Engineering Thermodynamics/First Law

Energy
Energy as described on Wikipedia is "the property that must be transferred to an object in order to perform work on, or to heat, the object". Energy is a conserved quantity, the Law of Conservation of Energy on Wikipedia states that "energy can be converted in form, but not created or destroyed".

Common forms of energy in physics are potential and kinetic energy. The potential energy is usually the energy due to matter having certain position (configuration) in a field, commonly the gravitational field of Earth. Kinetic energy is the energy due to motion relative to a frame of reference. In thermodynamics, we deal with mainly work and heat, which are different manifestations of the energy in the universe.

Work
Work is said to be done by a system if the effect on the surroundings can be reduced solely to that of lifting a weight. Work is only ever done at the boundary of a system. Again, we use the intuitive definition of work, and this will be complete only with the statement of the second law of thermodynamics.

Consider a piston-cylinder arrangement as found in automobile engines. When the gas in the cylinder expands, pushing the piston outwards, it does work on the surroundings. In this case work done is mechanical. But how about other forms of energy like heat? The answer is that heat cannot be completely converted into work, with no other change, due to the second law of thermodynamics.

In the case of the piston-cylinder system, the work done during a cycle is given by W, where W = &minus;&int; F dx = &minus;&int; p dV, where F = p A, and p is the pressure on the inside of the piston (note the minus sign in this relationship). In other words, the work done is the area under the p-V diagram. Here, F is the external opposing force, which is equal and opposite to that exerted by the system. A corollary of the above statement is that a system undergoing free expansion does no work. The above definition of work will only hold for the quasi-static case, when the work done is reversible work.



A consequence of the above statement is that work done is not a state function, since it depends on the path (which curve you consider for integration from state 1 to 2). For a system in a cycle which has states 1 and 2, the work done depends on the path taken during the cycle. If, in the cycle, the movement from 1 to 2 is along A and the return is along C, then the work done is the lightly shaded area. However, if the system returns to 1 via the path B, then the work done is larger, and is equal to the sum of the two areas.



The above image shows a typical indicator diagram as output by an automobile engine. The shaded region is proportional to the work done by the engine, and the volume V in the x-axis is obtained from the piston displacement, while the y-axis is from the pressure inside the cylinder. The work done in a cycle is given by W, where

$$ W = -\oint p dV $$

Work done by the system is negative, and work done on the system is positive, by the convention used in this book.

Flow Energy
So far we have looked at the work done to compress fluid in a system. Suppose we have to introduce some amount of fluid into the system at a pressure p. Remember from the definition of the system that matter can enter or leave an open system. Consider a small amount of fluid of mass dm with volume dV entering the system. Suppose the area of cross section at the entrance is A. Then the distance the force pA has to push is dx = dV/A. Thus, the work done to introduce a small amount of fluid is given by pdV, and the work done per unit mass is pv, where v = dV/dm is the specific volume. This value of pv is called the flow energy.

Examples of Work
The amount of work done in a process depends on the irreversibilities present. A complete discussion of the irreversibilities is only possible after the discussion of the second law. The equations given above will give the values of work for quasi-static processes, and many real world processes can be approximated by this process. However, note that work is only done if there is an opposing force in the boundary, and that a volume change is not strictly required.

Work in a Polytropic Process
Consider a polytropic process pVn=C, where C is a constant. If the system changes its states from 1 to 2, the work done is given by

$$ W = -\int_{V_1}^{V_2}\frac{C}{V^n}dV = \frac{p_2V_2 - p_1V_1}{1 - n} $$

And additionally, if n=1

$$ W = -\int_{V_1}^{V_2}\frac{C}{V^n}dV = -C \ln \frac{V_2}{V_1} $$

Heat
Before thermodynamics was an established science, the popular theory was that heat was a fluid, called caloric, that was stored in a body. Thus, it was thought that a hot body transferred heat to a cold body by transferring some of this fluid to it. However, this was soon disproved by showing that heat was generated when drilling bores of guns, where both the drill and the barrel were initially cold.

Heat is the energy exchanged due to a temperature difference. As with work, heat is defined at the boundary of a system and is a path function. Heat rejected by the system is negative, while the heat absorbed by the system is positive.

Specific Heat
The specific heat of a substance is the amount of heat required for a unit rise in the temperature in a unit mass of the material. If this quantity is to be of any use, the amount of heat transferred should be a linear function of temperature. This is certainly true for ideal gases. This is also true for many metals and also for real gases under certain conditions. In general, we can only talk about the average specific heat, cav = Q/m&Delta;T. Since it was customary to give the specific heat as a property in describing a material, methods of analysis came to rely on it for routine calculations. However, since it is only constant for some materials, older calculations became very convoluted for newer materials. For instance, for finding the amount of heat transferred, it would have been simple to give a chart of Q(&Delta;T) for that material. However, following convention, the tables of cav(&Delta;T) were given, so that a double iterative solution over cav and T was required.

Calculating specific heat requires us to specify what we do with Volume and Pressure when we change temperature. When Volume is fixed, it is called specific heat at constant volume (Cv). When Pressure is fixed, it is called specific heat at constant pressure (Cp).

Latent Heat
It can be seen that the specific heat as defined above will be infinitely large for a phase change, where heat is transferred without any change in temperature. Thus, it is much more useful to define a quantity called latent heat, which is the amount of energy required to change the phase of a unit mass of a substance at the phase change temperature.

