Friction velocity, calculation

Example 8 Compressible Flow with Friction Losses Calculate the discharge rate of air to the atmosphere from a reservoir at 10 Pa gauge and 20 G through 10 m of straight 2-in Schedule 40 steel pipe (inside diameter = 0.0525 m), and 3 standard radius, flanged 90 elhows. Assume 0.5 velocity heads lost for the elhows. 

Water is flowing through a 45° pipe bend at a rate of 200 gpm and exits into the atmosphere. The inlet to the bend is 1 in. inside diameter, and the exit is 1 in. in diameter. The friction loss in the bend can be characterized by a loss coefficient of 0.3 (based on the inlet velocity). Calculate the net force (magnitude and direction) transmitted to the flange holding the pipe section in place. 

Figure 24.8 Concentration of Xo estimate the dispersion coefficient Edjs we need the lateral turbulent diffusivity Ey measure t a)3 s a"ion°A (26 Tm (see Ecl- 24 45) which in tum is calculated from the friction velocity, u. Problem downstream of spill) and (b) 24.4 deals with the calculation of Edis. As it turns out, a realistic value which agrees...

The calculation of the resistance coefficients can be accomplished in the frame of the Monin-Obukhov similarity theory (Monin and Yaglom, 1971). The genuine flux quantities are the friction velocity u and the scale functions 0 and referring to temperature and humidity. The turbulent momentum flux f, the sensible heat flux the mass flux from evaporation and condensation and the corresponding latent heat flux are... 

In the early FVM program codes, all the equilibrium variable values at the grid point, P, were simply calculated from the given parameterizations. However, in these steady state program codes the friction velocity is not known a priori but is an outcome of the iterative solution algorithm where the boundary values are coupled through the governing transport equations. 

Calculating the friction velocity from the log-law (1.373), and the nonequilibrium velocity scale from (1.428), yields ... 

The accommodation coefficient or represents the fraction of the gas molecules that leave the surface in equilibrium with the surface. The fraction I — cr is specularly reflected such that the velocity normal to the surface is reversed. As in the case of Stokes law, the drag is proportional to the velocity of the spheres. However, for the free molecule range, the friction coefficient is proportional to dj whereas in the continuum regime dp ip), it is proportional to dp. The coefficient a must, in general, be evaluated experimentally but is usually near 0.9 for momentum transfer (values differ for heat and mass transfer). The friction coefficient calculated from (2.19) is only 1% of that from Stokes law for a 20-A particle. 

For laminar flow, the characteristic time of the fluid phase Tf can be deflned as the ratio between a characteristic velocity Uf and a characteristic dimension L. For example, in the case of channel flows confined within two parallel plates, L can be taken equal to the distance between the plates, whereas Uf can be the friction velocity. Another common choice is to base this calculation on the viscous scale, by dividing the kinematic viscosity of the fluid phase by the friction velocity squared. For turbulent flow, Tf is usually assumed to be the Kolmogorov time scale in the fluid phase. The dusty-gas model can be applied only when the particle relaxation time tends to zero (i.e. Stp 1). Under these conditions, Eq. (5.105) yields fluid flow. This typically happens when particles are very small and/or the continuous phase is highly viscous and/or the disperse-to-primary-phase density ratio is very small. The dusty-gas model assumes that there is only one particle velocity field, which is identical to that of the fluid. With this approach, preferential accumulation and segregation effects are clearly not predicted since particles are transported as scalars in the continuous phase. If the system is very dilute (one-way coupling), the properties of the continuous phase (i.e. density and viscosity) are assumed to be equal to those of the fluid. If the solid-particle concentration starts to have an influence on the fluid phase (two-way coupling), a modified density and viscosity for the continuous phase are generally introduced in Eq. (4.92). 

We will now develop an expression for the sonic speed experienced in the throat/outlet of a convergent-only nozzle and at the throat of a convergent-divergent nozzle when the expansion is frictionally resisted. Sonic conditions will exist in the throat when the velocity calculated by applying equation (14.45) to the convergent section of the nozzle has reached the local speed of sound, i.e. ... 

The damping of the vibration due to fluid friction is calculated from the drag on a cylinder in steady motion with the mean velocity tia = 4Sf, where S is the deflection and / is the frequency of vibration. The ratio of damping to critical damping becomes ... 

EXAMPLE 2.7-6. Rate of Flow from Pressure Measurements A liquid with a constant density p kg/m is flowing at an unknown velocity u, m/s through a horizontal pipe of cross-sectional area >4, m at a pressure Pi N/m, and then it passes to a section of the pipe in which the area is reduced gradually to A2 m and the pressure is P2. Assuming no friction losses, calculate the velocity u, and 02 if the pressure difference (p, — P2) is measured. 

The tests were conducted on aqueous solutions of SML/ESMIS mixtures at the concentrations of 0.01%, 0.1%, 1%, and 4 wt%. A T-11 tribometer (pin-on-disc) produced by ITeE Poland (Institute for Sustainable Technologies, National Research Institute, Radom, Poland) was used in the tests. The pins used were 3.0 mm in diameter and were made of LH15 steel or polyamide-6, while the discs were 25 mm in diameter and were made of polyamide-6. Before the tests, all components of the friction couples were thoroughly chemically cleaned. An ultrasonic cleaner was employed. The steel elements were cleaned in n-hexane, acetone, ethyl alcohol, and distilled water. The polyamide-6 elements were cleaned in ethyl alcohol and distilled water. After the cleaning process all friction-couple components were dried (50°C, 30 min). The time of the test run was 900 s, the friction-couple load was 10 N, and the sliding velocity was 0.1 m/s. The coefficient of friction was calculated from friction force measurements using... 

Theories from two extremely different points of view have been proposed to determine the relation between 0,9 and velocity U. They differ in the way the friction is calculated. In hydrodynamic theories, the dissipation is assumed to take place very close to the TCL mainly through viscous forces on the other hand, in the molecular model proposed by Blake the dissipation is mostly associated with molecules hopping at the molecular edge of the contact line. 

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