22-Mec-B2 Environmental Control in Buildings · May 2018

Question 3 of 8: Terminal reheat and VAV with discriminator control (20 marks)

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. Eight problems, three hours, open book. Problem 1 carries 30 marks, Problem 2 carries 10 marks and Problems 3 to 8 carry 20 marks each; candidates answer any five, and indicate their choice on the cover of the first workbook. Psychrometric charts (SI and inch-pound) and an R-134a pressure–enthalpy diagram are appended to the paper. All eight problems are solved below, because the set as a whole is the study resource.

Reference texts.

ASHRAE, Handbook — Fundamentals (2021): Ch.1 psychrometrics, Ch.16 ventilation and infiltration, Ch.18 heating and cooling load calculations, Ch.21 duct design, Ch.26 heat, air and moisture transmission.
McQuiston, Parker and Spitler, Heating, Ventilating and Air Conditioning: Analysis and Design, 6th ed. — loads, psychrometric processes, duct and air distribution design.
W. P. Jones, Air Conditioning Engineering, 5th ed. — the percentage-saturation convention, apparatus dew point and coil by-pass factor, face-and-by-pass plant.
Shan K. Wang, Handbook of Air Conditioning and Refrigeration, 2nd ed. — single-duct reheat and VAV systems, discriminator (zone-demand) control.
ASHRAE Standard 55-2023, Thermal Environmental Conditions for Human Occupancy; ASHRAE Standard 62.1-2022, Ventilation for Acceptable Indoor Air Quality.
National Building Code of Canada 2020 and NRCan / Environment and Climate Change Canada climatic design data for the Canadian design conditions used in Problems 5 and 7.

Check: two readings taken from the printed paper.

(1) The length label on the Problem 6 duct sketch is printed as “L₁ = =100 ft == 6ft”. The equation editor has dropped the symbols after each equals sign; the pattern “something = something = 100 ft” and “something = something = 6 ft” means four named lengths in two equal pairs, and the sketch shows exactly four duct runs. The solution therefore takes L₁ = L₂ = 100 ft (plenum to tee, and tee to elbow) and L₃ = L₄ = 6 ft (the two drops to the ceiling diffusers), and states the reading as an assumption under cover-page instruction 1. Only the pressure totals in parts (c) and (d) depend on it; the duct diameters in part (a) do not.

(2) Problem 1 gives the outdoor air as “percentage saturation 50 %” but the room as “RH 50 %”. These are different quantities and the difference is deliberate: percentage saturation is $\mu = W/W_{s}$, relative humidity is $\phi = p_{w}/p_{ws}$. At 26 °C the 50 % saturation state is 50.8 % RH, so treating them as interchangeable shifts the outdoor humidity ratio by about 0.2 g/kg.

Question 3: Terminal reheat and VAV with discriminator control (20 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. Two zones served from one air-handling unit, both held at the same dry-bulb temperature, with the loads at the design hour well below the design capacity.

Given data, Problem 3
Quantity	Zone 1	Zone 2
Zone dry-bulb setpoint	75 °F	75 °F
Design supply air flow	3,200 cfm	2,000 cfm
Sensible cooling load at the hour	60,000 Btu/h	30,000 Btu/h
Design cold-deck temperature	55 °F dry bulb
Air density (dry air assumed)	0.075 lbm/ft³
VAV minimum flow setting, part (b)	20 % of design

Find. The reheat energy required in each zone at that hour, (a) for a constant-volume system with terminal reheat and (b) for a VAV system with a 20 % minimum, both under discriminator control.

Approach. Discriminator (zone-demand) control resets the cold-deck temperature until the zone with the greatest demand is just satisfied with no reheat at all; every other zone then reheats the difference. So find each zone's required supply temperature, take the lowest as the reset value, and price the reheat from there.

Problem 3 (a): the constant-volume plant with terminal reheat. Discriminator control resets the cold deck until the hungriest zone (Zone 1) needs no reheat at all.

