External Flows

Flow of fluid over the surface is characterized by the formation of a hydro-dynamic or velocity boundary layer, which is defined as the thin layer of fluid over which the velocity of the fluid varies from no-slip zero velocity at the surface to the outer stream velocity over the thickness of the boundary layer y = 5. Because of the effect of viscosity, fluid flow slows down near the stationary solid surface and maintains the no-slip fluid-solid interface boundary conditions. The flow is assumed to be viscous within the boundary layer and inviscid outside the boimdary layer. The boundary layer thickness increases in the downstream x-direction and results in a varying x-component velocity profile u(y) as shown in the figure. 

Hydrodynamic boundary layer for flow over a flat plate. 

The fluid shear stress at the wall is given by Newton s law as 

The skin friction decreases in the flow direction as the boundary layer thickness increases in the downstream x-direction. The wall shear stress and hence the skin friction can be obtained from the known velocity field, which is defined by the continuity and momentum equations of fluid motion. The skin frictions are generally expressed in the form of a correlation as a function of characteristics flow Reynolds number as 

External flow is characterized as laminar or turbulent on the basis of the critical Reynolds number  

We learned in Chapter 1 that the heat transfer involved with thermal systems is composed of a convective internal resistance, conductive resistance(s), and a convective external resistance, 

Since the flow conditions strongly influence the heat transfer (recall Nu Re112 for laminar flows and Nu Re0- for turbulent flows over a flat plate not involving liquid metals), let us recall from fluid mechanics the results of experimental observations on flow around a cylinder, which are sketched in Kg. 6.7. We learn from this figure that the flow around a cylinder may assume different forms, depending on the Reynolds number. The separation, beginning with the second sketch from the top, is associated 

For most problems of technological importance, however, we usually need the average heat transfer coefficient. For example, the experimental data for air flowing 

For flow over spheres, experimental data is correlated by 

In the next section a procedure for the evaluation of the heat transfer coefficient is given in terms of five computational steps suitable to one of the correlations of Table 62. 

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Flow Past Deformable Bodies. The flow of fluids past deformable surfaces is often important, eg, contact of Hquids with gas bubbles or with drops of another Hquid. Proper description of the flow must allow for both the deformation of these bodies from their shapes in the absence of flow and for the internal circulations that may be set up within the drops or bubbles in response to the external flow. DeformabiUty is related to the interfacial tension and density difference between the phases internal circulation is related to the drop viscosity. A proper description of the flow involves not only the Reynolds number, dFp/p., but also other dimensionless groups, eg, the viscosity ratio, 1 /p En tvos number (En ), Api5 /o and the Morton number (Mo),giJ.iAp/plG (6). 

Correlations for Convective Heat Transfer. In the design or sizing of a heat exchanger, the heat-transfer coefficients on the inner and outer walls of the tube and the friction coefficient in the tube must be calculated. Summaries of the various correlations for convective heat-transfer coefficients for internal and external flows are given in Tables 3 and 4, respectively, in terms of the Nusselt number. In addition, the friction coefficient is given for the deterrnination of the pumping requirement. 

Table 4. Correlations for Convective Heat-Transfer and Friction Coefficients for External Flow...

In chemical process applications, one-dimensional gas flows through nozzles or orifices and in pipelines are the most important apphcations of compressible flow. Multidimensional external flows are of interest mainly in aerodynamic applications. 

This equation describes the change of population in a well-mixed system and is often used to model fully mixed crystallization and precipitation processes. If the system is imperfectly mixed, however, then the more complicated equation 2.88 can be used provided that the external flow field can be calculated e.g. by use of CFD (see later). 

Externally pressurized gas journal bearings have the same principle of operation as hydrostatic liquid-lubricated bearings. Any clear gas can be used, but many of the design charts are based on air. There are three forms of external flow restrictors in use with these bearings pocketed (simple) orifice, unpocketed (annular) orifice, and slot. 

Wu WT, Yang YM (1992) Enhanced boiling heat transfer by surfactant additives. In Pool and External Flow Boiling. ASME, New York, pp 361-366... 

This chapter considers the first group of instabilities and introduces the analysis of processes implying an interaction with external flow-field perturbahons. This is exemplified by investigations of coupling between pressure waves and plane flames and also between an external acceleration field and flame fronts. The coupling between flow perturbations and flames giving rise to heat release unsteadiness and coupling with acoushc modes is considered in Chapter 5.2, which deals with the relationship between perturbed flame dynamics and radiated acoustic field, a fundamental process of thermo-acoustic instabilities. 

