Environmental Engineer's Mathematics Handbook

Frank R. Spellman, Nancy E. Whiting

• Publisher: CRC Press
• Year: 2004
• ISBN:
  9781135460822
• Edition: 1
• Topic: Mathématiques
• Subtopic: Probabilités et statistiques

Advanced mathematics used in engineering is studied here in this text which examines the relationship between the principles in natural processes and those employed in engineered processes. The text covers principles, practices and the mathematics involved in the design and operation of environmental engineering works. It also presents engineering

PART I
Fundamental Computation and Modeling

CHAPTER 1
Conversion Factors and SI Units

1.1 INTRODUCTION

The units most commonly used by environmental engineering professionals are based on the complicated English System of Weights and Measures. However, bench work is usually based on the metric system or the International System of Units (SI) because of the convenient relationship among milliliters (mL), cubic centimeters (cm³), and grams (g).

The SI is a modernized version of the metric system established by international agreement. The metric system of measurement was developed during the French Revolution and was first promoted in the U.S. in 1866. In 1902, proposed congressional legislation requiring the U.S. government to use the metric system exclusively was defeated by a single vote. Although we use both systems in this text, SI provides a logical and interconnected framework for all measurements in engineering, science, industry, and commerce. The metric system is much simpler to use than the existing English system because all its units of measurement are divisible by 10.

Before we list the various conversion factors commonly used in environmental engineering, we describe the prefixes commonly used in the SI system. These prefixes are based on the power 10. For example, a “kilo” means 1000g, and a “centimeter” means 1/100 of 1m. The 20SI prefixes used to form decimal multiples and submultiples of SI units are given in Table 1.1.

Note that the kilogram is the only SI unit with a prefix as part of its name and symbol. Because multiple prefixes are not used, in the case of the kilogram the prefix names of Table 1.1 are used with the unit name “gram” and the prefix symbols are used with the unit symbol “g.” With this exception, any SI prefix may be used with any SI unit, including the degree Celsius and its symbol °C.

Example 1.1

10⁻⁶kg=1mg (1 milligram), but not 10⁻⁶kg=1μkg (1 microkilogram)

Example 1.2

Consider the height of the Washington Monument. We may write h_w=169,000mm=16,900cm= 169m=0.169km, using the millimeter (SI prefix “milli,” symbol “m”); centimeter (SI prefix “centi,” symbol “c”); or kilometer (SI prefix “kilo,” symbol “k”).

1.2 CONVERSION FACTORS

Conversion factors are given in alphabetical order in Table 1.2 and in unit category listing order in Table 1.3.

Table 1.1 SI Prefixes

Table 1.2 Alphabetical Listing of Conversion Factors

Table 1.3 Conversion Factors by Unit Category

Example 1.3

Problem:

Find degrees in Celsius of water at 72°F.

Solution:

1.3 CONVERSION FACTORS: PRACTICAL EXAMPLES

Sometimes we must convert between different units. Suppose that a 60-in. piece of pipe is attached to an existing 6-ft piece of pipe. Joined together, how long are they? Obviously, we cannot find the answer to this question by adding 60 to 6. Why?—because the two lengths are given in different units. Before we can add the two lengths, we must convert one of them to the units of the other. Then, when we have two lengths in the same units, we can add them.

To perform this conversion, we need a conversion factor. In this case, we need to know how many inches make up a foot—that is, 12in. is 1ft. Knowing this, we can perform the calculation in two steps:

1. 60in. is really 60/12=5ft
2. 5ft+6ft=11ft

From this example, we see that a conversion factor changes known quantities in one unit of measure to an equivalent quantity in another unit of measure.

In making the conversion from one unit to another, we must know two things:

• The exact number that relates the two units
• Whether to multiply or divide by that number

When conversions are necessary, confusion over whether to multiply or divide is common; however, the number that relates the two units is usually known and thus is not a problem. Understanding the proper methodology—the “mechanics”—to use for various operations requires practice and common sense.

Along with using the proper mechanics (and practice and common sense) in making conversions, probably the easiest and fastest method of converting units is to use a conversion table. The simplest conversions require that the measurement be multiplied or divided by a constant value. For instance, if the depth of wet cement in a form is 0.85ft, multiplying by 12in./ft converts the measured depth to inches (10.2in.). Likewise, if the depth of the cement in the form is measured as 16in., dividing by 12in./ft converts the depth measurement to feet (1.33ft).

1.3.1 Weight, Concentration, and Flow

Using Table 1.4 to convert from one unit expression to another and vice versa is good practice. However, in making conversions to solve process computations in water treatment operations, for example, we must be familiar with conversion calculations based upon a relationship among weight, flow or volume, and concentration. The basic relationship is

Weight=Concentration×Flow or Volume×Factor
(1.1)

Table 1.5 summarizes weight, volume, and concentration calculations. With practice, many of these calculations become second nature to users.

Table 1.4 Conversion Table

Table 1.5 Weight, Volume, and Concentration Calculations

The following conversion factors are used extensively in environmental engineering (water and wastewater operations):

• 7.48gal/ft³
• 3.785L/gal
• 454g/lb
• 1000mL/L
• 1000mg/g
• 1ft³/sec (cfs)=0.6465MGD (million gallons per day)

Key point: Density (also called specific weight) is mass per unit volume and may be registered as pounds per cubic foot; pounds per gallon; grams per milliliter; or grams per cubic meter. If we take a fixed volume container, fill it with a fluid, and weigh it, we can determine density of the fluid (after subtracting the weight of the container).

• 8.34lb/gal (water)—(density=8.34lb/gal)
• 1mL of water weighs 1g—(density=1g/mL)
• 62.4lb/ft³ (water)—(density=8.34lb/gal)
• 8.34lb/gal=milligrams per liter (converts dosage in milligrams per liter into pounds per day per million gallons per day)

Example: 1mg/L×10MGD×8.3=83.4lb/day

• 1psi=2.31ft of water (head)
• 1 foot head=0.433psi
• °F=9/5(°C+32)
• °C=5/9(°F−32)
• Average water usage: 100 gal/capita/day (gpcd)
• Persons per single-family residence: 3.7

1.3.2 Water/Wastewater Conversion Examples

Use Table 1.4 and Table 1.5 to make the conversions indicated in the following example problems. Other conversions are presen...