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Assignment 2: (Laboratory Project) Due Date: Friday 25
th September 2015, 4:00 pm
Please prepare MS Word document or print to pdf and submit online through learning at Griffith
Weighting: 12.5% of the total marks for this course Please complete the laboratory exercises and submit a report combining labs 1 and 2. The report should be produced with a word processor such as Microsoft Word. Although laboratory work will be done in groups, each student must submit their own individual laboratory report.
Laboratory 1: Internal combustion engine efficiency You are part of an engineering team that is developing a trickle irrigation system for agricultural purposes. The system consists of a 20000 L water tank on a stand which is 15 m high and a petrol or diesel powered pump to pump the water from a dam up into the elevated tank (Fig. 1). Your task is to report on the expected performance (from a thermodynamics point of view) of some available motor and pump combinations for the system. You must base your conclusions on experimental data you have collected from the laboratory. The range of choices under consideration has been narrowed to three pumps and two engines. One of the engines is a 232 cc four- stroke diesel motor and the other is a 172 cc four-stroke petrol motor. These engines are located in the mechanical engineering lab at Griffith University along with test beds and dynamometers which you will use to measure their performance and simulate the load that would be placed on the motor by the pump.
Another engineer on the team has analized the piping system in Fig. 1 (you will learn how to do this when you study fluid mechanics) and has found that if the flow rate, Q (m
3 /s) is decided then the required difference in head (meters of water) between the
15 m
2 m
Pump and motor
20 kL
Dam
Fig. 1 System in which motor is to be used
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inlet and outlet of the pump follows the following equation (Note: multiply hpump by (ρg) to get pressure difference between pump outlet and inlet in pascals):
2
4
2.1 17 Q
D h
pump += (1)
This equation is usually called the ‘system curve’. It is unique for each pipe system. It is different depending on the length and diameter of the pipe, the pipe fittings selected for the system and the difference in elevation between the supply water and tank. D is the inlet diameter of the pipe (in meters) and Q is the flow rate in m
3 /s. Q
2 /D
4 appears
in the equation because losses in a piping system are usually proportional to the square of the water velocity inside the pipe. The engineer suggests a pipe with inside diameter D = 35 mm = 0.035 m. Fig. 2 shows the characteristic perfomance curves for three different pumps being considered: Pump A, Pump B and Pump C supplied by the manufacturer. The curves show the difference in head supplied by the pump for any given flow rate. Efficiencies listed in Fig. 2 are mechanical efficiencies for the pump. If you plot Eq. (1) on the graph shown in Fig. 2, the points where Eq. (1) intercepts the characteristic curves for pumps A, B and C will show the required operating conditions for your motor.
Flow rate
P u
m p
h e
a d
Pump B
Pump C
Efficiency
Centrifugal Pump Characteristic Curves at 2700 rpm
Pump A
Fig. 2 Pump characteristic curves
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Using Fig. 2, Eq. 1 and the equipment available for testing the internal combustion engines in the lab, you are required to supply the following information:
1. The noise in dB made by each of the motors (measured in the laboratory). Note also
the location where the noise measurement was made.
2. The duty points for pumps A, B and C (i.e flow rate, head and pump efficiency
corresponding to where Eq. (1) intersects the characteristic curves shown in Fig. 2).
3. The power (and torque) required to turn the shaft of the pump at each of the three duty
points.
4. Whether or not the engines in the lab are capable of delivering enough power to run
the pumps at each of the duty points.
5. The thermal efficiency of the engines at each of the three duty points at the specified
rpm (measured in the laboratory)
6. Time needed to fill the 20000 L water tank for each pump/motor combination.
7. No. of litres of fuel (and approximate cost) required to fill the 20000 L tank for each
pump/motor combination?
8. The increase in potential energy of 20000 L of water as moves from the dam to the
tank.
9. The energy used by the motor in filling the water tank for each pump/motor
combination.
10. The overall thermal efficiency of the system (use lower heating value for the fuel).
11. The air/fuel ratio at the conditions tested.
12. The number of kg of carbon dioxide released to the atmosphere by filling the tank for
each pump/motor combination.
13. A recommendation as to the best pump/motor combination for this purpose and an
explanation of why you are making this recommendation.