Fix the sensible-heat coefficient from the stated density. For dry air at the stated density, $$\dot{Q}_{s} = \dot{V}\,\rho\,c_{p}\,\Delta t = \dot{V}\,(60)(0.075)(0.24)\,\Delta t = 1.08\,\dot{V}\,\Delta t$$ with $\dot V$ in cfm and $\Delta t$ in °F. The familiar 1.08 is therefore not a memorised constant here but a direct consequence of the density the question supplies.
Part (a) — what supply temperature does each zone want? At constant volume the flow is fixed at its design value, so each zone needs the temperature difference that carries its own load: $$\Delta t_{1} = \frac{60{,}000}{1.08 \times 3{,}200} = 17.36\ ^\circ\text{F} \quad\Longrightarrow\quad t_{s,1} = 75 - 17.36 = 57.64\ ^\circ\text{F}$$ $$\Delta t_{2} = \frac{30{,}000}{1.08 \times 2{,}000} = 13.89\ ^\circ\text{F} \quad\Longrightarrow\quad t_{s,2} = 75 - 13.89 = 61.11\ ^\circ\text{F}$$ Zone 1 is the hungrier of the two — not because its load is larger in absolute terms, but because its load per unit of air is larger.
Apply the discriminator. One cold deck serves both zones, so it must be cold enough for the most demanding of them. The controller resets it upward from the 55 °F design value to the lowest of the required temperatures: $$t_{\text{deck}} = \min\,(57.64,\ 61.11) = \boxed{57.64\ ^\circ\text{F}}$$ Zone 1 is then the control zone and takes no reheat at all.
Part (a) — price the reheat. Zone 2 receives air at 57.64 °F when it wanted 61.11 °F, and its terminal box makes up the difference at the fixed design flow: $$\dot{Q}_{rh,2} = 1.08 \times 2{,}000 \times (61.11 - 57.64) = \boxed{7{,}500\ \text{Btu/h}}, \qquad \dot{Q}_{rh,1} = \boxed{0}$$ The same calculation is worth doing without the reset, to see what the discriminator buys: at a fixed 55 °F deck the two boxes would burn $1.08(3{,}200)(2.64) = 9{,}120$ and $1.08(2{,}000)(6.11) = 13{,}200$ Btu/h, a total of 22,320 Btu/h. Resetting the deck cuts the reheat by 14,820 Btu/h, or 66 %, and cuts the coil load by the same amount again.
Part (b) — VAV, first at the design deck temperature. A VAV box throttles the flow instead of reheating it, so at 55 °F each zone would draw $$\dot{V}_{1} = \frac{60{,}000}{1.08(75-55)} = 2{,}778\ \text{cfm},\qquad \dot{V}_{2} = \frac{30{,}000}{1.08(75-55)} = 1{,}389\ \text{cfm}$$ against minimum positions of $0.20(3{,}200) = 640$ and $0.20(2{,}000) = 400$ cfm. Both required flows are far above their minima, so neither box is forced open beyond what its zone needs.
Part (b) — apply the discriminator to the VAV plant. Here the discriminator resets the deck upward until the neediest box reaches full flow. Zone 1 reaches its 3,200 cfm design flow at exactly the 57.64 °F found in step 3, so the reset is the same, and Zone 2 then throttles to $$\dot{V}_{2} = \frac{30{,}000}{1.08\,(75 - 57.64)} = 1{,}600\ \text{cfm}$$ which is four times its 400 cfm minimum.
Part (b) — the answer, and where it would change. Since neither box is on its minimum stop, neither has more air than its load can absorb, and $$\boxed{\dot{Q}_{rh,1} = \dot{Q}_{rh,2} = 0\ \text{Btu/h}}$$ Reheat in a VAV system appears only once a box bottoms out on its minimum position. At the 55 °F design deck that happens below $1.08(640)(20) = 13{,}824$ Btu/h in Zone 1 and $1.08(400)(20) = 8{,}640$ Btu/h in Zone 2 — roughly 23 % and 29 % of the loads at this hour, so both zones have a wide margin before any energy is thrown away.
Interpret the comparison. The same two zones at the same hour need 7,500 Btu/h of reheat as a constant-volume plant and none at all as a VAV plant, and the VAV fan is moving 4,800 cfm rather than 5,200 cfm. That is the entire case for VAV in one line: constant-volume reheat pays twice for part load, once at the coil and once at the reheater, while VAV simply stops delivering air it does not need.

Problem 3 — results
Part	Quantity	Zone 1	Zone 2
—	Required supply temperature	57.64 °F	61.11 °F
—	Cold deck after discriminator reset	57.64 °F (Zone 1 is the control zone)
(a)	Reheat, constant volume	0 Btu/h	7,500 Btu/h
(a)	Reheat if the deck were held at 55 °F	9,120 Btu/h	13,200 Btu/h
(b)	VAV flow at the reset deck	3,200 cfm (full)	1,600 cfm
(b)	Minimum flow setting	640 cfm	400 cfm
(b)	Reheat, VAV	0 Btu/h	0 Btu/h

Topic. Multi-zone air systems: single-duct constant-volume terminal reheat, variable-air-volume with a minimum stop, and cold-deck reset by discriminator (zone-demand) control.

Key relations. $\dot{Q}_{s} = 60\rho c_{p}\dot{V}\Delta t = 1.08\,\dot{V}\,\Delta t$ for standard air; required supply temperature $t_{s,i} = t_{z} - \dot{Q}_{i}/(1.08\,\dot{V}_{i})$; reset $t_{\text{deck}} = \min_{i} t_{s,i}$; reheat $\dot{Q}_{rh,i} = 1.08\,\dot{V}_{i}\,(t_{s,i} - t_{\text{deck}})$; VAV flow $\dot{V}_{i} = \dot{Q}_{i}/[1.08(t_{z} - t_{\text{deck}})]$ bounded below by the minimum position.

Why this works. Every zone on a common duct sees the same supply temperature, so the only free variables are that one temperature and, in a VAV system, each zone's flow. Discriminator control exploits the fact that reheat is pure waste: pushing the deck to the highest temperature that still satisfies the worst zone drives that zone's reheat to zero and minimises everyone else's, without any zone losing control.

Common pitfall. Resetting the deck to the highest required temperature, which starves the control zone, or comparing the loads rather than the loads per cfm — Zone 1 has twice the load of Zone 2 but is the control zone because of its temperature difference, not its kilowatts. In part (b) the second pitfall is to assume reheat is always needed; it appears only when a box hits its minimum stop.

Source. Shan K. Wang, Handbook of Air Conditioning and Refrigeration, 2nd ed., Ch.21 (single-duct constant-volume and VAV systems, zone-demand control); McQuiston, Parker and Spitler, HVAC Analysis and Design, 6th ed., Ch.3 and Ch.13.

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