Steady-State

Let's consider two bodies at different temperatures (T_a and T_b) separated by a conductive medium, if  T_a > T_b  then there will be a transfer of heat from A to B until a thermal equilibrium is reached.  Let's imagine that at the time t an amount of heat q is transferred from A to B with a consequent change of T_a and T_b, we can write:
 

Qta = Qa - q Qtb = Qb + q

where  Qa and Qb are the initial heat content of A and B respectively, and Qta and Qtb are the heat of A and B at the time t.

If at the same time t, the same amount of heat is added to A and removed from B, we have

Q_ta = Q_a - q + q = Q_a
Q_tb = Q_b + q - q = Q_b

Therefore, in this process, the heat content and the temperature of A and B remain unchanged. For an external observer, since there is no change in the temperature of A and B, the system appears to be in "equilibrium". This apparent equilibrium, obtained by adding and removing heat at the same rate, is referred to as steady-state. Let's consider another different example, the transformation of a product A to a product B by means of an hypothetical device:
 

Input  ==============> A

System transformation occurs here.

Output  =============> B

If we continuously remove B from system (i.e., B does not accumulate in the system) and continuously add A to the system we realize a steady state process.

Driving forces and resistances

It has been shown by Fourier that the rate of heat transfer between two bodies at differents temperatures can be calculated as follows:

A q = k ----Dt Dx

where q = heat flow rate in J/s (W, watts) A = the area perpendicular to the flow in m2 Dt = the temperature gradient (oK or oC) Dx = the distance between the two area at different temperatures (m) k = the thermal conductivity i.e., the heat flow rate when A, Dt and Dx are unitary which depends on the material forming the system.

The difference of temperatures (Dt) is known as the driving force of the process. In fact if no temperature difference exists no exchange of heat is possible.

Many processes involve the flow of a quantity under the action of a driving force or potential, for examples:

From the equation of Fourier, results that the rate of heat transfer is directly proportional to Dt (the driving force) and directly proportional to kA/Dx. This last term depends on the particular system we are analyzing and thus it is a measure of the ability of a given system to transfer heat. Of course its inverse Dx/kA is a measure of the resistance offered by the system to the heat transfer. Indicating with R (thermal resistance) the term Dx/Ka, the Fourier's law can be rewritten as

q = Dt/R

In general, processes involving the flow of a quantity under the action of a driving force or potential can be written as the ratio of the driving force with a resistance. For example, equ 2 is identical to the expression of Ohms' law (I = DV/R) which describes the flow of electrical charges under the action of a potential difference.

Resistance arrangements

Serie. In fig are represented two resistance in serie. Under steady-state conditions, the heat (or other) entering the resistances must equals the heat leaving them. This implies that the flow of heat through R_a must match the flow of heat through R_b, thus

t1-t2t2-t3 q = -------- = -------- RaRb

t1 - t2 = qRa

t2 - t3 = qRb

by adding the last two expressions above and solving for q we get

t1 - t2 + t2 - t3 = qRa + qRb = t1 - t3 = q(Ra + Rb)

t1-t3t1-t3 q = --------- = ------- Ra + RbRt

Therefore, the total resistance (R_t), for an arrangement in series, is the sum of the single resistances.

Parallel. In this case (see fig), under steady-state conditions, the heat leaving the resistances must be the sum of heat entering the single resistances:

DTDT qt = qa +  qb = -------- + ------ RaRb

11 qt =  DT (---- + -------) RaRb