Temperature profile similarity solution

Boundary layer similarity solution treatments have been used extensively to develop analytical models for CVD processes (2fl.). These have been useful In correlating experimental observations (e.g. fi.). However, because of the oversimplified fiow description they cannot be used to extrapolate to new process conditions or for reactor design. Moreover, they cannot predict transverse variations In film thickness which may occur even In the absence of secondary fiows because of the presence of side walls. Two-dimensional fully parabolized transport equations have been used to predict velocity, concentration and temperature profiles along the length of horizontal reactors for SI CVD (17,30- 32). Although these models are detailed, they can neither capture the effect of buoyancy driven secondary fiows or transverse thickness variations caused by the side walls. Thus, large scale simulation of 3D models are needed to obtain a realistic picture of horizontal reactor performance. 

This type of solution is called a similarity solution where the affected domain of the problem is proportional to yfat (the growing penetration depth), and the dimensionless temperature profile is similar in time and identical in the i) variable. Fluid dynamic boundary layer problems have this same character. 

Having established the form of the velocity profile, attention can now be turned to the determination of the temperature profile, i.e., to the solution of Eq. (3.3). Because the wall temperature is uniform in the situation being considered, it is logical to assume that the temperature profiles are similar in the same sense as the velocity profiles. For this reason, the following dimensionless temperature function is introduced... 

Similarity solutions for a few cases of flow over a flat plate where the plate temperature varies with x in a prescribed manna can also be obtained. In an such cases the solution for the velocity profile is, of course, not affected by the boundary condition... 

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]. 

Having established that similarity solutions for the velocity profile can be found for certain flows involving a varying ffeestream velocity, attention must now be turned to the solutions of the energy equation corresponding to these velocity solutions. The temperature is expressed in terms of the same nondimensional variable that was used in obtaining the flat plate solution, i.e., in terms of 8 = (Tw - T)f(Tw -Tt) and it is assumed that 0 is also a function of ij alone. Attention is restricted to flow over isothermal surfaces, i.e., with Tw a constant, and T, of course, is also constant. 

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. 

It seems reasonable to assume that similar temperature profiles will also exist when viscous dissipation is important. Attention will first be given to the adiabatic wall case. If the wall is adiabatic and viscous dissipation is neglected, then the solution to the energy equation will be T = Ti everywhere in die flow. However, when viscous dissipation effects are important, the work done by the viscous forces leads to a rise in fluid temperature in the fluid. This temperature will be related to the kinetic energy of the fluid in the freestream flow, i.e., will be related to u /2cp. For this reason, the similarity profiles in the adiabatic wall case when viscous dissipation is important are assumed to have the form ... 

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. 

Infrared measurements of extent of cure under conditions similar to the TICA experiments were conducted. ATS was cast from methylene chloride solution onto KBr windows and, after vacuum evaporation of all solvent, the KBr windows were put into the Rheometrics RMS environmental chamber and were subjected to a temperature profile under nitrogen identical to the mechanical measurement experiments (2 C/min). The windows were removed one at a time at various temperatures and IR spectra were taken at room temperature. 

Temperature profiles for flow over an isothermal flat plate are similar, just like the velocity profiles, and thus we expect a similarity solution for temperature to exist. Further, the thickness of the thermal boundar y layer is proportional to /i. T/V,just like the thickness of the velocity boundary layer, and thus the similarity variable is also t), and 0 = 6(ri). Using thechain rule and substituting the It and tt e.xpres ions from Eqs. 6-46 and 6—47 into the energy equation gives... 

Here, the last two equations define the flow rate and the mean residence time, respectively. This formulation is an optimal control problem, where the control profiles are q a), f(a), and r(a). The solution to this problem will give us a lower bound on the objective function for the nonisothermal reactor network along with the optimal temperature and mixing profiles. Similar to the isothermal formulation (P3), we discretize (P6) based on orthogonal collocation (Cuthrell and Biegler, 1987) on finite elements, as the differential equations can no longer be solved offline. This type of discretization leads to a reactor network more... 

This completes the similarity solution for arbitrary X(x). The temperature profile given by (11-48) is plotted in Fig. 11-4. 

Let us consider the case of an arbitrary viscoplastic fluid with yield stress To (similar results for nonlinear viscous fluids correspond to to = 0). To obtain the temperature profile, we proceed as follows. First, in the near-wall shear region 0 < < h- ho, where ho = toL/AP, we solve Eq. (6.5.2) with the boundary conditions (6.5.1). Then in the quasisolid region h - ho 2 < h, we solve Eq. (6.5.2) with - 0 under the boundary condition (6.5.3). Finally, we match the two solutions on the common boundary = ho. This procedure results in the following temperature distribution in the channel ... 

The equations to be solved are similar to those in the previous section with some minor differences due the change in geometry (parallel-plate microchannel versus microtube). In the solution, slip boundary conditions given in Eqs. (I) and (2) are applied and finite element method is used to solve for the velocity profile and the temperature distribution. Then, from the temperature profile, the local Nu is determined. 

If attention is restricted to the vicinity of the bifurcation point, then a nonlinear perturbation analysis can be developed for describing analytically the nature of the pulsating mode [111]. In effect, the difference between A and its bifurcation value is treated as a small parameter, say e, and oscillatory solutions for temperature profiles are calculated as perturbations about the steady solution in the form of a power series in /e. The departure of the oscillation frequency from its value at bifurcation is expressed in the same type of series. The methods of analysis possess a qualitative similarity to those of the shock-instability analysis discussed at the end of the previous section. The results exhibit the same general behavior that was found from the numerical integrations [109] for conditions near bifurcation. 

Comparison of this model with plug flow continuum models has shown in a limited number of cases a remarkable similarity in the concentration and temperature profiles. In fact, it has been shown that the general features of the solution are very similar to experimental results obtained by Padberg and Wicke (1967). 

The CCD is the second most important microstructural distribution in polyolefins. Differently from the MWD, the CCD carmot be determined directly only the distribution of crystallization temperatures (CTD) in solution can be measured and one can try to relate this distribution to the CCD using a calibration curve. Two techniques are commonly used to determine the CTD or CCD of polyolefins TREF and Crystaf. Both operate based on the same principle chains with more defects (more comonomer molecules or stereo-and/or regioirregularities) have lower crystallization temperatures than chains with fewer defects. Figure 2.11 compares the TREF and Crystaf profiles of an ethylene/1-butene copolymer made with a heterogeneous Ziegler-Natta catalyst. Notice that they have very similar shapes the Crystaf curve is shifted toward lower temperatures because it is measured as the polymer chains crystallize, while the TREF curve is determined as the polymer chains dissolve (melt) and are eluted from the TREF column, as explained in the next few paragraphs. 

Surface-directed spinodal decomposition was first observed in an isotopic polymer blend (Jones et al. 1991) thin films of a mixture of poly(ethylene-propylene) and its deuterated analogue were annealed below the upper critical solution temperature and the depth profiles measured using forward recoil spectrometry, to reveal oscillatory profiles similar to those sketched in figure 5.30. Similar results have now been obtained for a number of other polymer blends, including polystyrene with partially brominated polyst)u-ene (Bruder and Brenn 1992), polystyrene with poly(a-methyl styrene) (Geoghegan et al. 1995) and polystyrene with tetramethylbisphenol-A polycarbonate (Kim et al. 1994), suggesting that the phenomenon is rather general. 

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