chapter1 thermodynamics

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Engineering ThermodynamicsLecture Notes

Chapter 1(Draft )

Wayne HackerCopyrightWayne Hacker 2009. All rights reserved.


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Thermodynamics 232 Lecture notes CopyrightWayne Hacker 2009. All rights reserved.1

Contents

1 Introductory concepts, definitions, and principles 2

1.1 What is thermodynamics? . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Artificial subdivisions of thermodynamics . . . . . . . . . . . . . . 3

1.1.2 Engineering Thermodynamics . . . . . . . . . . . . . . . . . . . . 4

1.2 What is Energy? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Defining thermodynamic systems . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 What is a system? . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . 9

1.3.3 Property, state, and process . . . . . . . . . . . . . . . . . . . . . 9

1.3.4 Extensive and intensive properties . . . . . . . . . . . . . . . . . . 11

1.3.5 Equilibrium and quasi-equilibrium states . . . . . . . . . . . . . . 12

1.4 Physical quantities, dimensions, and units . . . . . . . . . . . . . . . . . 13

1.4.1 Dimensional consistency . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 Density, specific volume, and specific weight . . . . . . . . . . . . . . . . 19

1.6 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.7 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.7.1 Conversion formulas . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.8 Static fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.9 Introduction to the 4 laws of thermodynamics . . . . . . . . . . . . . . . 22


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1 Introductory concepts, definitions, and principles

About the Notes

The purpose of the exercises

No set of class notes would be complete without a set of exercises to accompany it. I viewthe exercises as mental exercises in thermodynamics. You would not dare set foot on afootball field, wrestling mat, or track for a competitive event without doing exercises onyour own time. You would then go to every practice (class lecture) before the big game(the exams). While this analogy is not perfect, you get my point.

Disclaimer

This set of notes are not finished numerous topics are not fully developed. I am still newat teaching thermodynamics and need a summer to think out my notes more carefully.Unfortunately, my home department is physics and the needs of the physics departmenttake precedence over this class. That is to say, with my other class teaching obligations,I simply dont have enough spare time in my life at this point to make this set of notesinto something good. Think of these notes as a random set of collected ramblings, ascreed.

1.1 What is thermodynamics?

Crudely, thermodynamics is the study of interaction between matter and energy. But thisis almost a vacuous statement since it hard to imagine any event in nature that does notdeal with the interaction between energy and matter. Today thermodynamics is a verybroad field with applications in every area of science from biology to physics. However, itwasnt always this way. The field had humble beginnings as inventors struggled to figureout how to turn heat energy into a form of energy that can do mechanical work.

Thermodynamics did not emerge as a science until the late 1600 hundreds (and somewould say much later) in the study of the first steam engines. The name comes fromthe Greek words Therme (heat) and dynamis (dynamite/power), and was first used by

Lord Kelvin (formerly William Thomson) in an 1849 publication. The first textbook waswritten in 1859 by William Rankine, for whom the Rankine temperature scale is namedafter.

Although most people associate thermodynamics as the study of heat, it has a long andinteresting history with the study of both heat and lack of heat (cold). One of the mainareas of research in modern thermodynamics is the study of cold. In fact, it could beargued that Michael Faraday, the famous 19th century scientist, was one of the originalfounders of man-made refrigerators. It was his experiments (and failures) with ammoniathat lead him to realize that by compressing and releasing gas in the right order, thatyou could cool the surroundings. He even went so far as to suggest ways that compressed


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gas could be used to cool food and other perishables and how developing this technologywould have great potential industrial applications . Fortunately, he was far too interestedin science to waist time on such a mundane endeavor.

From a practical engineering point of view, thermodynamics is the study of the transfor-

mation process of converting work into heat and heat back into work, and the efficiencywith which these two processes can occur.

1.1.1 Artificial subdivisions of thermodynamics

There are two basic areas of thermodynamics: microscopic and macroscopic.

Microscopic study: This area focuses on microscopic issues. Its main focus is thestructure of matter and the interaction of molecules in collisions. It is known asstatistical physics, or statistical thermodynamics 1. Topics studied in this disciplineinclude, but are not limited to: lasers, plasmas, high-speed gases, rarified gases(such as in the outer atmosphere), turbulence in fluids.

Macroscopic study: This area focuses on macroscopic (bulk) energy flow. It is typ-ically referred to as Classical Thermodynamics, or Engineering Thermodynamics.This field focuses on

the transfer of energy

the transformation of energy

the storage of energy

We will not study the details of molecular motion in this course; however, we will notignore them either. While it is true that thermodynamics has its historic roots in exper-imentation and observation (the macroscopic world), it is also a fact that modern-daythermodynamics owes much of its success to the study of the behavior of the moleculescomposing the matter under scrutiny. That said, the macroscopic study is theoretically

justified by the continuum hypothesis, an assumption used to avoid direct dealings withmolecular interactions. Mathematically, the continuum hypothesis says that if we startedwith a fixed amount of matter and kept cutting it in half, then the parts would look justlike the whole. We know that this is not the case. There is a limit where if we go below

it, we will be at the molecular level with lots of empty space. It is at these small lengthscales that we need the methods of statistical physics to justify the continuum hypothesis.

