Engineering ThermodynamicsChapter 2: The first law of thermodynamics

Homework Set 2

Wayne HackerCopyright Wayne Hacker 2009. All rights reserved.

Homework set 2 Copyright Wayne Hacker 2009. All rights reserved. 1

Contents

1 The first law of thermodynamics 2

1.1 Work, power, torque, and energy . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Single and Multi-step processes . . . . . . . . . . . . . . . . . . . 5

1.1.2 Power and refrigeration cycles . . . . . . . . . . . . . . . . . . . . 7

Homework set 2 Copyright Wayne Hacker 2009. All rights reserved. 2

1 The first law of thermodynamics

1.1 Work, power, torque, and energy

Problem 1. (conservation of M.E.)

Ricky, the 5 kg snowboarding raccoon, is on his snow-board atop a frictionless ice-covered quarter-pipe withradius 3 meters (see figure at right). If he starts fromrest at the top of the quarter-pipe, what is his speed atthe bottom of the quarter-pipe? For ease of calculation,assume that g = 10 m/s2. Round your answer to thenearest 0.1 m/s.

(a) 7.0 m/s (b) 7.7 m/s(c) 8.5 m/s (d) 9.3 m/s(e) None of these

x Raccoon: 5 kg

3 m

3 m

Problem 2. (moment of inertia)

A disk-shaped flywheel of radius R, thickness h, and uniform density is rotating withan angular velocity rad/s. Using the definition for momentum of inertia

I V r2dV,

show thatIflywheel = piR

4/2

and the kinetic energy of the flywheel is given by

KEflywheel = I2/2 .

Problem 3. (kinetic and potential energy)

Two different masses m1 and m2 are dropped from rest at the same initial elevation.Ignoring the air resistance, show that the particles have the same speed when they hitthe ground.

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Definition 1. The drag force on a blunt body moving through air at a subsonic highspeed is given by

Fdrag =1

2CdAsairv

2 ,

where Cd is known as the coefficient of drag, As is the shadow area of the object, air isthe air density, and v is the velocity of the object.

Problem 4. (power/drag)

Find the power needed by a vehicle to move at a constant speed v against the force ofgravity. Assume the vehicle is moving at a fast enough speed for the above drag-forceformula to apply.

Problem 5. (power/torque)

The driveshaft of a certain alternator is turning at 300 RMP. It is driven by a belt runningfrom a pulley with a radius of 0.15 m. The net force applied by the belt on the pulley is2000 N. Determine the torque applied by the belt on the pulley (in Nm) and the powertransmitted (in kW).

Problem 6. (work done by a gas)

Consider a gas in a closed piston-cylinder assembly that is undergoing a constant-pressureprocess. The mass of the gas is mgas =

14

kg, the pressure is P = 5 bar, and the initialspecific volume is vi =

15

m3/kg. If the work done by the piston on the system is W = 15kJ, what is the final volume of the gas?

Problem 7. (work done by a gas)

Consider a gas in a closed piston-cylinder assembly. The gas is expands under a polytropicprocess with PV = const from an initial state with pressure Pi = 500 kPa and volumeVi = 1/10 m3 to a final state with pressure Pf = 100 kPa. Determine the work done bythe system on the environment. Also, draw the P -V diagram for the process.

Problem 8. (First law of thermo)

Each line in the table below corresponds to a closed system undergoing a process. Usingthe formula for the first law of thermodynamics Ef Ei E = QW determine themissing component values in each row.

Process Q W Ei Ef Ea +50 20 +50b +50 +20 +20c 40 +60 +20d 90 +50 0e +50 +20 100

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Problem 9. (first law related rate form/time-dependent systems)

Consider a gas in a closed piston-cylinder assembly. The system of gas undergoes aprocess during which there is energy transfer from the system by heat at a constant rateof 10 kW into the environment, and where the work done on the system by the pistondepends on time according to the equation

P = W ={8t 0 < t 1 hours8 t > 1 hours

where t is time (in hours) and W is in kW.

