T-S Diagrams: Sketching and Interpreting

The Big Idea: Heat Transfer as an Area

So, what is a T-s diagram? It's simply a graph where the vertical axis is temperature ($T$) and the horizontal axis is specific entropy ($s$).

Why this particular combination of properties? It comes from the very definition of entropy. Remember our foundational equation for a reversible process?

$dS=\frac{\delta Q_{rev}}{T}$

Let's just rearrange this with a little algebra:

$\delta Q_{rev}=TdS$

Now, if you want the total heat transferred during a process from state 1 to state 2, you just sum up all those little bits of heat. As you know from your calculus background, summing up tiny pieces is what integration is all about:

$Q_{rev}={\int }_1^{2}TdS$

This equation holds a beautiful secret. An integral of a function is simply the area under the curve of that function. This means the area under a process path on a T-s diagram is equal to the heat transferred during that reversible process.

This is a massive insight! We can now see heat transfer just by looking at an area on a graph.

What about a full cycle? Remember the First Law for a cycle: $W_{net}=Q_{net}$. The net heat transfer ($Q_{net}$) is the heat added minus the heat rejected. On a T-s diagram, this will be the area enclosed by the cycle's path. So, the area enclosed by a cycle on a T-s diagram represents the net work done by the cycle.

Mapping the Territory: The T-s Diagram Landscape

Before we start drawing paths, let's get familiar with the landscape. Just like P-V and T-v diagrams, the T-s diagram has distinct regions that tell us the phase of a substance.

This diagram also shows lines of constant pressure (isobars) and constant specific volume (isochors). We'll look at these more closely now.

Drawing Our Fundamental Processes

Let's take our four basic reversible processes and see how they look on this new map.

1. Isothermal Process (Constant Temperature):
   If the temperature is constant, you're just moving along a constant $T$ value. This is a horizontal line.

   • Moving to the right ($s$ increases): Heat is being added to the system.
   • Moving to the left ($s$ decreases): Heat is being removed from the system.

2. Isentropic Process (Constant Entropy):
   "Isentropic" literally means constant entropy. If entropy ($s$) is constant, you're just moving along a constant $s$ value. This is a vertical line.

   • Moving up ($T$ increases): This represents isentropic compression (like in an ideal compressor or pump).
   • Moving down ($T$ decreases): This represents isentropic expansion (like in an ideal turbine).

3. Isobaric Process (Constant Pressure):
   This is a bit more interesting. Look at the isobars in the diagram above. They are upward-curving lines.

   • Intuition: To keep pressure constant while you add heat, both the temperature and entropy must increase.
   • The Math: For an ideal gas at constant pressure, we found $\Delta s=C_p\ln (T_2/T_1)$. The relationship is logarithmic, not linear, which gives it the curve. The slope of an isobar on a T-s diagram is given by $(\partial T/\partial s{)}_P=T/C_p$. As temperature $T$ increases, the slope increases, which is why the line gets steeper and curves upward.
   • Also notice that higher pressure isobars are shifted to the right of lower pressure ones.

4. Isochoric Process (Constant Volume):
   This also appears as an upward-curving line.

   • The Math: For an ideal gas, $\Delta s=C_v\ln (T_2/T_1)$. The slope of an isochor is $(\partial T/\partial s{)}_v=T/C_v$.
   • Key Difference: Since $C_p$ is always greater than $C_v$, the slope of a constant volume line ($T/C_v$) is steeper than the slope of a constant pressure line ($T/C_p$) at the same point.

Visualizing the King of Cycles: The Carnot Cycle

Now for the payoff. Let's sketch the Carnot cycle, the most efficient cycle possible. It consists of:

1. Reversible isothermal heat addition.
2. Reversible adiabatic (isentropic) expansion.
3. Reversible isothermal heat rejection.
4. Reversible adiabatic (isentropic) compression.

