Question 6 of 8: Equal-friction duct design for a conference centre (20 marks)

Question 6: Equal-friction duct design for a conference centre (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. A two-room branch system fed from a plenum through an abrupt entrance, a round diverging tee and a 90° pleated elbow.

Find. (a) the standard round diameter of each of the four sections; (b) the total pressure loss and NC rating of each ceiling diffuser; (c) the total pressure loss of each complete run from plenum to diffuser outlet; and (d) whether a balancing damper is needed and where.

Approach. Size every section at the same friction rate, round each up to the next standard diameter, check the resulting velocities against the limits for a quiet space, select diffusers on neck velocity for an acceptable NC, then add friction and fitting losses along each run and compare the two totals.

1. Part (a) — set up the equal-friction sizing. The Darcy relation in the units of a duct chart is $$\frac{\Delta p_{f}}{100\ \text{ft}} = f\,\frac{100}{D}\, \left(\frac{V}{4005}\right)^{2}$$ with $D$ in feet, $V$ in fpm and $\Delta p$ in inches of water, the friction factor $f$ coming from Colebrook at $\text{Re} = VD/(60\nu)$, $\nu = 1.57 \times 10^{-4}\ \text{ft}^{2}/\text{s}$, and $V = \dot{V}/(\pi D^{2}/4)$. Setting the left-hand side to 0.08 and solving for $D$ gives the theoretical diameter of each section; the equal-friction method then keeps that same 0.08 everywhere, which is what makes the system self-balancing in the first approximation.
2. Part (a) — the four sections. Solving section by section and rounding up to the next standard round size (so the actual friction rate never exceeds the design value):

   Equal-friction sizing at 0.08 in. wg per 100 ft

   Section	cfm	Theoretical D, in.	Standard D, in.	Actual V, fpm	Actual rate, in. wg / 100 ft	Section loss, in. wg
   1	450	10.58	11	682	0.0662	0.0662
   2	200	7.82	8	573	0.0715	0.0715
   3	250	8.50	9	566	0.0604	0.0036
   4	200	7.82	8	573	0.0715	0.0043

   Rounding up drops the realised friction rate below 0.08 in every section, by between 11 and 25 %, which is the normal and harmless consequence of a discrete size range.
3. Check the limiting velocities, as the note demands. A conference centre is an acoustically sensitive occupancy, for which the recommended maxima in a low-velocity system are about 1,000 fpm in the main duct and 600 fpm in the branches: $$V_{1} = 682\ \text{fpm} \lt 1{,}000, \qquad V_{2} = 573, \quad V_{3} = 566, \quad V_{4} = 573\ \text{fpm} \lt 600$$ Every section passes, so the friction-rate sizing governs throughout and no section has to be upsized on noise grounds. Had the design rate been higher, the branch velocities would have broken the 600 fpm limit first, and the velocity check — not the friction rate — would have set those sizes.
4. Part (b) — select the ceiling diffusers. A round ceiling diffuser is characterised by its neck area, and both the pressure loss and the sound generation rise steeply with neck velocity: $$V_{n} = \frac{\dot{V}}{A_{n}}, \qquad \Delta p_{t} = \zeta_{d}\left(\frac{V_{n}}{4005}\right)^{2}$$ Taking $\zeta_{d} = 1.6$ velocity heads and the catalogue trend $\text{NC} = 20 + 30\log_{10}(V_{n}/600)$ — both representative of published round-ceiling-diffuser data — and testing the standard neck sizes:

   Diffuser selection, evaluated at each standard neck size

   Room	cfm	Neck, in.	Vn, fpm	Δpt, in. wg	NC	Verdict
   Room 1	250	6	1,273	0.162	30	too noisy, too much loss
   		8	716	0.051	22	acceptable
   		10	458	0.021	16	selected
   		12	318	0.010	12	oversized, poor throw
   Room 2	200	6	1,019	0.104	27	too noisy
   		8	573	0.033	19	selected
   		10	367	0.013	14	oversized, poor throw

