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Thermodynamics Summary
1. Thermodynamics Definitions
1.1 Introduction to Thermodynamics
Thermodynamics is the science of energy. There are two kinds of thermodynamics. When applying classical thermodynamics, we do not consider individual atoms. We only look at a so-called contin- uum. In statistical thermodynamics we do deal with large groups of particles. In this summary we will generally discuss classical thermodynamics though.
1.2 Systems and their Properties
A system is defined as a quantity of matter or a region in space chosen for study. The mass/region outside the system is called the surroundings. The surface in between is the boundary. The boundary can be fixed or movable. A system can also be closed or open. In a closed system no mass can cross the boundary, while in an open system (also called a control volume) mass exchange with the surroundings is possible.
A characteristic of a system is called a property. There are multiple types of properties. Intensive properties do not directly depend on the amount of matter in the system (like temperature and pressure). Extensive properties like mass and volume do directly depend on the number of particles. When extensive properties are calculated per unit mass, then they are specific properties. These specific properties are usually denoted by small letters. Examples are the specific volume v = ρ−^1 (where ρ is the density) and the specific weight γ = ρg.
1.3 Equilibrium States
In thermodynamics, we consider the state or condition of a system. We usually deal with equilibrium states. A system in equilibrium experiences no changes when it is isolated from its surroundings. A system is in thermal equilibrium if the temperature is the same throughout the system. It is in mechanical equilibrium if the pressure stays constant. There are also more complicated types of equilibrium. Only if all the above equilibrium states are present, a system is in so-called thermodynamic equilibrium.
But how do we express the state of a system? We do this by using properties. The state postulate says that the state of a simple compressible system is specified by two independent intensive properties. This postulate requires some clarification. A system is compressible if the density isn’t constant throughout the system. A system is a simple system if electrical effects, magnetic effects, gravitational effects, and so on, are not present. (Otherwise an additional property is needed for every effect that is present.) Also two properties are independent if one property can be varied while the other is held constant.
1.4 Processes
A process is the change of a system from one state to another. The path is the series of states through which the system passes. A system has undergone a cycle if the initial and final states of the process are
the same.
There are multiple types of processes. When the system remains very close to an equilibrium state at all times, we have a quasi-static process. In a steady-state flow process a fluid flows through a control volume steadily. The term steady indicates nothing changes in time.
Also the prefix ”iso” can be used to indicate certain properties stay constant. In an isothermal process the temperature T stays constant. In an isobaric process the pressure P remains the same. Finally in adiabatic processes there is no heat exchange with the surroundings.
1.5 Pressure
Pressure is a normal force exerted per unit area. It is the same in all directions, and is therefore a scalar quantity. The absolute pressure is the pressure measured relative to a vacuum. Pressure is often measured with respect to atmospheric pressure. We then speak of gage pressure. In thermodynamics, the absolute pressure (denoted by P ) is almost always used though.
The pressure varies with the height z, due to gravity. How this occurs can be derived by looking at a small piece of fluid/gas. Doing this will result in
dP dz
= −ρg = −γ. (1.5.1)
This principle is used in a manometer. This is a device that measures pressure differences. It consists of a U-tube with a fluid, connecting two parts A and B. When a pressure difference is present between A and B, then the fluid levels hA and hB will not be equal. The relation between the pressures in A and B is then given by PA − PB = ρg (hB − hA) = γ (hB − hA). (1.5.2)
2.4 Energy Transfer
Energy can be transferred between systems in multiple ways. Heat is the form of energy that is transferred between systems due to a temperature difference. If there is no heat transfer in a process, then the process is an adiabatic process. Heat can be transferred by three mechanisms. In conduction the transfer occurs due to interaction between adjacent particles. Convection is the heat transfer between a solid surface and an adjacent moving fluid. Finally radiation is the transfer of energy due to the emission of electromagnetic waves.
The work W is the energy transfer caused by a force acting through a distance. Work done per unit time is called power. Usually the amount of work performed can be calculated using
W =
1
F ds, (2.4.1)
where s is the distance moved. But sometimes there isn’t a clear force. When a shaft is rotating, due to a torque T , the amount of work done is equal to
Wshaf t =
1
T dφ, (2.4.2)
where φ is the angle over which the shaft is rotated. Another example is a spring. Here the force depends on the displacement, according to F = kx. So now we have as work done
Wspring =
1
kx dx =
k
x^22 − x^21
The last way in which energy can be transferred is by a mass flow. Let’s consider the case where an amount of mass enters the system. This mass has internal energy. Therefore the total energy of the system rises. Identically, when mass is removed, the total energy decreases.