Heat transfer modes

Problem 10. (conduction)

A 0.2 meter-thick flat concrete wall separates two rooms of different temperatures. Thetemperature of the two rooms is held steady and it sets up a steady, linear, temperaturedistribute throughout the thickness of the inside of the wall. If the energy transfer rate,via heat conduction, through an area of one-square-meter of wall is 0.15 kW, then whatis the temperature difference across the wall?

Problem 11. (convection)

A flat surface having an area of 2 m2 and a temperature of 350 K is cooled convectivelyby a gas at 300 K. Using data from the table given in the book, determine the largest andsmallest heat transfer rates, in kW, that might be encountered for both free and forcedconvection.

Problem 12. (radiation)

A 1-cm-radius surface of a circular disk at temperature T = 1000 K emits thermalradiation at a rate 15 W. What is the emissivity of the surface? Assuming constantemissivity, plot the rate of radiant emission, in W, for surface temperatures ranging from0 to 2000 K. The Stefan-Boltzmann constant is = 5.67 108 W/m2 K4.

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1.1.1 Single and Multi-step processes

Problem 13. (single-step processes)

A gas contained within a closed, frictionless, piston-cylinder assembly is shown in figure1 below. We take the positive x-axis pointing upward. Initially, the piston face in contactwith the gas is at x = 0, and the spring exerts no force on the piston (i.e., the springis in equilibrium neither stretch nor compressed). As a result of heat transfer, the gasexpands, raising the piston until it hits the stops. At this point the face is located atx = Xstop, and the moment the piston touches the stops the heat source is turned off. Thisends the process. The force exerted by the spring on the piston as the gas expands varieslinearly with x according to Hookes law: |Fspring| = k|x|, where the spring constant kis assumed to be known. The other known parameters are: gravity g, the atmosphericpressure Patm, the mass of the piston mp and the surface area of the bottom and top ofthe piston Ap.

(i) Derive a formula for the initial pressure of the gas.

(ii) Derive a formula for the work done by the gas on the piston.

(iii) If the specific internal energies of the gas at the initial and final states are known tobe ui and uf , respectively, calculate the heat transfer Q over this process.

Figure 1: A closed, frictionless, piston-cylinder assembly with a spring and stabilizersattached to it. Initially the spring is uncompressed. As the gas is heated, the gasexpands and the rising piston begins to compress the spring. The piston continues torise until it hits the stoppers.

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Background: In this problem we are going to model a process that occurs naturallyin the atmosphere. Our system will be a small parcel of air. A parcel of air is the fluidequivalent to a point particle in mechanics. The collection of air molecules is consideredto be much smaller than the characteristic length scales occurring in the problem, butat the same time large enough to contain enough molecules so that the collection ofmolecules behaves as a continuous fluid, not a discrete swarm of molecules. Basically, weassume that there are enough molecules in the parcel of fluid that the collection behavesas a mini thermodynamic system. Parcels of air are conceptual constructs. They arefundamentally open systems: they hold on to their thermodynamic properties, but nottheir molecules! Molecules continuously enter and leave the parcel of fluid, but sincemolecules are not unique, we assume that the ones leaving the parcel are similar to theones entering the parcel. Parcels of air are fundamentally open systems.

Under the right conditions, we can use a closed system to model an open system. Onesuch process is in the case of strong atmospheric convection where a parcel of air is liftedso quickly that it doesnt have a chance to come into equilibrium with its surroundings(the environment). In this case the heat transfer between the parcel and the environ-ment is negligible. Such a process with negligible heat transfer is said to be adiabatic(impassable) without heat transfer. In this case, the only way for the parcel of air toeither gain or lose energy is through work due to pressure on the imaginary boundaryof the air parcel. Well use a perfectly-insulated, closed, frictionless, piston-cylinder as-sembly to model our air parcel. This apparatus does everything we want: it isolates thegas contained in the system, it does not allow heat transfer, and it only allows work totransfer energy to the gas via work done by pressure. Thus, once again the concept ofa closed, frictionless, piston-cylinder assembly has come in as a handy tool for studyingthe behavior of thermodynamic systems.

