Clausius-Clapeyron Relation Validation

Overview

This document validates the analytical derivatives of the Clausius-Clapeyron relation by comparing them with finite difference derivatives computed using ForwardDiff.jl. The Clausius-Clapeyron relation describes how the saturation vapor pressure changes with temperature during phase transitions.

Purpose

The validation ensures that our analytical implementation of the Clausius-Clapeyron derivatives is mathematically correct and numerically accurate. This is critical for:

• Saturation adjustment algorithms that rely on accurate derivatives
• Numerical stability in thermodynamic calculations
• Physical consistency of the thermodynamic formulation

Implementation

The validation compares two approaches:

1. Analytical derivatives: Direct computation using the derived mathematical expressions
2. Finite difference derivatives: Numerical approximation using ForwardDiff.jl

include("Clausius_Clapeyron.jl")

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Results

The plot shows:

• Top panel: Comparison of analytical vs. finite difference derivatives
• Middle panel: Error between the two methods
• Bottom panel: Saturation vapor pressure as a function of temperature

Interpretation

• Agreement: The analytical and finite difference derivatives should closely match
• Error analysis: The middle panel shows the difference between methods
• Physical range: The temperature range covers typical atmospheric conditions
• Validation: Small errors indicate correct implementation