Engineering Calculations
3.11 Problem Using the data from Example 3.10, estimate the highest expected BOD to occur once every 30 days Use grouped data analysis.
0.001 0.005 0.01 0.02 0.05 0.1 0.2 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.98 0.99 0.998 0.999 0.9999 0 10 20 30 40 50 60 70 80 Effluent BOD (mg/L)
Cumulative fraction of effluent BOD less than P
Problems 105 Solution The first step is to define the groups of BOD values. In this case a 10 mg/L step is chosen simply because it is convenient. The number of data points falling into each group is noted(r).
Effluent Number of Days BOD Mean of BOD Falls into
(mg/L) Group Group (r) r P= r/n 0–9 5 0 0 0 10–19 15 1 1 0.06 20–29 25 6 7 0.39 30–39 35 5 12 0.67 40–49 45 2 14 0.78 50–59 55 2 16 0.89 60–69 65 2 18 1.00
The number of data points are progressively summed(r), and this value is then divided by the total data points, n= 18. These now become single points on a graph of mean BOD group vs P, as in Figure 3.7. The probabilities can be read off the curve as before. As in Example 3.10, if we want to know the worst effluent quality in 30 days, we use
P29/30= 0.967 and read 66 mg/L BOD.
PROBLEMS
3.1 The following table shows the frequency distri- bution of influent biochemical oxygen demand (BOD) in a wastewater treatment plant:
No. of BOD (mg/L) Samples 110–129 18 130–149 31 150–169 23 170–189 20 190–209 7 210–229 1 230–249 2
Compute the following:
a. P10 (probability that 10% of the time the BOD will be less than this value)
b. P50 c. P95
3.2 The following are characteristics of annual precipitation for various cities.
Annual Rainfall Years ¯x s
of Record (in/yr) (in/yr)
Cheyenne, WY 70 14.61 3.61
Pueblo, CO 52 11.51 5.29
Kansas City, MO 63 36.10 6.64
a. Which city’s rainfall is most variable? b. If it is assumed that rainfall data are nor-
mally distributed, how many years in the next 50 would we expect rainfall to be above 20 in in each city?
c. Assuming a normal distribution, how many years in the next 50 would we expect rainfall to be below 12 in in each city?
106 Chapter 3 Engineering Calculations
d. What is the probability of the annual rainfall exceeding 36.1 in in Kansas City?
3.3 Find the won/lost records for your college bas- ketball team for the past 10 years. Calculate the winning percentage for each year. How often can the team be expected to have an equal number of wins and losses (play at least 500 ball)?
3.4 A normal distribution has a standard deviation of 30 and a mean of 20. Find the probability that
a. x≥ 80 b. x≤ 80 c. x≤ −70 d. 50≤ x ≤ 80
Graphical means are recommended.
3.5 A lake has the following dissolved oxygen readings: Month DO (mg/L) 1 12 2 11 3 10 4 9 5 10 6 8 7 9 8 6 9 10 10 11 11 10 12 11
a. What is the average DO? b. What is the standard deviation?
c. What fraction of time can the DO be expected to be above 10 mg/L?
d. What fraction of time will the DO be less than 8 mg/L?
3.6 Here are the return periods and flood levels for Morgan Creek:
Return Period Flood level (years) (m3/s) 1 108 5 142 10 157 20 176 40 196 60 208 100 224
Estimate the flood level for a return period of 500 yr.
3.7 The number of students taking a course for the past five years is as follows:
Number of Year Students 1 13 2 18 3 17 4 15 5 20
How many years in the next 20 years should the class be 10 students or smaller?
3.8 How much beer is consumed in the United States every day?
3.9 Coal has about 2% sulfur. How many kilo- grams of sulfur would be emitted from coal- fired power plants in the United States in 1 yr? (U.S. electricity production is about 0.28 × 1012 watts, and coal has an energy content of about 30× 106J/kg. Note that 1 J= 1 W-sec.) 3.10 How many automobile tires are sold annu- ally in the United States? Make reasonable assumptions.
3.11 The EPA expresses the emission rate of gaseous pollutants from cars in terms of grams per mile (g/mi). What do you think of this constructed unit of measurement?
3.12 One gram of table salt is put into an 8-oz glass, and the glass is filled with tap water. What is the concentration of salt in milligrams per
Problems 107 liter? Why might you not be so sure about your
answer?
