Module 1: Basic Concepts - Dimensional Calculations

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Practice Problems

Objective

Solve problems using dimensional calculations.

Many parameters of interest in air pollution and science have characteristics called dimensions. Dimensions include length, mass, time, force, energy, and temperature. The values of these dimensions are expressed in units. For example, the length, width, and height of your computer monitor can be expressed in inches. The volume of your monitor can be expressed in units of cubic inches (length cubed). The time it takes to read this module can be expressed in minutes or hours. These units are very important. Consider the following example.

What is the volume of a pipe that is 10 ft long and has an inside diameter of 4 in.? See Figure 1.
Figure 1 Dimensions of Pipe

Volume of cylinder = area of circle

length = (

r²)

length

: Where: r = radius of circle

Answer to Question

In this section you will learn the importance of including the units along with the numerical values when solving problems. This is important since the air pollution field overlaps many scientific disciplines and, therefore, has many systems of units. You will often need to convert data from one type of unit to another. Carrying units, which is one of the basic steps in proper problem solving, always provides a means to identify mistakes caused by inverting ratios, failing to convert from one unit of measurement to another, and other simple errors.

TIP: Always include the units for all dimensional parameters in your calculations. This prevents mistakes and allows you to solve problems logically. The dimensional units can be handled identically to algebraic units. A calculation that includes the units along with numerical values is called a dimensional calculation.

Example Problem 1.
A Dimensional Calculation

A particle is moving at 20,000 cm/sec. How fast is it moving in miles per hour?

Dimensional calculations organize the units in a logical way, which makes it easy to convert from one unit to another. For example, note the relationships between the values in the numerator and the denominator: 1 in. = 2.54 cm, 1 ft = 12 in., etc. Most of the units cancel each other out, leaving the answer expressed in miles per hour.

Many of the mistakes made in completing permit applications, in evaluating emission test data, and in analyzing air pollution control system performance can be traced to simple errors resulting from the failure to carry the units. These errors are becoming more common due to the ease of performing mathematical operations on calculators and minicomputers. You can guard against these errors by including units in the documentation that accompanies these electronic programs.

Practice Problems
Dimensional Calculations

Instructions:: Complete the Practice Problems before proceeding to the next lesson. Click on the button below.

Practice Problems

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Module 1: Basic Concepts - Dimensional Calculations

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