Fluid Flow and Momentum Transfer

There exists a conspicuous analogy between heat transfer and mass transfer. Hence, Equation 2.1 can be rewritten as 

The negative sign indicates that momentum is transferred down the velocity gradient. The proportionality constant n (Pa s) is called molecular viscosity or simply viscosity, which is an intensive property. The unit of viscosity in CGS (centimeter-gram-second) units is called poise (gcm s ). From Equation 2.4 we obtain 

Fermentation broths - that is, fermentation medium containing microorganisms - often behave as non-Newtonian liquids, and in many cases their apparent viscosities vary with time, notably during the course of fermentation. 

Fluids that show elasticity to some extent are termed viscoelastic fluids, and some polymer solutions demonstrate such behavior. Elasticity is the tendency of a substance or body to return to its original form, after the applied stress that caused strain (i.e., a relative volumetric change in the case of a polymer solution) has been removed. The elastic modulus (Pa) is the ratio of the applied stress (Pa) to strain (-). The relaxation time (s) of a viscoelastic fluid is defined as the ratio of its viscosity (Pa s) to its elastic modulus. 

The following experimental data were obtained with use of a rotational viscometer for an aqueous solution of carboxylmethyl cellulose (CMG) containing 1.3 g CMC per 100 cm solution. 

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In laminar flow, heat transfer occurs only by conduction, as there are no eddies to carry heat by convection across an isothermal surface. The problem is amenable to mathematical analysis based on the partial differential equations for continuity, momentum, and energy. Such treatments are beyond the scope of this book and are given in standard treatises on heat transfer, Mathematical solutions depend on the boundary conditions established to define the conditions of fluid flow and heat transfer. When the fluid approaches the heating surface, it may have an already completed hydrodynamic boundary layer or a partially developed one. Or the fluid may approach the heating surface at a uniform velocity, and both boimdary layers may be initiated at the same time. A simple flow situation where the velocity is assumed constant in all cross sections and tube lengths is called... 

When a liquid is forced through a microchannel under hydrostatic pressure, the ions in the mobile region of the electrodischarge layer (EDL) are carried toward one end. The motion of the ions in the diffuse mobile layer affects the bulk of the liquid flow via momentum transfer due to viscosity. In macroscale flows, the interfacial electrokinetic effects are negligible since the thickness of EDL is very small compared to the hydraulic diameter of the duct. However, in microscale flow, the EDL thickness is comparable to the hydraulic diameter, and the EDL effects must be considered during the analysis of fluid flow and heat transfer. The EDL effect on the flow depends on the Debye number defined as De = =. When De 1, EDL... 

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

There are many standard texts of fluid flow, e.g. Coulson Richardson,1 Kay and Neddermann2 and Massey.3 Perry4,5 is also a useful reference source of methods and data. Schaschke6 presents a large number of useful worked examples in fluid mechanics. In many recent texts, fluid mechanics or momentum transfer has been treated in parallel with the two other transport or transfer processes, heat and mass transfer. The classic text here is Bird, Stewart and Lightfoot.7... 

Computational fluid dynamics involves the analysis of fluid flow and related phenomena such as heat and/or mass transfer, mixing, and chemical reaction using numerical solution methods. Usually the domain of interest is divided into a large number of control volumes (or computational cells or elements) which have a relatively small size in comparison with the macroscopic volume of the domain of interest. For each control volume a discrete representation of the relevant conservation equations is made after which an iterative solution procedure is invoked to obtain the solution of the nonlinear equations. Due to the advent of high-speed digital computers and the availability of powerful numerical algorithms the CFD approach has become feasible. CFD can be seen as a hybrid branch of mechanics and mathematics. CFD is based on the conservation laws for mass, momentum, and (thermal) energy, which can be expressed as follows ... 

