The three laws of thermodynamics as noted below are very simple relatively obvious statements based on observations of the laws of nature. They are however of fundamental importance to all physical and chemical processes.
When two objects are separately in thermodynamic equilibrium with a third they are
in equilibrium with each other.
Objects in thermodynamic equilibrium are at the same temperature.
The measuring of temperature with thermometers is really in accordance with this law.
This law expresses the general law of conservation of energy. and states that heat and work are mutually convertible
Heat In = Work Out over complete cycle
or Sum (d Q ) = sum (d W )
Over a complete cycle the algebraic sum of the quantities of heat supplied to a system is equal to the algebraic sum
quantities of work performed by the system i.e.
In a cyclic process any property of the system are the same at the end of a cycle as at the beginning. Throughout the path of a cycle (δQ  δW) represents a change in the total stored internal energy property of the system δE . The basic energy equation results from this
δQ = δE + δW
The total stored internal energy E includes for various forms of energy including

Note: In classical thermodynamics as applicable to mechanical engineering the atomic energy and the chemical energy are not considered....
E = U + P.E + K.E + S.E
P.E = total potential energy, K.E = total kinetic energy, S.E = total surface energy . The intrinsic internal energy U is the total internal energy minus the energies of motion, gravitational, magnetic and surface forces energies . The first law can be written using U as
δQ = δU + δW
This is termed the restricted energy conservation equation for a system. U is
dependent on temperature and is not dependent
on pressure or volume.
Examples of various non flow processes using the restricted energy conservation equation NonFlow Processes
Steady flow equation notes...Steady Flow Equation
This law is derived from the whole field of physical experience. Although it is not possible to prove
the law or deduce it from other laws , no exception to it has yet been found.
This law in its simplest states that heat can only flow from hot to cold and not vice
versa. In terms of real thermodynamic engine cycles the law states that the gross heat
supplied to a system in a complete cycle must exceed the work done by the system. Therefore heat must
be rejected. The thermal efficiency of an heat engine must be less than 100%.
The maximum theoretical efficiency achievable is by use of the Carnot cycle. This is
based on reversible cycles using ideal gases ref Carnot Cycle
There are a number of sources for this law each providing a different interpretation.

Some simple conclusions resulting from this law are..

The Clausius inequality is a corollary of the second law and states that if a system completes a cycle
then.
This principle can be tested by considering a system A which completes a complete cycle
receiving heat Q_{1} from heat source R1 and heat Q_{2} from source R2. In accordance with the first law the work done by A over
a complete cycle = Q_{1} + Q_{2}. This system can now be isolated from the heat sources.
A second system C is provided which then completes a Carnot cycle which is
arranged to reject Q_{1} to the heat source R1 and receive heat Q_{2}from R2. The carnot engine
can be adjusted to match the conditions by setting of the isotherms and if necessary by having multiple cycles
The notes below show that for a simple system the Clausius Inequality conforms to the Second
Law of thermodynamics.
The case of a more complicated cycle (A) with variable temperatures and consequent heat flows
over the cycle is simply dealt with by using more and more matching Carnot cycles i.e. infinite of matching Carnot cycles with dQ heat transfer...
The value (dQ/T ) is a measure of the value of a property of the system called the entropy. Entropy is defined as
Notes on entropy are found on webpage Entropy