Mollier Diagram: Understanding Its Fundamental Equation

The Mollier diagram, also known as an enthalpy-entropy chart or h-s diagram, is a crucial thermodynamic tool used extensively in engineering, especially for analyzing power cycles involving substances like steam. It graphically plots specific enthalpy (h) on the y-axis against specific entropy (s) on the x-axis. To fully grasp and utilize this diagram, it's essential to understand the fundamental thermodynamic equation from which it is derived.

Thermodynamic Basis for the Mollier Diagram

The equation that forms the basis of the Mollier diagram is derived by combining the first and second laws of thermodynamics with the definition of enthalpy. Let's trace the derivation steps:

First Law of Thermodynamics and Heat

The first law of thermodynamics, for a closed system undergoing a reversible process, states that the heat added to the system ($\text{dQ}$) is used to change its internal energy ($\text{dU}$) and perform work ($\text{dW}$).

• Heat supplied: $\text{dQ}$
• Change in internal energy: $\text{dU}$
• Work done by the system: $\text{dW} = \text{PdV}$ (where $\text{P}$ is pressure and $\text{V}$ is specific volume)

Thus, the first law can be expressed as:

$\text{dQ} = \text{dU} + \text{PdV} \quad \text{(Equation 1)}$

Second Law of Thermodynamics and Entropy

The second law of thermodynamics defines the relationship between heat transfer, temperature, and entropy. For a reversible process, the heat transfer ($\text{dQ}$) is given by:

$\text{dQ} = \text{Tds} \quad \text{(Equation 2)}$

Here, $\text{T}$ is the absolute temperature and $\text{ds}$ is the change in specific entropy.

Combining the Fundamental Laws

By equating the expressions for $\text{dQ}$ from Equation 1 and Equation 2, we obtain a key fundamental thermodynamic relation:

$\text{Tds} = \text{dU} + \text{PdV} \quad \text{(Equation 3)}$

Enthalpy Definition and Differential

Enthalpy ($\text{h}$) is a thermodynamic property that is often used in energy balance equations, especially for open systems. It is defined as:

$\text{h} = \text{U} + \text{Pv} \quad \text{(Equation 4)}$

To find the differential change in enthalpy ($\text{dh}$), we differentiate Equation 4:

$\text{dh} = \text{dU} + \text{d(Pv)}$

Using the product rule for differentiation ($\text{d(xy)} = \text{xdy} + \text{ydx}$), we get:

$\text{dh} = \text{dU} + \text{PdV} + \text{vdP} \quad \text{(Equation 5)}$

Deriving the Mollier Diagram Equation

Now, we can substitute the term $(\text{dU} + \text{PdV})$ from Equation 3 into Equation 5:

From Equation 3, we know that $\text{dU} + \text{PdV} = \text{Tds}$.

Substituting this into Equation 5 yields:

$\text{dh} = (\text{Tds}) + \text{vdP}$

Rearranging this equation to isolate $\text{Tds}$ gives the specific form relevant to the Mollier diagram:

$\text{Tds} = \text{dh} - \text{vdP}$

This thermodynamic equation, $\text{Tds} = \text{dh} - \text{vdP}$, directly relates changes in temperature, specific entropy, specific enthalpy, and pressure, along with specific volume. It is a fundamental relation that underpins the Mollier diagram, which plots enthalpy versus entropy. For example, in an isentropic process (where $\text{ds} = 0$, meaning constant entropy), this equation simplifies to $\text{dh} = \text{vdP}$. This particular relationship is highly useful for analyzing processes like ideal expansions or compressions on the Mollier diagram. Therefore, this equation serves as the core mathematical foundation for the construction and interpretation of the Mollier diagram.