COURSE DESCRIPTION This course provides an introduction to the most powerful engineering principles you will ever learn - Thermodynamics: the science of transferring energy from one place or form to another place or form. We will introduce the tools you need to analyze energy systems from solar panels, to engines, to insulated coffee mugs. More specifically, we will cover the topics of mass and energy conservation principles; first law analysis of control mass and control volume systems; properties and behavior of pure substances; and applications to thermodynamic systems operating at steady state conditions. COURSE FORMAT The class consists of lecture videos, which average 8 to 12 minutes in length. The videos include integrated In-Video Quiz questions. There are also quizzes at the end of each section, which include problems to practice your analytical skills that are not part of video lectures. There are no exams. GRADING POLICY Each question is worth 1 point. A correct answer is worth +1 point. An incorrect answer is worth 0 points. There is no partial credit. You can attempt each quiz up to three times every 8 hours, with an unlimited number of total attempts. The number of questions that need to be answered correctly to pass are displayed at the beginning of each quiz. Following the Mastery Learning model, students must pass all 8 practice quizzes with a score of 80% or higher in order to complete the course. ESTIMATED WORKLOAD If you follow the suggested deadlines, lectures and quizzes will each take approximately ~3 hours per week each, for a total of ~6 hours per week. TARGET AUDIENCE Basic undergraduate engineering or science student. FREQUENTLY ASKED QUESTIONS - What are the prerequisites for taking this course? An introductory background (high school or first year college level) in chemistry, physics, and calculus will help you be successful in this class. -What will this class prepare me for in the academic world? Thermodynamics is a prerequisite for many follow-on courses, like heat transfer, internal combustion engines, propulsion, and gas dynamics, to name a few. -What will this class prepare me for in the real world? Energy is one of the top challenges we face as a global society. Energy demands are deeply tied to the other major challenges of clean water, health, food resources, and poverty. Understanding how energy systems work is key to understanding how to meet all these needs around the world. Because energy demands are only increasing, this course also provides the foundation for many rewarding professional careers.
03.03 - The Incompressible Substance and the Ideal Gas Models for Equations of State
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Introduction to Thermodynamics: Transferring Energy from Here to There
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University of Michigan
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Week 3
In this module, we introduce our first abstract concepts of thermodynamics properties – including the specific heats, internal energy, and enthalpy. It will take some time for you to become familiar with what these properties represent and how we use these properties. For example, internal energy and enthalpy are related to temperature and pressure, but they are two distinct thermodynamic properties. One of the hardest concepts of thermodynamics is relating the independent thermodynamic properties to each other. We have to become experts at these state relations in order to be successful in our analysis of energy systems. There are several common approximations, including the ideal gas model, which we will use in this class. The key to determining thermodynamic properties is practice, practice, practice! Do as many examples as you can.
Rencontrer les enseignants
Margaret Wooldridge, Ph.D.Arthur F. Thurnau Professor
Mechanical Engineering, Aerospace Engineering
Kay, welcome back. So, hopefully you realized if you're in
the superheat region, things are pretty easy.
So temperature and pressure are in fact independent in the superheat region.
So, sure, if we have T and P, we can define any other piece of information we
need for the thermodynamic properties. Now, within the saturation region, this
might have given you a little bit of scratching your head to think about it.
If you have pressure and internal energy, can you fully define the state of the
system? The answer is yes.
Pressure and internal energy are independent in the saturation region.
More specifically, what you would do is you would go to the tables or the online
calculator or you know, whatever the reference is.
But here I'll show you graphically what you would be doing.
Is you know the pressure who that's going to define our pressure, whatever
let's call this P, state P1 here. You would go look up for that pressure.
Remember there is a unique internal energy for the saturated liquid and
there's a unique internal energy for the saturated gas.
So, what you now have is the internal energy, remember is a function of the
quality of the mixture. So one minus the quality times uf plus x
times here and now you have this information.
That's what was given to you, you won. And you have this information from having
looked that up at the state, from the tables or from the graph or wherever you
got that information. And that would allow you to extract or
calculate what the quality is for the mixture.
Once you have the quality you can calculate anything else that you need.
So, the key here is that you take pressure and internal energy that gives
you quality. And from the quality plus pressure or
whatever else you're going to use, you can find anything else, whether it's
enthalpy, specific volume density, et cetera et cetera.
