Similarity solutions natural convection

A numerical solution to the laminar boundary layer equations for natural convection can be obtained using basically the same method as applied to forced convection in Chapter 3. Because the details are similar to those given in Chapter 3, they will not be repeated here. 

Eqs. (8.120) and (8.121) represent the limiting boundary layer solution for natural convective flow through a vertical plane duct. For the particular case of Pr = 0.7, the similarity solution for natural convective boundary layer flow on a vertical plate... 

Using the similarity solution results, derive an expression for the maximum velocity in the natural convective boundary layer on a vertical flat plate. At what position in the boundary layer does this maximum velocity occur ... 

It will be seen from the results given by the similarity solution that the velocities are very low in natural convective boundary layers in fluids with high Prandtl numbers. In such circumstances, the inertia terms (i.e., the convective terms) in the momentum equation are negligible and the boundary layer momentum equation for a vertical surface effectively is ... 

In the case of airlift reactors, the flow pattern may be similar to that in bubble columns or closer to that two-phase flow in pipes (when the internal circulation is good), in which case the use of suitable correlations developed for pipes may be justified [55]. Blakebrough et al. studied the heat transfer characteristics of systems with microorganisms in an external loop airlift reactor and reported an increase in the rate of heat transfer [56], In an analytical study, Kawase and Kumagai [57] invoked the similarity between gas sparged pneumatic bioreactors and turbulent natural convection to develop a semi-theoretical framework for the prediction of Nusselt number in bubble columns and airlift reactors the predictions were in fair agreement with the limited experimental results [7,58] for polymer solutions and particulate slurries. 

Conduction is treated from both the analytical and the numerical viewpoint, so that the reader is afforded the insight which is gained from analytical solutions as well as the important tools of numerical analysis which must often be used in practice. A similar procedure is followed in the presentation of convection heat transfer. An integral analysis of both free- and forced-convection boundary layers is used to present a physical picture of the convection process. From this physical description inferences may be drawn which naturally lead to the presentation of empirical and practical relations for calculating convection heat-transfer coefficients. Because it provides an easier instruction vehicle than other methods, the radiation-network method is used extensively in the introduction of analysis of radiation systems, while a more generalized formulation is given later. 

In all early experiments including the one by Poll (1979), existence of attachment-line vortical structures is well established. It is thus natural to investigate the sub-critical instability by looking at the role of convecting vortical structures in explaining LEG from the solution of two-dimensional Navier-Stokes equation in the attachment-line plane itself, similar to the vortex-induced instability problem studied in Lim et al. (2004) and Sengupta et al. (2003) for zero pressure gradient flow. 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

The high sensitivity of the Allendoerfer cell makes it of great value in the detection of unstable radicals but, for the study of the kinetics and mechanism of radical decay, the use of a hydrodynamic flow is required. The use of a controlled, defined, and laminar flow of solution past the electrode allows the criteria of mechanism to be established from the solution of the appropriate convective diffusion equation. The uncertain hydrodynamics of earlier in-situ cells employing flow, e.g. Dohrmann [42-45] and Kastening [40, 41], makes such a computational process uncertain and difficult. Similarly, the complex flow between helical electrode surface and internal wall of the quartz cell in the Allendoerfer cell [54, 55] means that the nature of the flow cannot be predicted and so the convective diffusion equation cannot be readily written down, let alone solved Such problems are not experienced by the channel electrode [59], which has well-defined hydrodynamic properties. Compton and Coles [60] adopted the channel electrode as an in-situ ESR cell. 

Ultrafiltration is a membrane process whose nature lies between nanofiltration and microfiltration. The pore sizes of the membranes used range from 0.05 um (on the microfiltration side) to 1 am (on the nanofiltration side). Ultrafiltration is typically used to retain macromolecules and colloids from a solution, the lower limit being solutes with molecular weights of a few thousand Daltons. Ultrafiltration and microfiltration membranes can both be considered as porous membranes where rejection is determined mainly by the size and shape of the solutes relative to the pore size in the membrane and where the transport of solvent is directly proportional to the applied pressure. Such convective solvent flow through a porous membrane can be described by the Kozeny-Carman equation (see eq. VI - 27) for example. In fact both microfiltration and ultrafiltration involve similar membrane processes based on the same separation principle. However, an important difference is that ultrafiltration membranes have an asymmetric structure with a much denser toplayer (smaller pore size and lower surface porosity) and consequently a much higher hydrodynamic resistance. 

A natural early goal of FP researchers was to develop a reactor in which the monomer-initiator solution was pumped in such that the product would continuously flow out, without the input of heat. Attempts were made with reactors of tylindrical and spherical geometries. Zhizhin and Segal performed a linear stability analysis of a reactor consisting of two concentric cylinders. A radial, axisymmetric front was supposed because the monomer/initiator would be pumped through the permeable inner cylinder. The viscous reacted polymer was supposed to flow out through the outer permeable cylinder. No buoyancy-driven convection was included. They found that if the resistance of the outer boundary was small, the front would become hydrodynamicaUy unstable. They also considered a reactor with concentric spheres and found similar results. 

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