Tube flow Newtonian fluids

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

A significant heat-transfer enhancement can be obtained when a nonckcular tube is used together with a non-Newtonian fluid. This heat-transfer enhancement is attributed to both the secondary flow at the corner of the nonckcular tube (23,24) and to the temperature-dependent non-Newtonian viscosity (25). Using an aqueous solution of polyacrjiamide the laminar heat transfer can be increased by about 300% in a rectangular duct over the value of water (23). 

For non-Newtonian fluids in slow flow, friclion loss across a square-woven or fuU-twill-woven screen can be estimated by considering the screen as a set of parallel tubes, each of diameter equal to the average minimal opening between achacent wires, and length twice the diameter, without entrance effects (Carley and Smith, Polym. Eng. Set., 18, 408-415 [1978]). For screen stacks, the losses of individual screens should be summed. 

Consider a steady unidimensional flow in a tubular reaetor as shown in Figure 8-21 in the absenee of either radial or longitudinal diffusion. The veloeity u(r) is the parabolie distribution for a Newtonian fluid at eonstant viseosity, with the fluid in the eenter of the tube spending the shortest time in the reaetor. 

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... 

In the steady flow of a Newtonian fluid through a long uniform circular tube, if ARe < 2000 the flow is laminar and the fluid elements move in smooth straight parallel lines. Under these conditions, it is known that the relationship between the flow rate and the pressure drop in the pipe does not depend upon the fluid density or the pipe wall material. 

As will be shown later, the velocity profile for a Newtonian fluid in laminar flow in a circular tube is parabolic. When this is introduced into Eq. (5-38), the result is a = 2. For highly turbulent flow, the profile is much flatter and a 1.06, although for practical applications it is usually assumed that a = 1 for turbulent flow. 

Example 5-2 Kinetic Energy Correction Factor for Laminar Flow of a Newtonian Fluid. We will show later that the velocity profile for the laminar flow of a Newtonian fluid in fully developed flow in a circular tube is parabolic. Because the velocity is zero at the wall of the tube and maximum in the center, the equation for the profile is... 

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. 

This result can also be derived by equating the shear stress for a Newtonian fluid, Eq. (6-9), to the expression obtained from the momentum balance for tube flow, Eq. (6-4), and integrating to obtain the velocity profile ... 

All models for turbulent flows are semiempirical in nature, so it is necessary to rely upon empirical observations (e.g., data) for a quantitative description of friction loss in such flows. For Newtonian fluids in long tubes, we have shown from dimensional analysis that the friction factor should be a unique function of the Reynolds number and the relative roughness of the tube wall. This result has been used to correlate a wide range of measurements for a range of tube sizes, with a variety of fluids, and for a wide range of flow rates in terms of a generalized plot of/ versus /VRe- with e/D as a parameter. This correlation, shown in Fig. 6-4, is called a Moody diagram. 

The model for turbulent drag reduction developed by Darby and Chang (1984) and later modified by Darby and Pivsa-Art (1991) shows that for smooth tubes the friction factor versus Reynolds number relationship for Newtonian fluids (e.g., the Colebrook or Churchill equation) may also be used for drag-reducing flows, provided (1) the Reynolds number is defined with respect to the properties (e.g., viscosity) of the Newtonian solvent and (3) the Fanning friction factor is modified as follows ... 

Show how the Hagen-Poiseuille equation for the steady laminar flow of a Newtonian fluid in a uniform cylindrical tube can be derived starting from the general microscopic equations of motion (e.g., the continuity and momentum equations). 

The Hagen-Poiseuille equation [Eq. (6-11)] describes the laminar flow of a Newtonian fluid in a tube. Since a Newtonian fluid is defined by the relation r = fiy, rearrange the Hagen-Poiseuille equation to show that the shear rate at the tube wall for a Newtonian fluid is given by yw = 4Q/nR3 = 8 V/D. 