We will try to balance the these two approaches by avoiding the technical mathematicsassociated with statistical physics, but at the same time not forgetting that all matteris composed of atoms and molecules, and action of these molecules necessarily affectsthe behavior of the substance. Moreover, without studying the molecular behavior wecannot understand the concepts behind internal energy. We will therefore be forced towalk a proverbial tightrope between the macroscopic and microscopic worlds. Be careful,dont fall off!

1A common sub-division of statistical physics used by engineers is known as kinetic theory


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1.1.2 Engineering Thermodynamics

Engineering thermodynamics, as a discipline, can be subdivided into three broad areasof study:

the transformation of energy from one form into another

(e.g., a car skids to a stop kinetic energywork heat)

the transfer of energy across boundaries

(heat + piston-cylinder container of waterwork move piston)

the storage of energy in molecules (heat, internal energy)

Of course, in practice these studies overlap.

The engineers objective in studying thermodynamics is the analysis and design of large-scale systems such as power plants, solar farms, air-conditioning systems, etc. Someexamples of areas of interest involving thermodynamics for engineers are listed below

large-scale energy storage

how energy is transferred through heat and work

how energy transforms from one form of energy into another(e.g., heat mechanical work)

the economic impact of various materials used for heat insulators and conductors

Example 1. Which of the following situations are related to the main focus of the studyof thermodynamics.(a) storing the energy made by a windmill(b) converting the energy stored in coal into work via a steam engine(c) determining the energy loss due to friction in the moving parts of an engine(d) how energy is bought and sold on the open market

Solution: Statements (a)-(c) are activities that engineers participate in.


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1.2 What is Energy?

Since thermodynamics involves the study of energy we should start with a little intro-duction to energy. Energy comes in many forms. It can be thought of as the ability tocause change. Here are a few common forms of energy:

kinetic energy [energy due to motion]

potential energy [energy due to position, or the configuration of a system]

internal energy [energy stored in the moleculesboth potential and kineticassociatedwith temperature]

chemical energy [energy due to chemical composition (i.e., energy due to the ar-rangement of molecules with respect to other molecules in the system)]

nuclear energy [subatomic energy (i.e, energy stored in the nucleus of an atom)]

In order for energy to change from one form to another it must go through some sort oftransition. The transition process is a mechanism for converting one form of energy toanother. Here are two conversion mechanisms:

work [mechanical energy in transit]

heat [molecular energy in transit]

Think of these mechanisms as energy in transit. Do not confuse these concepts with theconcept of energy. Conversion mechanisms differs from energy in a fundamental way. Theenergy stored in a system is independent of the way it goes into the object; whereas, thework done on an object (i.e., the amount of energy added or subtracted from a system)depends on the process (the path) used to transfer the energy into or out of the system,except when the net force doing the work is a conservative force. Mathematically, energyin a system can be defined in terms of an exact differential operator; whereas work andheat can only be described as 1-forms (inexact differential operators).

The main forms of energy that we will be interested in are: internal energy, chemical(phase transitions), potential, and kinetic.

Example 2. If heat is added to a system and the temperature of a system increases,without knowing anything else, which form of energy will be definitely increase?(a) The kinetic energy of the system(b) The potential energy of the system(c) The work done by the system(d) The internal energy (i.e., the molecular energy) of the system


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1.3 Defining thermodynamic systems

1.3.1 What is a system?

A first step in any analysis is to describe precisely what is to be studied. If you cant dothis, then you have no hope of solving the problem. The subject under investigationis known as the system. In a broad sense, a system is whatever we choose to study.That said, a more precise definition of system is needed if we are to conduct any kind ofquantitative rigorous analysis of a system.

For pedagogical reasons, it is advantageous to start our study of systems by giving ageneral all-encompassing definition of a system together with definitions of the systemssurroundings and boundary, and then to make a further refinement that distinguishesbetween two basic types of systems: closed and open. We will find that we gain insightinto the concept of a system by making this further decomposition of the concept of asystem into systems based on observed mass, and systems based observed space.

Definition: A thermodynamic system is defined as a quantity of matter or a region inspace chosen for study.

For brevity, we will use the verbiage system in place of thermodynamic system from hereforward.

Definition: Everything external to the system is the defined to be the systems sur-roundings.

Definition: The area separating the system from its surrounding is the boundary of thesystem. The boundary may be at rest or in motion.

The boundary is the collection of points that is in contact with both the system and itssurroundings. It is a surface, and since a surface is a two-dimensional object, it has zerovolume.

