Most aspects of fan engineering are concerned in some way with the flow of fluids. Consequently, many fundamentals of flow will be discussed in this chapter to provide a basis for the detailed discussions in subsequent chapters. The first part of this chapter deals with flow under more or less ideal conditions. While it does not ignore friction, it does not deal with it directly. The second part examines the effects of viscosity and other factors that determine the resistance to flow. The third part gives means for determining the frictional losses in the various elements of a duct system. The last partcovers the measurement of pressure and flow.
Principles of Fluid Flow
The general principles that govern fluid flow are discussed in the following sections. Only the integral equations have been developed. Differential equations are more suitable for examining some flow-related
questions; if this is so, refer to a text on fluid mechanics.
A mathematical model is necessary if a flow situation is to be analyzed quantitatively. The model must be capable of representing any variable that changes significantly during the flow. (Flow variables include the position, velocity, acceleration, pressure, temperature, density, enthalpy, and entropy of the fluid.) The changes predicted by the model must be consistent with the physical laws that govern fluid flow.
These laws are:
1) the general law regarding conservation of mass,
2) the first law of thermodynamics regarding conservation of energy,
3) Newton's second law of motion regarding force and momentum, and
4) the second law of thermodynamics regarding entropy.
In most fluid mechanics models, the fluid is assumed to be a continuum. That is, the velocities and other properties are assumed to vary continuously throughout the fluid. Such an assumption could not be justified for highly rarefied gas flows, but these usually do not occur in fan engineering. In the mathematical formulation of the physical laws governing fluid flow, coordinates must be assigned to points in space. In one method of modeling, coordinates that are a function of time are also assigned to identifiable particles (or portions) of the fluid. This Lagrangian approach requires that the two sets of coordinates be distinguished from each other and leads to complex equations. It is more usual in fluid mechanics to use the Eulerian approach, which leads to simpler equations. In this method, the flow field is described; that is, the fluid properties are specified at points in space by properly constructed equations. The properties can vary with time at any point in space, if that is a condition of modeling. And it is not necessary to assign coordinates to individual portions of the fluid. Any mathematical model that is capable of representing the most general
flow situation will be more complex than necessary for many fan-engineering applications. Accordingly, various assumptions can be made to simplify the model. However, caution is advisable in using simplified models until the assumptions have been verified for the situation being modeled. The more
common assumptions are discussed below.
1) All real flows are three-dimensional to some extent, but it is not always necessary to use a three-dimensional model to describe the flow. It may be possible to analyze some flow situations with either a one-dimensional or a two-dimensional flow model. In a one-dimensional flow model, the changes
in variables perpendicular to the main flow are taken into account by using average values. Only one dimension or coordinate is required to establish position along the flow path. Pipe flow, for example, usually can be treated as one-dimensional by using the proper averages across the section for the variables. In two-dimensional flow, just two independent coordinates are required to describe the flow field. For instance, the flow across a wing can be analyzed on a two-dimensional basis by using a correction factor to account for the three-dimensional effects at the tips. A three-dimensional model considers each variable to be a function of three space coordinates and time.
2) All real flows probably have some unsteadiness. That is, the properties at a point vary with time. However, the time variations may be small and centered around a constant value for each property. In such cases, the temporal average values can be used in the model, and the flow can be called steady. However, in turbulence studies, certain effects of these variations cannot be disregarded.
3) All flows are influenced by gravitational forces. However, in some flows, particularly those involving gases, the effects of gravity are negligible compared to the effects of other forces to which the fluid is exposed. If so, the gravity terms can be omitted from the equations in the model.
4) All fluids are compressible, especially gases. The effects of compressibility on liquids can be ignored, except for situations involving very large or sudden changes in pressure. The effects can also be ignored in many calculations involving gases when the pressure changes and Mach numbers are small. Using a compressibility factor may be the most convenient way to deal with compressibility when it cannot be ignored. Otherwise, the appropriate compressible-flow equation must be used.
5) All real flows involve friction. Often, however, it is convenient to consider the flow frictionless or ideal. The analysis of some flows may be divided into two parts: that in a boundary layer and that outside such a
boundary layer, with the latter considered ideal.
6) Other effects that are almost always ignored in fan engineering applications are those due to surface tension, buoyancy, and Coriolis forces.The number of equations required to model a flow situation must equal the number of unknowns. Basically, a continuity equation and one or moreequations of motion will be needed. The latter will provide a mechanical energy balance. If thermodynamic effects are significant, the general energy equation will be needed. An equation of state will be required when an explicit relationship among pressure, temperature, and density is necessary In these discussions, reference is made to a system,
as opposed to a control volume and its associated control surface. A system is a definite mass of material that can be distinguished from its surroundings. A control volume is a region in space and is signified by the abbreviation c.v. This region is usually fixed, but it may be moving in space. The control surface is the boundary of the control volume and is signified by theabbreviation c.s. For the control volume equations, the flow enters the control volume through the entrance area A1 on the control surface and leaves through the exit area A2 . If there is more than one entrance or more than one exit, the equations must be modified. Each discussion contains a system equation in differential form to illustrate the underlying physical law and at
least one version of a control volume equation in integral form to provide the basis for the simplified equations that are commonly used. Vector notation (denoted by symbols with superior arrows) is used in the general equations to convey the three-dimensional aspects without necessitating three equations.Vector quantities include velocity vector G V , area vector GA, force vector GF,surface force vector GFs , body force per unit volume vector. GB, radius or position vector Gr , and surface torque vector GTs . Most of the scalar quantitiesare defined as they appear; but time t, mass density r , and volume รน V should be noted here.
In most fan engineering applications, the reader is not expected to useeither the general system equations or the general control volume equations.These equations, however, do illustrate what has been omitted from the
simplified equations derived from them.
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