Flows mean fluid velocity

Where r is the fluid density, u is the mean fluid velocity, and E, is the pressure loss coefficient. The latter coefficient is different for a diffuser and a nozzle, so that the pressure loss depends on the direction of the fluid flow. After a first version... 

The following correlation for pressure drop describes flow perpendicular to pipe bundles for Re < 25xx/(x-l), where x is the ratio of pipe spacing to pipe diameter, v means the mean fluid velocity over the bundle front surface, 1 - the pipe bundle length in the flow direction, and d - the pipe diameter. In our case, we have two bundles in series with x=2 and with 1/d = 1 at a distance of more than 2xd, and d=d= le-4 m, so we can write here for the pressure drop ... 

Consider fully developed flow in a circular tube with a uniform heat transfer rate at the wall. If the mean fluid velocity in the tube is 1 m/s, find the value of the heat transfer coefficient for the following fluids ... 

The column section can be calculated as the ratio of the mean internal flow rate to mean fluid velocity. To keep a safety margin, one can replace the mean internal flow rate by the flow rate in zone I, which is greatest. Consequently, one obtains... 

Flow characteristics in a mixing vessel can influence process performance. The impeller is a device which imparts motion to the medium in which it operates. The characteristics of the flow which are of greatest interest are the mean fluid velocity at all points within the fluid and the turbulent fluctuations superimposed on the mean velocity. Paul and Treybal ( ) have discussed how the detailed flow characteristics can influence process performance. This paper will show how impeller style can influence the flow characteristics. 

Reynolds number (Re) Ratio of inertial forces to viscous forces. Describes laminar and turbulent flow. Re = v.L/v (where o5 is mean fluid velocity, L is length, and v is kinematic fluid viscosity). 

The Monte Carlo approach was extended to reactors with a plug flow macro-mixing RTD by Rattan and Adler [124]. Here, the coalescing fluid elements are moved through the reactor at a speed corresponding to the constant mean fluid velocity. Rattan and Adler [124] were able to simulate experimental results of Vassilatos and Toor [125]. The coalescence frequency was found from data for extremely rapid reactions, where the observed rate is essentially completely controlled by the micromixing in this situation with a flat velocity profile. Then, these coalescence rates were used to predict the experimental results for rapid and slow reactions taking place in the same equipment. 

Owing to the small channel diameters in MSR, laminar flow can be considered in which the fluid flows in parallel layers without lateral mixing. This situation occurs when the ratio between inertial forces to viscous forces is relatively low. The ratio is characterized by the Reynolds number, which is deflned as follows for circular tubes Re = u-d -pg)ft, where u is the mean fluid velocity. Laminar flow is stable for Re < 2000. At higher Reynolds number, inertia forces become dominant, producing flow instabilities such as eddies and vortices, and the flow becomes turbulent. 

Since the inlet flow rates are divided into sub-flows, the mean fluid velocity in the channels depends on the channel number and diameter, as follows ... 

The mean fluid velocity for the fully developed flow is given as... 

One of the few cases of practical interest which can be evaluated analytically is the laminar flow in the channel of rectangular cross section 2b x 2c driven by the hydrostatic pressure difference (see Fig. 10). As discussed in Ref [77] the stationary velocity profile is established for the distance from the inlet larger than 0.16 Foo/v (where Foo is the mean fluid velocity at the entrance to the channel). Then, the inertia term in the Navier-Stokes equation vanishes and Eq. (81) becomes linear. Its solution gives the following expression for the fluid velocity vector directed along the x axis [77] ... 

S-6.2.6 Particle Tracks. Whenever the discrete phase model is used (Section 5-2.2.2), particle tracks can be used to illustrate the trajectories of the particles, bubbles, or droplets. Trajectories can usually be displayed in a number of ways. For example, lines can be colored by the time of the trajectory or temperature of the particle itself. In addition to lines, ribbons and tubes can generally be used. The tracks can be computed and displayed using the mean fluid velocities, or in the case of turbulent flows, using random fluctuations in the mean fluid velocities as well. These stochastic tracks often give a more realistic picture of the extent to which the particles reach all comers of the solution domain than do tracks computed from the mean velocities alone. 

Fig. 32 Snapshots of vesicles in capillary flow, with bending rigidity K/k T = 20 and capillary radius / cap = 1-4/fo- a Fluid vesicle with discoidal shape at the mean fluid velocity v T/ffcap =41, both in side and top views, b Elastic vesicle (RBC model) with parachute shape at t m r/Rcap — 218 (with shear modulus nRl/ksT = 110). The blue arrows represent the velocity field of the solvent, c Elastic vesicle with shpper-like shape at v r/Rcap = 80 (with iiRl/k T = 110). The inside and outside of the membrane are depicted in red and green, respectively. The upper front quarter of the vesicle in (b) and the front half of the vesicle in (c) are removed to allow for a look into the interior, the black circles indicate the lines where the membrane has been cut in this procedure. Thick black lines indicate the walls of the cylindrical capillary. From [187]...

Here f denotes the fraction of molecules diffusely scattered at the surface and I is the mean free path. If distance is measured on a scale whose unit is comparable with the dimensions of the flow channel and is some suitable characteristic fluid velocity, such as the center-line velocity, then dv/dx v and f <<1. Provided a significant proportion of incident molecules are scattered diffusely at the wall, so that f is not too small, it then follows from (4.8) that G l, and hence from (4.7) that V v° at the wall. Consequently a good approximation to the correct boundary condition is obtained by setting v = 0 at the wall. ... 

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

Piping systems should be designed for an economic flow velocity. For relatively clean fluids, a recommended velocity range where minimum corrosion can be expected is 2 to 10 fps. If piping bores exist, maximum fluid velocities may have a mean velocity of 3 fps for a 3/8-in. bore to 10 fps for an 8-in.-diameter bore. Higher flow velocities are not uncommon in situations that require uniform, constant oxygen supply to form protective films on active/passive metals. 

Detemiine tlie mean (superficial) fluid velocity, u, as tlie volumetric flowrate divided by die flow channel cross-section. 

From Figure 26.7 it can be seen that for equal duties and flows the temperature difference for countercurrent flow is lower at the steam inlet than at the outlet, with most of the steam condensation taking place in the lower half of the plate. The reverse holds tme for co-current flow. In this case, most of the steam condenses in the top half of the plate, the mean vapor velocity is lower and a reduction in pressure drop of between 10-40 per cent occurs. This difference in pressure drop becomes lower for duties where the final approach temperature between the steam and process fluid becomes larger. 

Elutriation differs from sedimentation in that fluid moves vertically upwards and thereby carries with it all particles whose settling velocity by gravity is less than the fluid velocity. In practice, complications are introduced by such factors as the non-uniformity of the fluid velocity across a section of an elutriating tube, the influence of the walls of the tube, and the effect of eddies in the flow. In consequence, any assumption that the separated particle size corresponds to the mean velocity of fluid flow is only approximately true it also requires an infinite time to effect complete separation. This method is predicated on the assumption that Stokes law relating the free-falling velocity of a spherical particle to its density and diameter, and to the density and viscosity of the medium is valid... 

Equation 12.37 can be used in order to calculate the friction factor

turbulent flow of fluid in a pipe, It is first necessary to obtain an expression for the mean velocity u of the fluid from the relation ... 

A power law fluid is flowing under laminar conditions through a pipe of circular cross-section. At what radial position is the fluid velocity equal to the mean velocity in the pipe Where does this occur for a fluid with an n-value of 0.2 ... 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

---