Psychrometrics from Dry-Bulb and Wet-Bulb Temperatures: 04-BS-10, December 2015, Q5

Question 5: Psychrometrics from Dry-Bulb and Wet-Bulb Temperatures (15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. $T_{db}=25\ ^\circ$C, $T_{wb}=15\ ^\circ$C, $P=1$ atm $=101.325$ kPa.

Find. (a) $W$ [kg water/kg dry air]; (b) $\phi$; (c) $T_{dp}$ [$^\circ$C].

Approach

The wet-bulb temperature fixes the state of moist air along an adiabatic-saturation line through the given dry-bulb temperature: air at $(T_{db},\phi)$ cooled adiabatically by evaporating water into it reaches saturation exactly at $T_{wb}$. Rather than iterating the adiabatic-saturation energy balance by hand, evaluate the ASHRAE humid-air property state directly from the $(T_{db},P,T_{wb})$ triple, then read off specific humidity and relative humidity, and finally invert to the dew point (the temperature at which the same humidity ratio would be saturated).

Specific humidity (part a). From the moist-air state at $T_{db}=25\ ^\circ$C, $T_{wb}=15\ ^\circ$C, $P=101.325$ kPa: $$W=\boxed{0.006560\ \text{kg water/kg dry air}}.$$
Relative humidity (part b). At the same state: $$\phi=\boxed{0.3322\ (33.2\%)}.$$
Dew-point temperature (part c). The temperature at which air holding the same $W$ would be exactly saturated: $$T_{dp}=\boxed{7.73\ ^\circ\text{C}}.$$ As expected, $T_{dp}

Quantity	Result
(a) $W$	0.006560 kg/kg dry air
(b) $\phi$	0.3322 (33.2%)
(c) $T_{dp}$	7.73°C

Concept & Theory

Topic. Psychrometric state determination from a measured dry-bulb/wet-bulb pair, distinct from every other moist-air question in this subject (which give relative humidity directly or ask about mixing/heating processes).

Key relations. Adiabatic-saturation relation linking $T_{wb}$ to $W$: $W_1=\dfrac{c_p(T_{wb}-T_{db})+W_{sat}(T_{wb})h_{fg}(T_{wb})}{h_g(T_{db})-h_f(T_{wb})}$; $\phi=P_v/P_{sat}(T_{db})$; $T_{dp}$ solves $P_v=P_{sat}(T_{dp})$ for the same vapor pressure.

Why this works. A wet-bulb thermometer measures the temperature reached when water evaporates adiabatically into the surrounding air until it saturates — that evaporative-cooling process conserves the adiabatic-saturation enthalpy, giving one equation relating the unknown $W_1$ to the two measured temperatures, without needing $\phi$ as an independent input.

Common pitfall. Confusing wet-bulb temperature with dew-point temperature — they are only equal at saturation ($\phi=100\%$); for unsaturated air, $T_{dp}

Source. Cengel & Boles, Thermodynamics: An Engineering Approach, 8th ed., Ch. 14 (Psychrometrics); ASHRAE Fundamentals Handbook, humid-air property correlations.

Question 5 of 9: Psychrometrics from Dry-Bulb and Wet-Bulb Temperatures

Question 5: Psychrometrics from Dry-Bulb and Wet-Bulb Temperatures (15 marks)

Approach