Converting Between the SI and USCS
3.6 E STIMATION IN E NGINEERING
107
Discussion
First, we evaluate the order of magnitude of the solution. The acceleration is not large, which is not expected since the scale reading is not signifi cantly dif- ferent from the person’s weight. Second, we revisit our assumptions to make sure they are reasonable. All the assumptions are very logical. The person, scale, and elevator may undergo some motion relative to one other in reality, but the impact on the analysis would be minimal. Third, we draw conclusions from the solution and explain its physical meaning. The acceleration is nega- tive, indicating a downward acceleration, which aligns with the scale reading.
When an elevator begins to accelerate downward, the passengers temporarily feel lighter. Their mass does not change since gravity did not change. However, their perceived weight has changed, which is what the scale measures.
Note that the same analysis can be done using the SI. First, we convert the scale reading to newtons
W 5 (140 lb)
(
4.45 N _______ 1 lb)
5 623 lb ? N __
lb 5 623 N
Since this is less than the person’s actual weight of 687 N, we conclude that the elevator is accelerating downward. Solving for acceleration gives
a 5 ____ 兺F
m 5
(
623 N ______________ 2 687 N70 kg
)
5 2_______ 64 N
70 kg 5 20.91
(
kg ? m ______ kg__ s2)
5 20.91 m __s2
Using the conversion 1 ft 5 0.3048 m from Table 3.6, this solution can be converted to 2.9 ft/s2, which matches our previous analysis.
a 5 22.9 ft __
s2 Example 3.7 continued
leading to a result that is accurate enough for the task at hand. If the accuracy needs to be increased later on, for instance, as a design becomes fi nalized, then they need to incorporate more physical phenomena or details about the geometry, and the equations to be solved would likewise become more complicated.
Given that some imperfections and uncertainty are always present in real hardware, engineers often make order-of-magnitude estimates. Early in the design process, for instance, order-of-magnitude approximations are used to evaluate potential design options for their feasibility. Some examples are estimating the weight of a structure or the amount of power that a machine produces or consumes. Those estimates, made quickly, are helpful to focus ideas and narrow down the options available for a design before signifi cant effort has been put into fi guring out details.
Engineers make order-of-magnitude estimates while fully aware of the approximations involved and recognizing that reasonable approximations are going to be necessary to reach an answer. In fact, the term “order of magnitude” implies that the quantities considered in the calculation (and the ultimate answer) are accurate to perhaps a factor of 10. A calculation of this type might estimate the force carried by a certain bolted connection to be 1000 lb, implying that the force probably isn’t as low as 100 lb or as great as 10,000 lb, but it certainly could be 800 lb or 3000 lb. At fi rst glance, that range might appear to be quite wide, but the estimate is nevertheless useful because it places a bound on how large the force could be. The estimate also provides a starting point for any subsequent, and presumably more detailed, calculations that a mechanical engineer would need to make. Calculations of this type are educated estimates, admittedly imperfect and imprecise, but better than nothing. These calculations are sometimes described as being made on the back of an envelope because they can be performed quickly and informally.
Order-of-magnitude estimates are made when engineers in a design process begin assigning numerical values to dimensions, weights, material properties, temperatures, pressures, and other parameters. You should recognize that those values will be refi ned as information is gathered, the analysis improves, and the design becomes better defi ned. The following examples show some applications of order-of-magnitude calculations and the thought processes behind making estimates.
Order-of-magnitude estimates
Back-of-an-envelope calculations
Focus On importance of estimations
On April 20, 2010, an explosion destroyed Transocean’s Deepwater Horizon oil drilling rig in the Gulf of Mexico, killing 11 people, injuring 17 others, and creating the largest accidental marine oil spill in history. It was not until July 15 that the leak was stopped, but only after 120–180
million gallons of oil had spilled into the gulf and British Petroleum had spent over US$10 billion on the cleanup. During the initial release, multiple fluids flowed simultaneously out of the well, including seawater, mud, oil, and gas. Engineers quickly started creating an analytical model to
109
estimate future fl ow rates of these fl uids. Their approach was to calibrate their model based on the initial fl uid fl ow rates and pressures and then use it to predict future fl ow rates. They began by making assumptions about the following critical issues:
• Only one-fi fth of the total area of usable oil would be used in the calculations
• The remainder of the usable oil may have been isolated by cement, or the fl ow rate may have been limited by fl ow restrictions in the well • The approximate timing of actual prevention
events that occurred
Their computational fl uid dynamics calculations resulted in a solution that matched the real-time fl ow rates and pressures and aligned with accounts from eyewitnesses. In Figure 3.8, a graph from the Deepwater Horizon Accident Investigation Report shows the model that was created. The two lines that start close to each other on the left represent the actual and modeled drill pipe pressure. They are a very close matchup until the explosion
(vertical dotted line). Therefore, the developed model matches the actual events leading up to the explosion. The remaining three curves represent the modeled (predicted) fl ow rates of mud and water (the lower curve that starts on the left), oil (the shorter curve that starts increasing from zero right before the explosion), and gas (the taller curve that starts increasing from zero right before the explosion). The predicted fl ow rate curves clearly taper off to zero, but this did not occur. The fl ow of these fl uids continued, and so the engineers had to continue to develop models and solutions to help understand the accident and its future impact.
