In this lecture we're going to move our attention to the conservation of energy equation that
is a statement of the first law of thermodynamics.
For simplified flows where compressibility
and transfer and not considered the fluid is assumed to be isothermal and then the conservation
of energy equation is not required.
While, in problems that involve heat transfer or
compressible flows this equation let us obtain the temperature of the fluid, and, if the
density is a function of temperature we will also need an additional equation, namely in
equation of state, to close our problem.
Let's start with stating the first law of thermodynamics: the total energy change in
a system equals the difference between the heat transferred to the system and the work
done by the system on its surroundings.
The Lagrangian form of this law is what we see
here, where the Q term represents the heat transfer and the W term represents the work.
Next we’re going to analyze in detail each of these terms and see how we can mathematically
represent them.
First let's take a look at the energy and let's introduce the concept of energy per
unit mass.
The total energy is the combination of three forms of energy: the internal energy,
that is a function of temperature - the kinetic energy, that is a function of the fluid velocity
- and last the potential energy, that is proportional to the fluid local elevation in the direction
of the acceleration vector acting on the fluid.
In this way we can write the total energy
as an integral volume integral of the energy per unit mass or in terms of the three forms
of energy.
In order to close the energy equation we need an equation of state and some thermodynamic
relations.
The equation of state connects the density to pressure and temperature.
The
most common example is the ideal gas law where the density is to P over RT where P is the
pressure, T is the temperature and R is the gas constant specific to the fluid.
The ideal
gas law applies to most gases at standard conditions like air or carbon dioxide.
In
case of non-ideal gas behavior we need to state a different equation that is typically
referred as the real gas equation of state.
This equation can become very complex and
we're not going to consider this behavior for this course.
In addition to the equation of state we also need to introduce some thermodynamic relations
such as the internal energy, enthalpy, the specific heats at constant volume and at constant
pressure and their ratio.
These relations are very important for compressible flows,
but for the purpose of this course we will not consider this topic in more detaille.
The rate of change of the total energy can also be impacted by processes that are not
represented by internal, kinetic and potential energies.
In order to take into account these
effects we can add a source term to the equation.
Typical sources can be radiation effects when
we are dealing with very high temperatures or thermal heating from electrical current
flows.
Let's now analyze the other terms the two remaining terms in the equation.
First is
the Q term, the heat transfer.
Heat transfer can be defined as the heat flux across the
boundary surface of the system.
The primary mechanism of heat transfer is conduction - Introducing
the Furier’s Law of conduction we can describe the heat flux term as the thermal conductivity
times the temperature gradient.
The last term remaining is the work.
The work is done by the system on its surroundings
and can be divided in two parts.
One is the pressure work done by the pressure acting
normal to the system boundaries.
As an example we can consider the piston of an internal
combustion engine, where the pressure of the fluid, expanding after combustion, pushes
the piston back along the cylinder creating the rotating motion of the crankshaft.
The
other form of work is the viscous work that is done by the action of viscous shear stresses
that we can consider as a viscous dissipation term.
The viscous work can be also seen as
a friction work this transformed into heat.
For high velocities flows this term can create
a significant heat generation on the surface of a body.
For example imagine a reentry capsule.
The capsule will heat up re-entering in the atmosphere due to its high velocity.
Part
of this heating is due to the friction between the surface of the capsule and the air.
Now we have a mathematical description of all the terms in our equation.
So let's start
and derive the Eulerian differential form.
First we apply the Reynolds transport theorem
to convert the equation from the Lagrangian framework into the Eulerian framework.
Then
we substitute the right-hand side terms with the expression we defined before, and we apply
the divergence theorem in order to convert all the surface integrals into volume integrals.
Now we can assume that our control volume is completely arbitrary so the sum of all
the integrands must be zero, to satisfy the conservation equation.
In this way we can
achieve the differential form of the conservation of energy equation.
For low speed flows, the
total energy per unit mass can be directly related to temperature, so the energy equation
can be recast into an equation just for temperature.
The set of equations that we derived so far with an equation of state and suitable initial
and boundary conditions will allow us to model an entire flow field and solve for the unknown
flow variables, for a single phase compressible fluid with no advanced physics such as combustion
or radiation.
In the later lectures we will see how to impose suitable modeling assumptions
to obtain mathematical models that can be solved.