Momentum fluid flow, transfer

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. 

As noted previously, mixing in highly viscous liquids is slow both at the molecular scale, on account of the low values of diffusivity, as well as at the macroscopic scale, due to poor bulk flow. Whereas in low viscosity liquids momentum can be transferred from a rotating impeller through a relatively large body of fluid, in highly viscous liquids only... 

Because the mechanisms governing mass transfer are similar to those involved in both heat transfer by conduction and convection and in momentum transfer (fluid flow), quantitative relations exist between the three processes, and these are discussed in Chapter 12. There is generally more published information available on heat transfer than on mass transfer, and these relationships often therefore provide a useful means of estimating mass transfer coefficients. 

Consider the fluid to be flowing in a direction parallel to the surface (X-direction) and for momentum and heat transfer to be taking place in a direction at right angles to the surface (T-direction positive away from surface). 

Obtain the Taylor-Prandtl modification of the Reynolds analogy between momentum and heat transfer and write down the corresponding analogy for mass transfer. For a particular system, a mass transfer coefficient of 8,71 x 10 8 m/s and a heat transfer coefficient of 2730 W/m2 K were measured for similar flow conditions. Calculate the ratio of the velocity in the fluid where the laminar sub layer terminates, to the stream velocity. 

The momentum equation (the Navier-Stokes equation) for fluid flow (De Groot and Mazur, 1962) is complicated and difficult to solve. It is the subject of fluid mechanics and dynamics and is not covered in this book. When fluid flow is discussed in this book, the focus is on the effect of the flow (such as a flow of constant velocity, or boundary flow) on mass transfer, not the dynamics of the flow itself. 

In chemical engineering, the terms transfer of heat, mass, and momentum are referred to as the transport phenomena. The heating or cooling of fluids is a case of heat transfer, a good example of mass transfer being the transfer of oxygen from air into the culture media in an aerobic fermentor. When a fluid flows through a conduit, its pressure drops because of friction due to transfer of momentum, as shown later. 

The transfer of heat and/or mass in turbulent flow occurs mainly by eddy activity, namely the motion of gross fluid elements that carry heat and/or mass. Transfer by heat conduction and/or molecular diffusion is much smaller compared to that by eddy activity. In contrast, heat and/or mass transfer across the laminar sublayer near a wall, in which no velocity component normal to the wall exists, occurs solely by conduction and/or molecular diffusion. A similar statement holds for momentum transfer. Figure 2.5 shows the temperature profile for the case of heat transfer from a metal wall to a fluid flowing along the wall in turbulent flow. The temperature gradient in the laminar sublayer is linear and steep, because heat transfer across the laminar sublayer is solely by conduction and the thermal conductivities of fluids are much smaller those of metals. The temperature gradient in the turbulent core is much smaller, as heat transfer occurs mainly by convection - that is, by... 

Because the role of convection is to transfer information in the direction of the flow, the derivative dV/dz must be approximated by a difference formula that uses information that is upstream of the flow. If the derivative is approximated as (Vj - Vj- )/dz when the velocity is negative, then the derivative communicates information ahead of the flow, which is physically unrealistic. Moreover, and importantly, such a downwind difference formula can cause severe numerical instabilities. From the point of view of a control volume, recall the origin of the convective terms in the substantial derivative. They represent the mass, momentum, or energy that is carried into or out of the control volume from the surrounding regions with the fluid flow. Thus the term must have a directional behavior that depends on the local fluid velocity. 

The engineering science of transport phenomena as formulated by Bird, Stewart, and Lightfoot (1) deals with the transfer of momentum, energy, and mass, and provides the tools for solving problems involving fluid flow, heat transfer, and diffusion. It is founded on the great principles of conservation of mass, momentum (Newton s second law), and energy (the first law of thermodynamics).1 These conservation principles can be expressed in mathematical equations in either macroscopic form or microscopic form. 

Fluid flow, also known as fluid mechanics, momentum transfer and momentum transport, is a wide-ranging subject, fundamental to many aspects of chemical engineering. It is impossible to cover the whole field here, so we will concentrate upon a few aspects of the subject. 

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... 

Most physical phenomena in fires can be described as the transfer of heat and mass and chemical reactions in either gas or condensed phases. The physics of the fluid flow are governed by the conservation equations of mass, momentum, and enthalpy ... 

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 ... 

In this section we present the governing equations for the analysis of microchannel heat transfer in two-dimensional fluid flow. For steady two-dimensional and incompressible flow with constant thermophysical properties, the continuity, momentum and energy equations... 

The simulation program AIOLOS is developed for the numerical calculation of three-dimensional, stationary, turbulent and reacting flows in pulverised coal-fired utility boilers. AIOLOS contains submodels treating fluid flow, turbulence, combustion and heat transfer. In these submodels equations for calculating the conservation of mass, momentum and energy are solved, presupposing high Reynolds-numbers and steady-state flow conditions. It is assumed that the flow field is weakly compressible which means that the density depends only on temperature and fluid composition but not on pressure. 

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