The first law of thermodynamics "The conservation of energy" states that energy can be neither created or destroyed. Mass and energy may be converted one to the other.
In developing an equation that applies to the general application of that law we must consider different forms of energy.
Mechanical Potential Energy - is the energy a mass has due to gravitational attraction. On earth it is due to it's position or elevation. A weight at some distance above the floor will give up a certain amount of energy upon impact if it is allowed to fall freely to the floor. That body has w weight due to it's mass and is z unit's distance above the floor the energy is w z when z is in feet and m in pounds. The product m z is in ft-lb. Thus a difference in elevation of z units produces a potential energy of m z work units.
PE = m z
Mechanical Kinetic Energy - is due to a moving mass. A body or mass m of a substance moving with some velocity Vs possesses a certain amount kinetic energy (with the earth as the reference body) equal to
KE = m Vs / (2 g) work units
Where m is the weight of the body Vs it's velocity and g is the unresisted acceleration due to the force of gravity of the earth. 32.17 ft/sec^2. This is applicable to moving fluids (liquid or gas) as well as to rigid bodies.
Internal Energy - All mater can store energy. The kinetic theory of gases states that the impact of moving molecules upon a surface produces the pressure exerted by an inclosed gas. This energy is a function of the temperature of the gas or heat content stored within. I would say that, like other forms of electromagnetic energy of which heat is one, the energy is stored in atoms. The way photons are stored by atoms. The elections change orbit. When an atom absorbs a photon an electron changes to a higher energy level orbit. When it drops back it releases a photon. These atoms of higher energy increase there for simplicity say field size. taking up more space. This comes from quantum mechanics theory. I haven't read up on the application to thermodynamics. This is just my application of that theory to gasses. It more then likely has been done but I haven't looked for that information. That is the way a laser works. You pump atoms with energy and promote a cascade directional release of photons. But as we lack a specific formula for internal energy we call it U or w u. U is the relative internal energy of a mass m of a substance. Here u is a state variable of the working substance. Enthalpy h of a substance is related to its internal energy u pressure p and specific volume v.
u = h - p v
For a mass m of the substance
U = m u = m h - m p v
Flow.Work - is the energy to maintain flow. It a pipe or a duct filed with a substance there must be a pressure maintained on the up stream end to maintain flow against the resistance presented by the downstream column of fluid. The work done pushing the fluid through a length L of the pipe of duct is the force times the length L. The force is pressure p times the cross sectional area of the pipe or duct. The work is then (p A) (L) = (p) (A L) = p V where V is the volume of fluid pushed through the pipe or duct.
FW = p V
Work - is the mechanical energy produced by or applied to the process. Work can be measured it various points of a machine and in so doing will reflect various losses due to friction, heat loss etc. In order to simplify our approach we will make no accounting for these losses. We are here interested in the maxim potential work. Losses will then be measured against this maximum. W is the work put into or extracted from the process or cycle.
Transferred Heat - the last energy considered in our quest of motive power. There are other forms of energy that we will not enter into. The word heat alone is used to name the stored energy in a body and sometimes to designate a mode of transfer of energy. There are three primary modes of heat transfer. Convection heat transfer is due to the circulation of a moving fluid carrying the heat. Conduction heat transfer is due to the direct contact of the conducting bodies. Conduction and convection are closely related. The actual transfer of heat in convection is due to conduction at contact points. It is the nonstatic nature of the conduction point that distinguishes convection as a special case. The third type of heat transfer is radiant heat transfer. Radiant heat transfer occurs through electromagnetic waves or photons. In the study of heat transfer their different formula developed to address these cases. Right now in the development of or formula model of the first law the type of heat transfer is unimportant. We are only interested in the quantity of heat transferred.
Other forms of energy are atomic, chemical, electrical light or photonic. Chemical is of interest only in the combustion process within a boiler and is separated for simplicity.
We are now able to make up the general energy equation. In the equation all energies must be balanced. We will divide the equation between heat transfer and work with the other forms of energy on both sides.
PE1 + KE1 + U1 + FW1 + Q = PE2 + KE2 + U2 + FW2 + W
Or
m z1 + m Vs1 / (2 g) + U1 + m p1 V1 + Q = m z2 + m Vs2 / (2 g) + U2 + m p2 V2 + W
And for a unit mass of substance
z1 + Vs1 / (2 g) + U1 + p1 V1 + Q = z2 + Vs2 / (2 g) + U2 + p2 V2 + W
There can be a change in potential energy PE kinetic energy KE internal energy U and flow work FW. Heat Q may be transferred in out of the process and work W may be done to the system or by the system but the net total must be zero.
In the analyses of steam power we can usually disregard potential energy PE, kinetic energy KE, and flow work FW reducing the general energy equation for our steam power analyses to:
U1 + Q = U2 + W
From this we derive the special relation of an adiabatic process. In an adiabatic process no heat is transferred in or out of the working substance.
Q = 0 PW = U2 - U1 An adiabatic process is important to our analyses of steam power. In the engine (expander) work is done by the working substance while no heat is added or removed from the substance. The process of expanding steam in the engine is an adiabatic process and the work done by that expansion may be computed by the above formula.