Question 1:

The largest hydroelectric plant in the UK is found in Northumberland, located at the Kielder reservoir. During

its operation, the minimum flowrate discharge of water runs at Q=1.32 m³/s (cumec), however throughout

the year, as the reservoir level increases, the flow rate (Q) must be increased to different levels. One of the

standard increases, is to change the flow rate from 3.5 m³/s to 7 m/s. Figure 1 shows a change in the flow

rate ejection for the hydroelectric dam over a 210s period, represented in cm³/s for this 3.5 to 7 cumec flow

rate change.

Flow rate(t)= (-3.19892 10³) (t) + (2.77042 105) () + (-0.00928) (+³)+(1.49974) (t*) +

(-1.18729= 10²) (³) + (4.25940 10³) (²) + (-5.42866 104) (t) + (3.50829 10°) (Egn. 1)

From this derived flow rate information using equation 1, numerical integration can be used to calculate the

area under the curve and therefore calculate the volume of water that is discharged through the hydroelectric

plant over the timeframe.

Use the composite Trapezoidal rule numerical method of numerical integration taught during the

module to determine the volume of water discharged during the 210 seconds timeframe represented in

Figure 1 and Equation 1. You must obtain an approximation of the volume in m³ (SI units), with a relative

error of less than 0.00002%, when the analytical value is NOT known.

During the numerical integration calculations, if the relative error is not reached during a loop, double the

number of separations used over the timespan in the calculations for the following calculation cycle.

(i) In the command window, display the integral value calculated for volume, the number of

sections used in the numerical integration, and the relative error produced for each looped

calculation using 'fprintf" and associated commands.

Produce a single figure with two subplots, (1) showing the flow rate (m³/s) vs. time (t) of the

flow rate equation in one plot at a suitable accuracy, and (2) a cumulative volume (m³) graph of

the quantity of ejected water over time (s) in the second subplot.

Produce a figure showing the total volume calculated against the number of separations used in

each numerical integration calculation; use a logarithmic x-axis scale on the resulting plot.