Definition: A closed system is a definite quantity of matter contained within someclosed surface. A closed system is sometimes referred to as a control mass because thematter composing the system is assumed known for all time.

Thus, a closed system consists of a fixed amount of mass. That is, no mass can everenter or leave the system. This means no mass can ever cross the boundary of a closed

system.

A closed system is appropriate for systems in some sort of enclosure, such as a gas beingcompressed by a piston in a closed cylinder. If we choose the gas as our system, then thegas cannot escape, although energy could escape through the piston walls.

Although mass cannot leave a closed system, energy can escape in the form of heat orwork. A special type of a closed system that does not allow the escape of energy isknown as an isolated system. An isolated system is necessarily a closed system (sincemass is energy, and if energy cannot cross the boundary, then neither can mass), buta closed system need not be isolated since energy can be transferred across a boundary


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in the form of heat without any mass transfer. This is related to the idea that a wavecan occur in a medium without any net mass transfer. For example, waves in the oceancan transfer energy long distances, but the particles composing the wave at any givenmoment in time are only displaced over a very short distance.

Definition: A system is said to be isolated if no energy is transferred across the bound-aries.

Definition: An open system is a definite fixed location in space. The system is calledopen because mass may flow in or out of the system. An open system is sometimesreferred to as a control volume because the location composing the system is assumedknown for all time.

The surface surrounding the control volume is sometimes known as a control surface.The control surface can be along a real surface of the system or it can be an imaginarysurface chosen for convenience.

Choosing a system boundary in practice is somewhat of an art. In general, choosing asystems boundary depends on two factors: the objective of the analysis, and what isknown about the system at various locations, such as at an intake valve where we wouldhave explicit information about heat flow at the boundaries.

As a general rule of thumb, a control volume is used when studying things like turbines,pumps, hot water heaters, or any device where mass flows through a piece of machinery.Although we could in principle keep track of the mass of a fluid as it flows through apump, in practice this method would become unwieldily. It turns out that when onewishes to analyze a piece of machinery like a turbine, it is just easier to fix a location inspace around the turbine and monitor the flow in and out of it. Notice that the turbine

could be on a jet and be moving through space. In this case we would just choose aframe of reference moving with the system.

In summary, in a closed system the same matter (mass) is being studied. It follows thatthere can be no transfer of matter across the boundary of the system; whereas with anopen system there must be mass flow across the boundary, otherwise it would be a closedsystem (by definition). In general, a control mass can change shape and volume, but acontrol volume cannot. Notice that in both types of systems (open and closed) there canbe a flow of energy in or out of the system. In fact, in an open system, just as there mustbe a mass flow, there must also be a flow of energy in and/or out of the system.

Comments:

We will use a movable piston-cylinder assembly as our standard closed-system withmovable boundaries example.

If you have a flow through a device, you have an open system. Use a control volumeapproach.

In practice, most control volumes have fixed boundaries because the investigatorgets to choose the boundary.


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Example 3. (Defining Systems)Which statement best describes the concept of a system?(a) the subject of the analysis(b) an object with fixed set of molecules(c) a fluid-solid mixture

(d) an object that radiates heat into its environment

Solution: Only (a) describes the concept of a system. The other answers are particularsystems.

Example 4. (Systems Surroundings)Consider a refrigerator in a kitchen. Take the refrigerator and everything in it to be oursystem. Which best describes the systems surroundings?(a) All of the air in the kitchen.(b) Any one standing in the kitchen.(c) The air inside the refrigerator.

(d) Everything in the universe external to the system.Solution: By definition of surroundings, everything that is not in the system is outsideit. The answer is (d).

Example 5. (Systems Boundary)Consider a refrigerator in a kitchen. Take the refrigerator and everything in it to be oursystem. Which best describes the systems boundary?(a) All of the air in the kitchen.(b) The thin region separating the system from everything else.(c) The air inside the refrigerator.(d) The outer walls of the refrigerator.

Solution: In theory (b), in practice the outer wall (d)

Example 6. (Closed systems)Which of the following statements are true?(a) A closed system is necessarily an isolated system(b) An isolated system is necessarily a closed system(c) A system cannot be both closed and isolated.(d) The concepts of closed and isolated in regards to a system are independent concepts.

Solution: Only statement (b) is true.

Example 7. (Control volume)Circle all of the true statements:(a) closed system = control mass(b) closed system = control volume(c) open system = control mass(d) open system = control volume

Solution: Only statements (a) and (d) are true.


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Example 8. (Control volume)Which of the following situations would be best suited for using a control volume in thethermodynamic analysis of the system? Do not worry about the details of the analysis.(a) compression of air in a cylinder(b) expansion of gases in a cylinder after a combustion

(c) the air in a balloon(d) filling a bike tire with air from a compressor

Solution: Systems (a)-(c) are closed systems, so youd want to use control mass forthem. Only system (d) is an open system with a fixed boundary.

