculate the pressure drop (AP) per... 

Correlations for Convective Heat Transfer. In the design or sizing of a heat exchanger, the heat-transfer coefficients on the inner and outer walls of the tube and the friction coefficient in the tube must be calculated. Summaries of the various correlations for convective heat-transfer coefficients for internal and external flows are given in Tables 3 and 4, respectively, in terms of the Nusselt number. In addition, the friction coefficient is given for the deterrnination of the pumping requirement. 

Table 3. Correlations for Convective Heat-Transfer and Friction Coefficients for Circular Tube Flow ...

Table 5. Correlations for Heat-Transfer and Darcy Friction Coefficients for Noncircular Laminar Duct Flow ...

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