Shear stress in a pipe

Consider steady, fully developed flow in a straight pipe of length L and internal diameter d,. As shown in Example 1.8, a force balance on a cylindrical element of the fluid can be written as 

A special case of equation 2.4 is the shear stress tw at the wall 

Equation 2.5 shows that the value of tw can be determined if the pressure gradient is measured this is how values of the friction factor discussed in Section 2.3 have been found. Alternatively, if tw can be predicted, the pressure drop can be calculated. 

From equations 2.4 and 2.5, the shear stress distribution can be written as 

The shear stress varies linearly from zero at the centre-line to a maximum value tw at the pipe wall. 

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It was shown above that the limiting c.d. increases with velocity raised to the 0.8 power and the pipe diameter raised to the -0.2 power for piping corrosion rates that are controlled by mass transport. In contrast, it is evident that the shear stress increases with the fluid velocity raised to the 1.75 power and the pipe diameter raised to the -0.2 power. Thus equality of shear stress does not give equality of mass transfer rates. In both cases corrosion is enhanced in pipes of smaller diameter for the same solution velocity. Such a relationship can be rationalized based on the effect of pipe diameter on the thickness of the mass transport and hydrodynamic boundary layers for a given fixed geometry. Cameron and Chiu (19) have derived similar expressions for defining the rotating cylinder rotation rate required to match the shear stress in a pipe for the case of velocity-... 

The mean energy dissipation rate can be calculated directly from the energy spectrum using (1.332). Moreover, for isotropic turbulence, the energy dissipation rate and the micro scale are related as expressed by (1.323). Caution is required using the dissipation rate calculated from a turbulence model like the k-e model as in this model the dissipation quantity is merely a tuning variable for the shear stresses in a pipe and not necessarily a true physical dissipation rate. The latter approach requires validation for any applications. 

A high pressure steam pipe is 150 mm inside diameter and 200 mm outside diameter. If the steam pressure is 200 bar, what will be the maximum shear stress in the pipe wall ... 

Returning to the problem of the blocked pipe, consider the flow in the cylindrical coordinate system whose Oz axis, which coincides with the axis of the pipe (see Figure 1.3 of Chapter 1), is oriented in the direction of the flow. By considering the non-zero component(s) in the velocity vector, verify that the only non-zero term in the strain rate tensor is Drz. Set out the form of this term explicitly and explain why ) 5 0. Explain also why the only non-zero shear stress in the pipe is Trz, and why Vrz <0. It will be deduced there from that the rheological relation for the Bingham fluid can be written, for the flow of that fluid in a pipe, as ... 

Flow Along Smooth Surfaces. When the flow is entirely parallel to a smooth surface, eg, in a pipe far from the entrance, only the shear stresses contribute to the drag the normal stresses are directed perpendicular to the flow (see Piping systems). The shear stress is usually expressed in terms of a dimensionless friction factor ... 

A Newtonian fluid is one in which, provided that the temperature and pressure remain constant, the shear rate increases linearly with shear stress over a wide range of shear rates. As the shear stress tends to retard the fluid near the centre of the pipe and accelerate the slow moving fluid towards the walls, at any radius within the pipe it is acting simultaneously in a negative direction on the fast moving fluid and in the positive direction on the slow moving fluid. In strict terms equation 3.3 should be written with the incorporation... 

For the flow of a fluid in a pipe of length / and diameter d, the total frictional force at the walls is the product of the shear stress R and the surface area of the pipe (Rndl). This frictional force results in a change in pressure A Pf so that for a horizontal pipe ... 

From equation 3.8 and Figure 3.32a it is seen that the shear stress at a radius, v in a pipe of radius r is given by ... 

When the fluid behaviour can be described by a power-law, the apparent viscosity for a shear-thinning fluid will be a minimum at the wall where the shear stress is a maximum, and will rise to a theoretical value of infinity at the pipe axis where the shear stress is zero. On the other hand, for a shear-thickening fluid the apparent viscosity will fall to zero at the pipe axis. It is apparent, therefore, that there will be some error in applying the power-law near the pipe axis since all real fluids have a limiting viscosity po at zero shear stress. The procedure is exactly analogous to that used for the Newtonian fluid, except that the power-law relation is used to relate shear stress to shear rate, as opposed to the simple Newtonian equation. 

