Interconversions between SI and the cgs system are simple. Both use the same standards for time, temperature, and the mole, and only the decimal conversions defined by Eqs.
Both SI and the fps system also use the second as the standard for time; the three conversion factors defined for mass, length, and temperature by Eqs. In conversions involving g- in fps units, the use of the exact numerical ratio 9. Solution From Eqs. From Eqs. Although conversion factors may be calculated as needed, it is more efficient to use tables of the common factors. A table for the factors used in this book is given in Appendix 1 0. Many such equations contain terms that represent properties of substances, and these are introduced as needed.
All new quantities are measured in combinations of units already defined, and all arz expressible as functions of the five base units for mass, length, time, temperature, and the mole. Precision of calculations In the above discussion, the values of experimental constants are given with the maximum number of significant digits consistent with present estimates of the precision with which they are known, and all digits in the values of defined constants are retained.
In practice, such extreme precision is seldom necessary, and defined and experimental constants can be truncated to the number of digits appropriate to the problem at hand, although the advent of the digital computers makes it possible to retain maximum precision at small cost. The engineer should use judgment in setting a suitable level of precision for the particular problem to be solved. General equations Except for the appearance of the proportionality factors g. In this text, equations are written for SI units, with a reminder to use g.
Introduction Definitions and Principles 11 Dimensionless equations and consistent units Equations derived directly from the basic laws of the physical sciences consist of terms that either have the same units or can be written in the same units by usingthe definitions of derived quantities to express complex units in termsof the five base ones.
Equations meeting this requirement are called dimensionally homogeneous equations. When such an equation is divided by any one of its terms, all units in each term cancel and only numerical magnitudes remain, These equations are called dimensionless equations. A dimensionally homogeneous equation can be used as it stands with any set of units provided that the same units for the five base units are used throughout. Units meeting this requirement are called consistent units.
No conversion factors are needed when consistent units are used. Dividing the equation by Z gives 0. A combination of variables for which all dimensions cancel in this manner is called a dimensionless group.
The numerical value of a dimensionless group for given values of the quantities contained in it is independent of the unitsused, provided they are consistent. Both terms on the right-hand side of Eq. Dimensional equations Equations derived by empirical methods, in which experimental results are correlatedby empirical equations without regard to dimensional consistency, usually are not dimensionally homogeneous and contain terms in several different units, Equations of this type are dimensional equations, or dimensionally nonhomogeneous equations.
In these equations there is no advantage in using consistent units, and two or more length units, for example, inches and feet, or two or more time units, for example, seconds and minutes, may appear in the same equation. For example, a formula for the rate of heat loss from a horizontal pipe to the atmosphere by conduction and convection is 0. Quantities substituted in Eq, 0. If otherunits are to be used, the coefficient must be changed, To express AT in degrees Celsius, for example, the numerical coefficient must be changed to 0.
In this book all equations are dimensionally homogeneous unless otherwise noted. Problems of this type are especially common in fluid-flow, heat-flow, and diffusion operations. One method of attacking a problem for which no mathematical equation can be derived is that of empirical experimentation.
For example, the pressure loss from friction in a long, round, straight, smooth pipe depends on all these variables: the length and diameter of the pipe, the flow rate of the liquid, and the density and viscosity of the liquid. If any one of these variables is changed, the pressure drop also changes. The empirical method of obtaining an equation relating these factors to pressure drop requires that the effect of each separate variable be determined in turn by systematically varying that variable while Keeping all others constant.
The procedure is laborious, and it is difficult to organize or correlate the results s0 obtained into a useful relationship for calculations. There exists a method intermediate between formal mathematical development and a completely empirical study. It is based on the fact that if a theoretical equation does exist among the variables affecting a physical process, that equation must be dimensionally homogeneous. Because of this requirement, it is possible to group many factors into a smaller number of dimensionless groups of variables.
The groups themselves rather than the separate factors appear in the final equation. This method is called dimensional analysis, which is an algebraic treatment of the symbols for units considered independently of magnitude.
It drastically simplifies the task of fitting experimental data to design equations; it is also usefil in checking the consistency of the units in equations, in converting units, and in the scale-up of data obtained in model test units to predict the performance of full-scale equipment. In making a dimensional analysis, the variables thought to be important are chosen and their dimensions tabulated.
If the physical laws that would be involved in a mathematical solution are known, the choice of variables is relatively eesy. The fundamental differential equations of fluid flow, for example, combined with the laws of heat conduction and diffusion, suffice to establish the dimensions and dimensionless groups appropriate to a large number of chemical engineering problems.
In other situations the choice of variables may be speculative, and testing of the resulting relationships may be needed to establish whether some variables were left out or whether some of those chosen are not needed.