3.13 Metal concentrations in wastewater sludges are often expressed in terms of grams of metal per kilogram of total dry solids. A wet sludge has a solids concentration of 200,000 mg/L, and 8000 mg/L of these solids are zinc.
a. What is the concentration of zinc as g Zn/g dry solids?
b. If this sludge is to be applied to farm- land and spread on pastures used by cows, explain how you might monitor the project with regard to the potential environmental or health effects caused by the presence of the zinc.
3.14 One gram of pepper is placed into a 100-mL beaker, and the beaker is filled with water to the 100-mL mark. What is the concentration of pepper in mg/L?(What should you assume in this problem? What is the density of pepper? What is the volume of 1 g of pepper?)
3.15 A wastewater treatment plant receives 10 mgd of flow. This wastewater has a solids concen- tration of 192 mg/L. How many pounds of solids enter the plant every day?
3.16 Ten grams of plastic beads with a density of 1.2 g/cm3 are added to 500 mL of an organic solvent with a density of 0.8 g/cm3. What is the concentration of the plastic beads in mg/L? (This is a curve ball!)
3.17 A stream flowing at 60 gal/min carries a sedi- ment load of 2000 mg/L. What is the sediment load in kilograms per day?
3.18 A water treatment plant produces water that has an arsenic concentration of 0.05 mg/L. a. If the average consumption of water is 2 gal
of water each day, how much arsenic is each person drinking?
b. If you are the water treatment plant operator and the chairman of the town council asks you during a public hearing whether there
is arsenic in the town’s drinking water, how do you respond?
c. Analyze your answer to part b. from a moral standpoint. Did you do the right thing by answering as you did? Why do you think so?
3.19 A power plant emits 120 lb of flyash per hour up the stack. The flow rate of the hot gases in the stack is 25 ft3/sec. What is the concentration of the flyash in micrograms per cubic meter?
3.20 If electricity costs $0.05/kW· h, how much does your university pay annually on its power bill due to the use of desk lamps? (You need to make several assumptions here specific to your university.)
3.21 If tuition were to be paid in the form of tickets to lectures, how much would each ticket to one lecture cost at your university?
3.22 A university has about 12,600 undergraduate students, and graduates 2,250 students each year. Suppose everyone enrolled at the univer- sity graduates eventually. What is the approxi- mate retention time at the university?
3.23 A water treatment plant has 6 settling tanks that operate in parallel (the flow gets split into six equal flow streams), and each tank has a volume of 600 m3. If the flow to the plant is 10 mgd, what is the retention time in each of the settling tanks? If, instead, the tanks operated in series (the entire flow goes first through one tank, then the second, and so on), what would be the retention time in each tank?
3.24 Some of the most confusing statistics are in sports (see, for example, Figure 3.8). Define the following:
a. ERA b. RBI
c. slugging average d. average hang time e. on base percentage
108 Chapter 3 Engineering Calculations
Figure 3.8 Example of sports statistics. See Problem 3.24. (SHOE-NEW BUSINESS MACNELLY. DISTRIBUTED BY KING FEATURES SYNDICATE.)
f. average points per game (ice hockey) g. others?
3.25 You are working as a recruiter for your consult- ing engineering firm, and an applicant asks you the following question:
If I work with your firm and I find that I object, on ethical grounds, to a project the firm has been hired to do, can I request not to work on that project, without penalizing my future career with the firm?
Give three responses you might make to such a question, and then respond to these statements according to what you believe the student interviewee would think. Would he/she believe you? Why or why not?
3.26 How should Chris report the estimate of the construction cost? Why?
a. $56,078.34 b. $56,078 c. $56,080 d. $56,000
3.27 Why is the number of significant figures in your final answer important? In other words, what does the number of significant figures in your final answer reflect?
3.28 Example 3.7 mentions precision. What is the difference between precision and accuracy? 3.29 Should you round throughout a calculation or
End Notes 109 END NOTES
1. “Mars Climate Orbiter,” Microsoft Encarta Online Encyclopedia 2008. Accessed at http://encarta.msn.com on 8 December, 2008. 2. Articles on Mars Climate Orbiter. U.S.
National Aeronautics and Space Administration. Accessed at http://mars.jpl.nasa.gov/msp98/ orbiter/ on 8 December, 2008.
3. As observed by John Hart in a wonderful book on environmental calculations: Consider a
Spherical Cow. 1985. Los Altos, CA: William
Kaufmann, Inc.
4. This example is adapted from Middlebrooks, E. Joe. 1976. Statistical Calculations. Ann Arbor, MI: Ann Arbor Science Pub.