Fluids whose behaviour can be approximated by the power-law or Bingham-plastic equation are essentially special cases, and frequently the rheology may be very much more complex so that it may not be possible to fit simple algebraic equations to the flow curves. It is therefore desirable to adopt a more general approach for time-independent fluids in fully-developed flow which is now introduced. For a more detailed treatment and for examples of its application, reference should be made to more specialist sources/14-17) If the shear stress is a function of the shear rate, it is possible to invert the relation to give the shear rate, y = —dux/ds, as a function of the shear stress, where the negative sign is included here because velocity decreases from the pipe centre outwards. 

Equation 3.152 provides a method of determining the relationship between pressure gradient and mean velocity of flow in a pipe for fluids whose rheological properties may be expressed in the form of an explicit relation for shear rate as a function of shear stress. 

The expressions for the shear stress at the walls, the thickness of the laminar sub-layer, and the velocity at the outer edge of the laminar sub-layer may be applied to the turbulent flow of a fluid in a pipe. It is convenient to express these relations in terms of the mean velocity in the pipe, the pipe diameter, and the Reynolds group with respect to the mean velocity and diameter. 

Close to the wall of a pipe, the effect of the curvature of the wall has been neglected and the shear stress in the fluid has been taken to be independent of the distance from the wall. However, this assumption is not justified near the centre of the pipe. 

A Bingham plastic material is flowing under streamline conditions in a pipe of circular cross-section. What are the conditions for one half of the total flow to be within the central core across which the velocity profile is fiat The shear stress acting within die fluid Ry varies with velocity gradient du,/dy according to the relation ... 

In general, the velocity profile will be curved but as equation 1.33 contains only the local velocity gradient it can be applied in these cases also. An example is shown in Figure 1.13. Clearly, as the velocity profile is curved, the velocity gradient is different at different values of y and by equation 1.32 the shear stress r must vary withy. Flows generated by the application of a pressure difference, for example over the length of a pipe, have curved velocity profiles. In the case of flow in a pipe or tube it is natural to use a cylindrical coordinate system as shown in Figure 1.14. 

This simple force balance has provided an extremely important result the wall shear stress for flow in a pipe can be determined from the frictional component of the pressure drop. In practice it is desirable to use the conditions in Example 1.7 so that the frictional component is the only component of the total pressure drop, which can be measured directly. 

When analysing simple flow problems such as laminar flow in a pipe, where the form of the velocity profile and the directions in which the shear stresses act are already known, no formal sign convention for the stress components is required. In these cases, force balances can be written with the shear forces incorporated according to the directions in which the shear stresses physically act, as was done in Examples 1.7 and 1.8. However, in order to derive general equations for an arbitrary flow field it is necessary to adopt a formal sign convention for the stress components. 

In the case of laminar flow in a pipe, work is done by the shear stress component rTX and the rate of doing work is the viscous dissipation rate, that is the conversion of kinetic energy into internal energy. The rate of viscous dissipation per unit volume at a point, is given by... 

Viscous and turbulent contributions to the total shear stress for flow in a pipe... 

In the case of non-Newtonian flow, it is necessary to use an appropriate apparent viscosity. Although the apparent viscosity (ia is defined by equation 1.71 in the same way as for a Newtonian fluid, it no longer has the same fundamental significance and other, equally valid, definitions of apparent viscosities may be made. In flow in a pipe, where the shear stress varies with radial location, the value of fxa varies. As pointed out in Example 3.1, it is the conditions near the pipe wall that are most important. The value of /j.a evaluated at the wall is given by... 

The velocity profile for steady, fully developed, laminar flow in a pipe can be determined easily by the same method as that used in Example 1.9 but using the equation of a power law fluid instead of Newton s law of viscosity. The shear stress distribution is given by... 

As the shear stress for flow in a pipe varies from zero at the centre-line to a maximum at the wall, genuine flow, ie deformation, of a Bingham plastic occurs only in that part of the cross section where the shear stress is greater than the yield stress ry. In the part where r< rv the material remains as a solid plug and is transported by the genuinely flowing outer material. 

Laminar Newtonian flow in a pipe shear stress and 38... 

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