ENERGY AND HEATING
is endowed with an amount of energy, Ut, which is a function of state (no claim is made that Ut is the total energy of the body!). Conversely, kinetic energy is not a function of state. Ut is an absolutely additive function of mass (i.e. it is an extensive quantity).
U≡ INTERNAL ENERGY DENSITY
Energy can flow from one body to another by two mechanisms:
INSTANTANEOUS RATE OF ENERGY INFLUX INTO THE BODY DUE TO HEATING
There are two mechanism of heating:
where is the conductive heat flux vector and is the inward pointing area vector.
where Q is the local rate of radiation heat inflow per unit mass.
and Q are ‘local‘ quantities. is not a function of state. It is defined as:
In fact, as we will see, is a function of local state, σ, while Q is not
Power, P, is the rate at which a force is doing work acting on a material point, v which moves at velocity:
There are different types of forces:
acting on the surface of the body, whose power is indicated as Pt,E
acting directly on elementary masses within the body, whose power is indicated as Pt,B
Here is the field acceleration (e.g. acceleration of gravity). If , with φ independent of time (e.g. φ = g.h) , then:
Here we have used the notation:
to indicate the substantial derivative. We finally obtain:
since, from Reynolds’ Transport Theorem (see note 1 at the end of lecture) we have:
Should not admit a time-independent potential, one could not even define a potential energy but the power of body forces would still be defined by:
We pass now to define the Kinetic Energy, Kt:
A fundamental theorem of classical mechanics states that for a rigid body, or for any body which is instantaneously undergoing a rigid motion, we have:
For more details on that see note 2 at the end of the lecture. The previous equation is a purely mechanical result: it does not derive from any principle of energy conservation! This equation applies only to rigid body motions. In fact, rigid body motions can never result in the transformation of energy absorbed as heat into works, and viceversa. Rigid body motions are thermodynamically degenerate.
Equation (1) refers only to the nonpolar case. An analogous rotational contribution arises for the polar case, when also power related to work of distributed body couples have to be considered. In fact, with reference to rotational motions, it can be written:
For a body which is NOT undergoing a rigid body motion, we have:
In the previous equation Wt is the ‘net’ rate of work done on the body. It is worth noting that body forces contribute nothing to Wt (see note 2 at the end of the lecture): the work of body forces goes entirely into increasing Kt. The following expression can be adopted for Wt:
where w is the ‘net’ rate of work per unit mass done on a differential body element of mass dm.
It is important to underline here that both Wt and Qt are NOT functions of state. In fact dWt and dQt, defined respectively as
are not exact differential forms. We recall here that in order the differential form to be ‘exact’, it should happen that:
THE REYNOLD’S TRANSPORT THEOREM
On the material derivative of a volume integral
We illustrate here how to evaluate the material derivative of a quantity which consists in an integral extended to a volume, i.e.:
Here the integral is evaluated on the volume occupied by the body . If we the integral has to be evaluated on a volume V which is fixed in the space with time, we have:
However, if both the integrand and the integration volume, containing a fixed mass, are changing with time, expression (α) is no more valid. Now the problem can be stated as follows: we need to evaluate the material derivative of a volume integral which measures the rate of change of the total amount of a quantity transported by a certain amount of mass. In the following , where A is a quantity per unit mass while ψ is a quantity per unit volume.
Which can be recast in the following form:
which is one of the forms of the Reynolds’ Transport Theorem.
it follows that:
Now, it can be demonstrated that, in view of the local form of the differential mass balance, we have:
Consequently, one can put eq. n1.1 it the form
which is another form of the Reynold’s transport theorem. It is worth noting that, instead, we have:
KINETIC ENERGY AND THE POWER OF BODY AND CONTACT FORCES FOR DEFORMABLE BODIES
In the case of a deformable body, the following relationships hold between the power of body and contact forces and kinetic energy:
where is the stress tensor.
we obtain from eq. n2.2:
Hence we finally obtain:
We note here that, in the case of fluids, the term contains two distinct contributions. In fact, the total stress tensor can be put in the form:
where P* is the thermodynamic pressure while is the dynamical stress tensor, i.e. that part of the stress tensor which arises when the fluid is in an out-of-equilibrium state. Consequently we have that:
In the previous expression:
represents the rate of reversibile increase of internal energy per unit volume due to isotropic compression of the fluid.
represents the rate of increase of internal energy due to the dynamical part of the stress tensor. As an example, in the case of a viscoelastic fluid, this term include i) a rate of irreversible increase of internal energy due to viscous dissipation (related to changes of both shape and volume of the fluid element) and ii) a rate of reversible increase of internal energy due to elastic deformation of the fluid element.