Chapter 7 -- Conservation of Energy

 

 

 

Updated ---  28 October 2006

 

 

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Hint

Answer

Update Date

7.1

Consider the lunar module as the system for both Parts (a) and (b). Both approaches to the solution should give the same answer. Be very careful with signs. The first answer may seem strange since it says that the answer does not depend upon the direction of motion as long as the vertical speed of the lunar module is 3 m/s. 

To clarify this, think about the change in elevation it would take for the module to reach the ground. When the module is traveling up at 3 m/s, it will keep going up until it slows to zero velocity and then returns to its initial elevation (and coincidentally the same 3 m/s velocity) but this time it will be moving in the opposite direction. This is exactly the same case as occurs when you start with a 3 m/s velocity moving towards the moon. Thus it will take more time for the lunar module to land if it initially travelling up; however, the distance it can safely "fall" is the same. This also explains why it can safely fall a greater distance if their is no initial velocity.

If V1 = 3 m/s ↑ or ↓, then
z1-z2 = 4.894 m

If V1 = 0 m/s then
z1-z2 = 7.645 m

19Oct05

7.2

 

 

 

7.3

 

 

 

7.4

 

 

 

7.5

 

 

 

7.6

(a) Apply the definition of mechanical work.

(b) Apply the energy balance to a system that includes the collar and the spring. Based on an examination of this system assume no heat transfer and negligible changes in internal energy. (If you somehow forgot to include the spring V2 = 4.2 m/s). Remember that spring energy is related to the spring extension from the unextended length, i.e. in State 1 the spring already has energy stored in it since it is extended 0.5 m. Why is it easier to put the spring inside the system?

(a)  W1-2,in = 64.95 N-m

(b) V2 = 4.927 m/s

19Oct05

7.7

 

 

 

7.8

Reference in Problem is incorrect. It should be to Problem 7.7 The answer to this problem depends on how you interpret "the last 5 feet of the ramp." The answer reported here assumes that the package travels a total of ten feet before it stops and that the last five of these ten feet has the friction strip.

mk = 1.17

19Oct05

7.9

(a) What would be wrong with including the piston inside the system?

(b)  Apply the defining equation for PdV work into the system. Work each process individually and see how P and V are related so that you can perform the integration. (See section on PdV work in text.)

(c)  Sketch the three processes on a P-V graph to scale. The areas under each curve represents the absolute value of the work for the process. The enclosed area represents the net work into the system for the process.

(d)  The net work in is just the sum of the work for each process.   (For kicks, write the closed system, finite-time form of the energy balance for the three stage process. Because this is a cycle the final and intial states are identical and Efinal  = Einitial.  What does this say about the net heat transfer into the system for this process.?

(a) System should be the gas.

(b) W1-2,in = 128.8 kJ;
W2-3,in = -320kJ; W3-1,in = 0

(d) Wnet,in -191.2 kJ

19Oct05

7.10

Application of finite-time form of conservation of energy plus the definitions of kinetic energy and gravitational potential energy. Careful with units.

DEk = 186.5 ft-lbf
DEG = -745.9 ft-lbf
DU = 0.462 Btu

01Nov05

7.11

(b) Application of rate form of conservation of energy to a closed system and understanding of dE/dt.
(c)  Application of finite-time form of conservation of energy to a closed system and integration of heat transfer rate and power over a time interval.

(b) dE/dt = 2.0 kW at 1.2 h and 0 kW at 2.4 h.
(c)  DE = 5.00 kW-h = 18,000 kJ

19Oct05

7.12

System boundary should surround the gas and "cut" the paddle wheel shaft. Use definition of PdV work and finite time form of conservation of energy.

2790 J

19Oct05

7.13

(a) Treat the gas as your system. Now use PdV work relation to find the work during the change in state. BUT how do you find P. Look at the piston as a system for Cons. of Linear Momentum and assume that acceleration of the system is negligible. How is pressure inside the gas acting on the left side of the piston related to the Patm and the force of the spring acting against the exterior of the piston? Now if you think about the mechanical work done by the piston against Patm and Fspring you can find where the energy goes -- into the spring or into the atmosphere.