This response time should be compared to the turbulent eddy lifetime to estimate whether the drops will follow the turbulent flow. The timescale for the large turbulent eddies can be estimated from the turbulent kinetic energy k and the rate of dissipation e, Xc = 30-50 ms, for most chemical reactors. The Stokes number is an estimation of the effect of external flow on the particle movement, St = r /tc. If the Stokes number is above 1, the particles will have some random movement that increases the probability for coalescence. If St 1, the drops move with the turbulent eddies, and the rates of collisions and coalescence are very small. Coalescence will mainly be seen in shear layers at a high volume fraction of the dispersed phase. 

Schematic view of an arrangement that allows a sustained redox reaction accompanied by an external flow of electrons.

Once a collision occurs, the liquid between the drops is squeezed, forming a film. As the drops are continually squeezed by the external flow field, the drops rotate as a dumbbell and the film drains. At some distance h0, the drops begin to influence each other and their rate of approach, dh/ dt, decreases and is now governed by the rate of film drainage. 

Bar-Cohen, A., 1992, Hysteresis Phenomena at the Onset of Nucleate Boiling, Engineering Foundation Conf. on Pool and External Flow Boiling, Santa Barbara, CA, pp. 1-14. (4)... 

Dhir, V. K., 1990, Nucleate and Transition Boiling Heat Transfer under Pool and External Flow Conditions, Proc. 9th Int. Heat Transfer Conf, vol. 1, pp. 129 155 see also Int. J. Heat Fluid Flow 12(4) 290. (2)... 

Dhir, V. K., 1992, Some Observations from Maximum Heat Flux Data Obtained on Surfaces Having Different Degrees of Wettability, in Pool and External Flow Boiling, V. K. Dhir and A. E. Bergles, Eds., pp. 185-192, ASME, New York. (2)... 

Let us consider the problem specification discussed in Section II,C. For a network with M external flows (inputs and outputs), the specification introduces the following additional equations ... 

One of the interesting consequences of changing specifications is the effect on the equation structure. With formulations C and D, the occurrence matrix is symmetric. But if external flows, w, and model parameters, / , are introduced as the unknown variables, the symmetry may be destroyed. One way of preserving the local symmetry is to augment the system of equations and to bifurcate the variables in terms of state and design variables (M2). 

A well-known class of techniques for reducing the number of iterates is the use of tearing (L4). We shall illustrate this procedure by way of an example taken from Carnahan and Christensen (C3). Let us consider the two-loop network shown in Fig. 5 and assume that formulation A is used. To abbreviate the notation let us denote the material balance around vertex i [Eq. (35)] by fi = 0 and the model of the element [Eq. (36)] by fu — 0. Then assuming all external flows and one vertex pressure, p, say, are specified, we have a set of 12 equations that must be solved simultaneously. But if we now assume a value for ql2, the remaining equations may be solved sequentially one at a time to yield the variables in the following... 

More recently, Cheng (C7) showed that to apply tearing effectively one must take into consideration the topology of the network and the nature of problem specifications. For instance, if the network consists of a number of cyclic subnetworks imbedded in an acyclic framework and if all the external flows and one reference pressure are specified, the flows and the pressures external to the cyclic subnetworks may be computed sequentially and only... 

The solid line shows the flow inside the valve when external flow is through the core, and the dashed line represents the flow during an experiment that uses the capillary tube in place of the core. 

Oscillatory shear experiments are the preferred method to study the rheological behavior due to particle interactions because they directly probe these interactions without the influence of the external flow field as encountered in steady shear experiments. However, phenomena that arise due to the external flow, such as shear thickening, can only be investigated in steady shear experiments. Additionally, the analysis is complicated by the different response of the material to shear and extensional flow. For example, very strong deviations from Trouton s ratio (extensional viscosity is three times the shear viscosity) were found for suspensions [113]. 

Drag coefficient Cd c - Fd Cd 1PV2A Fd =drag force A = area normal to flow (Drag stress)/ ( momentum flux) External flows... 

Reynolds number flows /vRe N -°Vp /vRe — pV2 pV/D AQp izDp PV2 Tw/8 Pipe flow rw =wall stress (inertial momentum flux)/ (viscous momentum flux) Pipe/internal flows (Equivalent forms for external flows)... 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

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