Watershed Abstractions, Hydrograph Shape and a Pump Impeller Diameter Change: 18-Env-A2 Hydrology and Municipal Hydraulics Engineering, May 2016, Q4

Problem 4: Watershed Abstractions, Hydrograph Shape and a Pump Impeller Diameter Change (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) Watershed Abstractions Determining the Runoff Hydrograph

Three important abstractions subtract from gross rainfall to leave the "rainfall excess" that actually becomes direct runoff and shapes the hydrograph: interception — rainfall caught on vegetation canopy that evaporates without ever reaching the ground; depression storage — water ponding in small surface depressions that must fill before overland flow can begin; and infiltration — water entering the soil profile, governed by soil type, antecedent moisture and land cover, typically the largest of the three abstractions in most storm events. Because these abstractions are satisfied first, a hydrograph's rising limb only begins once their combined capacity is exceeded, which is why a dry, permeable watershed produces a delayed, low peak while a saturated or impervious one produces an almost immediate, high peak for the same storm.

(ii) Watershed Factors Influencing Hydrograph Shape

Watershed factors affecting hydrograph shape
Factor	Description	Effect on peak flow	Effect on time to peak
Watershed shape & drainage-area size	An elongated watershed spreads tributary flow paths over a wide range of travel times; a compact, fan-shaped watershed concentrates them.	Fan-shaped → higher, sharper peak (flows arrive together); elongated → lower, more attenuated peak.	Fan-shaped → shorter time to peak; elongated → longer time to peak.
Slope & imperviousness (land cover)	Steeper slopes and greater impervious cover increase overland-flow velocity and reduce infiltration losses.	Steeper/more impervious → higher peak flow (more rainfall excess, faster delivery).	Steeper/more impervious → shorter time to peak (faster travel time).

(iii) Impeller Diameter Change: New Operating Point via the Pump Affinity Laws

Given.

Given data
Quantity	Symbol	Value
Original impeller diameter	$D_1$	100 mm
New impeller diameter	$D_2$	140 mm
Speed	$N$	constant (same casing/motor)

Check: the pump chart gives axis ranges (head 0–45 m, capacity 0–0.030 m³/s) and iso-efficiency contours up to 77% at the family's overall centre, but not a digitized curve. The $D_1=100$ mm curve, a smaller trim within the family, is read to peak on the chart's own ~66% efficiency contour at $Q_1\approx0.0108$ m³/s, $H_1\approx12.5$ m — treat these two readings as the input to the calculation, ± chart-reading precision; the percent-capacity-improvement result below does not depend on this reading at all.

Find. The new BEP capacity $Q_2$, head $H_2$, brake power $P_2$, expected efficiency $\eta_2$, and the % capacity improvement.

Approach. At constant speed, geometrically similar impeller trims within the same casing obey the pump affinity laws: $Q\propto D$, $H\propto D^2$, $P\propto D^3$, with efficiency approximately unchanged for a modest trim ratio.

Diameter ratio. $$r=\frac{D_2}{D_1}=\frac{140}{100}=\boxed{1.40}.$$
New capacity (affinity law, $Q\propto D$). $$Q_2=Q_1\,r=0.0108\times1.40=\boxed{0.01512\ \text{m}^3/\text{s}}\ (15.1\ \text{L/s}).$$
New head (affinity law, $H\propto D^2$). $$H_2=H_1\,r^2=12.5\times(1.40)^2=\boxed{24.5\ \text{m}}.$$
Efficiency and brake power. Efficiency is assumed essentially unchanged for a geometrically similar trim, $\eta_2\approx\eta_1=\boxed{66\%}$. Brake power follows from $P=\rho g Q H/\eta$: $$P_2=\frac{\rho g Q_2 H_2}{\eta_2}=\frac{1000\times9.81\times0.01512\times24.5}{0.66}=\boxed{5.51\ \text{kW}}.$$
Percent capacity improvement. This follows directly from the diameter ratio alone, independent of the assumed BEP reading: $$\%\ \text{improvement}=(r-1)\times100\%=\boxed{40.0\%}.$$

Quantity	Value
New optimum capacity, $Q_2$	0.01512 m³/s (15.1 L/s)
New head, $H_2$	24.5 m
New brake power, $P_2$	5.51 kW
Expected efficiency, $\eta_2$	≈66% (unchanged from $D_1$)
% capacity improvement	40.0%

Impeller-trim family: reading the 100 mm curve's operating point and projecting to 140 mm along the affinity parabola through the origin.

Concept & Theory

Topic. Watershed abstractions and hydrograph shaping factors; centrifugal-pump affinity laws for an impeller diameter change.

Key relations. $Q_2=Q_1(D_2/D_1)$; $H_2=H_1(D_2/D_1)^2$; $P_2=P_1(D_2/D_1)^3$; $P=\rho gQH/\eta$.

Why this works. The affinity laws follow from dimensional/geometric similarity between two impeller trims running in the same casing at the same speed — velocity triangles at the impeller tip scale linearly with $D$, so flow (a velocity×area) scales as $D$, head (a velocity-squared term) as $D^2$, and power (the product of flow and head, with density and $g$ constant) as $D^3$.

Common pitfall. Applying the affinity laws across a change in speed and a change in diameter simultaneously without separating the two effects; assuming efficiency is exactly constant for a large trim ratio, when in practice larger trims (here 40%) see a modest efficiency penalty that a real pump curve (rather than the ideal affinity projection) would show.

Source. Walski, Advanced Water Distribution Modeling and Management, pump chapter; Linsley, Hydrology for Engineers, runoff and hydrograph chapters.

Question 4 of 7: Watershed Abstractions, Hydrograph Shape and a Pump Impeller Diameter Change

Problem 4: Watershed Abstractions, Hydrograph Shape and a Pump Impeller Diameter Change (20 marks)

(i) Watershed Abstractions Determining the Runoff Hydrograph

(ii) Watershed Factors Influencing Hydrograph Shape

(iii) Impeller Diameter Change: New Operating Point via the Pump Affinity Laws