Mean flow velocity

The components of G (which, as was noted above, are taken with respect to the mean flow velocity) and g, which enter Eq. (1-93), are now rewritten in terms of and ... 

Since the expression in Eq. (1-93), which must be used in Eq. (1-89), involves a nonintegral power of g, the evaluation of the % and x integrations is best carried out in a -dependent coordinate -system. We may form dimensionless coordinates G0,g0 in terms of the G,g system used in Eq. (1-93) (where, again, G is defined with respect to the mean flow velocity ) ... 

Miyauchi and Vermeulen (M7, M8) have presented a mathematical analysis of the effect upon equipment performance of axial mixing in two-phase continuous flow operations, such as absorption and extraction. Their solutions are based, in one case, upon a simplified diffusion model that assumes a mean axial dispersion coefficient and a mean flow velocity for... 

This system produces a steady laminar flow with a flat velocity profile at the burner exit for mean flow velocities up to 5m/s. Velocity fluctuations at the burner outlet are reduced to low levels as v /v< 0.01 on the central axis for free jet injection conditions. The burner is fed with a mixture of methane and air. Experiments-described in what follows are carried out at fixed equivalence ratios. Flow perturbations are produced by the loudspeaker driven by an amplifier, which is fed by a sinusoidal signal s)mthesizer. Velocity perturbations measured by laser doppler velocimetry (LDV) on the burner symmetry axis above the nozzle exit plane are also purely sinusoidal and their spectral... 

Trendelenberg position, in order to help perfuse the ischemic brain regions. Wojner-Alexander et studied 20 stroke patients with transcranial Doppler (TCD) of the middle cerebral artery (MCA), measuring mean flow velocities (MFV) at 30°, 15°, and 0° head of bed elevation. The MFV improved in all patients by lowering the head of bed, and three patients demonstrated immediate neurological improvement when lowered to 0°. 

Magnitude of velocity Line velocity Typical velocity scale Mean flow velocity Elow velocity vector... 

R hydraulic radius (m) s slope (m m-1) ii mean flow velocity (m s-1)... 

Thus, this equation has indeed the form of Eq. 1 with the exponent n = 1/4. Note that the shear velocity for river flow can be calculated either from the slope and geometry of the river bed (Eq. 20-33a) or from the mean flow velocity u (Eq. 20-3 3b). 

Once the mathematical description of dispersion has been clarified, we are left with the task of quantifying the dispersion coefficient, Eiis. Obviously, Edh depends on the characteristics of the flow field, particularly on the velocity shear, dvx/dy and dvx /dz. As it turns out, the shear is directly related to the mean flow velocity vx. In addition, the probability that the water parcels change between different streamlines must also influence dispersion. This probability must be related to the turbulent diffusivity perpendicular to the flow, that is, to vertical and lateral diffusion. At this point it is essential to know whether the lateral and vertical extension of the system is finite or whether the flow is virtually unlimited. For the former (a situation typical for river flow), the dispersion coefficient is proportional to (vx )2 ... 

Illustrative Example 24.1 Mean Flow Velocity of River G for Different Discharge Rates... 

Although it seems natural to formulate the dynamic equations of a chemical in a river in terms of the Langrangian picture, the field data are usually made in the Eulerian reference system. In this system we consider the changes at a fixed point in space, for instance, at a fixed river cross section located atxQ. In Eq. 22-6 we adopted the Eulerian system and found that this representation combines the influence from in-situ reactions (the Langrangian picture) with the influence from transport. The latter appears in the additional advective transport term -udCJdx, where the mean flow velocity ... 

The ratio between mean flow velocity, u, and friction velocity, u, is called a. According to Eqs. 24-4 and 24-5, a is related to the friction factor f ... 

Note that due to Eq. 24-10 the mean flow velocity u x) is hidden in the function t (x). To make this more explicit, we assume u(x) = constant = u. Then Eq. 24-10 becomes ... 

After division by the mean flow velocity u we get the corresponding s-values. The results are summarized in the following table. 

Give a qualitative explanation for the physical meaning of the hydraulic radius Rb and why it is relevant for the mean flow velocity of a river, u. ... 

The flow rate of water through the porous medium per unit total (bulk) area perpendicular to the direction of flow, the so-called specific discharge q, is related to the effective mean flow velocity in the pores along the x-axis, u, by... 

Specific Discharge q, Effective Mean Flow Velocity u, and Travel Time fw for Different Flow Regimes in Groundwater System S... 

The following table summarizes the characteristic numbers for all cases. The mean flow velocities, u, are taken from Illustrative Example 25.1. 

Explain the difference between specific discharge and effective mean flow velocity. What is effective porosity ... 

You are responsible for the safe operation of a drinking-water supply system that gets its raw water from a well located close to a river. From tracer experiments you know that the effective mean flow velocity is u =3 m d l and that the distance along the streamline from the point of infiltration to the well is x = 18 m. The dispersivity of the aquifer for this distance of flow is aL = 5 m. In order to be prepared for a possible pollution event in the river you are interested in the following questions ... 

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