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ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2018/s4/2/solution

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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S4 / 2 / Solution by

Codex  0 2026-10-25

A pumped-storage plant moves water uphill using electrical power, then returns it through a turbine and generator when electricity is needed. The upper and lower reservoirs set the usable volume and head; pumps, penstocks, valves and reversible pump-turbines set the power and flow. The plant stores energy rather than generating net energy over a cycle.

Figure 1.
Gravitational storage separates reservoir energy capacity from pump and turbine power
. An original functional diagram shows an upper and a lower reservoir connected through a penstock and reversible pump-turbine. Charging uses grid electricity to lift water; discharge returns gravitational potential energy through the generator. **Details**.

For constant head, the gravitational energy is

E_{g} = ρ V g h

. With the stated generation efficiency,

E_{usable} = 0.89 (1000) (1.4 \times 1 0^{7}) (9.81) (400) ≃ 4.89 \times 1 0^{13} J = 13.6 GWh .

(1)

3

GW,

P = η ρ g Q h

gives

Q ≃ 8.59 \times 1 0^{2} m^{3} s^{- 1}, t_{full} = E_{usable} / P ≃ 4.53 h .

(2)

Head variation, dead volume and hydraulic losses beyond the stipulated efficiency have been neglected.

To deliver

9

kWh with the same head and efficiency needs

V ≃ 9.3 m^{3}

of water, roughly a

2.1

m cube, together with a

400

m elevation difference and a lower reservoir. At a

10

m head it would need about

370 m^{3}

. This contrasts with the small fuel volume: the useful energy density of lifted water depends on its external head, not its chemical composition.

Pumped storage offers long service life, fast dispatch, substantial power, and separate choices of storage volume and turbine rating. Its drawbacks include geographic constraints, construction cost and time, ecological impacts, evaporation and water requirements, plus round-trip losses. The quoted generation efficiency alone is not the round-trip efficiency because pumping also consumes more than the stored potential energy.

The rock cylinder has mass

ρ_{r} π r^{2} (2 r)

and rises through

r

. Equating its useful energy to the reservoir result cancels the common efficiency and gravity:

2 π ρ_{r} r^{4} = ρ_{w} Vh, r = [\frac{1000 ( 1.4 \times 1 0 ^{7} ) ( 400 )}{2 π ( 3000 )}]^{1/4} ≃ 131 m .

(3)

Its height is about

262

m and its lift about

131

m, ignoring buoyancy as instructed. Stored energy scales as

r^{4}

, excavated volume roughly as

r^{3}

, and lined area as

r^{2}

. If excavation dominates cost, cost per unit stored energy decreases approximately as

r^{- 1}

. Seals, increasing pressure, geology, leakage, supporting structure and power equipment can change that scaling; a volume law is not a complete cost model.

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