2.5 Mechanical Energy
Mechanical energy is the form of energy that can be converted to work completely. While kinetic and potential energy are types of mechanical energy, thermal energy is not.
Now let’s consider pressure. Pressure itself isn’t an energy. But it can produce work, called flow work. The magnitude of flow work (per unit mass) can be found using P/ρ. The mechanical energy per unit mass therefore becomes
emech =
P
ρ
ν^2 + gz. (2.5.1)
2.6 Performance
The efficiency of a system is defined as
η = efficiency =
Desired output Required input
Loss Required input
We can think of awfully many types of efficiencies. Examples are the combustion efficiency and the mechanical efficiency. These are defined as
ηcombustion = Heat released during combustion Heating value of the fuel burned
, ηmechanical = Mechanical energy output Mechanical energy input
Sometimes multiple efficiencies need to be combined (multiplied) to find the overall efficiency of a system. This efficiency is an indication of how well the system performs its job.
- Pure Substances
3.1 Introduction to Pure Substances
A pure substance is a substance that has a fixed chemical composition. This can be a mixture of elements, as long as the mixture is homogeneous.
Pure substances can come in various phases. In a solid, the molecules are arranged in a three-dimensional pattern (the lattice) that repeats through the material. In a liquid the molecules can move and rotate freely, but are still quite close together. Finally, in a gas, the molecules are far apart from each other.
When dealing with pure substances, often the quantity u+P v is encountered. This quantity has therefore been given its own name. The enthalpy h is thus defined as
h = u + P v. (3.1.1)
3.2 Vaporizing a Liquid
Let’s suppose we want to vaporize a liquid to its gas form. Initially the liquid is not about to vaporize. We say it is a compressed liquid or a subcooled liquid. We then add heat. After a while, the liquid will be about to vaporize. We now call it a saturated liquid. When part of the liquid has vaporized, and part has not, we have a saturated liquid-vapor mixture. Finally, when all the liquid has vaporized and we only have vapor left, we have a saturated vapor. This vapor it still about to condense. If we heat it further, then it won’t be about to condense any more. We are then dealing with a superheated vapor. The heat which is absorbed to make the phase change is called the latent heat.
At a given pressure, the temperature at which pure substances change phase is called the saturation temperature Tsat. Identically, for a given temperature, the pressure at which the phase changes is called saturation pressure Psat. A plot of these two is called a liquid-vapor saturation curve. Such a plot is shown in figure 3.1.
Figure 3.1: The liquid vapor saturation curve for water.
line connecting the saturated vapor points is called the saturated vapor line. The point where these two lines meet is the critical point. Corresponding to this point are the critical temperature Tcr , the critical pressure Pcr and the critical specific volume vcr.
The P − v diagram is very similar to the T − v diagram. But although the v increases with increasing T , the specific volume v decreases for increasing P. So while the lines in figure 3.2 go upward, in a P − v diagram they go downward.
3.5 P − T Diagrams
Given the pressure and temperature of a substance, its phase can be derived. This is done using P − T diagrams. Such a diagram is shown in figure 3.3.
Figure 3.3: Diagram showing the phase of a pure substance, given its pressure and temperature.
So we see that at low pressures it is possible for a solid to change to a gas without becoming liquid. Also, at high pressures (higher than the critical pressure), a liquid can turn to a gas without any clear transition. But maybe the most interesting point in figure 3.3 is the triple point. This is the point where all three phases can coexist in equilibrium.
Now we have discussed T − v, P − v and P − T diagrams. However, there are also P − v − T diagrams. These are three-dimensional diagrams, consisting of a so-called P − v − T surface. Using this surface, it is usually possible to determine one of the three properties, once the other two are known.
3.6 Property Equations
It’s nice to have graphs showing the relation between properties. But sometimes it’s more convenient to have an equation that does the same thing. Any equation that relates the pressure, temperature and specific volume of a substance is called an equation of state. The problem is that there is no equation of state that always works.
The most well-known equation of state is the ideal-gas equation of state (or ideal-gas relation), stating that P v = RT. (3.6.1)
Here R is the gas constant, which differs per gas. For high specific volume gases this equation is rather accurate. However, as the state of the gas becomes closer to the saturated vapor line, this equation loses its accuracy.