1.3.2 The continuum hypothesis

The key to the success of the macroscopic approach is the continuum hypothesis. It isa well-established fact that all matter is composed of atoms and molecules. Thus air,

water, honey, and metal are necessarily discrete even though they feel continuous to thetouch. In fact, there is more space in a full glass of water than there is matter!

Fortunately we dont need to track each molecule in order to make predictions aboutthe behavior of various properties of a system. For example, if we are interested in thepressure in a tire, we dont have to worry about how each molecule of air is interactingwith its neighbors and with the walls of the tire; all we need is a pressure gage (i.e., atire gage).

For definiteness, we will demonstrate how this hypothesis is applied to one of the prop-erties of a gas. In particular, well use density for its straight-forward graphical interpre-

tation.Example 9 (continuum). Start with a fixed collection of gas in a closed chamber.FINISH

1.3.3 Property, state, and process

Matter may exist in one or more of several phases: solid, liquid, gas, or plasma. We willnot be interested in plasmas in this course, and we will do very little with the solid phaseof materials.

Definition: For a given quantity of matter, a phase is a state where all of the matterhas the same chemical composition throughout. Matter that is in the same phase ishomogeneous.

Example 10 (phase). Water can exist as a solid (ice), liquid, or gas. It can also existin mixed phases, such as an ice cube in water.


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Definition: A property is any quantity which serves to describe a system. For ourpurposes, a property is any macroscopic characteristic of a system that can be assigned anumerical value at a given time without knowledge of the previous history of the behaviorof the system.

Clearly, work cannot be a property of a system, since it is something that is done on orby a system, and not something that the system possess.

List of Examples 1. (properties) Here is a partial list of some typical system prop-erties:

mass

volume

pressure

density

temperature

energy

viscosity

modulus of elasticity

thermal expansion coefficient

velocity

elevation

electrical resistivity

Definition: The state of a system is the condition of the system as determined by itsproperties.

Definition: A process is a transformation from one state to another. Whenever anyproperty of a system undergoes a change (e.g, a change in pressure), then by definitionthe state changes, and the system is said to have undergone a process (also called atransformation in older physics books).

Definition: If none of a systems properties change with time, then the system is saidto be in steady state.

Example 11. True or False: A system is in steady state if its properties are independentof time.Solution: True.


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1.3.4 Extensive and intensive properties

Thermodynamic properties can be placed into two general classes: extensive and inten-sive.

Definition: A property of a system is called extensive if its value for the overall systemis the sum of the values of the parts to which the system has been divided into.

Some examples of extensive properties are: mass, volume, and energy

Extensive properties, as the name suggests, depend on the extent (size) of the system.

Definition: A property of a system is called intensive if its value is independent ofthe extent (size) of the system, and may vary from place to place and from moment tomoment.

Some examples of intensive properties are: density, specific volume, pressure, and tem-

perature.

An easy test for whether a property is extensive or intensive is to imagine a fixed amountof matter and ask if you cut the matter into two pieces would the property in questionremain unchanged.

Example 12. (extensive-intensive test)Consider a fixed amount of gas in a closed insulated container. Assume the temperature,pressure, and density of the gas were uniform throughout the volume. If we cut thecontainer in half by magically inserting an insulated impermeable wall that did notdisturb the gas, then the temperature of the two halves would not change, the pressure

would not change, and the density would not change; but the volume would be half thatof the original volume, the total mass would also be cut in half, and the total energywould be cut in half as well.

Intensive properties may be functions of position and time; whereas extensive propertiescan only be functions of time.

Example 13. (extensive properties)Which of the following is not an extensive property?

(a) Kinetic Energy (b) Momentum(c) Mass (d) Density

(e) None of theseSolution: Density is an intensive property. Answer (d).

Example 14. (intensive properties)Which of the following is not an intensive property?

(a) Velocity (b) Volume(c) Pressure (d) Temperature(e) None of these

Solution: Volume is an extensive property. Answer (b).


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1.3.5 Equilibrium and quasi-equilibrium states

A system is in equilibrium if at each moment in time its properties are approximatelyindependent of space. A system undergoing a process (i.e., a transformation) is said to bein quasi-equilibrium if it is approximately in equilibrium at each point in time. Without

going into detail at this point in time, a quasi-equilibrium process is one where at eachmoment in time the system is arbitrarily close to being in equilibrium. This means thatthe systems properties are almost homogeneous through out the system.

