Difficulty: Medium
Correct Answer: smaller than
Explanation:
Introduction / Context:
In gas–vapor systems (most commonly air–water), several humidity measures exist. Two often-confused quantities are relative humidity (relative saturation) and percentage saturation. Plant operators, HVAC engineers, and dryer designers must distinguish these to read charts and size equipment correctly.
Given Data / Assumptions:
Concept / Approach:
Because the presence of vapor reduces the partial pressure available for the dry gas, the humidity ratio at saturation differs from the simple ratio implied by φ. The rigorous definition yields μ < φ for all unsaturated mixtures. The two values coincide only in the dilute-vapor limit (very low humidity) or at saturation where both equal 100%.
Step-by-Step Solution:
Relative humidity: φ = p_v / p_v^sat.Humidity ratio at state: W = 0.622 p_v / (P − p_v); at saturation: W_s = 0.622 p_v^sat / (P − p_v^sat).Percentage saturation: μ = 100 × (W / W_s).Algebra shows μ = 100 × φ × (P − p_v^sat)/(P − p_v) < 100 × φ for unsaturated states (p_v < p_v^sat), thus μ < φ.
Verification / Alternative check:
On a psychrometric chart, lines of constant relative humidity lie above those of constant percentage saturation except at saturation where they meet.
Why Other Options Are Wrong:
Higher than / Equal to: contradicts the mathematical relationship and chart geometry.“Either (a) or (b)”: not correct for standard air–water systems at fixed T and P below saturation.
Common Pitfalls:
Using φ and μ interchangeably; forgetting the denominator correction (P − p_v) vs (P − p_v^sat).
Final Answer:
smaller than
Discussion & Comments