(b) Definition of shaft work in terms of torque and angle of rotation

(c) Too answer (c) you must apply the cons. of energy equation to a system. What's inside this system? If you include part of the stirring device inside your system, you must assume what about its mass or internal energy change to get ΔU for the gass.

(a) Win = -115.6 J for the gas
31.2% against spring and rest against atmospheric pressure.

(b) Wshaft, in = 427.3 J

(c) ΔU = 311.7 J for gas

19Oct05

7.14

 

 

 

7.15

Application of rate form of conservation of mass and conservation of energy to the compressor. Note that many of the assumptions given in the problem are best applied after you apply conservation of mass and collect the mass flow rate terms.

(b) 4.2
(c) 2.74 kW = 3.76 hp

19Oct05

7.16

 

 

 

7.17

A pretty standard COE problem. Note, to solve for Part c) you will need a system boundary that cuts through the shaft. Use the internet to find any unit conversions you may need.

a) 194000 Btu/h

b) 258 amps

c) 111.8 ft.lb

 

7.18

(a)  Apply conservation of mass to the entire system of heat exchanger and turbine.
(b)  Try applying conservation of energy and mass to just the heat exchanger, if insufficient equations then also try applying it to the turbine. (Alternatively you might be able to apply it to the entire system, but how can you do this to find the heat transfer for the heat exchanger?)

(a)  800 kg/s
(b)  3163 MW
(c)  1790 kN-m

19Oct05

7.19

(a) Strategy is to apply the rate form of the conservation of energy and mass equations. Use the open system around the gas turbine to find the shaft power out of the system. Use the closed system around the generator to find the electrical power out. (You could also find the electrical power out by using a combined open system that includes both systems.)

(b)  Use the definition of shaft power to relate the torque to the rotational speed. Most common source of error is the unit conversion from "rpm" , revolutions per minute, to "rad/s", radians per second.

(a) Shaft power out = 800 Btu/s
Electrical power out = 780.8 kW(b) Torque = 3302 ft-lbf 
@ 1800 rpm (188.5 rad/s)

19Oct05

7.20

(a)  Application of rate-form of conservation of energy and mass to the correct system. Carefully consider your options. Where are the boundaries on which I know something? How can I relate the given information to the change in specific enthalpy across the air compressor for the air?
(b)  Same thing as (a).
(c)  Application of definition of shaft power. (This does not require a system selection as the shaft power is evaluated at the boundary.)

(a)  703.5 kg/min
(b)  187.6 kW
(c)  2986 N-m

24Oct06

7.21

 

 

 

7.22

First--- Error in Problem Statement. It should say repeat 7.21 (not 8.21)
(a)  Ideal gas.
(b)  Definition of PdV work, Yup, ya' gotta perform the integral!
(c)  Application of conservation of energy (finite-time form). Use PV1.3 relation to find P2 and then use PV=mRT to find temperature which you will need to evaluate changes in internal energy.

(a)  0.1169 gram
(b)  28.87 J
(c)  ???

19Oct05

7.23

 

 

 

7.24

Two possible systems: volume occupied by water in the tank OR volume occupied by water and heating element in the tank. One of these systems has electric power transfers and no heat transfer and the other has only heat transfer with no electric power. If you pick the one with no electric power transfer, then you need a second system to actually find the electric power input. To find the resistance just remember Ohms Law V=IR and the definition of electric power IV where V is now voltage difference.

(a) 0.4157 lbm/s
(b) 35.1 kW
(c) 1.38 ohm

19Oct05

7.25

Apply problem information and definitions to answer questions where possible.

If you need to relate input and output energy transfers then you must apply the general energy equation to the appropriate system.

(g)  Here you must look as the contents of the tank. Determine the increase of internal energy for this system during the time interval. Now using the constitutive relation relating u and T, relate ΔT to Δu.

(a) 120 rpm and 100 rpm
(b)  9.520 hp
(c) 9.520 hp
(d) 10.58 hp
(e)  10.58 hp
(f)   30.87 ft-lbf
(g)  0.231oR

19Oct05

7.26

 

 

 

7.27

What should your system be if you want to know the heat transfer rate from the pump by itself? What's the appropriate substance model for liquid water and how can you use it to relate changes in enthalpy to changes in temperature and pressure.