For example, if a system such as a piston-cylinder container is filled with an explosivegas and the gas is ignited, then that process cannot be a quasi-equilibrium process sincethe pressure for example could not possibly be the same throughout the cylinder sincethere would be shock waves everywhere! On the other hand, if the piston cylinder deviceis a syringe that is slowly being compressed, then this process could be described as aquasi-equilibrium process since the gas inside the cylinder would have time to adjust tothe increasing pressure and every point inside the syringe would approximately be at the

same pressure. Its as though the air inside the syringe were acting like a memory foamthat was uniformly being compressed by the piston.

You have undoubtedly see the classic case of a stable system as being a ball in thebottom of a bowl, or two blocks that are touching one another that are both at thesame temperature. In this case we would say that there exists a temperature equilibriumbetween two bodies.

Example 15. Two metal blocks, one at 50, and the other at 0 are set next to eachother in a perfectly-insulted box at time t = 0. At this instant are the blocks in thermalequilibrium? Now suppose you wait a long time. Are they in equilibrium now?

Solution: The blocks are not initially in thermal equilibrium. The hotter block will cooland the colder block will warm. Eventually, the two blocks will come into equilibriumand be at the same temperature. We cannot say what this temperature will be withoutmore detail about the blocks.

Example 16. In a quasi-equilibrium process, the pressure in a system(a) is held constant throughout the entire process(b) is spatially independent (uniform throughout the system) at each moment in time(c) increases if volume increases(d) always varies with temperature

Solution: Answer (b).

Example 17. Which of the following process is a quasi-equilibrium process?(a) the stirring and mixing of cold creamer in hot coffee(b) a balloon bursting(c) combustion(d) the slow compression of air in a cylinder

Solution: Answer (d).


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1.4 Physical quantities, dimensions, and units

Definition: A dimension is a name given to any measurable physical quantity.

Thus any physical quantity that you can think of can be characterized by dimensions.

Some examples of common physical quantities are: mass, length, time, velocity, temper-ature, and pressure.

The set of all measurable physical quantities (dimensions) are divided into two categories:fundamental and derived, which are sometimes referred to as primary and secondary. Justwhat constitutes a fundamental dimension and a derived one is a matter of convenience,tradition, and personal preference. There are no divine dimensions! One could builda physical system with only one fundamental dimension, time. It would make workingwith such a system difficult, but it could be done.

Once a set of fundamental quantities is chosen, all other quantities are described in

terms of these quantities using definitions and mathematical representations of physicallaws. Thus derived quantities are just that, they can be expressed entirely in terms ofthe fundamental physical quantities. Of the set of all physical dimensions, the deriveddimensions are merely the ones that didnt make it into the collection of fundamentalquantities, they are the leftovers. In the language of mathematics: ifSdim is the set ofall dimensions and Sfund the set of fundamental dimensions, then the complement of thefundamental set Scfund = Sdim Sfund is the set of derived dimensions.

Most modern systems of dimensions and corresponding units take mass, length, andtime (MLT) as the fundamental dimensions. Although, the old-english system tookforce, mass, length, and time as (FMLT) the fundamental dimensions. If this isnt

confusing enough, there is a system referred to as the absolute english system that takesforce, length, and time (FLT) as the fundamental/primary dimensions and treats massas a secondary unit.

We now list the fundamental dimensions that were chosen for the SI system and theirabbreviations:

The Fundamental Dimensions:

Mass M

Length L

Time T


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1.4.1 Dimensional consistency

A formula that is not dimensionally homogenous, is not correct. A formula that isdimensionally homogenous, is not necessarily correct. Thus,

formula dimensionally inconsistent (inhomogeneous) formula is wrong

formula dimensionally consistent formula is correct

In order to check physical equations for dimensional consistency we need a mathematicaltool (operator) known as the dimensionator.

Definition: The Dimensionator [ ] is the take the fundamental dimensions of op-erator.

Example 18. Apply the dimensionator to find the dimensions of the following funda-

mental quantities of Mass M, Length L, and Time T:

d = distance; [d] = L

x = distance; [x] = L

R = radius; [R] = L

h = height; [h] = L

m = mass; [m] = M

t = time; [t] = T

Example 19. Apply the dimensionator to find the dimensions of the following derivedquantities in terms of the fundamental quantities:

v = velocity; [v] = L/T

r = rate (speed); [r] = L/T

A = area; [A] = L2

V = volume; [V] = L

3

In the following examples determine which of the formulas could not be correct because itviolates the consistency-of-dimensions principle discussed above. Do not worry about theorigin or application of the formulas: just focus on the issue of consistency of units. Thatis, based solely on consistency of units in an equation, determine if the given formulacould be correct.

Warning: Do not let the subscripts confuse you. The subscripts are only used to labelvariables, they are dimensionless numbers and letters and do not affect the outcome ofdimensionator one bit.


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Example 20. d = rt

Solution: [d] = [rt] = [r][t] L =?=

L

T

T = L dimensionally consistent

Example 21. v = x + t

Solution: [v] = [x + t] = [x] + [t] L

T=?= L + T dimensionally inconsistent

Example 22. Convert the expression v2, where is density = mass/volume, into thefundamental dimensions: L, T, and M.

