Newtonian flow rate equation

Another approach to optimization of the screw geometry for melt conveying can he made by using the Newtonian flow rate equation with correction factors for pseudoplastic behavior see Eq. 7.291. If the flight width is neglected this equation can be written as ... 

For optimization of just the helix angle or the channel depth, the results from the Newtonian flow rate equation with correction factors for pseudo-plastic behavior, Eq. 8.54, are more accurate than the results from the modified Newtonian analysis, Eq. 8.36. It should again be noted that simultaneous optimization only makes sense for relatively large positive pressure gradients. When the pressure gradient is negative, the output increases monotonicaiiy with channel depth. In this case, there is no optimum channei depth. 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

Extensive experiments were carried out to investigate the extrusion process. The first attempt to verify experimentally the flow rate equation for Newtonian fluids was made by Rowell and Finlayson [7]. The first experiments to verify the previously discussed velocity profiles for Newtonian fluids were performed by... 

Melt rheometers either impose a fixed flow rate and measure the pressure drop across a die, or, as in the melt flow indexer, impose a fixed pressure and measure the flow rate. Equation (B.5) gives the shear stress, but Eq. (B.IO) requires knowledge of n to calculate the shear strain rate. It is conventional to plot shear stress data against the apparent shear rate y, calculated using n = 1 (assuming Newtonian behaviour). If the data is used subsequently to compute the pressure drop in a cylindrical die, there will be no error. However, if a flow curve determined with a cylindrical die is used to predict... 

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. Hence, formulation of mathematical models for these processes can be based on the equations of non-Newtonian fluid mechanics. The general equations of non-Newtonian fluid mechanics provide expressions in terms of velocity, pressure, stress, rate of strain and temperature in a flow domain. These equations are derived on the basis of physical laws and... 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

Capillary viscometers are useful for measuring precise viscosities of a large number of fluids, ranging from dilute polymer solutions to polymer melts. Shear rates vary widely and depend on the instmments and the Hquid being studied. The shear rate at the capillary wall for a Newtonian fluid may be calculated from equation 18, where Q is the volumetric flow rate and r the radius of the capillary the shear stress at the wall is = r Ap/2L. 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

For Newtonian fluids the dynamic viscosity is constant (Equation 2-57), for power-law fluids the dynamic viscosity varies with shear rate (Equation 2-58), and for Bingham plastic fluids flow occurs only after some minimum shear stress, called the yield stress, is imposed (Equation 2-59). 

In order to overcome the shortcomings of the power-law model, several alternative forms of equation between shear rate and shear stress have been proposed. These are all more complex involving three or more parameters. Reference should be made to specialist works on non-Newtonian flow 14-171 for details of these Constitutive Equations. 

By equating the shear stress from Eqs. (6-42) and (6-4), solving for the velocity gradient, and introducing the result into Eq. (6-7) (as was done for the Newtonian fluid), the flow rate is found to be... 

Like the von Karman equation, this equation is implicit in/. Equation (6-46) can be applied to any non-Newtonian fluid if the parameter n is interpreted to be the point slope of the shear stress versus shear rate plot from (laminar) viscosity measurements, at the wall shear stress (or shear rate) corresponding to the conditions of interest in turbulent flow. However, it is not a simple matter to acquire the needed data over the appropriate range or to solve the equation for / for a given flow rate and pipe diameter, in turbulent flow. 

We will use the Bernoulli equation in the form of Eq. (6-67) for analyzing pipe flows, and we will use the total volumetric flow rate (Q) as the flow variable instead of the velocity, because this is the usual measure of capacity in a pipeline. For Newtonian fluids, the problem thus reduces to a relation between the three dimensionless variables ... 

The inclusion of significant fitting friction loss in piping systems requires a somewhat different procedure for the solution of flow problems than that which was used in the absence of fitting losses in Chapter 6. We will consider the same classes of problems as before, i.e. unknown driving force, unknown flow rate, and unknown diameter for Newtonian, power law, and Bingham plastics. The governing equation, as before, is the Bernoulli equation, written in the form... 

If the velocity had the uniform value u, the momentum flow rate would be mfpu2. Thus for laminar flow of a Newtonian fluid in a pipe the momentum flow rate is greater by a factor of 4/3 than it would be if the same fluid with the same mass flow rate had a uniform velocity. This difference is analogous to the different values of a in Bernoulli s equation (equation 1.14). 

In the case of laminar flow of a Newtonian fluid in a pipe, the velocity profile is given by equation 1.54 so the volumetric flow rate is... 

Equation 3.29 is helpful in showing how the value of the correction factor in the Rabinowitsch-Mooney equation corresponds to different types of flow behaviour. For a Newtonian fluid, n = 1 and therefore the correction factor has the value unity. Shear thinning behaviour corresponds to < 1 and consequently the correction factor has values greater than unity, showing that the wall shear rate yw is of greater magnitude than the value for Newtonian flow. Similarly, for shear thickening behaviour, yw is of a... 

Runnels and Eyman [41] report a tribological analysis of CMP in which a fluid-flow-induced stress distribution across the entire wafer surface is examined. Fundamentally, the model seeks to determine if hydroplaning of the wafer occurs by consideration of the fluid film between wafer and pad, in this case on a wafer scale. The thickness of the (slurry) fluid film is a key parameter, and depends on wafer curvature, slurry viscosity, and rotation speed. The traditional Preston equation R = KPV, where R is removal rate, P is pressure, and V is relative velocity, is modified to R = k ar, where a and T are the magnitudes of normal and shear stress, respectively. Fluid mechanic calculations are undertaken to determine contributions to these stresses based on how the slurry flows macroscopically, and how pressure is distributed across the entire wafer. Navier-Stokes equations for incompressible Newtonian flow (constant viscosity) are solved on a three-dimensional mesh ... 

Concerning the proportionality constant, it involves quantities difficult to determine experimentally, such as the thickness and rate of advance of the craze-bulk interface, the coefficients ef and ne in the constitutive equation of non-Newtonian flow ... 

Thus, we have uz = uz(r), ur = ug = 0 and p = p(z). With this type of velocity field, the only non-vanishing component of the rate-of-deformation tensor is the zr-component. It follows that for the generalized Newtonian flow, rzr is the only nonzero component of the viscous stress, and that Tzr = rZT r). The -momentum equation is then reduced to,... 

To show how the above equations are used, let us consider a disc-shaped cavity of R =150 mm, a gate radius, n, of 5 mm, and a cavity thickness of 2 mm, i.e., h =1 mm. Assuming a Newtonian viscosity /. =6,400 Pa-s and constant volumetric flow rate Q =50 cm3/s predict the position of the flow front, r2, as a function of time, as well as the pressure distribution inside the disc mold. 

In Chapter 5 of this book we derived the equations that govern the pressure flow between two parallel discs for a Newtonian fluid. In a similar fashion, we can derive the equations that govern flow rate, gate pressure, and pressure distributions for disc-shaped cavities filling with a shear thinning fluid. For the equations presented in this section, we assumed a power law viscosity. For the velocity distribution we have... 

Derive the equation for the cumulative residence time distribution, F(t), for the fluid driven by pressure flow inside a slit. Assume a volumetric flow rate of Q and a Newtonian viscosity of /t. Use the notation used in the schematic of Fig. 6.78. 

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