Shear stress tube wall

Other errors, which could influence the results obtained, are, for example, wall effects ( slipping ), the dissipation of heat, and the increase in temperature due to shear. In a tube, the viscosity of a flowing medium is less near the tube walls compared to the center. This is due to the occurrence of shear stress and wall friction and has to be minimized by the correct choice of the tube diameter. In most cases, an increase in tube diameter reduces the influence of wall slip on the flow rate measured, but for Newtonian materials of low viscosity, a large tube diameter could be the cause of turbulent flow. ° When investigating suspensions with tube viscometers, constrictions can lead to inhomogeneous particle distributions and blockage. Due to the influence of temperature on viscosity (see Section Influence Factors on the Viscosity ), heat dissipated must be removed instantaneously, and temperature increase due to shear must be prevented under all circumstances. This is mainly a constructional problem of rheometers. Technically, the problem is easier to control in tube rheometers than in rotating instruments, in particular, the concentric cylinder viscometers. ... 

As will be shown later, a momentum (force) balance on the fluid in the tube provides a relationships between the shear stress at the tube wall (rw) and the measured pressure drop ... 

The foregoing procedure can be used to solve a variety of steady, fully developed laminar flow problems, such as flow in a tube or in a slit between parallel walls, for Newtonian or non-Newtonian fluids. However, if the flow is turbulent, the turbulent eddies transport momentum in three dimensions within the flow field, which contributes additional momentum flux components to the shear stress terms in the momentum equation. The resulting equations cannot be solved exactly for such flows, and methods for treating turbulent flows will be considered in Chapter 6. 

Tr = particle-wall shear stress in the draft tube... 

The flow rate-pressure drop measurements shown in Table 3.1 were made in a horizontal tube having an internal diameter d, = 6 mm, the pressure drop being measured between two tappings 2.00 m apart. The density of the fluid, p, was 870 kg/m3. Determine the wall shear stress-flow characteristic curve and the shear stress-true shear rate curve for this material. 

In order to determine whether slip occurs with a particular material, it is essential to make measurements with tubes of various diameters. In equation 3.66, the value of the integral term is a function of the wall shear stress only. Thus, in the absence of wall slip, the flow characteristic 8 u/dt is a unique function of tw. However, if slip occurs, the term 8vjd will be different for different values of d, at the same value of tu., as shown in Figure 3.11. It is clear from equation 3.66 that for a given value of the slip velocity vs, the effect of slip is greater in tubes of smaller diameter. If the effect of slip is dominant, that is the bulk of the material experiences negligible shearing, then it can be seen from equation 3.66 that on a plot of... 

Mooney s method has been modified in various ways to allow for the observation that, with many suspensions, the slip velocity depends on the tube diameter as well as the wall shear stress. Jastrzebski (1967) deduced that, for certain kaolinite-water suspensions, vs was inversely proportional to d Thus a modified slip coefficient Cj may be defined by... 

The science that deals with the deformation and flow of matter is called rheology. An important rheological concept is the shear force, sometimes called the shear stress, or the force that causes a layer of a fluid material to flow over a layer of stationary material. The rate at which a layer of a fluid material flows over a layer of stationary material is called the shear rate. A fluid flowing through a tube, for example, would be the fluid material, while the tube wall would be the stationary material. An important rheological measurement that is closely related to the resistance to flow is called viscosity. The technical definition of viscosity is the ratio of shear stress to shear rate ... 

Apart from the short beam shear test, which measures the interlaminar shear properties, many different specimen geometry and loading configurations are available in the literature for the translaminar or in-plane strength measurements. These include the losipescu shear test, the 45°]5 tensile test, the [10°] off-axis tensile test, the rail-shear tests, the cross-beam sandwich test and the thin-walled tube torsion test. Since the state of shear stress in the test areas of the specimens is seldom pure or uniform in most of these techniques, the results obtained are likely to be inconsistent. In addition to the above shear tests, the transverse tension test is another simple popular method to assess the bond quality of bulk composites. Some of these methods are more widely used than others due to their simplicity in specimen preparation and data reduction methodology. 

The quasi-steady laminar model is now employed to describe the heat transfer near the wall. Note that while the shear stress at the wall can be related easily to the pressure drop for the flow in a tube, it is more difficult to establish a relation between these two quantities for a packed or fluidized bed. However, while for the flow in a tube the dissipated energy is not uniform over the section... 

For the turbulent motion in a tube, the mass transfer coefficient k is proportional to the diffusion coefficient at the power of 2/3. It is easy to realize by inspection that this value of the exponent is a result of the linear dependence of the tangential velocity component on the distance y from the wall. For the turbulent motion in a tube, the shear stress t r0 = const near the wall, whereas for turbulent separated flows, the shear stress is small at the wall near the separation point (becoming zero at this point) and depends on the distance to the wall. Thus, the tangential velocity component has, in the latter case, no longer a linear dependence on y and a different exponent for the diffusion coefficient is expected. For separated flows, it is possible to write under certain conditions that [90]... 

Mooney clearly showed that the relationship between the shear stress at the wall of a pipe or tube, DAP/4L, and the term 8V/D is independent of the diameter of the tube in laminar flow. This statement is rigorously true for any kind of flow behavior in which the shearing rate is only a function of the applied shearing stress.1 This relationship between DAP/4L and 8VJD may be conveniently determined in a capillary-tube viscometer, for example. Once this has been done over the range of... 

It is true, therefore, for Newtonian, Bingham-plastic, pseudoplastic and dilatant fluids. The same relationship can possibly be extended to thixotropic and rheopectic fluids by evaluating the shear stress at the wall over a differential length of tube, i.e., by replacing D P/4L with DdP/idL. This term will vary with distance along the pipe, however, and as no evident means of developing this relationship has been... 

B. Determination of the relationship between shear stress at the wall of a round tube or pipe, DAP/4L, and the term 8V/D. 

Once the Reynolds number (based on the mean velocity) is known for a given tube and flow situation, the friction factor follows as / = Re//Re. From the friction factor the wall shear stress and pressure gradient are easily determined. 

J5 Consider the wall-injection problem in an axisymmetric setting, where a uniform injection velocity flows through the wall of a cylindrical tube. There is a mean velocity U that enters through one end of the tube. Following a procedure analogous to the flow-between-plates problem (Section 5.6), develop a solution for the velocity profiles and the wall shear stress as characterized by the product of a Reynolds number and a friction factor. 

Experimental measurements of the wall shear stress exerted by a falling liquid film have been reported for the cases of film flow outside a vertical tube (B14) and in a channel of variable slope (F7). In both cases the experimental results in the zone of smooth laminar flow were in agreement... 

Brauer (B14), 1956 Extensive experimental study of film flow outside tube 4.3X130 cm. films of water, water + surfactant, aqueous diethylene glycol solutions, kinematic viscosity 0.9-12.7 cs. Nr = 20-1800. Data on film thicknesses, waves, maximum and minimum thicknesses, characteristic Reynolds numbers of flow, onset of rippling and turbulence, wall shear stress, etc. 

Tube flow data for aqueous solutions of polyacrylamide (Separan AP-30) and carboxymethylcellulose (8,21) provide confirmation for all of Mooney s assumptions. The slip velocity was found to be invariant with tube length and was a function of the wall shear stress only, increasing with increase in the wall shear stress. 

In a pressure capillary viscometer, such as a rheological die, pressure is used to force fluid through a capillary tube at constant volumetric flow rate Q, as shown in Figure 6.4. The pressure difference AP is measured between points A and B spaced apart a distance 1 along the tube. The basic rheological equations are as follows for shear stress and shear rate taken to be very near the wall in a tube of radius r ... 

Upon exiting the die, the sheet extrudate will swell to a level determined by the polymer, the melt temperature, the die length-to-opening ratio, and the shear stress at the die walls. Additionally, flow instabilities will occur at values of the corrected shear stress at the wall, of the order of, but higher than 105 N/m2, as found by Vlachopoulos and Chan (58), who also concluded that, for PS, HDPE, and LDPE, the critical Sr in slits is 1.4 times higher than in tubes of circular cross section. Aside from these differences, the information presented in Section 12.1 and 12.2 applies to slit flow. 

For shear-thinning fluids, // —> oo a I zero shear stress and fi 0 at infinite shear stress. Paint often exhibits shear thinning behaviour as its apparent viscosity is very high while in the can and when just applied to a wall but its apparent viscosity is very low as the brush applies it to the surface when it flows readily to give an even film. Toothpaste remains in its tube and on the brush when not subjected to shear but when sheared, as it is when the tube is squeezed, it flows readily through the nozzle to the brush. 

The surface equipment, pipe, and tubing are of carbon steel. The wall shear stress of the liquid due to the high gas velocity is 90 N/m2. (a) What type of corrosion is encountered (b) What type of protection can be adopted economically (c) If a corrosion inhibitor is adopted, what type of selection tests should be used to choose the right chemical (Pou)... 

The yield stress of a foam depends to a considerable extent on the character of foam interaction with the tube walls or the cylindrical surface of the viscometer, used in the study of its rheological properties. At low flow rates and smooth tube walls the maximum shear stress of the foam layers contacting the wall can be less than the shear stress of the foam matrix (shear of bubble layers). Hence, the foam flow will occur as a movement of a continuous medium in a cylinder covered with a thin lubricating layer of thickness 2-10 pm [9,16], In this case t0 is ca. 1 Pa, that is, much less than its theoretical value. 

A study of the flow of a polyhedral foam in a regime of slip at the tube walls has been conducted [39]. It has been established that the rise in the dynamic viscosity of the foaming solution leads to diminishing the flow rate but to a much lesser extent at t0 = 1.25 Pa. Thus, a two fold increase in viscosity causes a 1.3 times decrease in the flow rate, while a 6 times increase in the dynamic viscosity only a 2.23 times decrease. This is probably related to the expanding of the effective thickness of the liquid layer 8 (ca. 3 times). The transition from plug flow (slip regime) to shear flow occurs at To = 9-10 Pa. This value of the shear stress is much smaller than the one obtained from Princen s formula for a two-dimensional foam (Eq. (8.18)) at a given expansion ratio and correlates well with To calculated from Eq. (8.24) and the experimental data of Thondavald and Lemlich [23],... 

An outline of the steps involved to derive the equations for shear rate and shear stress for fully developed flow in a tube is given in Appendix 3-B. The shear stress (ctw) is given by Equation (3.34) and the shear rate by Equation 3.35, where the subscript w is used to emphasize that the values obtained are those at the pipe wall. 

The shear stress at the tube wall is related to the slope of the velocity profile at the surface. Noting that the velocity profile remains unchanged in the hydrodynamically fully developed region, the wall shear stress also remains constant in that region. A similar argument can be given for Ihe heat transfer coefficient in the thermally fully developed region. 

---