CHEM 331 Physical Chemistry Fall Engines

Transcription

1 CHEM 331 Physical Chemistry Fall 2014 Engines We now have the First Law firmly established in the form U = Q + W. We have also introduced an alternative form of the First Law in terms of the enthalpy (H = U + PV) which is useful for analyzing thermochemical measurements. We now return to the task at hand, establishing the third of the four laws of thermodynamics. The Second Law is stated in terms of the efficiency of heat engines. Historically this is how the enunciation of this law came about and it is still the easiest conceptual route to the Law. So, we will continue our quasi-historical discussion of the laws of thermodynamics by first considering the construction of heat engines. This will lead us to a statement of the Second Law of Thermodynamics. The engine we wish to build and analyze will operate by allowing for the expansion of a Hot gas in a piston, which will result in the production of work. This process converts heat into work. We use a gas as a working fluid in the piston simply because the expansion of liquids and solids is minimal and the much greater expansion of gases will help us to maximize the work output of the piston. This is one the few nods we will make toward practicality; however, you should understand that this in no way limits the ultimate generality of our results. Heat is used to raise the temperature of the gas in the piston's cylinder. The hot gas now imparts an increased force on the piston. This allows for its expansion, which, via a coupling mechanism, allows the expansion to do work. To maximize the work obtained from the expansion we will do it reversibly. Further, we will keep the gases hot by continuously supplying heat to them via a temperature reservoir; meaning the process is isothermal. Now an actual engine must be able to do this expansion again. In order to make the engine cyclic, we must now push the piston back in. This is where the rub comes in. As we have seen,

2 the minimum amount of work required to push the piston back in is numerically equivalent to the amount of work we just obtained in the expansion. This is a problem! We can keep operating the engine cyclically, but we are not obtaining any net work from it. A problem indeed. Of the course, the answer is to simply cool the gas. It will require less work to push the piston back in if the gas is cold and we will then have a net production of work from the heat input. We are not out of the woods yet. But we are getting there. Now, how to cool the gas? One simple way is to let the gas undergo an adiabatic expansion. As Joule demonstrated, gases tend to cool during an expansion. If the piston is isolated adiabatically, heat cannot flow in and re-heat them. The previously hot gas is now cold. So, let us compare the changes in state of a gas when it undergoes a reversible isothermal expansion versus when a reversible adiabatic expansion is carried-out. We will make the additional assumption that our working fluid is an Ideal Gas. Again, in no way does this affect the generality of our results, it simply simplifies the calculations. Isothermal Expansion We will start by analyzing the isothermal reversible expansion of an Ideal Gas. This is represented in the indicator diagram below. State A (T i, V i, P i ) State B (T f, V f, P f )

3 Since the Process is isothermal, we have: T f = T i And, by Boyle's Law: P i V i = P f V f or P f = So, in the isothermal expansion of an Ideal Gas, its state changes according to: State A State B V i, T i, P i V f, T i, P f = Now, to our thermodynamic quantities: U = 0 W = Because the gas is Ideal and by Joule's free expansion experiment. = Because the process is reversible. = Because the gas is Ideal.

4 = Because the process is isothermal. = - RT Q = - W By the First Law. Adiabatic Expansion Next we analyze the reversible adiabatic expansion of an Ideal Gas. Again, the indicator diagram for the process is represented below. State A (T i, V i, P i ) State B (T f, V f, P f ) You should note that the adiabat provides for a steeper curve in the indicator diagram, meaning less work will be produced for a given expansion, than for the case of an isothermal expansion. This is perfectly reasonable. As the gas cools, it will have less kick against the piston, meaning it will exert less pressure and hence less work will be obtained. Now to the change of state of the gas. Here the temperature changes as the expansion occurs. So we will need to determine the final temperature of the gas in State B. We do this by starting with the First Law: du = Q + W = W Because the process is adiabatic.

5 = - P dv Because the process is reversible. = Because the gas is Ideal. Also, du = C v dt So, C v dt = Separating variables, we obtain: Assuming C v is relatively constant and integrating, we get: Now we manipulate: Thus,, having defined = Undoing the logarithms and inverting the volume term: Once T f has been obtained, P f is easily calculated using the Ideal Gas Law: P f = P i It should be noted that simple manipulation of the above relationships for a reversible adiabatic expansion of an Ideal Gas, yield the following state relationships:

6 T i V i f V f -1 T i P i 1- = T f P f 1- P i V i = P f V f So, in the adiabatic expansion of an Ideal Gas, its state changes according to: State A State B V i, T i, P i V f,, P f = P i Turning to our thermodynamic quantities: Q = 0 Because the process is adiabatic. U = Because the gas is Ideal and by Joule's free expansion expt. = C v T If C v is relatively constant. W = U By the First Law. Summary of Expansion Results For comparative and summative purposes, the above results are presented below. Isothermal Expansion Adiabatic Expansion

