Sensible Heating of Humid Air: 04-BS-10, May 2018, Q8

Question 8: Sensible Heating of Humid Air (15 marks)

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Given. Inlet: $P=95$ kPa, $T_1=15\ ^\circ$C, $\phi_1=30\%$, $\dot V_1=4$ m³/min. Exit: $T_2=25\ ^\circ$C. Sensible heating only (no moisture added or removed).

Find. (a) $\dot Q$ [kW]; (b) $\phi_2$.

Approach

A heating coil with no condensation or humidification keeps the humidity ratio $W$ (mass of water vapor per mass of dry air) constant across the section, so the exit state is fixed by $T_2$ and $W_2=W_1$. Convert the given volumetric flow rate to a dry-air mass flow rate using the inlet humid-air specific volume, then take the enthalpy difference at constant $W$ for the heat rate; the exit relative humidity follows directly once $T_2$ and $W_1$ are known.

Inlet humid-air properties. At $P=95$ kPa, $T_1=15\ ^\circ$C, $\phi_1=0.30$: $$W_1=0.003382\ \text{kg water/kg dry air},\qquad h_1=23.65\ \text{kJ/kg dry air},\qquad v_1=0.8750\ \text{m}^3/\text{kg dry air}.$$
Dry-air mass flow rate. $$\dot m_{da}=\frac{\dot V_1}{v_1}=\frac{4/60\ \text{m}^3/\text{s}}{0.8750} =\boxed{0.0762\ \text{kg dry air/s}}.$$
Exit state at constant humidity ratio. With no moisture added, $W_2=W_1= 0.003382$. At $P=95$ kPa, $T_2=25\ ^\circ$C, $W_2=0.003382$: $$h_2=33.78\ \text{kJ/kg dry air},\qquad \phi_2=\boxed{0.1614\ (16.1\%)}.$$ The relative humidity drops even though no moisture is removed, because the saturation pressure of water rises sharply between 15°C and 25°C while the actual vapor pressure (set by $W_1$) stays fixed.
Heat transfer rate (part a). $$\dot Q=\dot m_{da}(h_2-h_1)=0.0762\times(33.78-23.65)=0.0762\times10.13 =\boxed{0.771\ \text{kW}}.$$

Quantity	Result
(a) $\dot Q$	0.771 kW
(b) $\phi_2$	0.1614 (16.1%)

Concept & Theory

Topic. Sensible (dry-bulb-only) heating of humid air in a heating coil, with no moisture addition or removal.

Key relations. Humidity ratio conserved across a sensible-only process: $W_2=W_1$. Dry-air mass flow from volumetric flow and specific volume: $\dot m_{da}=\dot V/v$. Heat rate: $\dot Q=\dot m_{da}(h_2-h_1)$, both enthalpies evaluated per unit mass of dry air at the same $W$.

Why this works. All psychrometric mass/energy balances are conventionally done per unit mass of dry air (which is conserved through the process) rather than per unit mass of the moist-air mixture (whose total mass would change if any moisture were added or removed); since this coil adds no moisture, $W$ (and hence the dry-air basis) stays fixed and only sensible heat crosses the boundary.

Common pitfall. Assuming relative humidity must rise or stay constant when air is heated — because $\phi=P_v/P_{sat}(T)$ and $P_{sat}$ increases steeply with $T$ while $P_v$ (set by the fixed $W$) does not change, $\phi$ always drops under sensible heating, which is exactly why heated air feels "drier."

Source. Cengel & Boles, Thermodynamics: An Engineering Approach, 8th ed., Ch. 14 (Psychrometrics, heating and cooling of humid air).

Question 8 of 9: Sensible Heating of Humid Air

Question 8: Sensible Heating of Humid Air (15 marks)

Approach