Many times, technical problem solving is an inexact science, and a rough estimate is the best engineers can do. Models that seem to match actual data do not effectively predict future performance. In this example, with such a dynamic fl ow environment being modeled by the engineers, the range of uncertainty is very high, and the models at best can provide only an estimation of what happened.
3.6 Estimation in Engineering
21:20 180 160 140 120 100 80 60 40 20 0
6,000
Time
Flow Rate (bpm & mmscfd) Pressure (psi)
5,000
4,000
3,000
2,000
1,000 0
Recorded data ends/main power loss
and explosion Peak gas rate
(165 mmscfd)
Divert to MGS and close BOP
Flow rates to zero/
BOP closed and riser evacuated Actual drill pipe pressure (psi)
Modeled drill pipe pressure (psi) Modeled gas flow rate (mmscfd) Modeled oil flow rate (stb/min) Modeled mud+water flow rate (stb/min)
Time of data loss
(pressure and flow rate measure at the diverter)
21:25 21:30 21:35 21:40 21:45 21:50 21:55 22:00 22:05 22:10
Figure 3.8
Fluid fl ow modeling and prediction.From BP, Deepwater Horizon Accident Investigation Report, September 8, 2010, p. 13 at http://
www.bp.com/liveassets/bp_internet/globalbp/globalbp_uk_english/incident_response/STAGING/
local_assets/downloads_pdfs/Deepwater_Horizon_Accident_Investigation_Report.pdf.
Commercial jet aircraft have pressurized cabins because they travel at high altitude where the atmosphere is thin. At the cruising altitude of 30,000 ft, the outside atmospheric pressure is only about 30% of the sea-level value.
The cabin is pressurized to the equivalent of a mountaintop where the air pressure is about 70% that at sea level. Estimate the force that is applied to the door of the aircraft’s main cabin by this pressure imbalance. Treat the following information as “given” when making the order-of-magnitude estimate: (1) The air pressure at sea level is 14.7 psi, and (2) the force F on the door is the product of the door’s area A and the pressure difference Δp according to the expression F 5 AΔp.
Approach
We are tasked with approximating the amount of force exerted on the inte- rior of an aircraft door during fl ight. The pressure information is given, but we have to make some assumptions about the door and cabin surroundings.
We assume that:
• The size of the aircraft’s door is approximately 6 3 3 ft, or 18 ft2 • We can neglect the fact that the door is not precisely rectangular • We can neglect the fact that the door is curved to blend with the shape
of the aircraft’s fuselage
• We do not have to account for small changes in pressure due to the movement of passengers inside the cabin during fl ight
We will fi rst calculate the pressure difference and then calculate the area of the door to fi nd the total force.
Solution
The net pressure acting on the door is the difference between air pressures inside and outside the aircraft.
Δp 5 (0.7 – 0.3)(14.7 psi) 5 5.88 psi
Because Δp has the units of pounds per square inch (Table 3.5), in order for the equation F 5 AΔp to be dimensionally consistent, the area must be con- verted to the units of square inches:
A (18 ft2)
(
12 in. ___ ft)
22592 (ft2)
(
in___ ft22)
2592 in2 Example 3.8 Aircraft’s Cabin Door
111
The total force acting on the door becomes
F 5 (2592 in2)(5.88 psi) ← 3 F 5 A Dp 4 5 15,240 (in2)
(
lb ___ in2)
5 15,240lb Discussion
First, we evaluate the order of magnitude of the solution. The forces created by pressure imbalances can be quite large when they act over large surfaces, even for seemingly small pressures. Therefore, our force seems quite reasonable.
Second, our assumptions greatly simplifi ed the problem. But since we need to estimate only the force, these assumptions are realistic. Third, recognizing the uncertainty in our estimate of the door’s area and the actual value of the pressure differential, we conclude that the pressure is in the range of 10,000–20,000 lb.
approximately 10,000–20,000 lb Example 3.8 continued
In an analysis of sustainable sources of energy, an engineer wants to estimate the amount of power that a person can produce. In particular, can a person who is riding an exercise bike power a television (or similar appliance or product) during the workout? Treat the following information as given when making the order-of-magnitude estimate: (1) An average LCD television consumes 110 W of electrical power. (2) A generator converts about 80% of the supplied mechanical power into electricity. (3) A mathematical expression for power P is
P 5 Fd ___
Dt
where F is the magnitude of a force, d is the distance over which it acts, and Δt is the time interval during which the force is applied.
Approach
We are tasked with estimating whether it is feasible for a person exercising to independently power a product that requires approximately 110 W. We fi rst make some assumptions to make this estimation:
• To estimate a person’s power output while exercising, we will make a comparison with the rate at which a person can climb a fl ight of stairs with the same level of effort
• We will assume that a fl ight of stairs has a 3-m rise and that it can be climbed by a 700-N person in under 10 s
Example 3.9 Human Power Generation
3.6 Estimation in Engineering