14. Any suggestions of how the efficiency of the system may be improved.
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Report Format
It is expected your report will be typed using a word processor such as Microsoft Word (apart from raw data and any hand written calculations that you may include in the appendix) and will contain the following sections: Title: Name & Date: Summary: Briefly describe (approximately half a page) the purpose of the report and the main findings of your investigation. Equipment: Include photograph(s) of the apparatus used Results and Discussion: You should summarize your results in a table such as is shown below: Table 1: Results. You should explain briefly how your calculations were done including any assumptions you have made. You should also include a graph of the experimental results from the lab showing the efficiencies of the engines for different loads (i.e. thermal efficiency of the engine (vertical axis) against power output (horizontal axis)). Conclusions and Recommendations: Here you should give a recommendation as to the best pump/motor combination for this purpose and an explanation of why you are making this recommendation. You should also include any suggestions of how the efficiency of the system may be improved. Appendix 1: Raw Data This section should contain the raw data you collected in the lab Appendix 2: Sample Calculations This section should contain one of the following:
• hand written sample calculations
• or a printout from a spreadsheet that you used to do your calculations
• or a printout of a computer program code (e.g. Matlab) that you made to do your calculations.
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Table 1: Results Engine: Diesel Petrol Noise (dB):
Pump: A B C A B C Flow rate (kg/s)*: (water)
Pump Head (m)*:
Pump Efficiency*:
RPM*:
Power Required from motor (kW)*:
Torque Required from motor (N.m)*:
Can engine deliver required power?
Thermal efficiency of engine (%)
Time needed to pump 20000 L of water
No. of litres of fuel required to pump 20000 L of water (L)
Cost of fuel required to pump 20000 L ($)
Increase in potential energy of 20000 L of water (kJ)
Energy used by motor to fill the tank (kJ)
Overall thermal efficiency of the system
Air/Fuel Ratio:
Mass of CO2 produced by pumping 20000 L of water (kg):
Recommended pump/motor combination:
*These correspond to the duty points and should be determined before doing the lab.
Intermediate Microeconomic Theory Fall 2015
Homework #1 - Part B
Choice:
Exercise 7: For this exercise replace A with the last digit and B with the second-to-last
digit of your ASU ID#. Assume preferences can be represented by the following utility function:
u(x1;x2) = (A+1)ln(x1)+ ln(x2) ;
a. Is the utility function monotonic? Justify. b. Determine the set of bundles that are ranked higher than the bundle
(x1;x2) = (10;10) c. Set up the utility maximization problem for the consumer, when facing
prices p1 = 6; p2 = B +1 and income m = 2520(A+2):
d. Solve the problem by �nding (x�1;x � 2) :
e. Graph the budget set, a couple of indi¤erence curves and the optimal choice.
Exercise 8: Assume preferences can be represented by the following utility function:
u(x1;x2) = �x12 +150x1 �2x22 +100x2 +x1 x2 a. Is the utility function monotonic? Justify. b. Obtain a bundle that is ranked higher than the bundle (x1;x2) =
(100;100) c. Set up the utility maximization problem for the consumer, when facing:
prices p1 = 2; p2 = 1 and income m = 30:
d. Solve the problem by �nding (x�1;x � 2) :
Exercise 9: Assume preferences can be represented by the following utility function:
u(x1;x2) = 4ln(x1)+ x2
a. Is the utility function monotonic? Justify. b. Set up the consumer�s utility maximization problem for prices p1; p2 and
income m (the general case) c. Solve the problem. You will obtain solutions x�1 (p1;p2;m) ;x
� 2 (p1;p2;m)
in terms of the parameters of the model (p1;p2;m) :
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Demand:
Exercise 10: You are the owner of a supermarket that wants to understand your client�s
preferences so that you can optimally price your products. You record a client�s purchases of two products x1 and x2 in 8 di¤erent occasions. The following table summarizes the results (for a similar exercise see Varian, ch.5, sec.4):
Obs. p1 p2 m x1 x2 1 3 2 94 12 29 2 2 2 106 15 38 3 2 3 95 10 25 4 1 3 170 20 50 5 3 1 83 15 38 6 2 1 135 30 75 7 4 1 143 22 55 8 1 2 168 28 70
a. Notice observations number 2 and 5. Quantities purchased are the same but prices are not. What does this mean in terms of the marginal rate of substitution at those quantities? b. Plot the 8 bundles purchased by the individual in a graph. c. Which type of preferences comes closer to describing this individual�s
behavior (Cobb-Douglas, Perfect Complements or Perfect Substitutes)? d. Write down a utility function that represents these preferences fairly well.
