Dimensions and units

Dimensions can be classified as either fundamental or derived. Fundamental dimensions cannot be expressed in terms of other dimensions and include length (L), time (t), temperature (T), mass (M), and/or force (F) (depending upon the system of dimensions used). Derived dimensions can be expressed in terms of fundamental dimensions, for example, area ([A] = L2), volume [V] = L3), energy ([F] = FL = ML2/t2), power ([HP] = FL/t = ML2/t3), viscosity ([/i] = Ft/L2 = M/Lt), etc.  

There are two systems of fundamental dimensions in use (with their associated units), which are referred to as scientific and engineering systems. These systems differ basically in the manner in which the dimensions of force is defined. In both systems, mass, length, and time are fundamental dimensions. Furthermore, Newton s second law provides a relation between the dimensions of force, mass, length, and time  

In scientific systems, this is accepted as the definition of force that is, force is a derived dimension, being identical to ML/t2. 

In engineering systems, however, force is considered in a more practical or pragmatic context as well. This is because the mass of a body is not usually measured directly but is instead determined by its weight (IF), i.e., the gravitational force resulting from the mutual attraction between two bodies of mass mx and m2  

The quantity g is called the acceleration due to gravity and is equal to m2G/r2. At sea level and 45° latitude on the Earth (i.e., the condition for standard gravity, gstd) the value of g is 32.174ft/s2 or 9.806m/s2. The value of g is obviously different on the moon (different r and m2) and varies slightly over the surface of the earth as well (since the radius of the earth varies with both elevation and latitude). 

98% sulphuric acid of viscosity 0.025 N s/m and density 1840 kg/m is pumped at 685 cm /s through a 25 mm line. Calculate the value of the Reynolds number. 

A boiler plant raises 5.2 kg/s of steam at 1825 kN/m pressure, using coal of calorific value 27.2 MJ/kg. If the boiler efficiency is 75%, how much coal is consumed per day If the steam is used to generate electricity, what is the power generation in kilowatts assuming a 20% conversion efficiency of the turbines and generators  

From the steam tables, in Appendix A2, Volnme 1, total enthalpy of steam at 1825 kN/m = 2798 kJ/kg. 

Neglecting the enthalpy of the feed water, this mnst be derived from the coal. With an efficiency of 75%, the heat provided by the coal = (14,550 x 100)/75 = 19,400 kW. For a calorific value of 27,200 kJ/kg, rate of coal consumption = (19,400/27,200) 

The power required by an agitator in a tank is a function of the following fonr variables  

subtract, multiply, and divide units associated with numbers. 

Convert one set of units in a function or equation into another equivalent set for mass, length, area, volume, time, energy, and force. 

Specify the basic and derived units in the SI and American engineering systems for mass, length, volume, density, and time, and their equivalences. 

Apply the concepts of dimensional consistency to determine the units of any term in a function. 

Dimensions are our basic concepts of measurement such as length, time, mass, temperature, and so on units are the means of expressing the dimensions, such 2isfeet or centimeters for length, or hours or seconds for time. Units are associated with some quantities you may have previously considered to be dimensionless. A good example is molecular weight, which is really the mass of one substance per mole of 

A measured or counted quantity has a numerical value (2.47) and a unit (whatever there are 2.47 of). It is useful in most engineering calculations—and essential in many—to write both the value and the unit of each quantity appearing in an equation  

Units can be treated like algebraic variables when quantities are added, subtracted, multiplied, or divided. The numerical values of two quantities may he added or subtracted only if the units are the same. 

On the other hand, numerical values and their corresponding units may always be combined by multiplication or division. 

Following a brief section on Units and Dimensions (Section 4.2), Sections 4.3 and 4.4 review some of the key physical and chemical properties, respectively. Three important conservation laws are presented in Section 4.5. Basic engineering principles are discussed in Section 4.6, to present a foundation for the theory underlying the proper design and operation of a chemical process. 

Because the six main topics are somewhat unrelated, this chapter admittedly lacks a certain cohesiveness. However all the material presented here will find use in later chapters. 

Most of the units used in tliis book are consistent with those adopted by the engineering profession in tlie United States. For engineering work, International SySterne (SI) and English units are most often employed in the United States, tlie English engineering units are generally used. Tliese systems of units are shown in Table 4.2.1.  

The authors gratefully acknowledge the assistance of Tara E. Fleck in researcliing, reviewing, and editing tliis cliapter. 

Converting a measurement from one unit to anotlier can conveniently be accomplished by using unit conversion factors, tliese factors are obtained from tlie simple equation that relates tlie two units numerically. The following is an example of a unit conversion factor 

Any physical quantity has dimensions - not just the usual dimensions of length and time, but others too. Base dimensions, listed in Table 5.1, are denoted by uppercase 

Each base dimension has a base unit. The base unit for each dimension depends on the system of units. Two systems dominate Systeme International (SI) and English. The base units are listed in Table 5.2. Any physical quantity can be expressed as a product of a number and a combination of base units. 

The existence of two systems of units causes frustration, tedium, and error. It is important to develop a rigorous method of converting between systems. The guidelines are  

From base units we create multiple units. Multiple units are, for example, milligram and kilometer. Why multiple units Because people like to use numbers between 0.1 and 100 and SI units are sometimes awkward. For example, consider length. One describes a marathon as a 10 kilometer run, as opposed to a 10,000 meter run. Similarly, when describing the distance between two atoms, it is convenient to say a C—H bond length is 0.11 nanometers, as opposed to 1.1 x 10 m. One describes a feature on a silicon wafer as approximately 1 jum, as opposed to 1 x 10 m. The common prefixes are listed in Table 5.3. For the dimension of time, multiple units 

The third and final class of units are derived units. Derived units are the products of base units and have special names. Some important derived units are listed in Table 5.4. 

length and time are commonly used primary units, other units being derived from them. Their dimensions are written as M, L and T respectively. Sometimes force is used as a primary unit. In the Systeme International d Unites, commonly known as the SI system of units, the primary units are the kilogramme kg, the metre m, and the second s. A number of derived units are listed in Table 1.1. 

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Dimensions given in brackets L = length, M = mass, t = time, T = temperature, C = charge, H = heat = thermal energy = ML2/t2. (See Chapter 2 for discussion of units and dimensions.)... 

The practical chemist working with vacuum systems has, in the past, used practical units, such as millimetres of mercury and atmospheres, for measuring the quality of the vacua which he has produced. However, for the modern chemist it is important to have a coherent system of units in which no numerical factors are inherent. Within the SI system, the unit of pressure is the pascal, which has the units and dimensions given below ... 

Label the axes (the vertical axis is the ordinate, the horizontal axis the abscissa) with both units and dimensions. 

In this section we outline the systems of units which are used throughout the book. One must be careful not to confuse the meaning of the terms units and dimensions. A dimension is a physical variable used to specify the behavior or nature of a particular system. For example, the length of a rod is a dimension of the rod. In like manner, the temperature of a gas may be considered one of the thermodynamic dimensions of the gas. When we say the rod is so many meters long, or the gas has a temperature of so many degrees Celsius, we have given the units with which we choose to measure the dimension. In our development of heat transfer we use the dimensions... 

Using the basic definitions of units and dimensions given in Sec. 1-5, arrive at expressions (a) to convert joules to British thermal units, (b) to convert dyne-centimeters to joules, (c) to convert British thermal units to calories. 

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