Question 1 of 8: Summer air conditioning of a clothing store

Question 1: Summer air conditioning of a clothing store (20 points)

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. A direct-expansion plant serving one retail zone, with the room and outdoor design states fixed and the load already split into room and outdoor-air components.

Find. The room and grand sensible heat factors, the room and coil apparatus dew points, the supply and mixed-air temperatures, the total supply air quantity, the rate of moisture removed at the coil, and the refrigeration duty in tons.

Approach. Fix O and R from the chart, form the room sensible heat factor to locate the room apparatus dew point, then form the effective sensible heat factor — which charges the bypassed share of the outdoor-air load to the room — to locate the coil apparatus dew point, and read the supply air quantity, supply temperature, mixing state, moisture removal and refrigeration duty from that geometry.

1. Fix the two design states on the chart. At $75^\circ\text{F}$ and 50 % RH the room humidity ratio and enthalpy are $W_r = 0.00924\ \text{lb}/\text{lb}$ and $h_r = 28.11\ \text{Btu}/\text{lb}$. At $100^\circ\text{F}$ DB with a $78^\circ\text{F}$ wet bulb the outdoor state is $W_o = 0.01560\ \text{lb}/\text{lb}$, $h_o = 41.25\ \text{Btu}/\text{lb}$. These two points anchor everything that follows.
2. (a) Form the sensible heat factors. The room condition line is set by the loads the room itself sees, $$\mathrm{RSHF}=\frac{\mathrm{RSH}}{\mathrm{RSH}+\mathrm{RLH}}=\frac{265{,}000}{265{,}000+71{,}500}=\boxed{0.788}$$ while the coil, which also handles the ventilation load, works on the grand sensible heat factor $\mathrm{GSHF}=355{,}500/529{,}500=0.671$. Both are quoted because the question asks for “the SHF” and the two describe different lines on the chart.
3. (b) Locate the room apparatus dew point. Drawing the RSHF line through R until it meets the saturation curve gives $$t_{\mathrm{ADP},\ \mathrm{room}} = \boxed{50.3^\circ\text{F}}$$ This is the coil surface temperature that a system handling only the room load would need.
4. Charge the bypassed outdoor air to the room. A fraction BF of the air passes the coil untreated, so that share of the outdoor-air load arrives in the space. The effective room loads are $\mathrm{ERSH} = \mathrm{RSH} + \mathrm{BF}\cdot\mathrm{OASH} = 265{,}000 + 0.084(90{,}500) = 272{,}602\ \text{Btu}/\text{hr}$ and $\mathrm{ERLH} = 71{,}500 + 0.084(102{,}500) = 80{,}110\ \text{Btu}/\text{hr}$, whence $\mathrm{ESHF} = 272{,}602/352{,}712 = 0.773$.
5. (f) Coil apparatus dew point. The ESHF line drawn through R meets saturation at $$t_{\mathrm{ADP},\ \mathrm{coil}} = \boxed{49.7^\circ\text{F}}$$ slightly below the room ADP, which is the whole point of the effective-factor construction: the coil must run a little colder than the room line alone suggests because part of the outdoor air slips past it.
6. (d) Supply air quantity. With the apparatus dew point known, the air quantity follows from the effective room sensible heat, $$\dot V = \frac{\mathrm{ERSH}}{1.10\,(1-\mathrm{BF})\,(t_r - t_{\mathrm{ADP}})} = \frac{272{,}602}{1.10(0.916)(75-49.72)} = \boxed{10{,}700\ \text{cfm}}$$ of which 3,500 cfm is outdoor air and the remaining 7,204 cfm is recirculated.
7. (c) Temperature of the air leaving the coil. The supply air must remove the room sensible heat between S and R, $$t_s = t_r - \frac{\mathrm{RSH}}{1.10\,\dot V} = 75 - \frac{265{,}000}{1.10(10{,}704)} = \boxed{52.5^\circ\text{F}}$$ As an independent check, the bypass-factor definition gives the same answer from the other direction: $t_s = t_{\mathrm{ADP}} + \mathrm{BF}\,(t_m - t_{\mathrm{ADP}}) = 49.72 + 0.084(83.17-49.72) = 52.5^\circ\text{F}$. The two routes agreeing is the proof that the ESHF chain has been built correctly.
8. (e) Mixed air entering the coil. Mixing 3,500 cfm of outdoor air with 7,204 cfm of return air, $$t_m = \frac{3{,}500(100)+7{,}204(75)}{10{,}704} = \boxed{83.2^\circ\text{F}}$$ with a mixed humidity ratio $W_m = 0.01132\ \text{lb}/\text{lb}$.
9. (g) Moisture removal. The supply humidity ratio follows from the room latent load, $W_s = W_r - \mathrm{RLH}/(4{,}840\,\dot V) = 0.00786\ \text{lb}/\text{lb}$. Standard air at 10,704 cfm carries $\dot m_{da} = 60(10{,}704)(0.075) = 48{,}168\ \text{lb}/\text{hr}$ of dry air, so the coil condenses $$\dot m_w = \dot m_{da}\,(W_m - W_s) = 48{,}168\,(0.01132-0.00786) = \boxed{167\ \text{lb}/\text{hr}}$$
10. (h) Refrigeration duty. The coil must handle the grand total heat, $\mathrm{GTH} = 355{,}500+174{,}000 = 529{,}500\ \text{Btu}/\text{hr}$, so $$\mathrm{Tons} = \frac{529{,}500}{12{,}000} = \boxed{44.1\ \text{tons}}$$

The design that emerges is conventional for a retail space of this size: a 44-ton direct-expansion plant, about 10,700 cfm of supply air of which roughly one third is outdoor air, and a coil surface running near 50°F. What the problem is really testing is the discipline of the effective sensible heat factor. A candidate who sizes the airflow from the room load alone, ignoring the bypassed outdoor air, arrives at about 9,900 cfm and a coil that cannot hold 50 % relative humidity in the space on a design day — a failure mode that shows up as summer humidity complaints rather than as a temperature complaint, and is correspondingly hard to diagnose after the fact.