Velocity and Temperature Profiles

Velocity profiles and temperature profiles in extruder dies are intimately related because of the high polymer melt viscosity and because the melt viscosity is temperature dependent. It is important to understand and appreciate this interrelationship in order to understand the die forming process and the variables that influence this process. The relationship between velocity and temperature profiles can be illustrated by considering the down-channel velocity profile in a circular die. Typical velocity profiles are shown in Fig. 7.106 for several values of the power law index. 

Velocity versus radius for power law fluid at various values of the power law index n 

As a result of the velocity gradients, there will be heat generation in the fluid from the viscous dissipation of energy. In rectilinear flow, the rate of energy dissipation per unit volume is given by  

This is a simpiiiied version of the general expression for energy dissipation, Eq. 5.5d. If the fluid can he described by the power law equation, then Eq. 7.436 becomes  

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This involves knowledge of chemistry, by the factors distinguishing the micro-kinetics of chemical reactions and macro-kinetics used to describe the physical transport phenomena. The complexity of the chemical system and insufficient knowledge of the details requires that reactions are lumped, and kinetics expressed with the aid of empirical rate constants. Physical effects in chemical reactors are difficult to eliminate from the chemical rate processes. Non-uniformities in the velocity, and temperature profiles, with interphase, intraparticle heat, and mass transfer tend to distort the kinetic data. These make the analyses and scale-up of a reactor more difficult. Reaction rate data obtained from laboratory studies without a proper account of the physical effects can produce erroneous rate expressions. Here, chemical reactor flow models using matliematical expressions show how physical... 

Fully developed velocity and temperature profiles (i.e. the limiting Nusselt case) ... 

Figure 5. Velocity and temperature profiles for the cut-off Lennard-Jones potential. The system consists of 1152 atoms enclosed in a box of side length equal to 32 6 6.

Solve the nondimensional steady-state problem and plot the radial nondimensional velocity and temperature profiles versus the nondimensional radius. 

Solve the transient problem and plot nondimensional velocity and temperature profiles at some representative nondimensional times. 

Fig. 6.14 Nondimensional velocity and temperature profiles for an ideal rotating disk. The temperature profiles, and hence the surface heat transfer, vary greatly with Prandtl number. These solutions were computed for an incompressible flow using constant properties. For the Pr = 0.1 case, the axial domain extended to z = 26.

Fig. 6.16 Nondimensional velocity and temperature profiles in a finite gap with a rotating surface. In all cases the Prandtl number is 0.7 and the forced-flow Reynolds number is Rey = 100. The profiles are illustrated for four values of the rotation Reynolds number Re = G1L2/v. The viscous boundary layers are close to the surface. With the exception of the axial velocity, the plots show the range 0 < z < 0.2, with the small insets illustrating the entire gap 0 < z < 1.

Fig. 7.1 Illustration of the velocity and temperature profiles in the entry region of a cylindrical channel. Gases enter the channel with uniform velocity and temperature profiles. The no-slip condition causes a zero velocity at the wall, and the heat transfer from the hot wall increases the gas temperature.

At the channel inlet, the radial-velocity v profile cannot be specified independently of the axial-velocity and temperature profiles. Explain why this is so. Explain the implications concerning consistent initial conditions from the differential-algebraic-equation perspective. Develop an algorithm to determine the consistent initial v profile, given the u profile. 

Develop a system of equations in nondimensional form that can be solved to determine nondimensional velocity and temperature profiles function nondimensional length (z = z/D) along the channel. Define Reynolds and Prandtl numbers in the ordinary way as... 

Fig. 17.4 Simulation of stoichiometric methane-air flames approaching a stagnation surface. The top panels show the axial velocity and temperature profiles. The lower panels show details of the species composition with the thin flame-front.

Systematically find the expression for the temperature distribution and the Nusselt number for laminar flow between two large parallel plates in the region of fully developed velocity and temperature profiles for a uniformly applied wall heat flux. 