Exercise 11: You record a client�s purchases of two products x1 and x2 in 8 di¤erent
occasions. The following table summarizes the results:
Obs. p1 p2 m x1 x2 1 2 2 20 4 6 2 2 2 40 4 16 3 2 2 30 4 11 4 1 3 51 12 13 5 1 3 48 12 12 6 1 3 63 12 17 7 2 1 70 2 66 8 2 1 50 2 46
Notice that for observations 1,2 and 3 even though prices did not change for di¤erent amounts of income spent by the client, he/she still purchased the same amount of good one in all three occasions. The same can be said for observations 4,5 and 6 and observations 7 and 8. a. Draw the Engel curves for good 1 and the income expansion paths for the
three sets of prices in the table.
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b. The preferences underlying this individual�s behavior can be represented by one of the three utility functions in Exercises 7,8 and 9. Find which one it is and explain why. c. Based on your answer to part b, complete the following table with your
predictions on the client�s purchases for the given prices and income:
Obs. p1 p2 m x1 x2 9 2 2 50 10 2 4 100 11 3 3 45 12 4 1 20
Slutsky Equation:
Exercise 12: Assume preferences can be represented by the following utility function:
u(x1;x2) = x1 x2 2
a. Is the utility function monotonic? Justify. b. Set up the consumer�s utility maximization problem for prices p1; p2 and
income m (the general case) c. Solve theproblem. Youwill obtaindemand functionsx�1 (p1;p2;m) ; x
� 2 (p1;p2;m)
in terms of the parameters (p1;p2;m) : Obtain price elasticity of demand for good one. Obtain income elasticity of demand for good 2. d. Assume that, originally, the consumer faces:
prices p1 = 2; p2 = 5 and income m = 30(A+1);
where A is the last digit of your ASU ID#. Now assume the price of good 1 increases to p;1 = 3: Obtain the income and substitution e¤ects for good 1 with Slutsky compensation (that is, compensating the individual so that it can still buy the old bundle at the new prices). e. Find the amount of compensation needed for Hicks compensation (that is,
compensating the individual so that he is indi¤erent to his old bundle). To do this plug the old bundle into the utility function to obtain the level of utility you want to acheive. Then plug the demand functions into the utility function. Then replace prices with new prices and equate the two utilities. By now you should have a function of income equal to a number. Solve for the appropriate income level. That is the compensation needed to make the individual indi¤erent to the old bundle. The amount of compensation needed should be lower than with Slutsky compensation, but because the price change is very small, there should be barely any di¤erence between the two. f. Graph your results in (e) by plotting the old and new indi¤erence curves,
the old, compensated and new budget sets and the old, compensated and new choices (quantities demanded).
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Exercise 13: For the following demand function:
x�1 (p1;p2;m) = 4
p1 (3+m) ;
for values of m > 1: a. Obtain Income elasticity of demand. Plot the Engel curve for p1 = 1: b. Is this a normal good? c. Assuming that preferences are monotonic (then the individual always
spends all its income), use the budget constraint to solve for x�2 (p1;p2;m). d. The consumer faces the following prices and income level:
prices p1 = 1; p2 = 1:5 and income m = 5:
Calculate the quantity demanded for goods 1 and 2 at these prices and this income level. e. Obtain income and substitution e¤ects with Slutsky compensation when
the price of good 1 drops to p;1 = 0:5
Exercise 14: Assume preferences can be represented by the following utility function:
u(x;y) = �x12 +100x1 +20x2
a. Is the utility function monotonic? Justify. b. Set up the consumer�s utility maximization problem for prices p1; p2 and
income m (the general case) c. Solve the problem. You will obtain demand functions x�1 (p1;p2;m) and
x�2 (p1;p2;m) in terms of the parameters (p1;p2;m) : d. Graph the demand function for good 1 when the price of good 2 is p2 = 2
and income is m = 200: e. Obtain the change in consumer surplus when the price of good 1 goes
from p1 = 2 to p01 = (B +7)=2; where B is the last digit of your ASU ID#. f. Again, assuming the price of good 1 increases to p01 = (B + 7)=2: Find
the Compensating and the Equivalent Variations g. For the same price increase, obtain the income and substitution e¤ects
on good 1, both with Slutsky and Hicks compensations.
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Intermediate Microeconomic Theory Fall 2015
StudentName:___________________ Student ID________
Homework 1 - Part B
Front Page
Selected Answers:
Exercise 7.d.
x�1 = ________ x � 2 = ________
Exercise 9.c.
x�2 = ___________
Exercise 12.d.
The income required to purchase the old bundle at the new prices is:
m0 = ___________
Exercise 13.e.
The income e¤ect with Slutsky compensation in terms of good one is:
x1(p 0 1;p2;m)�x1(p
0 1;p2;m
0) = ___________
Exercise 14.e.
The change in Consumer Surplus is:
�CS = ___________
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