Okay, so that's the process. So hopefully, we're getting a feel for
how these state relations work, how we can determine information from tables.
So the, now what I want to do is introduce two very common simplifications
of state relations. The first is the incompressible substance
model. This is only appropriate for solids and
liquids and the name says it all. If it's an incompressible substance I
can't compress it anymore. What I'm saying is that the specific
volume is a constant or if you prefer the density is a constant, the same thing.
In that case the internal energy is a function of, oh sorry, that shouldn't be
a little t that should be a capital T. Temperature only, okay?
Not time, temperature. Okay.
If I have that information, I can now go back and say, hey.
The definition of the specific heat. Remember the specific heat at constant
volume was defined as the partial derivative.
The specific, internal energy with respect to temperature for a constant
volume. But because we now know that the internal
energy's only a function of temperature. It's not a function of, of volume for
this approximation. It tells us mathematically that the
specific volu, the specific heat at constant volume is now the simple
derivative of the internal energy with respect to temperature.
Okay, so what does that mean. Well lets go back and look at CP.
CP this specific key that constant pressure, remember, was by definition the
partial derivative of the entropy with respect to temperature at constant
pressure. But we also know that entropy is by
definition the internal energy plus pressure times volume.
So we look at this expression, we say hey, but this, we know from our model
it's only a function of temperature. The internal energy is only a function of
temperature. The definition of this partial derivative
is with respect to constant pressure. So, this mathematically is treated as a
constant. And this density or specific volume is
prescribed as constant in this model. So, these are both constant.
So, hopefully, we realize that when we have two constants and you're trying to
take a partial derivative of those constants is that in fact they drop out.
And what we have is that the specific key to constant pressure, under these
circumstances is identical to the definition for the specific key, oh
sorry, no, let me erase one of those. This is, oop, took out too many.
the specific heat at constant pressure for the incompressible substance model is
equal to the definition of the specific unit constant value.
Okay? So, they're identically equal for this
model. So, what it's saying to us is that the CP
is equal to CV and so they can just drop the subscripts.
There is only one specific heat. That is necessary for us to describe an
incompressible substance or if you prefer we only need one specific heat to
describe liquids and solids. Alright, now having said that we can go
back and look at these definitions and descriptions for the internal energy.
And we can say hey, if I need to know what the change is between the initial
state and the final state for the specific, for the specific internal
energy. That's simply the integral from the
temperature at state one to the temperature at state two of the product
of the specific heat times, the differential step in temperature.
If we assume that the specific heat is constant, that the specific heat is not a
function of temperature, so if we assume C is not a function of temperature.
Then we can make the even further simplification that the change in the
internal energy is simply given by the product of the change in the temperature
of the states and the specific, times the specific heat of the mixture.
And you can see we went from these kind of abstract concepts of specific
heats, and partial derivatives, to something that looks really easy to use.
Okay? And you look at this and you say, hey,
because of the assumptions of the incompressible substance model, we can
take this information, go back to the definition of the entropy and we can say
hey the change in entropy. So the cost you know again by definition
the entropy is u plus pv so the change in entropy from the initial state to the
final state is simply the change in the, internal energy from the initial to the
final state. And then we have plus p2 v2 minus p1,
specific volume one. Except again, one of the fundamental
assumptions of the incompressible substance model is that there's only one
specific volume. And so I can simplify this into an
expression that says this. Okay.
So what we can see is that the change in the enthalpy is simply given by the
change in the internal energy. Which is simply given by the change in
the temperature and then we subtract off this piece of the the contribution from
the pressure and the density. And sometimes this portion will be really
small and it won't matter. So all we care about is this portion.
And what that's really telling us is that the change in the enthalpy and change in
the internal energy are the same. Okay, so, these are pretty powerful
approximations. But we need to remember this caveat, it
is for liquids and solids only. One of the most common mistakes I will
see is the inappropriate application of a model.
So you say ohp, I'm going to apply this in the superheat region.
no, you can't use that approximation in the superheat region, only for liquids
and solids. Another common mistake that I see, is
people will not you'll assume an incompressible substance.
And then, at some point, you'll go back and say, oh, I'm going to treat the
system as having a variable density. So you, you mix the the fundamental
assumptions of the model with the actual output.
So in other words you, again, you try not to make these mistakes but don't, don't
forget that if you're going to consider the system incompressible.
You have to assume the density is constant, the specific volume is constant
and the internal energy is a function of temperature only.