Evaluate the kinetic energy correction factor a in Bernoulli s equation for turbulent flow assuming that the 1/7 power law velocity profile [Eq. (6-36)] is valid. Repeat this for laminar flow of a Newtonian fluid in a tube, for which the velocity profile is parabolic. 

Plug flow is an idealization. Deviations arise with viscous or non-Newtonian fluids. A mathematically simple deviation from the plug flow pattern is that of power law fluids whose velocity in a tube depends on the radial position, /3 = r/R, according to the equation,... 

The second subscript N is a reminder that this is the wall shear rate for a Newtonian fluid. The quantity (8u/d,), or the equivalent form in equation 3.13, is known as the flow characteristic. It is a quantity that can be calculated for the flow of any fluid in a pipe or tube but it is only in the case of a Newtonian fluid in laminar flow that it is equal to the magnitude of the shear rate at the wall. 

The solution to the problem of determining the wall shear rate for a non-Newtonian fluid in laminar flow in a tube relies on equation 2.6. 

Recall that the wall shear rate for a Newtonian fluid in laminar flow in a tube is equal to —8w/d,. In the case of a non-Newtonian fluid in laminar flow, the flow characteristic is no longer equal to the magnitude of the wall shear rate. However, the flow characteristic is still related uniquely to tw because the value of the integral, and hence the right hand side of equation 3.17, is determined by the value of tw. 

The Hagen-Poiseuille equation for steady laminar flow of Newtonian fluids in pipes and tubes can be written as... 

The modelling of the flow of a non-Newtonian fluid through a packed bed follows a similar, though more complex, procedure to that adopted earlier in this chapter for the flow of a Newtonian fluid. It first involves a consideration of the flow through a cylindrical tube and then adapting this to the flow in the complex geometry existing in a packed bed. The procedure is described in detail elsewhere(24,25). 

The velocity profiles of pseudoplastic non-Newtonian fluids (Fig. 8) in laminar flow deviate from the Newtonian parabola in the same way as the velocity profile of Newtonian liquids changes when heat is being transferred to them (M4, p. 229), since in both cases the viscosity of the fluid is lower at the wall than at the center of the tube. For the Newtonian... 

Eigure 2.4a shows the velocity distribution in a steady isothermal laminar flow of an incompressible Newtonian fluid through a straight, round tube. The velocity distribution in laminar flow is parabolic and can be represented by... 

From the above relationships it can also be shown that the pressure drop AP (Pa) in the laminar flow of a Newtonian fluid ofviscosity // (Pa s) through a straight round tube of diameter d (m) and length L (m) at an average velocity of v (ms ) is given by Equation 2.9, which expresses the Hagen-Poiseuille law ... 

Derive an equation for the shear rate at the tube surface for laminar flow of Newtonian fluids through a tube of radius fj. 

Specific correlations of individual film coefficients necessarily are restricted in scope. Among the distinctions that are made are those of geometry, whether inside or outside of tubes for instance, or the shapes of the heat transfer surfaces free or forced convection laminar or turbulent flow liquids, gases, liquid metals, non-Newtonian fluids pure substances or mixtures completely or partially condensable air, water, refrigerants, or other specific substances fluidized or fixed particles combined convection and radiation and others. In spite of such qualifications, it should be... 

Michalik (M9), 1957 Study of hydraulics of laminar film flow of Newtonian fluid in tubes and on vertical plates. Optimum design parameters for wetted-wall columns derived. 

If one considers fluid flowing in a pipe, the situation is highly illustrative of the distinction between shear rate and flow rate. The flow rate is the volume of liquid discharged from the pipe over a period of time. The velocity of a Newtonian fluid in a pipe is a parabolic function of position. At the centerline the velocity is a maximum, while at the wall it is a minimum. The shear rate is effectively the slope of the parabolic function line, so it is a minimum at the centerline and a maximum at the wall. Because the shear rate in a pipe or capillary is a function of position, viscometers based around capillary flow are less useful for non-Newtonian materials. For this reason, rotational devices are often used in preference to capillary or tube viscometers. 

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