Solution: Start by writing each component and v in terms of the fundamental dimen-sions. = density = massvolume =

masslength3

and v = velocity = lengthtime . Using bracket notation:

[]

[v

2

] =

M

L3L2

T2 =

M

LT2 .

Example 23. (The three fundamental kinematic equations) Verify that the threefundamental equations are dimensionally consistent.

The general fundamental kinematic equations are:

Fundamental Equation 1: vf = vi + a(tf ti)

Fundamental Equation 2: v2f = v2i + 2a(xf xi)

Fundamental Equation 3: xf = xi + vi(tf ti) +1

2

a(tf ti)2

where a is acceleration, v is velocity, x is position, t is time, and the subscript i and fdenotes initial and final values, respectively. Typically, we take the initial time ti = 0since, in practice, when you time an experiment you always start with the stopwatch setto zero.

(a) Verify that the 1st fundamental equation is dimensionally consistent.

Solution: Start by rewriting the equation as v = at. Taking the units of theright-hand side: [at] = (L/T2)T]= L/T, the units for velocity.

(b) Verify that the 2nd fundamental equation is dimensionally consistent.

Solution: Start by rewriting the equation as v2f v2i = 2ax. Taking the units of the

right and left-hand side yields [v2f v2i ] = [2ax]], which leads to (L/T)

2 = (L/T2)L.

(c) Verify that the 3rd fundamental equation is dimensionally consistent.

Solution: Start by rewriting the equation as

x = vit +1

2a(t)2

t

x

t= vi +

1

2at

As weve already seen, at has units of velocity. Thus the equation is dimensionally selfconsistent.


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1.4.2 Units

In order to compare two similar quantities, such as length, we need to introduce a stan-dard. The arbitrary, but fixed, standardized dimensions of physical quantities are knownas units. For example, one way of measuring the length of an object is to compare it

to the standardized length of the foot, or the meter. Both the meter and the foot arearbitrary standards, but they are fixed standards, and thats the important part.

The units corresponding to the fundamental dimensions are known as Base Units. Theunits corresponding to the fundamental dimensions must be independent of one an-other, and they must be able to span the other physical quantities of interest in thefollowing sense: the units of all other quantities of interest must be able to be expressedin terms of these proposed fundamental units. Such a complete set is known as a basis 2

and the collection is typically referred to as the Fundamental Physical Quantities, or theFundamental Dimensions; however, the corresponding fundamental units are referred toas the Base Units. In the same spirit as the dimensionator operator will have need for

an operator that can express a physical quantity in terms of the base units.

Definition: The Uninator [ ] is the take the Base Units of operator.

The General Conference of Weights and Measures produced the SI system based on sixfundamental physical quantities (dimensions) and their corresponding units, which arelisted below.

The Base SI Units:

For Length the meter, denoted by m.

For Mass the kilogram, denoted by kg.

For Time the second, denoted by s.

For Temperature the kelvin, denoted by K (this is an absolute temperature).

For Electric Current the ampere, denoted by A.

For Luminous Intensity the candela, denoted by cd.

They also set down the following rules for abbreviations:

The degree symbol was to be dropped from the absolute temperature scale. Thisonly applies to SI units, although some people apply the rule to the Rankine tem-perature scale in the English system.

2The name is derived from the concept of a basis in linear algebra, although it is a bit more complicated

in this case since the combination of units is nonlinear. However, by taking the natural log of the physical

quantities one can reduce all of the physical quantities to a linear system. For more details, look up the

Buckingham Pi Theorem.


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All unit names are to be written without capitalization, even if they were derivedfrom proper names. However, the abbreviation was to be capitalized if the unitwas derived from a proper name.

The full name of a unit may be pluralized, but the abbreviation may not.

For example, the SI unit of temperature is the kelvin. It was named after Lord Kelvin,so its abbreviation K is capitalized. They probably didnt want the unit capitalized sothat there would be a clear distinction between Kelvin the scientist, and kelvin the unit.

All other units are referred to as derived units. For example, the SI unit for force, thenewton (N), can be expressed in terms of the base units using Newtons second law:

F = ma[ ]

[F] = [m][a] 1N = 1kgm

s2.

Such a technique of defining derived units in terms of base units using a physical equation

to connect the units is known as aliasing.

The English System or United States Customary System of units

The unit for length is the foot

The unit for mass is the slug

The unit for time is the second

The unit for mass in the old British system was the pound-mass, denoted lbm. The unit

for force, which was considered to be one of the fundamental units, was the pound-force,denoted lbf. The reason the pound was a fundamental unit was mostly historical. Thepound lb is an abbreviation for libra, a unit used for weight by the Romans. Today,we mostly just refer to the pound, but in thermodynamics there is a difference. Thepound-mass requires the conversion factor: gc = 32.174 lbm = 1 slug.