Chemical engineering processes involve the transport and transfer of momentum, energy, and mass. Momentum transfer is another word for fluid flow, and most chemical processes involve pumps and compressors, and perhaps centrifuges and cyclone separators. Energy transfer is used to heat reacting streams, cool products, and run distillation columns. Mass transfer involves the separation of a mixture of chemicals into separate streams, possibly nearly pure streams of one component. These subjects were unified in 1960 in the first edition of the classic book. Transport Phenomena (Bird et al., 2002). This chapter shows how to solve transport problems that are one-dimensional that is, the solution is a function of one spatial dimension. Chapters 10 and 11 treat two- and three-dimensional problems. The one-dimensional problems lead to differential equations, which are solved using the computer. 

Gas phase viscosity data, iTq, are used in the design of compressible fluid flow and unit operations. For example, the viscosity of a gas is required to determine the maximum permissible flow through a given process pipe size. Alternatively, the pressure loss of a given flowrate can be calculated. Viscosity data are needed for the design of process equipment involving heat, momentum, and mass transfer operations. The gas viscosity of mixtures is obtained from data for the individual components in the mixture. 

The prevalence of pipe flows in engineering (heating, cooling, power plants, water transport, etc.) makes pipe flow the most important application of internal flows. Because of this importance, there exist a number of correlations of experimental data on pipe flow. Before listing these correlations, however, let us recall Eq. (6.20), obtained from the analogy between heat and momentum transfer. All of the physical properties associated with the dimensionless numbers of this equation depend on the fluid temperature. Therefore a reference temperature is needed for the evaluation of the properties. A commonly used temperature for this purpose is the bulk temperature 7j, associated with the enthalpy flow in the first law (recall of Eq. (1.10)),... 

The situation is analogous to momentum flux, where the relative Importance of turbulent shear to viscous shear follows the same general pattern. Under certain ideal conditions, the correspondence between heat flow and momentum flow is exact, and at any specific value of rjr the ratio of heat transfer by conduction to that by turbulence equals the ratio of momentum flux by viscous forces to that by Reynolds stresses. In the general case, however, the correspondence is only approximate and may be greatly in error. The study of the relationship between heat and momentum flux for the entire spectrum of fluids leads to the so-called analogy theory, and the equations so derived are called analogy equations. A detailed treatment of the theory is beyond the scope of this book, but some of the more elementary relationships are considered. 

For a problem involving fluid flow and simultaneous heat and mass transfer, equations of continuity, momentum, energy, and chemical species (Eqs. 1.41, 1.44, 1.54, and 1.63) are a formidable set of partial differential equations. There are four independent variables three space coordinates (say, x, y, z) and a time coordinate t. 

Metzner and Friend [73] measured turbulent heat transfer rates with aqueous solutions of Carbopol, corn syrup, and slurries of Attagel in circular-tube flow. They developed a semi-theoretical correlation to predict the Stanton number for purely viscous fluids as a function of the friction factor and Prandtl number, applying Reichardt s general formulation for the analogy between heat and momentum transfer in turbulent flow ... 

Equation 20 shows that a porous medium is permeative, that is, a shear factor exists to account for the microscopic momentum loss. Our preliminary study recently reveals that, however, a porous medium is not only permeative but dispersive as well. The dispersivity of a porous medium has been traditionally characterized through heat transfer (in a single- or multifluid flow) and mass transfer (in a multifluid flow) studies. For an isothermal single-fluid flow, the dispersivity of a porous medium is characterized by a flow strength and a porous medium property-de-pendent apparent viscosity. For simplicity, we discuss the single-fluid flow behavior in this chapter without considering the dispersivity of the porous medium. 

In setting down the conservation equations, we considered only fluids of uniform and homogeneous composition. Here we examine how these conservation equations change when two or more species are present and when chemical reactions may also take place. In a multicomponent mixture a transfer of mass takes place whenever there is a spatial gradient in the mixture proportions, even in the absence of body forces that act differently upon different species. In fluid flows the mass transfer will generally be accompanied by a transport of momentum and may further be combined with a transport of heat. 

Gravitational settling of particle in fluid Flow of viscoelastic fluid Pressure and momentum in fluid Unsteady state heat transfer/mass transfer Fluid flow with free surface Gravitational settling of particle in fluid Heat transfer by natural convection Heat transfer to fluid in tube Flow of fluid exhibiting yield stress... 

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