Adiabatic Process
An adiabatic process is defined as one in which there is no heat transfer with the surroundings, that is, the change in amount of energy dQ=0 . A gas contained in an insulated vessel undergoes an adiabatic process. Adiabatic processes also take place even if the vessel is not insulated if the process is fast enough that there is not enough time for heat to escape (e.g. the transmission of sound through air). Adiabatic processes are also ideal approximations for many real processes, like expansion of a vapor in a turbine, where the heat loss is much smaller than the work done.

Joule Experiments




It is well known that heat and work both change the energy of a system. Joule conducted a series of experiments which showed the relationship between heat and work in a thermodynamic cycle for a system. He used a paddle to stir an insulated vessel filled with fluid. The amount of work done on the paddle was noted (the work was done by lowering a weight, so that work done = mgz). Later, this vessel was placed in a bath and cooled. The energy involved in increasing the temperature of the bath was shown to be equal to that supplied by the lowered weight. Joule also performed experiments where electrical work was converted to heat using a coil and obtained the same result.

Statement of the First Law for a Closed System
The first law states that when heat and work interactions take place between a closed system and the environment, the algebraic sum of the heat and work interactions for a cycle is zero.

Mathematically, this is equivalent to

dQ + dW = 0 for any cycle closed to mass flow

Q is the heat transferred, and W is the work done on or by the system. Since these are the only ways energy can be transferred, this implies that the total energy of the system in the cycle is a constant.

One consequence of the statement is that the total energy of the system is a property of the system. This leads us to the concept of internal energy.

Internal Energy
In thermodynamics, the internal energy is the energy of a system due to its temperature. The statement of first law refers to thermodynamic cycles. Using the concept of internal energy it is possible to state the first law for a non-cyclic process. Since the first law is another way of stating the conservation of energy, the energy of the system is the sum of the heat and work input, i.e., &Delta;E = Q + W. Here E represents the internal energy (U) of the system along with the kinetic energy (KE) and the potential energy (PE) and is called the total energy of the system. This is the statement of the first law for non-cyclic processes, as long as they are still closed to the flow of mass (E = U + KE + PE). The KE and PE terms are relative to an external reference point i.e. the system is the gas within a ball, the ball travels in a trajectory that varies in height H and velocity V and subsequently KE and PE with time, but this has no affect upon the energy of the gas molecules within the ball, which is dictated only by the internal energy of the system (U). Thermodynamics does not define the nature of the internal energy, but it can be rationalised using other theories (i.e. the gas kinetic theory), but in this case is due to the KE and PE of the gas molecules within the ball, not to be mistaken with the KE and PE of the ball itself.

For gases, the value of KE and PE is quite small, so the important term is the internal energy function U. In particular, since for an ideal gas the state can be specified using two variables, the state variable u is given by u(v, T), where v is the specific volume and T is the temperature.

Introducing this temperature dependence explicitly is important in many calculations. For this purpose, the constant-volume heat capacity is defined as follows: cv = (&part;u/&part;t)v, where cv is the specific heat at constant volume. A constant-pressure heat capacity will be defined later, and it is important to keep them straight. The important point here is that the other variable that U depends on "naturally" is v, so to isolate the temperature dependence of U you want to take the derivative at constant v.

Internal Energy for an Ideal Gas
In the previous section, the internal energy of an ideal gas was shown to be a function of both the volume and temperature. Joule performed an experiment where a gas at high pressure inside a bath at the same temperature was allowed to expand into a larger volume.



In the above image, two vessels, labeled A and B, are immersed in an insulated tank containing water. A thermometer is used to measure the temperature of the water in the tank. The two vessels A and B are connected by a tube, the flow through which is controlled by a stop. Initially, A contains gas at high pressure, while B is nearly empty. The stop is removed so that the vessels are connected and the final temperature of the bath is noted.

The temperature of the bath was unchanged at the end of the process, showing that the internal energy of an ideal gas was the function of temperature alone. Thus Joule's law is stated as (&part;u/&part;v)t = 0.

Enthalpy
According to the first law, dQ + dW = dE

If all the work is pressure volume work, then we have

dW = &minus; p dV

&rArr; dQ = dU + pdV = d(U + pV) - Vdp

&rArr; d(U + pV) = dQ + Vdp

We define H &equiv; U + pV as the enthalpy of the system, and h = u + pv is the specific enthalpy. In particular, for a constant pressure process,

&Delta;Q = &Delta;H

With arguments similar to that for cv, cp = (&part;h/&part;t)p. Since h, p, and t are state variables, cp is a state variable. As a corollary, for ideal gases, cp = cv + R, and for incompressible fluids, cp = cv

Throttling


Throttling is the process in which a fluid passing through a restriction loses pressure. It usually occurs when fluid passes through small orifices like porous plugs. The original throttling experiments were conducted by Joule and Thompson. As seen in the previous section, in adiabatic throttling the enthalpy is constant. What is significant is that for ideal gases, the enthalpy depends only on temperature, so that there is no temperature change, as there is no work done or heat supplied. However, for real gases, below a certain temperature, called the inversion point, the temperature drops with a drop in pressure, so that throttling causes cooling, i.e., p1 &lt; p2 &rArr; T1 &lt; T2. The amount of cooling produced is quantified by the Joule-Thomson coefficient &mu;JT = (&part;T/&part;p)h. For instance, the inversion temperature for air is about 400 °C.