7 Final State Final State V f V f T f = T i T f = P f = Thermodynamic Quantities P f = P i Thermodynamic Quantities U = 0 Q = 0 W = - RT Q = - W U = C v T W = U Now for a numeric example. Suppose we start with 1 mole of an Ideal Gas (C v = 3/2R) at 10 L, 244 K and 2 atm. In each case we will carry-out an expansion to 20 L. Results are: For the isothermal expansion: T f = 244 K P f = = 1 atm U = 0 W = = J Q = J For the adiabatic expansion: = = 5/3 = 1.67 T f = = 154 K P f = = 0.63 atm

8 Q = 0 U = (3/2) (8.314 ) (154 K K) = J W = J These results are summarized in the following indicator diagrams: Isothermal Expansion Adiabatic Expansion Well.. we now know how, at least in principle, to build an Engine. We let our hot gas expand reversibly and isothermally, extracting work, and then cool the gas by letting it expand reversibly and adiabatically. We then begin compressing the piston. First we do it isothermally and reversibly, so as to require the minimum amount of work. Then we compress the gas reversibly and adiabatically to heat the cold gas back to its original temperature. If done correctly, this cycle will return us to our initial state. And, on net, it will draw heat from a high temperature reservoir, do work and exhaust heat into a low temperature reservoir. This engine is referred to as a Carnot Engine; after Sadi Carnot, who first proposed it as a means of analyzing the work output of steam engines. The cyclic operation of the Carnot Engine is as illustrated below. Nicolas Leonard Sadi Carnot

9 The cycle involved is referred to as a Carnot Cycle. Carnot Cycle

10 Carnot was attempting to analyze the efficiencies of steam engines. An invention of the Industrial Revolution, the steam engine was first made practical by Thomas Newcomen in Early steam engines were of a double cylinder variety. Hot steam in one cylinder expands, pushing the piston out, exhausting the cold steam in an adjacent cylinder. The cylinders then reverse roles and the position of the piston is restored to its initial position. Cylinder 1 Exhaust Cold Steam Cylinder 2 Hot Steam Input A Mill Engine James Watt "About 1750, however, James Watt had added the major improvement of a condenser, thus allowing regular, reciprocating action and a great saving of heat energy. The early engines were slow and clumsy and it was fortunate that they were in fact well suited for an important application - that of pumping water out of mines." (A Textbook of Physical Chemistry by Arthur W. Adamson) The steam engine is best analyzed by considering the Rankine Cycle:

11 William John Macquorn Rankine Rankine Cycle The Rankine cycle closely describes the process by which steam-operated heat engines commonly found in thermal power generation plants generate power. The heat sources used in these power plants are usually nuclear fission or the combustion of fossil fuels such as coal, natural gas, and oil. The working fluid in a Rankine cycle follows a closed loop and is reused constantly. The water vapor with condensed droplets often seen billowing from power stations is created by the cooling systems (not directly from the closed-loop Rankine power cycle) and represents the means for (low temperature) waste heat to exit the system, allowing for the addition of (higher temperature) heat that can then be converted to useful work (power). This 'exhaust' heat is represented by the "Qout" flowing out of the lower side of the cycle shown in the T/s diagram [above]. Cooling towers operate as large heat exchangers by absorbing the latent heat of vaporization of the working fluid and simultaneously evaporating cooling water to the atmosphere. While many substances could be used as the working fluid in the Rankine cycle, water is usually the fluid of choice due to its favorable properties, such as its non-toxic and unreactive chemistry, abundance, and low cost, as well as its thermodynamic properties. By condensing the working steam vapor to a liquid the pressure at the turbine outlet is lowered and the energy required by the feed pump consumes only 1% to 3% of the turbine output power and these factors contribute to a higher efficiency for the cycle. The benefit of this is offset by the low temperatures of steam admitted to the turbine(s). The gasoline engine, an internal combustion engine, was invented by Nicholas Otto in Hot gases are produced as a result of combustion of a Light Hydrocarbon-Air mixture. These gases expand and force a piston outward. Once cooled, the gases are exhausted and a new Hydrocaron-Air mixture is admitted to the piston cylinder. The Otto Cycle approximately describes the operation of a four stroke gasoline engine.

12 pic/226592/gasoline-engine Nikolaus Otto During the Intake Stroke of the cycle, a Gas-Air mixture is admitted to the cylinder at constant pressure. This mixture is then compressed adiabatically during the Compression Stroke. A spark then ignites the mixture and the pressure spikes. Adiabatic expansion follows, producing the Power Stroke. Finally, the spent gases are exhausted and the cycle takes us back to our initial state; the Exhaust Stroke. For one last example, the Bryaton or Joule Cycle describes the operation of a Gas Turbine Engine.

13 "In a working engine air (and fuel) is compressed adiabatically, heated by a fuel combustion at constant pressure, expanded, and rejected to the atmosphere. The [last step in the cycle] occurs outside the engine, and a fresh charge of air is taken in to repeat the cycle." (Thermodynamics by H.B. Callen) During the expansion the hot gases turn a turbine. This is the equivalent of expanding a piston. We will next turn to analysis of the Carnot Cycle. This will lead us to a new state function, the Entropy (S), as well as the Second Law of Thermodynamics.