This equation can be made more accurate, using the compressibility factor Z, defined as
Z =
P v RT
Let’s also define the reduced pressure PR and the reduced temperature TR as
PR =
P
Pcr and TR =
T
Tcr
Now, for every type of gas, the factor Z only (approximately) depends on the reduced pressure and temperature. This principle is called the principle of corresponding states.
There are many (more complicated) equations of state. A few examples are the Van der Waals Equa- tions of State, the Beattie-Bridgeman Equation of State, the Benedict-Webb-Rubin Equation of State and the Virial Equation of State. All the above equations of state have certain coefficients which need to be determined by experiments. Generally we can say that the more coefficients an equation has, the more accurate it is.
For an ideal gas the internal energy u and the enthalpy h only depend on the temperature T. So the specific heats also depend on temperature only. Therefore we have
∆u = u 2 − u 1 =
1
cv (T ) dT and ∆h = h 2 − h 1 =
1
cP (T ) dT. (4.3.2)
Usually the functions for cv and cp are unknown. However, there are tables with their values for given temperatures. So, what we then do, is take the value of (for example) cv at T 1 and T 2 , take their average, and use that value to calculate ∆u. In an equation this becomes
∆u = cv (T 2 ) − cv (T 1 ) 2
(T 2 − T 1 ) = cv,avg (T 2 − T 1 ) and identically ∆h = cp,avg (T 2 − T 1 ). (4.3.3) There is an important relation between cp and cv. We know (from the definition of enthalpy and the perfect gas law) that dh = du + R dT. Differentiating with respect to temperature gives
cp = cv + R, (4.3.4)
where R is, as we already know, the gas constant for ideal gases. We can also define the specific heat ratio as γ = cp cv
Like the specific heats, also the specific heat ratio γ depends on the temperature T. The variation with temperature is very small though, so usually this ratio is assumed to be constant.
4.4 Incompressible Substances
An incompressible substance is a substance whose specific volume v is constant. Solids and liquids can be approximated as such substances. For such substances R = 0 and thus cp = cv = c. We now would like to know how these substances respond to changes. Or, to be more specifically, how does the enthalpy change during a process?
From the definition of enthalpy, we find that for incompressible substances
∆h = ∆u + v ∆P + P ∆v = ∆u + v ∆P = cavg ∆T + v ∆P. (4.4.1)
Note that the term P ∆v has disappeared, since v is assumed to be constant. For liquids, we can distinguish two special cases, being
- Constant pressure processes (∆P = 0) where ∆h = ∆u = cavg ∆T.
- Constant temperature processes (∆T = 0) where ∆h = v ∆P.
For solids the term v ∆P is insignificant, so all processes for solids can be approximated as constant pressure processes.
- Control Volumes
5.1 Conservation of Mass
In the previous chapter we looked at closed systems. Now we look at control volumes. A control volume is a fixed volume in space, where mass can move through. Just like we have applied conservation of energy in closed systems, we can apply the conservation of mass principle in control volumes. So we have
∆m = min − mout. (5.1.1)
The mass flow rate m˙ is the mass flowing through a cross-section per unit time. Suppose we have a pipe or duct. We can then determine the mass flow through this pipe using
m˙ =
Ac
ρνn dAc = ρνn,avg Ac, (5.1.2)
where Ac is the cross-sectional area of the pipe and νn is the velocity of the flow normal to this area. Identically, we can define the volume flow rate V˙ as
V˙ =
Ac
νn dAc = νn,avg Ac = ˙mv. (5.1.3)
We can now rewrite the conservation of mass principle as
dm dt
= ˙min − m˙out. (5.1.4)
5.2 Energy of a Flowing Fluid
The interesting thing about a control volume, is that mass can enter it. What we want to know is how much energy is added to the system when a piece of mass enters a control volume. First of all, we know that the specific energy e consists of specific internal energy u, specific kinetic energy V 2 /2 and specific potential energy gz. But there is more.
When a mass enters a control volume, something pushes it in. This pushing performs so-called flow work (also called flow energy), having magnitude
Wf low = P V, or, expressed per unit mass, wf low = P v. (5.2.1)
So we find that the total energy of a flowing fluid per unit mass is
θ = wf low + e = P v + u +
V 2 + gz = h +
V 2 + gz. (5.2.2)
5.3 Steady Flow Processes
During a steady flow process the properties inside the control volume do not change with time. Since the total mass then is constant, we have dm/dt = 0 and thus
m˙in = ˙mout. (5.3.1)
If the flow is also incompressible, then also
V˙in = V˙out. (5.3.2)
- Heat-Work Transformations
6.1 Reservoirs
In this chapter we’ll take a look at how we can transform heat into work, and vice verse. But before we can go into details, we have to make some definitions.