   ASHRAE recommends NC 25 to 30 for conference rooms, so both selections are comfortably inside the target with several decibels in hand for duct-borne fan noise; the 6 in. necks are rejected on both counts, and the very large necks are rejected because the throw collapses and the room stops being ventilated properly even though the noise is low.
5. Part (c) — total pressure loss, run to Room 1. The run comprises the abrupt entrance, 100 ft of 11 in. duct, the branch side of the diverging tee, 6 ft of 9 in. duct and the diffuser. Fitting losses are taken on the velocity pressure of the section in which the fitting sits, $p_{v} = (V/4005)^{2}$: $$\Delta p_{A} = C_{\text{ent}}p_{v,1} + \Delta p_{f,1} + C_{b}p_{v,3} + \Delta p_{f,3} + \Delta p_{t,\text{diff}}$$ $$= 0.50(0.0290) + 0.0662 + 0.50(0.0200) + 0.0036 + 0.0210 = \boxed{0.115\ \text{in. wg}}$$
6. Part (c) — total pressure loss, run to Room 2. This run passes straight through the tee, continues for another 100 ft, turns through the pleated elbow and drops 6 ft to its diffuser: $$\Delta p_{B} = C_{\text{ent}}p_{v,1} + \Delta p_{f,1} + C_{s}p_{v,2} + \Delta p_{f,2} + C_{\text{elb}}p_{v,4} + \Delta p_{f,4} + \Delta p_{t,\text{diff}}$$ $$= 0.0145 + 0.0662 + 0.0031 + 0.0715 + 0.0102 + 0.0043 + 0.0328 = \boxed{0.202\ \text{in. wg}}$$ Run B is therefore the index (critical) run, and 0.202 in. wg is the external static pressure the supply fan must develop for this branch system, before any allowance for the plenum, filters and coil upstream.
7. See where the difference comes from. Setting the two runs side by side:

   Component pressure losses, in. wg

   Component	Run to Room 1	Run to Room 2
   abrupt entrance (C = 0.50 on section 1)	0.0145	0.0145
   friction, section 1 (100 ft of 11 in.)	0.0662	0.0662
   tee: branch (C = 0.50) / straight (C = 0.15)	0.0100	0.0031
   friction, section 2 (100 ft of 8 in.)	—	0.0715
   90° pleated elbow (C = 0.50 on section 4)	—	0.0102
   friction, branch section (6 ft)	0.0036	0.0043
   ceiling diffuser, total pressure	0.0210	0.0328
   Total	0.115	0.202

   Almost the whole 0.087 in. wg difference is the second 100 ft of main duct plus its elbow; the two branches themselves are nearly identical. That is the structural weakness of the equal-friction method: it equalises the loss per foot, not the loss per run, so runs of unequal length never balance.
8. Part (d) — is a damper necessary, and where? Yes. Both diffusers discharge to the same room pressure, so without intervention the air divides itself to equalise the two path losses, not to deliver the design flows. With Run A 0.087 in. wg easier than Run B, Room 1 would receive substantially more than its 250 cfm and Room 2 correspondingly less. The damper goes in the easier run: an opposed-blade balancing damper in the section 3 branch, immediately downstream of the tee take-off and well upstream of the diffuser, set to dissipate $$\Delta p_{\text{damper}} = \Delta p_{B} - \Delta p_{A} = 0.202 - 0.115 = \boxed{0.087\ \text{in. wg}}$$ Two practical points go with that. Placing the damper at the take-off rather than at the diffuser neck keeps the throttling noise away from the occupied space and gives the disturbed flow several diameters to re-develop before it reaches the outlet; a damper integral to the diffuser would put the same 0.087 in. wg of regenerated noise directly above the conference table. And the index run — Run B — gets no damper at all beyond the diffuser's own volume-control pattern, since throttling it would only raise the fan pressure the whole system must work against.