--- title: "Supply air quantity for a conditioned room" subject: "Applied Thermodynamics" chapter: 4 key: "semester_3/applied_thermodynamics:room-supply-air-numerical" source: https://www.bctengineer.com/q/semester_3/applied_thermodynamics/room-supply-air-numerical site: BCT Engineering --- # Supply air quantity for a conditioned room Applied Thermodynamics · Chapter 4: Air-Conditioning - Asked in: year not given, 8 marks - Video: [Solve Classic Air Conditioning Problem Using the Sensible Heat Ratio Protractor](https://www.youtube.com/watch?v=mb5p55ruXeM) — Mechanical PE Exam Prep - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/applied_thermodynamics/room-supply-air-numerical - Whole chapter: https://www.bctengineer.com/md/semester_3/applied_thermodynamics/chapter-4 ## Question A room is to be maintained at 25 °C and 50% RH. The room sensible heat gain is 18 kW and the latent heat gain is 4.5 kW. Air is supplied at 14 °C. Determine (a) the room sensible heat factor, (b) the mass flow rate of supply air, (c) the specific humidity and relative humidity of the supply air, and (d) the volume flow rate of supply air. Take $p = 101.325$ kPa, $h_{fg} = 2501$ kJ/kg, $c_{pa} = 1.005$ kJ/kg K and $c_{pv} = 1.88$ kJ/kg K. The saturation pressure at 25 °C is 3.169 kPa. ## Short answer Room sensible heat factor is 0.80. Using sensible load and 11 K temperature difference, mass flow is 1.60 kg/s. From latent load, supply specific humidity is 0.00876 kg/kg (≈88% RH). Volume flow is about 1.32 m³/s (79 m³/min). ## Answer #### Room state $$ p_v = 0.5 \times 3.169 = 1.585\ \text{kPa},\qquad \omega_r = 0.622 \times \frac{1.585}{101.325 - 1.585} = 0.00988\ \text{kg/kg} $$ $$ h_r = 1.005 \times 25 + 0.00988(2501 + 47) = 50.3\ \text{kJ/kg} $$ #### (a) Room sensible heat factor $$ \text{RSHF} = \frac{Q_s}{Q_s + Q_l} = \frac{18}{18 + 4.5} = 0.80 $$ #### (b) Supply air mass flow (from the sensible load) $$ Q_s = \dot m_a\,c_{pm}(t_r - t_s),\qquad c_{pm} = 1.005 + 1.88\,\omega_s \approx 1.0215 $$ $$ \dot m_a = \frac{18}{1.0215 \times (25 - 14)} = 1.602\ \text{kg/s} = 5767\ \text{kg/h} $$ ($\omega_s$ is found in step (c), and the two steps are iterated; the values converge after two or three trials.) #### (c) Supply air humidity (from the latent load) $$ Q_l = \dot m_a(\omega_r - \omega_s)\,h_{fg} $$ $$ \omega_r - \omega_s = \frac{4.5}{1.602 \times 2501} = 1.123 \times 10^{-3} $$ $$ \omega_s = 0.00988 - 0.00112 = 0.00876\ \text{kg/kg} $$ Relative humidity at 14 °C: the vapour pressure of the supply air is $p_v = \omega_s p/(0.622 + \omega_s) = 1.407$ kPa, and the saturation pressure at 14 °C is about 1.598 kPa, so $$ \phi_s = \frac{p_v}{p_s} = \frac{1.407}{1.598} = 0.88\ \ (\approx 88\ \%) $$ The supply enthalpy is $h_s = 36.2$ kJ/kg. #### (d) Volume flow of supply air $$ v_s = \frac{0.287 \times 287.15\,(1 + 1.608 \times 0.00876)}{101.325} = 0.8248\ \text{m}^3/\text{kg} $$ $$ \dot V_s = 1.602 \times 0.8248 = 1.321\ \text{m}^3/\text{s}\ (\approx 79.3\ \text{m}^3/\text{min}) $$ **Answer:** (a) RSHF = 0.80; (b) 1.60 kg/s; (c) $\omega_s$ = 0.00876 kg/kg, RH about 88 %; (d) about 1.32 m³/s (79 m³/min). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. RSHF for sensible 18 kW and latent 4.5 kW is: A. 4.5/22.5 = 0.20 B. 18/22.5 = 0.80 C. 18/4.5 = 4 D. 22.5/18 = 1.25 Answer: B. RSHF = Q_s/(Q_s + Q_l) = 18/22.5 = 0.80. (Key point: Room SHF) 2. Supply air mass flow is determined from: A. The latent load alone B. The sensible load: m_a = Q_s / (c_pm (t_r - t_s)) C. Total load divided by h_fg D. Room volume only Answer: B. The sensible gain is carried by the temperature difference 25 - 14 = 11 K, giving about 1.60 kg/s. (Key point: Supply mass flow) 3. The supply specific humidity is found from: A. omega_s = omega_r + Q_l/(m_a h_fg) B. omega_s = omega_r - Q_l/(m_a h_fg), giving about 0.00876 C. omega_s = Q_s/Q_l D. omega_s = 0.622 x 14/101.325 Answer: B. Supply air must be drier than the room to absorb the latent gain: omega_r - omega_s = Q_l/(m_a h_fg) = 0.00112. (Key point: Supply humidity) 4. Why is the mass flow calculation iterated? A. c_pm depends on omega_s, which comes from the latent step that uses the mass flow B. Room temperature changes each trial C. p changes D. RSHF is unknown Answer: A. c_pm = 1.005 + 1.88 omega_s needs omega_s, which itself needs m_a, so the two steps converge after a few trials. 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