Assuming that the variables are related by a power series, in which the dimension of each term must be the same as that of the primary quantity, an exponential relationship is written in which the exponents relating to any given quantity for example, length must be the same on both sides of the equation. The relationship among the exponents is then found algebraically, as shown in Example 0.
A steady stream of liquid in turbulent flow is heated by passing it through a long, straight, heated pipe. The temperature of the pipe is assumed to be greater by a constant amount than the average temperature of the liquid. It is desired to find a relationship that can be used to predict the rate of heat transfer from the wall of the liquid. If a theoretical equation for this problem exists, it can be written in the general form Ads tse, ST 0.
Let the phrase the dimensions of be shown by the use of square brackets. Thus I refers to the dimension of length. Substituting the dimensions from Table 0. This gives the following set of equations: Exponents of : Ieetf 0. Five of the unknowns may be found in terms of the remaining two. The two letters to be retained must be chosen arbitrarily. The final result is equally valid for all choices, but for this problem it is customary to retain the exponents of the velocity u and the specific heat cp.
The letters b and e will be retained and the remaining five eliminated, as follows. Any function whatever of these. Let such a fanetion be 0. Correlating the experimental values of the three groups of variables of Eq. Is there any point where the Kelvin temperature is the same as the Rankine temperature?
Chapter 1 Fluid Flow The behavior of fluids is important to process engineering generally and constitutes one of the foundations for the study of unit operations. Fluid mechanics in turn is part of larger discipline called continuum mechanics, which also includes the study of stressed solids. Fluid mechanics has two branches important to the study of unit operations: fluid statics, which treats fluids in the equilibrium state of no shear stress, and fluid dynamics, which treats fluids when portions of the fluid are in motion relative to other parts.
Although gases and liquids consist of molecules, it is possible in most cases to treat them as continuous media for the purposes of fluid flow calculations. This treatment as a continuum is valid when the smallest volume of fluid contains a large enough number of molecules so that a statistical average is meaningful and the macroscopic properties of the fluid such as density, pressure, temperature, velocity and so on, vary smoothly or continuously from point to point.
This chapter deals with those areas of fluid mechanics that are important to unit operations. The choice of subject matter is but a sampling of the huge field of fluid mechanics generally. Section 1. An attempt to change the shape of a mass of fluid results in layers of fluid sliding over one another until a new shape is attained.
During the change in shape, shear stresses Shear is the lateral displacement of one layer of material relative to another layer by an external force. Shear stress is defined as the ratio of this force to the area of the layer. See section 1. A fluid in equilibrium is free from shear stresses.
Density At a given temperature and pressure, a fluid possesses a definite density, which in engineering practice is usually measured in kilograms per cubic meter. Although the density of all fluids depends on the temperature and pressure, the variation in density with changes in these variables may be small or large. If the density changes only slightly with moderate changes in temperature and pressure, the fluid is said to be incompressible; if the changes in density are significant, the fluid is said to be compressible.
Liquids are generally considered to be incompressible and gases compressible. The terms are relative, however, and the density of a liquid can change appreciably if pressure and temperature are changed over wide limits.
Also, gases subjected to small percentage changes in pressure and temperature act as incompressible fluids, and density changes under such conditions may be neglected without serious error. Pressure also exists at every point within a volume of fluid, It is a scalar quantity; at any given point its magnitude is the same in all directions, The pressure at a point within a volume of fluid will be designated as either an absolute pressure of a gage pressure.
Absolute pressure is measured relative to a perfect vacuum absolute zero pressure , whereas gage pressure is measured relative to the local atmospheric pressure. Thus, a gage pressure of zero corresponds to a pressure that is equal to the local atmospheric pressure. Absolute pressures are always positive, but gage pressures can be cither positive or negative depending on whether the pressure is above atmospheric pressure a positive value or below atmospheric pressure a negative value.
A negative gage pressure is also referred to as a suction or vacuum pressure. Itis to be noted that pressure differences are independent of the reference, so that no special notaticn is required in this case. In addition to the reference used for the pressure measurement, the units used to express the value are obviously of importance. Pressure is a force per unit area.
Pressure can also be expressed as the height of a column of liquid, the units will refer to the height of the column mm, m, etc. For example, standard atmospheric pressure can be expressed as mm Hig abs 1. Consider the vertical column of fluid shown in Fig. Assume the cross-sectional area of the column is S. At a height Z above the base of the column let the pressure be p and the density be p. However, it is ofien satisfactory for engineering calculations to consider p to be essentially constant.
The den: constant for incompressible fluids and, except for large changes in height, is nearly so for compressible fluids. Ruel Arila Jr. Sunil Yadav. Ibrahim Ibu. Juliana Gume.
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