The British/English system is no longer used in the UK. In fact, this system hasnt beenthe standard for a long time. This systems last strong hold is the US. However, withmore and more outsourcing the British system, like the US industrial base, is becomingobsolete. In this course we will give little attention to the United States CustomarySystem.

Example 24. Which of the following is not an acceptable extended SI unit? Recall:The SI system is the MKS system (Meter, Kilogram, Second), but well allow the ex-tended SI system to include the cgs system (centimeter, gram, second), but you cantmix these two systems.(a) distance measured in centimeters(b) pressure measured in newtons per square meter(c) volume measured in cubic meters(d) density measured in grams per cubic meter

Solution: Statement (d) is not correct, because you cant mix meters and grams. Eithergrams must be converted to kg, or meters to centimeters.


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Converting between different sets of units

We can convert between different sets of units by using conversion factors.

Example 25. You have a measuring stick that has centimeters and inches on it. You

notice that 1.00 in = 2.54 cm. Using only this information together with the relationship1 mile = 5,280 ft., find a relationship between miles and kilometers. That is, 1 mile =how many kilometers?

Solution: We will use a total of 5 conversion factors. That is, we will apply the one-factor 5 times (via multiplication).

1 mile = 1 mile 1 1 1 1 1

= 1 mile

5280ft

1mile

12in

1ft

2.54cm

1in

1m

100cm

1km

1000m

= 5280 12 2.54105

km

= 1.61 km

where the second equality follows from the one-factor. Thus, 1 mile = 1.61 km. Crudely,1mile 3

2km.

Example 26. An engine burns fuel at a rate of 11.2 g/s. What is the consumption ratein kg/hr? Round your answer to the nearest 0.1 kg/hr.

Solution: Use the conversion factors: 1000 g/kg and 3600 s/hr. To make the units

work, multiply by the second factor and the reciprocal of the first:

g

s

s

hr

g

kg

1

=g

s

s

hr

kg

g=

kg

hr

The engine consumes fuel at a rate of

11.2g

s

3600

s

hr

1 kg1000 g

= 40.3

kg

hr

Example 27. The pressure at the bottom of a swimming pool is 18 lb/in2. What is the

pressure in pascals (Pa = N/m2)? Round your answer to 2 significant figures.

Solution: We need to convert pounds to newtons, and inches to meters. Use theconversion factors: 4.45 N/lb and 2.54 cm/in. To make the units work, multiply by thefirst conversion factor, and divide by the square of the second, and lastly well need toconvert centimeters to meters. The pressure is

18lb

in2

4.45

N

lb

1 in

2.54 cm

2 100cm

1m

2= 12 104

N

m2= 12 104 Pa


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1.5 Density, specific volume, and specific weight

Definition: The density of a homogeneous substance is defined as its mass m dividedby its volume V. It is denoted by the Greek lower-case letter rho. In symbols,

mV

Definition: The specific volume v of a homogeneous substance is defined as its volumeV divided by its mass m. It is denoted by the lower-case letter v. In symbols,

v V

m=

1

Definition: The specific weight w of a homogeneous substance is defined as its weightW divided by the volume occupied by the substance. It is denoted by the lower-caseletter w. In symbols, starting from the equation for weight and dividing by volume gives

W = mg

V w

W

V=

mg

V=

m

Vg = g

Example 28. The mass of an unknown gas mixture in a room that is 4 m 3 m 5m is known to be 600 kg. What is the density and specific volume of the gas?

Solution: The density is found using the equation

= mV

= 600kg60 m3

= 10kg/m3 .

The specific volume is found using the equation

v =V

m=

60 m3

600kg=

1

10kg/m3 =

1

.

Example 29. Given that the density of water is approximately 1.0 103 kg/m3, whatis the specific weight of water?

Solution: Using the formula for specific weight

w = g 103kg

m3 10

m

s2= 104

kg

m2 s2

Example 30. Determine the fundamental dimensions for density = mass/volume, de-noted by .

Solution: Start by writing = density =mass

volume=

m

V, where m is mass and V is

volume. Using the dimensionator: [] =[m]

[V]=

M

L3.


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Example 31. The mass of an unknown gas mixture in a room that is 3 m 4 m 5m is known to be 500 kg. What is the density of the gas ( = m

V)?

Solution: From the definition of density and the volume of a rectangular room:

= MLWH

= 500kg(5m)(4m)(3m)

= 253

kgm3

= 8.33 kgm3

,

where M = mass, L = length, W = width, and H = height.

1.6 Pressure

Definition: The pressure P of a fluid is defined as the normal force Fn per unit areaA on a real or imaginary surface of the fluid.

P FnA

.