Okay. So that gives you an introduction to a
very common approximation for having the left side of our phase diagrams.
Now let's look at the super heat region or the right side of our phase diagrams.
And what we have now is an approximation that's called the Ideal Gas Model.
And I'm sure you have seen the ideal gas equation before.
It's usually introduced in typically physics and chemistry classes.
So hopefully, this is a little bit of a review for you.
If it's not, no worries and we'll go through this from the very basic steps.
So just like the incompressible substance model had two assumptions, that the
system had a constant density. And the internal energy was a function of
temperature only, so does a gas model, the ideal gas model.
The ideal gas model says this expresion has to be true, which is the pressure
times a specific volume has to be equal to this product of the gas constant,
that's what r is times temperature. So we're going to introduce the gas
constant and more more specifically this is the specific gas constant.
So you, likely have been introduced to the universal gas constant.
So the universal gas constant is denoted in thermodynamics in having the over bar
above the R. And it's the specific gas constant is
the, universal gas constant divided by the mole cur weight of the fluid.
Okay, so this, our bar, is the universal gas constant.
And it has a variety of different numbers based on what unit system that you're
using, and so we'll, we'll get a chance to see those in later examples.
But the universal gas constant it's name means everything.
It's universal it's the same number everywhere.
The specific gas constant, it's name means, mean everything.
It's very specific to the fluid you're using.
So if you're using air, it's a different number than the gas constant for water,
than the gas constant for nitrogen, than the gas constant for ammonia.
So, again one of the most common mistakes that people make with the ideal gas model
is there's probably a hundred different ways for you to write this expression.
I'll give you a few examples here in a moment.
Make sure you're consistent in each one. So here we have pressure times specific
volume. Well, I could also write that in terms of
[SOUND] density. Okay.
I can write this expression in terms of molar units.
So this over bar tells me that it's the universal gas constant.
It also means that that unit is on a per mole basis.
So the over bar means it's, that that unit or that variable is on a per mole
basis. Just like the lowercase means per mass
basis in thermodynamics, the over bar means at a per mole basis.
So I can do this. I can do this.
So just make sure you're using a consistent set of units on all of your
expressions. Okay?
You've also probably seen where you'd separate the total volume instead of
using specific volumes and specific densities, you use the total volume and
the mass. And one last representation, let me do
this, write the expression this way. because we are going to sneak in another
definition where we have 'n' is equal to the total number of moles in the system.
Okay. So again all these permutations and
combinations of how to write the ideas, ideal gas model.
Make sure that you are using the right one and you have consistence set of
units. Okay.
So just like we did before, we say we can look at these expressions and say, well,
if the internal energy is a function of temperature only.
Then that's going to give a simplification that I can invoke for
specific heats and all that sort of stuff.
But before we go there, what we want to understand is the fact that the ideal gas
model has we defined the compressibility factor that allows us to understand how
close we are. How good of an approximation it is that
we are treating the system as an ideal gas.
So it's a check. We have a check in the system and that's
the compressibility factor. So that's this Z.
So this is compressibility. Okay.
So the compressibility factor, you can see again is essentially pick any of
those forms of the ideal gas model and normalize it by itself.
So what we know is that as Z of the compressibility factor approaches 1, then
that means the ideal gas model is valid. as Z decreases and goes, becomes smaller
and smaller as Z approaches 0, we know that we're having what they would say
officially is that we'd have significant molecular interactions.
And so the molecules actually know each other are there as a consequence, the
ideal gas approximations are no longer valid.
So, we know as Z approaches 0, we have interaction.
And our ideal gas model. That interaction causes that
approximation of the ideal gas relation. And in terms of the functional dependence
of the internal energy and things like that to break down, so it's no longer
appropriate. There are some advanced pneumatic methods
that you can use the compressibility factor to correct your approximations and
we're not going to cover those in this class.
But what we should recognize is there are times when the ideal gas model is not
valid and the compressibility factor tells us when those times occur.
Now having said that I want you to think about by intuition we know that the ideal
gas model is typically valid in the super heat region.
So, I want you to visualize the pressure volume diagram and I want to you or try
to provide whichever one works for you. And I want you to tell me what pressure
in temperature conditions or temperature conditions would you expect the ideal gas
law to fail? So I'm talking like high, low, middle
things like that. So when would you expect the ideal gas
model to not be appropriate? And we'll start with that next time.
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