GGRA03H3 Lecture 6: Lecture 6.docx

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21 Mar 2012

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A new way of understanding and seeing cities and urban processes. Ecology is the study of the relations of living organisms to each other and to their surroundings. Ecology studies flow within particular ecosystems of nutrients, energy, waste and the subsystems that structure those flows cities are a special kind of ecosystem. What is the overall structure of urban ecosystems. Closed and open loops there is no waste in natural ecosystems seeing cities as ecosystems makes clear that cities are not closed loops cities import water, energy, food on a vast scale from a very long distances. Natural ecosystems contain multiple relatively stable nutrient loops. And export their wastes into global waste sinks. Most waste disposal solutions so far have involved finding better waste sinks, not reducing or re using industrial ecology seeks to close loops. At the global scale all loops are closed.

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Situations arise in engineering when we want to control the flow rate of a solution. This solution could be pumped into a holding tank, distillation column, heat exchanger, reactor, etc. This means that the flow rate becomes a function of time. When we turn on a pump, the flow rate is (for the most part) constant. The problem is that we may want the flow rate to be very fast at first, and then very slow towards the end of the process. This is especially true in certain chemical reactors where the concentration of reactants is always important. Control systems that could handle this type of situation are very expensive. Pumps (relatively speaking) are cheap. So, having a bank of pumps will usually solve the problem. The pumps are simply turned on or off to regulate the flow. Mathematically, this amounts to taking the volumetric flow rate function and finding a piecewise-linear approximation. Suppose that, because of the application we are running, we need a flow rate given by the function ,where t is in hours and v(t) is in 100 cubic feet per hour (e.g., v(0)=1 which means that at time t=0, we need a flow rate of 1*100=100 cubic feet per hour). Find a linear approximation and a piecewise-linear approximation using first two lines and then three lines. Think about what using more lines would do to the approximation. You will need to find suitable values of t to be able to find your approximation. As a further exercise, try to find values of t for your approximation that keeps the error (i.e., the difference between the approximation and the real function v(t) ) less than 0.1.Hint: to help with this, ask yourself this question: where would more lines be useful - where the function changes a lot or where the function changes a little?]

A magnetic field of 0.1T forces a proton beam of 1.5 mA to move ina circle of radius 0.1 m. The plane of the circle is perpendicularto the magnetic field.

47. Of the following, which is the best estimate of the speed of aproton in the beam as it moves in the circle?
(A) 10-2 m/s (B) 103 m/s (C) 106 m/s (D) 108 m/s (E) 1015 m/s

52. An ideal gas is initially in a state that corresponds to point1 on the graph above, where it has presÂ¬sure p1, volume V1, andtemperature T1. The gas undergoes an isothermal process representedby the curve shown, which takes it to a final state 3 attemperature T3. If T2 and T4 are the temperaÂ¬tures the gas wouldhave at points 2 and 4, respecÂ¬tively, which of the followingrelationships is true?
(A) T1 < T3 (B) T1 < T2 (C) T1 < T4 (D) T1 = T2 (E) T1 =T4

53. The absolute temperature of a sample of monatomic ideal gas isdoubled at constant volume. What effect, if any, does this have onthe pressure and density of the sample of gas?
Pressure Density
(A) Remains the same Remains the same
(B) Remains the same Doubles
(C) Doubles Remains the same
(D) Doubles Is multiplied by a factor of 4
(E) Is multiplied by a factor of 4 Doubles

62. A sample of an ideal gas is in a tank of constant volume. Thesample absorbs heat energy so that its temperature changes from 300K to 600 K. If v1 is the average speed of the gas molecules beforethe absorption of heat and v2 is their average speed after theabsorption of heat, what is the ratio v2/ v1 ?
(A) 1/2 (B) 1 (C) (D) 2 (E) 4

63. Two people of unequal mass are initially standing still on icewith negligible friction. They then simulÂ¬taneously push each otherhorizontally. Afterward, which of the following is true?
(A) The kinetic energies of the two people are equal.
(B) The speeds of the two people are equal.
(C) The momenta of the two people are of equal magnitude.
(D) The center of mass of the two person system moves in thedirection of the less massive person.
(E) The less massive person has a smaller initial acceleration thanthe more massive person.