(a)  2.80 kW out of the system
(b)  218.3 N-m
(c)  95.5 amps

19Oct05

7.28

 

 

 

7.29

Apply conservation of energy (finite-time form) to the gas in the system and assume the CO2 acts like an ideal gas with room-temperature specific heats. Also use definition of PdV work.

W1-2,in = -112.5 kJ
Q1-2,in = 895.9 kJ

19Oct05

7.30

All sorts of good stuff in this problem! You will need COM, ideal gas model, COE, constitutive relations, and definition of mass flow rate. Note that the air and the oil do not mix.

a) 93.8F

b) 5.85 ft2

c) 0.00267 ft2

 

7.31

Consider the air as a closed system. Is this a finite-time or rate problem? Is the air a steady-state system? What types of work transfers of energy are possible for the air? What can you say about the process that the air undergoes as its volume is decreased? What substance model can you use to relate changes in internal energy to changes in other properties, e.g. temperature and pressure?

Win =  16.64 kJ

24Oct05

7.32

Apply conservation of mass and energy to the pump assuming that the incompressible model describes the behavior of the liquid. Notice how even a small increase in temperature results in a tremendous increase in shaft power into the pump.

(a)  D2/D1 = 0.577
(b)  3.04 kW = 4.08 hp
(c)  5.43 kW = 7.29 hp

19Oct05

7.33

Is this a steady-state or a finite-time problem?  Is this an open or a closed system? What energy transfers are important for a nozzle? A nozzle is a device designed to increase the kinetic energy of a flowing fluid without any energy transfers across the non-flow boundaries of the device. Be VERY careful with units.

V2 = 333 m/s
A2/A1 = 0.2364

24Oct05

7.34

Same as Problem 7.33, but a different substance model. Be VERY careful with units.

(a) V2 = 20.0 m/s
(b) V2 = 17.82 m/s

19Oct05

7.35

(b)  Application of Ohm's law and definition of electric power
(c)  Application of energy balance to the battery assuming the battery is adiabatic during the process.
(d)  Apply Newton's law of cooling to predict the temperature of surface of the resistor along with the known heat transfer rate from the resistor.

(b)  0.0144 watts; 1.2 mA
(c)  -26.0 J
(d)  316 K or 43oC

19Oct05

7.36

(a)  Consider definition of average power for an ac system.
(b)  Consider application of conservation of energy to the transformer. (In case you are unfamiliar with a transformer it consists of an input electrical power connection and an output electrical power connection -- a box with a power cord in and a power cord out.)
(c)  Your answer depends on the number of surfaces that have convection heat transfer. (If you assume a six-sided cubic transformer, then the temperature is 83.3oC.)  If you also allowed thermal radiation (as would happen in most real transformers), you could lower the temperature from 95oC to 57.2oC.

(a)  1.046 amps
(b)  21.0 watts
(c)  95.0oC (5 sides with heat transfer)

19Oct05

7.37

Same approach as for 7.38 only this time you do not know the change in internal energy, work and heat transfer for each process. So you will have to these before you can find the cycle information. You have four processes you must consider. Treat helium as an ideal gas.

None available

I think its a refrigerator, what do you think?

19Oct05

7.38

(a)  Apply finite-time form of the conservation of energy to the gas and evaluate the energy transfers and changes for each process. It is suggested that you consider each process sequentially, i.e. 1-2, 2-3, 3-4, and 4-1. A correct solution should complete the table so that Qnet,in = Wnet,out and ∆Ucycle = 0.
(b) Examine the net work produced by the cycle.
(c)  Depending on your answer for (b) calculate the appropriate MOP. It cannot have two different MOP's. It's either a power cycle OR a refrigerator.

(a)  ???
(b)  ???

19Oct05

7.39

(a)  Apply finite-time form of the conservation of energy to the gas and evaluate the energy transfers and changes for each process. It is suggested that you consider each process sequentially, i.e. 1-2, 2-3, 3-4, and 4-1. A correct solution should complete the table so that Qnet,in = Wnet,out and ∆Ucycle = 0.
(b) Examine the net work produced by the cycle.
(c)  Depending on your answer for (b) calculate the appropriate MOP. It cannot have two different MOP's. It's either a power cycle OR a refrigerator.