A constant property fluid flows between two horizontal, semiinfinite, parallel plates, kept at a distance 2m apart. The upper plate is at a constant temperature Ti and the lower plate is at a constant temperature T2. Consider the fully developed velocity and temperature profiles region for laminar flow. Include viscous dissipation. Find the heat flux to each of the plates. 

The values of 0 (O) and f (0) are obtained from the solutions, and these are provided in Table 9.1. The velocity and temperature profiles, compared with the experiments of Schmidt and Beckmann, show good agreement at Pr = 0.733. For the dimensionless velocity, the maximum values of the distributions increase with decrease in Pr. For the dimensionless temperature, at any t the value of 0 increases with a decrease in Pr. 

For fully developed duct flows in which it can be assumed that fhe fluid properties are constant, the form of the velocity and temperature profiles do not change with distance along the duct, i.e., considering the variables as defined in Fig. 2.12. if the velocity and temperature profiles are expressed in the form... 

In the present case, the velocity and temperature profiles at the end of the plate, i.e., at x = 0.2 m, are required, so, using the value of v given previously, it follows that at die trailing edge of the plate ... 

In the preceding sections, the solution for boundary layer flow over a flat plate wav obtained by reducing the governing set of partial differential equations to a pair of ordinary differential equations. This was possible because the velocity and temperature profiles were similar in the sense that at all values of x, (u u ) and (Tw - T)f(Tw - T > were functions of a single variable, 17, alone. Now, for flow over a flat plate, the freestream velocity, u, is independent of x. The present section is concerned with a discussion of whether there are any flow situations in which the freestream velocity, u 1, varies with Jr and for which similarity solutions can still be found [1],[10]. 

It should be realized that there is no real purpose in comparing this velocity profile with that given by the exact similarity solution since the integral equation method does not seek to accurately predict the details of the velocity and temperature profiles. The method seeks rather, by satisfying conservation of mean momentum and energy, to predict with reasonable accuracy the overall features of the flow. 

In evaluating the integral, care has to be exercised because the velocity and temperature profiles are really discontinuously described, one relation being used inside the boundary layer and another outside the boundary layer. The velocity profile, for example, is actually described by ... 

When Pr is equal to 1, A is, of course, equal to 1 because the form of the assumed velocity and temperature profiles are identical and when Pr is equal to one the momentum and energy integral equations have the same form for flow over a flat plate. 

Now, for St < 5, i.e., A < 1. it was shown [see the derivation of Eq. (3.151)] that the energy integral equation becomes the following when third-order polynomials are assumed to describe the velocity and temperature profiles ... 

Air flows at a velocity of 9 m/s over a wide flat plate that has a length of 6 cm in the flow direction. The air ahead of the plate has a temperature of 10°C while the surface of the plate is kept at 70°C. Using the similarity solution results given in this chapter, plot the variation of local heat transfer rate in W/m2 along the plate and the velocity and temperature profiles in the boundary layer on the plate at a distance of 4 cm from the leading edge of the plate. Also calculate the mean heat transfer rate from the plate. 

Attention is being restricted to fully developed flow in the present section which means that the forms of the velocity and temperature profiles are not changing with distance along the pipe, i.e., that... 

In fully developed flow in a pipe and in a plane duct, as discussed above, the velocity and temperature profiles could be expressed in terms of a single cross stream coordinate, i.e., in terms of either r/R or y/W. In many other situations, however, the cross-sectional shape of the duct is such that the profiles will depend on two cross stream coordinates, e.g., consider fully developed flows in the ducts with the cross-sectional shapes shown in Fig. 4.7. 

The numerical solution to these equations for the case where both the velocity and temperature profiles are developing and where the wall temperature is uniform will be considered in the present section. 

The velocity and temperature profiles are assumed to be symmetrical about the center line of the pipe and the radial velocity component, v, is therefore zero on the center line. The boundary conditions on the solution on the center line are therefore ... 

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