Sanitary Sewer Design, Runoff Control, and Flood Frequency Analysis: 18-Env-A2 Hydrology and Municipal Hydraulics Engineering, December 2014, Q4

Problem 4: Sanitary Sewer Design, Runoff Control, and Flood Frequency Analysis (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.

(i) Sanitary Sewer Sizing at 80% Full

Given. Circular PVC sewer, partial flow:

Given data
Quantity	Symbol	Value
Peak design flow	$Q_d$	5 m³/s
Proportional depth	$y/D$	0.80
Bedding slope	$S$	0.05 (5%)
Manning's roughness (PVC)	$n$	0.013
Velocity limits	$V$	$0.8 < V < 7$ m/s

Find. The required pipe diameter $D$, and whether the resulting velocity satisfies the stated limits.

Approach. At $y/D=0.80$ the flow subtends a fixed central angle $\theta$; express the partial-flow area $A$, wetted perimeter $P$ and hydraulic radius $R$ in terms of $\theta$ and $D$, substitute into Manning's equation, and solve for the diameter that delivers $Q_d = 5$ m³/s.

Central angle at 80% full. For $y/D = 0.80$: $\theta = 2\cos^{-1}(1-2y/D) = 2\cos^{-1}(-0.60) = 4.4286\ \text{rad}$ (253.7°).
Partial-flow geometry (in terms of $D$). $$A = \frac{r^2}{2}(\theta-\sin\theta), \qquad P = r\theta, \qquad R = \frac{A}{P}, \qquad r=\frac{D}{2}.$$
Solve Manning's equation for $D$. $Q_d = \tfrac{1}{n}AR^{2/3}S^{1/2}$ is a monotonic function of $D$ alone (since $\theta$ is fixed); solving numerically for $Q_d=5$ m³/s gives $$D = \boxed{983\ \text{mm}}\ (0.9825\ \text{m}), \qquad A = 0.650\ \text{m}^2,\ R = 0.299\ \text{m}, \qquad V = Q_d/A = \boxed{7.69\ \text{m/s}}.$$
Check against the velocity limits and select a commercial size. $V=7.69$ m/s marginally exceeds the stated 7 m/s ceiling. Rounding up to the next standard commercial PVC size, $D=1000$ mm, does not fix this — at $y/D=0.80$ it carries $Q=5.24$ m³/s at $V=7.78$ m/s, an even higher velocity (partial-flow velocity is not monotonically decreasing with diameter at fixed $y/D$; it is essentially flat-to-rising in this size range). Because $Q_d$ and $S$ are fixed by the problem, no pipe-diameter choice alone brings $V$ under 7 m/s at this slope.

Check: the 5% bedding slope is unusually steep for a gravity sanitary sewer and is the root cause of the velocity exceedance — it is not a sizing error. In practice this reach would need a flatter grade over part of its length or an energy-dissipating drop structure/manhole to bring the operating velocity within the erosion/scour limit; select $D=1000$ mm (next standard size above the exact 983 mm) as the specified pipe and flag the velocity for the drop-structure design that follows.

Quantity	Value
Exact required diameter	983 mm (0.9825 m)
Velocity at exact diameter, 80% full	7.69 m/s (exceeds 7 m/s cap)
Specified commercial diameter	1000 mm
Velocity/capacity at 1000 mm, 80% full	7.78 m/s at 5.24 m³/s — still exceeds cap (flag for drop structure)

(ii) On-Site and Off-Site Stormwater Runoff Control

On-site vs. off-site control, 25-year municipal O&M perspective
System	Description	Advantage	Disadvantage
On-site (e.g. permeable pavement / bioretention at each lot)	Distributed source control installed on individual private parcels	Treats/attenuates runoff at the source, reducing peak flow and pollutant load entering the piped system and downstream infrastructure sizing	Hundreds of small facilities on private property are difficult for the municipality to inspect and maintain consistently over 25 years — clogging or failure on any one lot often goes unnoticed and the maintenance burden falls on many separate owners
Off-site (e.g. a centralized regional detention/wet pond)	A single municipally-owned facility serving the tributary catchment	One asset to inspect, budget and maintain over the design life (economy of scale, clear ownership and access)	Requires dedicated land and a conveyance network to bring flow to the facility, and provides no control at all until runoff has already left the individual sites (no source-level treatment)

(iii) Flood-Frequency Curve Fitting

The standard method is to treat the 15 annual-maximum instantaneous flows as a sample from a probability distribution and fit a flood-frequency distribution to them, most commonly the Log-Pearson Type III distribution (the method recommended in Canadian and US flood-frequency practice) or, for a more symmetric record, the Gumbel (Extreme Value Type I) distribution. The procedure is: (1) transform each annual maximum to $\log_{10}Q$; (2) compute the sample mean, standard deviation and skew of the log-transformed series; (3) read (or compute via the Wilson–Hilferty approximation) the frequency factor $K_T$ corresponding to the sample skew and the desired return period $T$; (4) recover the flood magnitude from $\log_{10}Q_T = \overline{\log Q} + K_T\,s_{\log Q}$. This lets the fitted curve be extrapolated beyond the 15-year record to estimate rare events (50- or 100-year floods) that have not necessarily been observed.

Illustrative application to the Wavy River record ($N=15$):

Log-Pearson III statistics (illustrative)
Statistic	Value
Mean of $\log_{10}Q$	2.743
Standard deviation of $\log_{10}Q$	0.1245
Skew coefficient, $C_s$	0.107
Frequency factor $K_{50}$ / $K_{100}$	2.111 / 2.405
Estimated $Q_{50}$	≈ 1013 m³/s
Estimated $Q_{100}$	≈ 1102 m³/s

Check: a 15-year record is short for extrapolating to 50- or 100-year events (the true sampling uncertainty on $Q_{50}$/$Q_{100}$ from only 15 points is large); these values illustrate the METHOD and should be treated as indicative, not design-final, without a longer record or regional skew adjustment.

Concept & Theory

Topic. Partial-flow (circular pipe) Manning design; on-site vs. off-site stormwater control; flood-frequency analysis (Log-Pearson III).

Key relations. Partial-flow geometry $A=\tfrac{r^2}{2}(\theta-\sin\theta)$, $P=r\theta$; Manning $Q=\tfrac{1}{n}AR^{2/3}S^{1/2}$; Log-Pearson III $\log Q_T = \overline{\log Q} + K_T s_{\log Q}$.

Why this works. All three parts are exercises in fitting a design decision to imperfect or extreme conditions: partial-flow design accepts that a sewer is not meant to run full, source vs. regional stormwater control trades distributed treatment against centralized maintainability, and flood-frequency analysis fits a statistical curve because the physical flood-generating process is too complex to model deterministically from 15 years of data alone.

Common pitfall. Assuming a larger pipe always reduces velocity at a fixed flow and fill ratio — it does not, because both $Q$ and $V$ grow together with $D$ for a fixed $y/D$; extrapolating a short flood record to a 100-year event without acknowledging the associated confidence interval.

Source. Metcalf & Eddy, Wastewater Engineering, sewer-design chapter; Linsley, Kohler & Paulhus, Hydrology for Engineers, ch. 6 (flood-frequency analysis).

Question 4 of 7: Sanitary Sewer Design, Runoff Control, and Flood Frequency Analysis

Problem 4: Sanitary Sewer Design, Runoff Control, and Flood Frequency Analysis (20 marks)

(i) Sanitary Sewer Sizing at 80% Full

(ii) On-Site and Off-Site Stormwater Runoff Control

(iii) Flood-Frequency Curve Fitting