FLUID FLOW- CENTRIFUGAL FAN DESIGN

Fluid Flow
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.

Mathematical Models
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 ù 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.

Properties of Air and Other Gases

Properties of Air and Other Gases

The thermodynamic and transport properties of gases and vapors are important in fan engineering. This chapter deals with the thermodynamic properties, especially pressure, temperature, humidity, density, and enthalpy Transport properties, such as viscosity, thermal conductivity, and diffusivity, are dealt with in subsequent chapters. The gaseous materials most frequently encountered in fan engineering are air and water vapor; accordingly, most of the data are for these substances. Some formulae have been written specifically
for these materials, but most are generalized to accommodate any gas.

Atmospheric Air
Atmospheric air is a mixture of dry air, water vapor, and impurities. Dry air is a mechanical mixture of gases, whose principal constituents are listed in Table 1.1. (The table values may be considered representative of the composition of normal outdoor air throughout the troposphere.) The amount of water vapor in atmospheric air will depend on weather conditions. The nature and amount of impurities in the atmosphere depend on the forces at work in producing and dispersing contaminants. Industrial, urban, rural, seaside, and
other areas have characteristic atmospheres due to differences in impurities.



Adapted from the data of J. A. Goff: "Standardization of Thermodynamic Properties of Moist
Air," Trans. ASHVE, vol. 55, 1949, pp. 462-464,
The reference for Table 1.1 lists neon, helium, krypton, hydrogen, xenon, ozone, and radon, totaling less than 0.0025 percent by volume, as the residual part of atmospheric air. ASHRAE1 also lists methane, nitrous oxide, sulfur-dioxide, nitrogen dioxide, ammonia, carbon monoxide, and iodine, totaling 0.0003 percent by volume, as constituents of normal, clean, dry atmospheric air. ASHRAE considers all these gases in the calculation of the apparent molecular weight of clean, dry atmospheric air and obtains a value of 28.9645. Rounding off and lumping the residuals with the nitrogen, as has been done in Table 1.1, yields an apparent molecular weight of 28.964.

Standard Atmosphere
In 1952, the National Advisory Committee for Aeronautics adopted the International Civil Aviation Organization's Standard Atmosphere. Portions of this Standard are given in Table 1.2. (The reference contains much more extensive data in both U.S. and metric units.) Temperatures t are based on 15°C at sea level and a lapse rate of 0.0065°C/m throughout the troposphere, and they are assumed to be constant throughout the stratosphere. The tropopause is considered to be at the level where the temperature becomes -56.50°C. Pressures p are based on 101.325 kPa at sea level, a gas constant of 287.04 J/kg-K, and the perfect gas laws. Densities r are based on the temperature and pressure at the altitude Z and the perfect gas laws. Absolute viscosities m , kinematic viscosities n , and speeds of sound c are based on relationships that will be explained in later sections dealing with these subjects.




Standard Air
In fan engineering, standard air is considered to be air with a density of  1.2 kg /m3 when SI units are employed, or 0.075 lbm/ft3 when U.S. customary units are used. These two values are not exact equivalents, but they are close enough for most fan engineering purposes. Neither do these values exactly
correspond to the sea level value given for the Aeronautical Standard in Table 1.2. Atmospheric air of the composition shown in Table 1.1 will have standard density at various combinations of pressure, temperature, and humidity. Two convenient combinations are shown in Table 1.3, one for dry air and another for moist air. Note that all the combinations listed in Table 1.3 utilize the standard barometric pressure at sea level.
The concept of standard air is useful in rating fans, ducts, and other air handling equipment. Often both duct losses and fan capabilities can be determined from standard air data and used without correction. Even when
the actual density is considerably different from standard air density, it is frequently more convenient to apply corrections to standard air data than it would be to publish separate data for each condition. A slightly different concept, that of standard, or normal, temperature and pressure (STP or NTP), is sometimes employed in specifications. The values for standard temperature and pressure may differ from those in Table 1.3. One should always verify the exact values of standard conditions before selecting a fan based on STP or NTP. See the discussion of conversion from standard, or normal, conditions to actual conditions


Molecular Weight
The molecular weight of a pure substance is the sum of the atomic weights of the atoms in a molecule of that substance. Water, for instance, has a molecular weight of 18.015 based on two atoms of hydrogen at 1.008 each and one atom of oxygen at 15.999, all on the carbon-12 scale. Because air is a mechanical mixture of gases, it does not have a true molecular weight. Dry air of the composition shown in Table 1.1 has an apparent molecular weight of 28.964. The apparent molecular weight M of any mixture of gases can be
calculated either from a volumetric analysis using

where f x is the volume or mass fraction of constituent x ; Mx . is the molecularweight of constituent x ; and n is the total number of constituents.

Examples 1.1 and 1.2 illustrate the use of these equations in calculating the apparent molecular weight of dry outdoor air of the composition shown in Table 1.1.



Differences in molecular weights for the same substance usually can be traced to either rounding off or to differences between the carbon-12 and the oxygen-16 scales. ASHRAE1 lists molecular- weights on the carbon-12 scale as 28.9645 for dry air and 18.015 34 for water. The previously used value of 28.966 for dry air was based on the oxygen-16 scale. Throughout the remainder of this handbook, a value of 28.965 will be used as the apparent molecular weight of dry air. A mole, abbreviated mol, is the base unit of substance in SI. As such, it is further defined as the amount of substance that contains as many elementary
entities as there are atoms in 12 grams of carbon-12. In fan engineering, the elementary entities of interest are molecules, and the usual units of mass are the kg or the lbm. One kg-mol of air will have a mass of 28.965 kg. One lbm-mol of air will have a mass of 28.965 lbm.

The unit of molecular weight is the kg/kg-mol in metric units, and the lbm/lbm-mol in U.S. units. The number of molecules in a kg-mol of any gas is 6.02252 ´1026 . There are 2.73177 ´1026 molecules in a lbm-mol of gas. The volume occupied by a mole of gas will depend on the unit of the mole and on the temperature and pressure. For a gas constituent, the mole fraction, volume fraction, and pressure fraction are equal.