We define a thermal energy reservoir (or just a reservoir) as a hypothetical body that can ab- sorb/supply heat without any change in temperature. A reservoir that supplies energy in the form of heat is called a source. A reservoir that absorbs it is a sink.
6.2 Heat Engines
Work can be transformed entirely into heat. However, transforming heat to work is a bit troubling. So let’s give that process a closer look. A device that converts heat to work is called a heat engine. A heat engine receives a heat QH from a high-temperature source. It converts part of the heat into work Wout. The remaining heat QL is dumped in a low-temperature sink. Using this data, we find that
Wout = QH − QL. (6.2.1)
The thermal efficiency ηth of a heat engine can now be found using
ηth =
Desired output Required input
Wout QH
QL
QH
So what we need to do is make sure we need to dump as few heat as possible to the low-temperature sink. If possible, we would of course prefer to have QL = 0. But sadly this is not possible. This is because the Kelvin-Plank statement says that it is impossible for any device to receive heat from a single reservoir and produce work. Therefore the fraction QL/QH should simply be minimized. How this is done is something we will look at later.
6.3 Refrigerators and Heat Pumps
We saw that a heat engine uses temperature to create work. On the other hand, we can also use work to regulate temperature. A refrigerator is a device that transfers heat from a low-temperature region to a high-temperature region. The Clausius statement states that this transfer won’t happen by itself. We therefore need to add work to get the desired effect.
We just saw that a heat engine produced work. A refrigerator, on the other hand, requires work Win. Also, the direction of heat transfer is reversed. An amount of heat QL now comes from the low-temperature source, after which an amount QH goes to the high-temperature sink. So this time
Win = QH − QL. (6.3.1)
To check how well a refrigerator functions, we now don’t use an efficiency. Instead, the coefficient of performance of a refrigerator is defined as
COPR =
Desired output Required input
QL
Win
QL
QH − QL
QH QL −^1
The reason for using an other term than the efficiency, is because the coefficient of performance can be bigger than 1, while an efficiency can not.
A device very similar to a refrigerator is a heat pump. In fact, refrigerators and heat pumps are the same, except for their goal. While refrigerators want to keep the low-temperature source cold, heat pumps want to keep the high-temperature sink warm. The coefficient of performance of a heat pump thus becomes
COPHP = Desired output Required input
QH
Win
QH
QH − QL
1 − Q QLH
= COPR + 1. (6.3.3)
6.4 Reversible Processes
A reversible process is a process that can be reversed without leaving any trace on the surroundings. Processes that are not reversible are called irreversible processes. Reversible processes actually do not occur in nature. They are simply idealizations of actual processes. Reversible processes are always more efficient than irreversible processes.
Factors that cause a process to be irreversible are called irreversibilities. The most well-known irre- versibility is friction. Also heat transfer over a finite temperature difference causes an irreversible process. A process is called internally reversible if no irreversibilities occur within the boundaries of the system during the process. Identically, a process is called externally reversible if no irreversibilities occur within the surroundings of the system. A process is totally reversible (or simply reversible) if it is both internally and externally irreversible.
6.5 The Carnot Cycle
We know that reversible processes are the most efficient processes. One example of a reversible process is the Carnot cycle, using a so-called Carnot heat engine. This cycle consists of four reversible steps.
First the heat engine is connected to an energy source at temperature TH , causing isothermal expan- sion. The energy source is then replaced by an insulation, causing adiabatic expansion. After that, the insulation is removed, and an energy sink at temperature TL is connected to the heat engine. This causes isothermal compression. Finally, the energy sink is once more replaced by insulation, causing adiabatic compression. This completes the cycle. In this cycle, heat has flowed from the energy source (at TH ) to the energy sink (at TL), producing work Wout.
The funny thing about the Carnot cycle is that it can be reversed. Now work is added to the system, to transport heat from the (new) energy source (at TL) to the (new) energy sink (at TH ). We then would have a Carnot refrigerator.
6.6 Quality of Energy
We could ask ourselves: How does the efficiency/coefficient of performance depend on the temperatures of the source and the sink? We know that the efficiency strongly depends on the factor QH /QL. For reversible processes, it can be shown that
QH QL
φ(TH ) φ(TL)
where φ(T ) is a certain unknown function. Usually this function is assumed to be φ(T ) = T , resulting in
QH QL
TH
TL
