Velocity profile, tube flow

If rheological flow curve data are available (as in Fig. 3-16), we can directly compute the pressure drop in tube flow as well as determine the velocity profile and flow rate. 

As will be illustrated in the next chapter, certain important calculations (e.g., the determination of velocity profiles and flow rates in tubes) are facilitated by flow equations that are explicit in z, that is, have the form y = y (t) or— j)(t). The power law can be written in either form. Some of the equations not given here are in or can be put in one of these forms. Unfortunately, they tend not to fit... 

This equation is appHcable for gases at velocities under 50 m/s. Above this velocity, gas compressibiUty must be considered. The pitot flow coefficient, C, for some designs in gas service, is close to 1.0 for Hquids the flow coefficient is dependent on the velocity profile and Reynolds number at the probe tip. The coefficient drops appreciably below 1.0 at Reynolds numbers (based on the tube diameter) below 500. 

Entrance flow is also accompanied by the growth of a boundary layer (Fig. 5b). As the boundary layer grows to fill the duct, the initially flat velocity profile is altered to yield the profile characteristic of steady-state flow in the downstream duct. For laminar flow in a tube, the distance required for the velocity at the center line to reach 99% of its asymptotic values is given by... 

In the simple Bunsen flame on a tube of circular cross-section, the stabilization depends on the velocity variation in the flow emerging from the tube. For laminar flow (paraboHc velocity profile) in a tube, the velocity at a radius r is given by equation 20 ... 

The dispersion that takes place in an open tube, as discussed in chapter 8, results from the parabolic velocity profile that occurs under conditions of Newtonian flow (i.e., when the velocity is significantly below that which produces turbulence). Under condition of Newtonian flow, the distribution of fluid velocity across the tube... 

When the Reynolds number is under 2000, it is shown empirically that the flow in a smooth tube is laminar. This flow has a parabolic velocity profile, as shown in Fig. 4.3. 

The calibration air flows through a thin tube. The probe is placed at the exit of the tube. When the tube is long enough and the tube flow is laminar, the reference velocity for calibration can be calculated from the theoretical, fully developed laminar velocity profile. 

Air at 323 K and 152 kN/m2 pressure flows through a duct of circular cross-section, diameter 0.5 m. In order to measure the flow rate of air, the velocity profile across a diameter of the duct is measured using a Pitot-static tube connected to a water manometer inclined at an angle of cos-1 0.1 to the vertical. The following... 

Hao et al. (2007) investigated the water flow in a glass tube with diameter of 230 Lim using micro particle velocimetry. The streamwise and mean velocity profile and turbulence intensities were measured at Reynolds number ranging from 1,540 to 2,960. Experimental results indicate that the transition from laminar to turbulent flow occurs at Re = 1,700—1,900 and the turbulence becomes fully developed at Re > 2,500. 

Figure 2.6 shows a typical temperature profile.t l The temperature boundary layer is similar to the velocity layer. The flowing gases heat rapidly as they come in contact with the hot surface of the tube, resulting in a steep temperature gradient. The average temperature increases toward downstream. 

Consider isothermal laminar flow of a Newtonian fluid in a circular tube of radius R, length L, and average fluid velocity u. When the viscosity is constant, the axial velocity profile is... 

Equation (8.9) can be applied to any reaction, even a complex reaction where ctbatch(t) must be determined by the simultaneous solution of many ODEs. The restrictions on Equation (8.9) are isothermal laminar flow in a circular tube with a parabolic velocity profile and negligible diffusion. 

Flow in a Tube. Laminar flow with a flat velocity profile and slip at the walls can occur when a viscous fluid is strongly heated at the walls or is highly non-Newtonian. It is sometimes called toothpaste flow. If you have ever used Stripe toothpaste, you will recognize that toothpaste flow is quite different than piston flow. Although Vflr) = u and z(7) = 1, there is little or no mixing in the radial direction, and what mixing there is occurs by diffusion. In this situation, the centerline is the critical location with respect to stability, and the stability criterion is... 

Equation (8.64) allows the shape of the velocity profile to be calculated (e.g., substitute ytr = constant and see what happens), but the magnitude of the velocity depends on the yet unknown value for dPjdz. As is often the case in hydrodynamic calculations, pressure drops are determined through the use of a continuity equation. Here, the continuity equation takes the form of a constant mass flow rate down the tube ... 

Once the velocity profile has been obtained, the shear rate is calculated. This is the most difficult step. To ensure that the viscosity is determined without any bias, no assumption is made regarding the constitutive behavior of the material. Every effort is made to obtain smooth, robust values of the shear rate without any bias towards a particular model of the flow behavior. Particularly near the tube center, the velocity profiles are distorted by the discrete nature of the information. The size of a pixel is defined by the velocity and spatial resolutions. These are given by... 

Velocity images and profiles at several selected heights are shown in Figure 4.3.6, where the noisy points in the images indicate the air space where a liquid signal was not detected. When the fluid is inside the glass pipette, the velocity profile is nearly Poiseuille and a non-slip boundary condition is almost achieved. This is consistent with one of the early tube flow reports that the 0.5% w/v solution of... 

Fig. 4.3.6 Velocity maps and profiles at differ- mark the NMR foldbacks from the stationary ent heights of the Fano column. The dark ring fluid at the inner surface of the fluid reservoir, surrounding the pipe at z= 1.5 mm (larger In the velocity profiles, the solid curves are the white arrow) is due to a layer of stationary fluid calculated Poiseuille profiles in tube flow, adhering to the pipe exterior following the Velocity images are reprinted from Ref. [20], dipping of the pipe into the reservoir at the with permission from Elsevier, start of the experiment. The small white arrows...

A velocity profile u(r) is obtained using MRI flow imaging and, with respect to radial position r, the values of shear rate y(r), ranging from zero at the tube center to a maximum at the tube wall, can be calculated from the velocity profile as local velocity gradients ... 

The velocity profile during slip flow in a cylindrical tube is shown in Figure 21. As in conventional fluid flow, the flow velocity in the z direction, u(r), is parabolic, but rather than reach zero at the tube wall, slip occurs, and the velocity at the wall is greater than zero. The velocity does not reach zero until distance h from the wall surface. The derivation of the mass flux equation proceeds along the same lines as the derivation of Poiseuille s law in conventional hydrodynamics, but in slip flow, u(r) = 0 at r = a + h instead of reaching zero at r = a. 

---