Example 32. Express pressure, denoted by P, in terms of the fundamental dimensions.For now, just take pressure to be force/area. Note: Using the mathematical expression

for Newtons second law weve seen that the dimensions of force are [F] = ML

T2.

Solution: Start by writing P = pressure = forcearea =FA

, where F is force and A is area.

Using the dimensionator: [P] =[F]

[A]=

[F]

L2= M

L

T2

1

L2

Example 33. Determine the expression for the unit of pressure (the pascal Pa) in termsof the SI units for pressure and area from the equation defining pressure P = F

A, where

F is force and A is area.

Solution: Using the uninator on the mathematical expression for pressure gives

[P] =

F

A

=

[F]

[A]=

N

m2 1 Pa =

N

m2

Example 34. Which one of the following expressions can be converted to the unit of a

joule (J = N m)?(a) Pa m2 *(b) Pa m3

(c) Pa/m2 (d) N/kg(e) None of these

Solution: Answer (b).

1.7 Temperature

Derive formulas from temperature scales.


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1.7.1 Conversion formulas

Between relative temperature scales:

TF =

9

5TC + 32 (Convert from degrees Fahrenheit to degrees Celsius)

TC =5

9(TF 32) (Convert from degrees Fahrenheit to degrees Celcius)

Between absolute temperature scales:

TK = TC + 273.15 (Convert from degrees Celsius to Kelvin )

TR = TF + 459.67 (Convert from degrees Fahrenheit to degrees Rankine)

Example 35. Convert 98F to C.(a) 32C (b) 208C(c) 20C *(d) 37C(e) None of these

Solution: TC =5

9(TF32) =

5

9(9832)C=

5

966C=

5

322C= (35+

5

3)C 37C

Example 36. Convert 20C to K.(a) 253 K *(b) 293 K(c) 68 K (d) 0 K(e) None of these

Solution: TK = TC + 273 K = (20 + 273) K = 293 K.


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1.8 Static fluids

Pascals law: Pressure applied to an enclosed fluid is transmitted undiminished to everyportion of the fluid and to the walls of the container.

Archimedess principle: A body immersed in a fluid is buoyed up with a force equalto the weight of the fluid displaced by the body.

Equations for static pressure:P = Patm + gh (Total Pressure)

Pgage = P Patm = gh (Gage Pressure)

1.9 Introduction to the 4 laws of thermodynamics

Of course no introductory chapter for thermodynamics would be complete without givingan overview of the four fundamental laws on thermodynamics.

Much of engineering thermodynamics is based on experimental observations that havebeen summarized and are expressed in mathematical statements known as physical laws.For completeness of this introductory discussion, these laws are listed below after givingand discussing the definition of the zeroth law. For technical reasons will not get to thefirst and second law until later in the course; furthermore, will never discuss the thirdlaw, which involves topics far beyond the scope of this course.

The zeroth law is defined in terms of the transitive property. Recall from algebra class:

The Transitive property: Ifa = b and b = c, then a = c.

The zeroth law of thermodynamics: If two systems A and C are each in ther-modynamic equilibrium with a third system B, then A and C are in thermodynamicequilibrium with each other.

Note: This law may seem obvious, but it is not. It allows us to classify it from otherphysical laws that do not satisfy a transitive law.

Example 37. Magnetic attraction does not satisfy a transitive law. Let A and C beiron nails and B be a magnet. Then B attracts A and B attracts C, but A and C do

not attract each other.


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The Four Fundamental Laws of Thermodynamics:

Zeroth Law (temperature satisfies a transitive property, which turns out to justifythe use of thermometers)

First Law (conservation of energy)

Second Law (asserts energy has a quality as well as quantity)

Third Law (says you cant reach absolute zero in finitely many steps)

A brief history of zeroth law:

The first and second laws emerged in the middle of the 19th century out of theworks of Rudolph Clausius, William Rankine, and Lord Kelvin.

In 1931 Robert Fowler realized that the fundamental physical principle that was tobecome the zeroth law could not be derived from the first or second laws, and waseven more fundamental than the first and second law.

Oh s**t, this posed a real problem for the scientific community!

One option would be to call the new law the first law, and rename the old first lawas the new second law, and so on, but this would only lead to confusion for futurescientist that would be reading old papers.

The only real option then was to name the new law the zeroth law.

About the 3rd Law:

The 3rd law comes from the way that various gases were supercooled into a liquidform on the race to absolute zero.

A liquid was cooled using a special apparatus, then that liquid was used to coola second liquid, which was used to cool a third, and so on until the desired gasliquefied, or the lab equipment exploded (the latter was usually the case).

The 3rd law states that if one used this procedure to reach absolute zero, then it

would take infinitely many steps.

Absolute zero does not mean that the atoms cease to move. It means that they arein a state of minimum energy. That is, they can only accept energy, they cannottransmit any energy.

chapter1 thermodynamics

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