**ENERGY AND HEATING**

is endowed with an amount of * energy*,

**U≡ INTERNAL ENERGY DENSITY**

Energy can flow from one body to another by two mechanisms:

**HEATING****WORK**

**HEATING**

** INSTANTANEOUS RATE OF ENERGY INFLUX INTO THE BODY DUE TO HEATING**

There are two mechanism of heating:

* CONDUCTION MECHANISM*:

where is the * conductive heat flux* vector and is the

* RADIATION MECHANISM*:

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

**FORCES**

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:

* CONTACT FORCES*:

acting on the surface of the body, whose power is indicated as

* BODY FORCES*:

acting directly on elementary masses within the body, whose power is indicated as

Here is the * field acceleration* (e.g. acceleration of gravity). If , with

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:

with

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:

**KINETIC ENERGY**

We pass now to define the * Kinetic Energy, K_{t}*:

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

Equation (1) refers only to the * nonpolar* case. An analogous rotational contribution arises for the

Where consequently:

For a body which is **NOT** undergoing a rigid body motion, we have:

In the previous equation * W_{t}* is the

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 * W_{t }*and

are not exact differential forms. We recall here that in order the differential form to be ‘exact’, it should happen that:

i.e.

**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

Which can be recast in the following form:

which is one of the forms of the **Reynolds’ Transport Theorem**.

Since

and

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.

Since

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

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

*1*. Body, state and constitutive behaviour

*3*. 1st and 2nd principles of thermodynamics. Integral and local forms. The concept of entropy

*4*. State and equilibrium. Part 1

*5*. State and equilibrium. Part 2

*6*. State and equilibrium. Part 3

*7*. State and equilibrium. Part 4

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion