## Sections on the page :- Internal Energy
- Heat
- Work
- Putting it together for closed system (constant mass)
- The control volume approach
The ## 1. Internal EnergyAll matter contains at a given temperature, pressure, etc. a certain amount
of energy which we call the The ways of changing the internal energy relevant to the analysis and simulating Stirling engines are 1) adding/removing heat and 2) performing physical work by compressing/expanding the working fluid (gas in most cases). In many text books and journal articles the symbol In text books or on the web the specific internal energy is listed as function of
pressure and
temperature for many different substances. Actually, what you find listed
are the changes with respect to some reference point which you can find
by looking for the point in the table where ## 2. HeatBy adding heat to a mass we are raising its temperature. But raising the temperature also means raising its internal energy, that is increasing the motion and oscillations, watch here on wikipedia, etc. of its molecules. In that sense mass can serve as a storage medium of energy because we can get back the amount of energy by lowering the mass' temperature (and warming something else). In the not so distant past we actually used the rise of temperature from 4 to 5°C of 1 gram of water at standard atmospheric pressure to define the calorie as unit of heat/energy, i.e. exactly 1 cal is needed for that task. Having agreed on that definition you can go into the laboratory and measure how many calories are needed to - for example - melt 1 gram of ice (about 80 cal). Such measurements are done routinely all over the world in devices called calorimeters. Except for the food industry and its costumers (us) the calorie as measuring unit has been replaced by the Joule :
The amount of heat needed to raise the temperature per unit of mass of
a given substance by one degree is referred to as the specific heat
capacity (often "specific heat" for short). But there
is a slight problem with this definition, which is particularly grave
when measuring the specific heat of a gas. The reason is that gases
want to expand with rising temperatures to quite a large extent in comparison
to liquids and solids. As a consequence we typically have two different
scenarios under which to measure the specific heat of gases. In one we
do not allow the gas to expand ( constant volume but pressure will rise );
for the other we allow the gas to expand freely ( constant pressure but
volume will rise). We address these two values as specific heat capacities
at constant volume ( symbol c ). Both carry the sam units, namely
energy [J] per unit mass [kg] and per degree temperature change [°K]._{p}
c The ratio of the specific heats is :
(1)
κ = c The value of κ is surprisingly large, about 1.67 for monoatomic gases like helium, argon, xenon etc. and about 1.4 for gases whose molecules consists of two atoms like nitrogen, oxygen and air. For solids and liquids the value of κ is so close to 1 that often only a single value is listed with the subscript omitted. The values of c
are constant for monoatomic gases over wide range of temperatures ( a few
thousand degrees) and pressures and the same holds true in good
approximation for other gases at least for the range of temperatures and
pressures of interest for Stirling engines._{v}## 3. WorkWhen you push an object across the floor you have to overcome the friction between the floor and that part of the object which is in contact with the floor. In physics we say that you performed work and calculate the amount of work by multiplying the force applied by the distance travelled. W = F * d The units of work are N*m. Note that torque has the same units, so don't confuse the two. The work (energy) you expand has to go somewhere according to the law of conservation of energy. It actually gets converted into heating up the floor and the object at point of contact (usually too small to detect but present nevertheless). You can visualize it by rubbing the palms of your hands together under pressure. A more relevant application is that of expanding/compressing
a gas inside a cylinder/piston arrangement. Imagine that the piston seales
perfectly without the slightest amount of friction so that it can move freely
without letting any of the gas at (2) dW = p * dV Observe that when the gas expands (positive dV) we gain work (dW positive) but the gas looses energy according to the law of conservation of energy. The opposite occurs when we compress the gas (negative dV). The work dW comes out to be negative and the gas gains energy.## 4. Putting it together for closed system (constant mass)We consider now the gas in a cylinder/piston arrangement but in addition to moving the piston by a small amount we also add asmall amount of heat dQ. Both effects will change the internal energy
of the gas by a small amount, dU.Conservation of energy now means : (3) dQ - p dV = dU The minus sign is a historical artifact based on the prior agreement that we count both, heat added and work extracted, as positive. Often Eq.(3) is written in an alternate form which you can obtain by dividing Eq.(3) by themass [kg] at hand.
(4) dq - p dv = du In this form it is easier to see how the specific heat capacity
dq = c and _{v} dTdv=0. Hence : c Or as the mathematician then would write it : (5)
_{v}
In mathematics-speak the symbol ∂ indicates partial derivative
and the vertical bar with the symbol near its bottom indicates the conditions,
here constant volume. We later, in Section 5, encounter a quantity called enthalpy, (6) h = u + p v We use this here to derive a counterpart to Eq.(5). dh = du + p dv + v dp Using this to eliminate dq + v dp = dh This equation we inspect now from the point of view of supplying heat at constant pressure that is :dq = c and
_{p} dTdp = 0. The results is :
c Or as the mathematician would write it : (7)
_{p}
## 5. The control volume approachThe previous sections on this page delt with a given constant mass. This scenario is easier to deal with but in itself not quite relevant to investigating Stirling engines. There each of the five sub-spaces ( compression , heater , regenerator, kooler, compression spaces ) looses and gains masses during a single cycle and for an analysis of the engine performance each of these sub-spaces has to be considered individually. The approach we are persuing here is called the control volume approach which considers what forms of energy cross the system boundary. In the figure we show a general space with :
m [kg] , mass Now, more or less at the same time, the piston is moved increasing
the volume by an amount dQ
is added and a mass dm of volume dV is pushed in across
the dashed boundary on the right side of the space.dm brings with it some internal energy dU = u dm
which depends on its temperature ( u = internal energy per unit mass).
But there is more to it; in order to
get dm inside the space you have to push aside gas already inside.
This adds energy to our control volume of the amount p dV = p v dm
similar to what happens when a piston moves (v= specific volume [m³/kg]).
Adding it all up :
(8)
dQ - p dV u + p v is called enthalpy for the letter h
is often used which is together with the internal energy listed in many
thermodynamic tables.
(9) h = u + p v And Eq.(8) becomes :(10)
dQ - p dV This equation together with Eqs.(5) and (7) serves as starting point for much of the detailed analysis of Stirling Engines. * d(p v) = p dv + v dp is actually
an approximation which gets more and more accurate the smaller |

Last revised: 12/12/14