Process 2-3
Qin = 10.08 kJ  Win = -2.73 kJ

For the complete cycle,
Wnet,in = -4.76 kJ

19Oct05

7.40

(a) Show your work. -- Definition of COP for a Heat Pump
(b)  Energy to run for one day found by Integrating the power required to run for one day.
(c)  Compute energy to run for 30 days.
(d)  Without a heat pump, you must buy all of the energy required to heat the house using electric resistance heating from the electric company. An electric resistance heater is like a heat pump with a COP =1

(a)  3.02
(b)  238,000 Btu
(c)  $167.52
(d)   ???

19Oct05

7.41

 

 

 

7.42

Apply Work-Energy Principle for a particle to the bucket. Pay careful attention to the work term and the fact that it is the integral of a dot product between the net surface force on the bucket and displacement of the bucket, F·ds. How are the vectors F and ds oriented with respect to each other?

4.05 m/s

19Oct05

7.43

The only (?) system for this problem is the package. It is also suggested that you consider the overall process 1 → 3 as the sum of subprocess 1→2 and subprocess 2→3. State 1 is the initial state of the package; State 2 is the state of the package as it reaches point B at the bottom of the ramp; and State 3 is the state of the package at Point C. For each subprocess, draw a free-body diagram showing all surface forces, then apply the work-energy principle (mechanical energy balance) to each subprocess. Be careful about the sign of the work term for 2→3.

d = 20.271 ft

19Oct05

7.44

(a) & (b) Apply the energy balance to a system consisting of the mass attached to the bungee cord and the bungee cord. Also be careful about calculating the deflection of the spring, i.e. the extension/compression of the spring from its unloaded length. Make the standard assumptions to recover the mechanical energy balance, i.e. no heat transfer and no changes in internal energy.

(c) Use Conservation of Linear Momentum OR if you know how velocity changes with position you might be able to differentiate that expression with respect to time.

(a)  533 lbf/ft
(b)  37.0 ft

19Oct05

7.45

Since the two springs initially support the large block, they have an initial deflection. Note that that you must treat this as a two step process. First an impact problem and then frictionless motion. Be careful to identify your systems from each step of the process. (Do you have to use the same system for each process?  NO!)

0.271 ft

19Oct05

7.46

 

 

 

7.47

 

 

 

7.48

At first look you might assume that this is only a MEB problem with a system including both blocks and the spring; however, it has the telltale impact (friction inside the system) that makes the MEB invalid. (Mechanical energy is not conserved in this process; however, the all inclusive energy is conserved.) Thus, this requires analysis as a two-step process:
1 --> 2  Using a closed system of the two blocks, apply the finite time form of conservation of linear momentum. In addition, you must assume that the time interval is so small that all external forces are negligible during this process. (The terms perfectly plastic or perfectly inelastic means that the two blocks stick together after they contact.) [For those of you who do not believe that you get two different answers using the MEB and CofLM, I suggest you solve for the velocity of the combined blocks after the impact using the MEB and compare your results with the CofLM result.]
2 --> 3  Now using a closed system consisting of the two blocks and the spring, you can apply the MEB for a finite time interval and solve for the final displacement. [Again this last portion could have also been done using CofLM for just the two blocks and then treat the spring as an external force that varies with position.]

(a) 1.00 m ;  (b)  0.817 m

19Oct05

7.49

 

 

 

7.50

 

 

 

7.51

 

 

 

7.52

 

(a) 900 K; (b) Win = 1050 kJ

28Oct05

 7.57

Never, ever, ever use COE during an impact! You can use COE before an impact, you can use COE after an impact, but never use COE during an impact! That is what the finite-form of COLM is for.

a) 19.7 m/s

b) 5.7 m

 

 7.60

The change in height between the inlet of the pump and the inlet of the hot water tank is not negligible. You will have to account for this change in specific potential energy in your equations.

a) 0.0359 kg/s

b) 7.94 W

c) 2.14 m2

 

7.69

See the hint for 7.57.

a) vb,before_impact = [2(mbgL + 1/2kd2)/(mb)]1/2

b) v = (mbvb,after impact)/(mb+mm)

 

7.71

You do not have any equations to compute enthalpy h at a specific state, you only have equations to compute the change in enthalpy from one state to another. Therefore, you must write your equations so they contain terms like (h1 - h3) and (h3 - h2). COM may be helpful.

a) 0.4 ft3/min