64. Two parallel conducting plates, separated by a distance d, areconnected to a battery of emf e. Which of the following is correctif the plate separaÂ¬tion is doubled while the battery remainsconnected?
(A) The electric charge on the plates is doubled.
(B) The electric charge on the plates is halved.
(C) The potential difference between the plates is doubled.
(D) The potential difference between the plates is halved
(E) The capacitance is unchanged.

68. A rod on a horizontal tabletop is pivoted at one end and isfree to rotate without friction about a vertical axis, as shownabove. A force F is applied at the other end, at an angle ? to therod. If F were to be applied perpendicular to the rod, at whatdistance from the axis should it be applied in order to proÂ¬ducethe same torque?
(A) L sin ? (B) L cos ? (C) L (D) L tan ? (E) L

70. A 4 ÂµF capacitor is charged to a potential differÂ¬ence of 100V. The electrical energy stored in the capacitor is
(A) 2 x 10-10 J (B) 2 x 10 8 J (C) 2 x 10 6 J (D) 2x 10-4 J (E) 2 x10-2 J

Vapor cycles
5EM - Thermodynamics
September 2010
1 The T-S diagram
The temperature-entropy diagram is an alternative to the well knownpressure-
volume diagram. Before using it to analyze different vapor cycles,we need
to understand it properly.
1. Plot the following transformations for ideal gases in a T-Sdiagram
(a) Isothermal expansion
(b) Isobaric expansion
(c) Isometric heating
(d) Reversible adiabatic expansion
(e) Irreversible adiabatic expansion
2. Repeat the last question, but this time do it for a realsubstance,
starting in the compressed liquid1 region and ending in thesuperheated
region, when possible. Indicate the boundaries of eachregion.
3. We know that under certain circumstances the area under theplot
in a P-V diagram represents the work performed during theprocess.
What does it represent in this case? Under what assumptions isyour
answer true? Compare examples 6.1 and 6.2.
2 Thermal machines
We understand by thermal machine a device capable of transformingther-
mal energy (heat) into mechanical energy (work). We already knowthat
these machines have a limited efficiency which, in the two-sourcesmodel, is
obtained by Carnotâs cycle.
(Compressed liquid is sometimes also called subcooled liquid)
2.1 Carnot cycle
This cycle, although efficient, is impractical. To begin with, theextreme
temperatures of the cycle should be within the sources temperaturesfor a
finite heat flow to exist.
1. Given the temperature of the sources, draw the cycle accordingto the
named limitation.
2. Show that this leads to lower efficiency than the theoreticallimit.
3. Assume that heat flow from/to the sources is proportional to thetem-
perature difference between them and the working fluid, anddemon-
strate that a higher power output is related to a decreasedefficiency.
2.2 Rankine cycle
Out of the several practical possibilities, Rankine cycle and itsvariations is
widely used, especially for electric power generation.
1. Draw the basic cycle in a T-S diagram and explain it.
2. What does the efficiency of the cycle depend on? Comparewith
Carnotâs.
3. What are the differences between the ideal cycle and the realone?
4. Describe how it is modified by superheating and/or reheating.Explain
te benefits.
2.3 Problems
After answering the previous questions solve the followingexercises from the
book: 8.2, 8.4, 8.5, 8.13, 8.19, 8.20
3 Frigorific machines
3.1 Reverse Carnot cycle
We have already studied Carnot cycle in detail as the ideal thermalmachine.
Now we will take is as the starting point to understand thecharacteristics
of frigorific machines.
1. Draw the ideal cycle in a T-S diagram and explain it.
2. Show that this cycle has a theoretical efficiency Ã =Tcold/Thot-Tcold
3. What are the difficulties in implementing such cycle in actualdevices?
3.2 Actual refrigerators
1. Why is the turbine replaced by an expansion valve in mostcases?
2. Plot the actual cycle and explain the differences with the idealcase.
3. How do cascade and multistage systems improve the performanceof
the refrigerator?
4. What aspects are important in the choice of a refrigerant fluid?Justify
3.3 Problems
After answering the previous questions solve the followingexercises from the
book: 10.2 10.7, 10.8, 10.9, 10.22, 10.23 10.1D

GGRA03H3 Lecture 6: Lecture 6.docx

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