On a T-s diagram, this translates to:

1. A horizontal line to the right.
2. A vertical line down.
3. A horizontal line to the left.
4. A vertical line up.

The result is a perfect rectangle!

This is incredibly elegant.

• Heat In ($Q_H$): The area under the top line (A-B). Area = $T_H\times (s_B-s_A)$.
• Heat Out ($Q_L$): The area under the bottom line (C-D). Area = $T_L\times (s_C-s_D)$.
• Net Work ($W_{net}$): The area of the rectangle itself. Area = $(T_H-T_L)\times (s_B-s_A)$.

You can even see the efficiency $\eta =W_{net}/Q_H$ as the ratio of the rectangle's area to the total area under the top line.

Sketching Real-World Cycles

The Carnot cycle is a perfect ideal. Real-world engine cycles try to get close, but they have to obey practical constraints. The following resource is fantastic for seeing how these real cycles look on a T-s diagram.

Let's recap what you saw in that reading, as it's the heart of today's lesson.

1. Steam Power Plant (Rankine Cycle)

• Process 2-3 (Boiler): This is where it really differs from Carnot. Instead of adding all heat at a single high temperature, we heat liquid water at constant pressure. This follows the curving isobar up to the boiling point, then horizontally across the dome, and then up the curving isobar again in the superheat region. This creates a "peak" rather than a flat top, and it's a major reason the cycle is less efficient than a Carnot cycle operating at the peak temperature.
• Process 3-4 (Turbine): In an ideal turbine, this would be a straight vertical line down (isentropic). But in reality, there's friction and turbulence. These are irreversibilities, and they always generate entropy. So, the real expansion path (shown as process 3-4 in the diagram) slants down and to the right. The entropy at the end of the expansion is higher than at the start.

2. Gas Turbine (Brayton Cycle)

• Process 2-3 (Combustor): Again, heat is added at constant pressure, not constant temperature. This follows the upward-curving isobar, moving far away from the horizontal line of a Carnot cycle.
• Processes 1-2 (Compressor) & 3-4 (Turbine): Both of these are supposed to be isentropic (vertical lines), but due to real-world irreversibilities, they both slant to the right, showing an increase in entropy.

3. Refrigeration Cycle

• This cycle runs counter-clockwise. This means it consumes work (the area inside the loop) to move heat from a cold place to a hot place.
• Process 3-4 (Throttling Valve): This is a huge source of deviation. Instead of an ideal isentropic expansion (a vertical line down) to recover work, a simple valve is used. This is an isenthalpic (constant enthalpy) process, which on a T-s diagram is a steeply sloped line that generates a lot of entropy. It's inefficient, but it's simple and cheap.

Practice Problems

Let's try sketching and interpreting a few scenarios.

Key Takeaways

Excellent work today! We've added a powerful new visual tool to our thermodynamics toolbox.

• A T-s diagram plots Temperature vs. specific entropy.
• Its greatest power is that the area under a reversible process curve is the heat transferred, and the area enclosed by a cycle is the net work.
• The four fundamental reversible processes have simple representations:
  • Isothermal = Horizontal line
  • Isentropic = Vertical line
  • Isobaric (constant P) = Upward-curving line
  • Isochoric (constant v) = A steeper upward-curving line
• The ideal Carnot cycle is a perfect rectangle on the T-s diagram.
• Real cycles like the Rankine and Brayton cycles deviate from this rectangle, and we can see exactly where and why (e.g., isobaric heating, irreversibilities in turbines/compressors). Irreversibilities always cause processes to shift to the right (entropy generation).

Preview of the Next Lesson

We just saw visually that a real turbine process ends at a different state than an ideal, isentropic one. We can see the difference on the T-s diagram. But how much different is it? How do we quantify the "goodness" of a real turbine or compressor compared to its ideal version?

In our next lesson, we will do exactly that by defining and applying the concept of Isentropic Efficiency. The T-s diagram will be our constant guide as we compare the actual work output/input to the ideal, isentropic case.