THE 1st LAW OF THERMODYNAMICS
INTEGRAL FORMS OF 1st LAW
We introduce here the following postulate of conservation of energy:
the net rate at which energy flows into a body must be equal to the rate at which energy is accumulated in the body.
This postulate can be put into the following mathematical form:
For a rigid body (or for a body undergoing a rigid motion) the work done results in accumulation of kinetic energy only (i.e. Wt = 0).
FOR A RIGID BODY:
→ there is no exchange between ‘thermal’and ‘kinetic’ energy!
To deal with the thermodynamics of real materials, one needs to consider the deformation (e.g. change in density) of bodies. There are alternative forms of (eq 1a):
In the cases in which the body forces can be expressed as gradient of a scalar quantity which is independent of time:
(if b=-gradΦ, with Φ independent of time).
LOCAL FORMS OF 1st LAW
Eqs. (1) can be integrated over any finite interval of time to obtain different macroscopic formulations. We can also obtain from the integral forms illustrated by eqs. (1) local forms of 1st law of thermodynamics. In fact, eq. (1a), which applies to a body of finite size, can be transformed into a field (or local) equation:
From the Gauss theorem:
where ρ is the local mass density. Equation (2) can be then recast to obtain:
which is a local form of 1st law.
Now a question arises:
is it also true that ?
The answer is YES, IF THE ‘PRINCIPLE OF LOCAL ACTION’ HOLDS TRUE.
Moreover, if we introduce the following definition:
we consequently derive that:
Here q is a local rate of heating per unit mass and is related to Qt by the following expression:
THE 2nd LAW OF THERMODYNAMICS
INTEGRAL FORMS OF 2nd LAW
The 2nd law is a formalization of the intuitive concept of irreversibility of natural phenomena. IT IS NOT A CONSERVATION PRINCIPLE! Conservation laws are invariant under a reversal of the direction of time and are expressed in the form of an equality: in fact, 2nd LAW IS NOT AN EQUALITY!
In the 2nd law entropy is involved: it is a ‘primitive’ concept. Given a body B, it is postulated that it possesses a total entropy, St, which:
The integral form of 2nd law is expressed as:
where, by definition, Jt is the instantaneous rate of supply of entropy and
1st and 2nd laws are coupled if, as postulated below, Jt is related to Qt.
By definition, we have:
and we postulate that:
where j is the entropy flux and J is the radiative entropy density. In these expressions T (temperature) is a ‘primitive’ (undefined) concept.
Based on eq. (6) and (7), the form of eq. (5a) for the 2nd law becomes:
In writing 1st and 2nd law we have introduced four ‘primitive concepts’: heat and energy in the 1st law and entropy and temperature in the 2nd law.
In the following we will introduce some special forms of the inequality (5).
LOCAL FORMS OF 2nd LAW
From eq. (5c) we obtain the local form of the 2nd law:
As for the case of U, if the principle of local action holds true,
Equation (8a) can be recast in the following form:
From the previous expression an alternative expression for the local form of the 2nd law is obtained:
Also for the local form of 2nd law we can introduce expressions for special cases.
LOCALLY ISOTHERMAL BODY
STATIONARY BODY AT STEADY STATE
From eq. (10) one can deduce that in the case of stationary heat conduction in stationary bodies the thermal flux vector must have a positive component in the direction of decreasing T.
Eq. (8b) can be put in another form by considering that :
This expression is the basis for the dissipation theory. If we consider a process (σt=P(t) or σ = P (t)) at every time the 2nd law (in each of the proposed forms) can be either satisfied as an INEQUALITY (in such a case the process is INSTANTANEOUSLY IRREVERSIBLE) or as an EQUALITY (in such a case the process is INSTANTANEOUSLY REVERSIBLE).
For instantaneously irreversible processes, we can define a LOCAL VELOCITY OF ENERGY DISSIPATION PER UNIT MASS, Z.
In fact, multiplying eq. (8c) by T, we obtain:
where with and
We define now the Helmholtz Free Energy, A:
If we consider the ISOTHERMAL CASE (i.e. ):
Moreover, if it is also we have
represents the rate of accumulation of elastic energy.
On the basis of previous expressions, in general, for an isothermal process at constant T, the work is the sum of the non-thermal dissipation rate and of the rate of accumulation of elastic energy.