Shear fluid, pressure drop

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

In order to select the pipe size, the pressure loss is calculated and velocity limitations are estabHshed. The most important equations for calculation of pressure drop for single-phase (Hquid or vapor) Newtonian fluids (viscosity independent of the rate of shear) are those for the deterrnination of the Reynolds number, and the head loss, (16—18). 

Economic Pipe Diameter, Laminar Flow Pipehnes for the transport of high-viscosity liquids are seldom designed purely on the basis of economics. More often, the size is dictated oy operability considerations such as available pressure drop, shear rate, or residence time distribution. Peters and Timmerhaus (ibid.. Chap. 10) provide an economic pipe diameter chart for laminar flow. For non-Newtouiau fluids, see SkeUand Non-Newtonian Flow and Heat Transfer, Chap. 7, Wiley, New York, 1967). 

Since it is recognised that the fluid is Non-Newtonian, this is often referred to as the apparent shear rate to differentiate it from the true shear rate. If the pressure drop, P, across the die is also measured then the shear stress, r, may be calculated from... 

The surface shear stress t is a consequence of the velocity difference between the metal surface and the fluid velocity. For tubular geometries it can be obtained from pressure drop measurements or calculated ... 

It is now convenient to relate the pressure drop due to fluid friction —APf to the shear stress R0, at the walls of a pipe. If Ry is the shear stress at a distance y from the wall of the pipe, the corresponding value at the wall is given by ... 

The pressure drop due to friction and the velocity distribution resulting from the shear stresses within a fluid in streamline Newtonian flow are considered for three cases (a) the... 

Thus, for a given fluid (Ry constant), the critical radius rt is determined entirely by the pressure drop and becomes progressively larger as the pressure drop is reduced. Flow-ceases when the shear stress at the wall R falls to a value equal to the yield stress Ry. 

A Newtonian liquid of viscosity 0.1 N s/m2 is flowing through a pipe of 25 mm diameter and 20 m in lenglh, and the pressure drop is 105 N/m2. As a result of a process change a small quantity of polymer is added to the liquid and this causes the liquid to exhibit non-Newtonian characteristics its rheology is described adequately by the power-law model and the flow index is 0.33. The apparent viscosity of the modified fluid is equal to ihc viscosity of the original liquid at a shear rate of 1000 s L... 

As indicated earlier, non-Newtonian characteristics have a much stronger influence on flow in the streamline flow region where viscous effects dominate than in turbulent flow where inertial forces are of prime importance. Furthermore, there is substantial evidence to the effect that for shear-thinning fluids, the standard friction chart tends to over-predict pressure drop if the Metzner and Reed Reynolds number Re R is used. Furthermore, laminar flow can persist for slightly higher Reynolds numbers than for Newtonian fluids. Overall, therefore, there is a factor of safety involved in treating the fluid as Newtonian when flow is expected to be turbulent. 

Because n < 1 for a shear thinning fluid, < 7 will be Jess than unity and a reduction in pressure drop occurs. The lower the value of n and the larger the value of b, the greater the effect will be. It will be noted that the effects of expansion of the air as the pressure falls have not been taken into account. 

The Martinelli correlations for void fraction and pressure drop are used because of their simplicity and wide range of applicability. France and Stein (6 ) discuss the method by which the Martinelli gradient for two-phase flow can be incorporated into a choked flow model. Because the Martinelli equation balances frictional shear stresses cuid pressure drop, it is important to provide a good viscosity model, especially for high viscosity and non-Newtonian fluids. 

There are commercially available in-line or on-line viscometer devices. In-line devices are installed directly in the process while on-line devices are used to analyze a side stream of the process. Most devices are based on measuring the pressure drop and flow rate through a capillary. The viscosity is either determined at a single shear rate or, at most, a few shear rates. Complex fluids, on the other hand, exhibit a viscosity that cannot be so easily characterized. In order to capture enough information that allows, for example, a molecular weight distribution to be inferred, it is necessary to determine the shear viscosity over reasonably wide ranges of shear rates. 

Other schemes have been proposed in which data are fit to a lower, even order polynomial [19] or to specific rheological models and the parameters in those models calculated [29]. This second approach can be justified in those cases when the range of behavior expected for the shear viscosity is limited. For example, if it is clear that power-law fluid behavior is expected over the shear rate range of interest, then it would be possible to calculate the power-law parameters directly from the velocity profile and pressure drop measurement using the theoretical velocity profile... 

For viscous fluids, the velocity may be constrained by the allowable pressure drop or shear degradation of the fluid (e.g. large molecules breaking down into smaller molecules from high shear rates). Typical values are given in Table 13.6. 

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 ... 

A coal slurry is found to behave as a power law fluid, with a flow index of 0.3, a specific gravity of 1.5. and an apparent viscosity of 70 cP at a shear rate of 100 s 1. What volumetric flow rate of this fluid would be required to reach turbulent flow in a 1/2 in. ID smooth pipe that is 15 ft long What is the pressure drop in the pipe (in psi) under these conditions ... 

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. 

The membrane viscometer must use a membrane with a sufficiently well-defined pore so that the flow of isolated polymer molecules in solution can be analyzed as Poiseuille flow in a long capillary, whose length/diameter is j 10. As such the viscosity, T, of a Newtonian fluid can be determined by measuring the pressure drop across a single pore of the membrane, knowing in advance the thickness, L, and cross section. A, of the membrane, the radius of the pore, Rj., the flow rate per pore, Q,, and the number of pores per unit area. N. The viscosity, the maximum shear stress, cr. and the velocity gradient, y, can be calculated from laboratory measurements of the above instrumental parameters where Qj =... 

In addition to the effect of the walls on the drag on the particle, the particle alters the shear on the duct. Consider a particle settling through a quiescent fluid (Fig. 9.1 with Uq = 0). Brenner (B3) showed that, for low particle Re with the particle small by comparison with the distance between particle and wall (i.e., / 1 — where = b/R), there is an excess pressure drop, AP, between points far below and far above the particle given by... 

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