Thermodynamics and Propulsion

6.3 Representation of Thermodynamic Processes in $T\textrm {-}s$ coordinates

It is often useful to plot the thermodynamic state transitions and the cycles in terms of temperature (or enthalpy) and entropy, $T$ , $S$ , rather than $P$ , $V$ . The maximum temperature is often the constraint on the process and the enthalpy changes show the work done or heat received directly, so that plotting in terms of these variables provides insight into the process. A Carnot cycle is shown below in these coordinates, in which it is a rectangle, with two horizontal, constant temperature legs. The other two legs are reversible and adiabatic, hence isentropic ( $dS = dQ_\textrm{rev}/T= 0$ ), and therefore vertical in $T$ -$s$ coordinates.

Figure 6.3: Carnot cycle in $T$ -$s$ coordinates

If the cycle is traversed clockwise, the heat added is

$$\textrm{Heat added: }Q_H =\int_a^b TdS =T_H (S_b -S_a)=T_H \Delta S.$$

The heat rejected (from $c$ to $d$ ) has magnitude $\vert Q_L\vert = T_L \Delta S$ .

The work done by the cycle can be found using the first law for a reversible process:

$$dU =dQ -dW$$

$$=TdS -dW \quad\textrm{(This form is only true for a reversible process)}.$$

We can integrate this last expression around the closed path traced out by the cycle:

$$\oint dU =\oint TdS -\oint dW.$$

However $dU$ is an exact differential and its integral around a closed contour is zero:

$$0 = \oint TdS - \oint dW.$$

The work done by the cycle, which is represented by the term $\oint dW$ , is equal to $\oint TdS$ , the area enclosed by the closed contour in the $T$ -$S$ plane. This area represents the difference between the heat absorbed ($\int TdS$ at the high temperature) and the heat rejected ($\int TdS$ at the low temperature). Finding the work done through evaluation of $\oint TdS$ is an alternative to computation of the work in a reversible cycle from $\oint PdV$ . Finally, although we have carried out the discussion in terms of the entropy, $S$ , all of the arguments carry over to the specific entropy, $s$ ; the work of the reversible cycle per unit mass is given by $\oint Tds$ .

Muddy Points

How does one interpret $h$ -$s$ diagrams? (MP 6.3)

Is it always OK to ``switch'' $T$ -$s$ and $h$ -$s$ diagram? (MP 6.4)

What is the best way to become comfortable with $T$ -$s$ diagrams? (MP 6.5)

What is a reversible adiabat physically? (MP 6.6)