Problem 14. (first law of thermo and potential energy)

Consider a gas of mass mgas in a perfectly-insulated, closed, frictionless, piston-cylinderassembly, where we take the gas to be our system. The apparatus is lifted from ground-level to a height y = h in the atmosphere. We assume that the apparatus is lifted so slowlythat the system is in a state of quasi-equilibrium throughout the entire process. This canbe accomplished by initially accelerating the apparatus from rest to a negligibly smallvelocity. This small time interval over which the system is accelerated can be neglected.Thus, the acceleration over most of the apparatuss path can be approximated as beingidentically zero.

During this process, the system experiences a change in external pressure from the en-vironment, since pressure decreases with height. Since the apparatus is insulated, thesystem is isolated, so there is no heat transfer between the system and the environment.Thus, the only mode of energy transfer to or from the system is through work done bypressure.

Assuming that the system of gas is in equilibrium with its environment at ground level,derive a formula for the change in the internal energy of the system as a function ofheight.

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Problem 15. (multi-step processes)

Air in a closed piston-cylinder assembly undergoes two processes in succession:

Process 1-2: The first process is a polytropic compression with n = 1.3. State1 has pressure P1 = 100 kPa, and specific volume v1 = 0.04 m

3/kg. State 2 hasspecific volume v2 = 0.02 m

3/kg.

Process 2-3: The second process is a constant-pressure process. The final state 3has specific volume v3 = v1.

Sketch the processes on a P -V diagram and determine the work per unit mass of air, inkJ/kg.

1.1.2 Power and refrigeration cycles

Problem 16. (hot body, cold body)

For each of the following systems identify what plays the role of the hot body and whatplays the role of the cold body for the appropriate schematic of the power cycle andrefrigeration cycle.

(a) Window air conditioner

(b) Window heat pump

(c) Steam engine of a train

Problem 17. (power)

For a power cycle with heat transfer Qin = 50 kJ and Qout = 35 kJ. Determine the workdone over one complete cycle Wcycle (in kJ) and the thermal efficiency .

Problem 18. (refrigeration)

For a refrigeration cycle with Qout = 500 kJ, and a coefficient of performance = 1.5,determine Qin and Wcycle (in kJ).

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Problem 19. (analyzing cycles)

The following table gives data (in kJ) for a gas (the system) in a closed, frictionless,piston-cylinder apparatus undergoing a thermodynamic cycle. The cycle consisting offour processes in succession. None of the processes involve a significant change in thesystems kinetic or potential energy, so these effects can be neglected. The first-law thenreduces to U = Q W . Use this information to complete the table and determine ifthe cycle is a power cycle or refrigeration cycle.

Process U Q W1-2 +600 6002-3 13003-4 700 04-1 +500 700

Problem 20. (analyzing cycles)

A gas in a closed, frictionless, piston-cylinder assembly undergoes a thermodynamic cycleconsisting of three processes:

process 1-2: A polytropic process PV = constant, where the gas undergoes a compressionand the system has no change in internal energy: U12 U2 U1 = 0. Also,State 1: P1 = 1 bar = 100 kPa, V1 = 1.6 m3,State 2: V2 = 0.2 m3.

process 2-3: Constant pressure expansion of the gas from to volume V2 to volume V3 = V1.process 3-1: Constant volume, U31 = U1 U3 = 1000 kJ

Assumption: We can neglect any change in the systems kinetic and potential energy. In thiscase, the 1st law of thermodynamics reduces to the equation U = QW .

(a) Find the work done over process 2-3: W23.

(b) Find the heat transfer Q23 for process 2-3, in kJ.

(c) Determine if the cycle is a power cycle or a refrigeration cycle.