__First Law
of Thermodynamics__

**Q = ****D****E + W**for a closed system,

where **Q**
is the heat transferred into the system, **W**
is the work done by, or taken out, of the system, and **D****E **is the increase in the
internal energy **E** of the system.

*Note that this
precludes the possibility of "perpetual motion" machines, whose work
output is always greater than the work obtained by converting the heat put into
them. At some point, there will be no more internal energy available to convert
to work.*

__Derivation
of the Energy Equation for a Control Volume__

We can
also relate the rates of heat transfer, energy change, and work as:

_{}

Consider each of these rates for a control volume in
a fluid:

_{}

^{where }_{}^{is the mass flow rate
out of the control surface through its surface, and e is the energy per unit
mass of the fluid. }

^{ }

_{}^{}

where _{}is specific
volume (volume per unit mass), and _{}is the power taken
out through shear forces and through rotating shafts.

^{ }

^{The
energy per unit mass of the fluid can be written as}

_{}^{
}

^{ }

^{where
e}_{int}^{ is internal energy, u is speed, and P.E. is potential energy per
unit mass. Thus, }

^{ }

_{}

where
the integral is taken over the surface of the control volume.Now

_{}
,the enthalpy per
unit mass.

_{}

_{}can be written as _{}, and_{}can be written as _{}

where
_{}is the component of
the velocity directed outward from the surface of the control volume.

_{}

Thus,

_{}

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