Ib Math Hl Coursework Flow Rate

Flow Rate Abstract: The aim of this paper is to analyse the flow rate of the Nolichucky River in Tennessee. Various mathematical models are compared so as to find the most appropriate one and then estimate some specific features such as the maximum, the mean and the rate of change. The data used throughout this analysis originates from the Nolichucky River in Tennessee between 27 October 2002 and 2 November 2002. The time is measured in hours past midnight, starting at 00:00 on 27 October and the flow is measured in cubic feet per second (cfs). Fig.1 shows the original data. Fig.1 Time Flow

0 440

6 450

12 480

18 570

24 680

30 800

36 980

42 1090

48 1520

54 1920

60 1670

66 1440

Time Flow

78 1300

84 1150

90 1060

96 970

102 900

108 850

114 800

120 780

126 740

132 710

138 680

144 660

72 1380

A scatter plot of the data illustrates the relationship between the points. The flow is represented on the y-axis and time on the x-axis. This is shown in fig.21

Fig.2 2500

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1500

1000(cfs) Flow

500

0 0

20

40

60

80

100

120

140

160

Time (hours)

It is important to know that a graph plotting flow rate over time is often called a Hydrograph. The graph shows that the flow rate was less then 500 cfs2 when something caused it to 1

Graph Plotted in Macintosh Microsoft Excel 2004.

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Cubic feet per second

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increase parabolically until reaching a maximum value 50 hours later. After this, the flow rate decreased rapidly for about 20 hours and then steadily fell towards its initial value during the next 60 hours. Before finding the line of best fit, it is possible to estimate the rate of change of the data. This is done using a numerical method and allows us to roughly illustrate the behaviour of the flow rate. Considering the first two points of the graph: (t = 0, flow = 440) and (t = 6, flow = 450) , the change dx of the x values is 6 - 0 = 6 and the change dy of the y values is 450 - 440 =10 . As a result, the rate of change of the data is

dy 10 = = 1.66 . The procedure is repeated for all dx 6

points of data and the results are plotted in Fig.3. (Note that the time value is the average between the two original points)

Fig.3 80 60 40 20 dy/dx 0 0

50

100

150

-20 -40 -60 Time

The rate of change of the flow rate is therefore positive and increasing during the first 52 hours and negative and decreasing for the rest of the time. In order to accurately process and analyse the data, it is necessary to find the lines of best fit, or the mathematical relationship between the points on the graph. Best-fit lines are estimated

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using technology in programs such as Microsoft Excel or the TI-84 GDC calculator. A straightforward linear correlation would not be accurate because the data shows distinct changes in patterns. Moreover, the graph does not resemble to any known polynomial function except the quartic, which is still far from the real correlation as shows model 1

Model 1 y = 5E-05x4 - 0.0132x 3 + 0.8078x2 + 6.2022x + 2500

2000

1500

Best-Fit line

1000(cfs) Flow

500

0 0

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60

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120

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Time (hours)

Model 1 shows the quartic best fit: y = 5 ´ 10- 5 x 4 - 0.0132x 3 + 0.807x 2 + 6.2022x + 346.64 . Note that even if the line follows the trends to a certain extent, it is still very far from the original values. Moreover, the line increases again when time equals around 130 hours whereas the data shows a continuous decrease.

It seems that there is no single function that would show behaviour similar to the original data. (i.e. increasing parabolically, reaching a maximum, decreasing parabolically and then decreasing steadily.) It is therefore necessary to split the data into two parts in order to produce more accurate best-fit lines.

Estimating from the graph, the maximum value takes place when time equals 54 hours hence points occurring before 54 hours belong to the increasing trend and points occurring after 54

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hours belong to the decreasing trend. The composite function representing the original data ì f (t)              for  0 £  t £ 54 ü 1 ý with one function effective for f î 2 (t)            for  144 ³ t ³ 54 þ

will be in the form of Flow = í

times before 54 hours and another for times after 54 hours (note that for more accuracy the point t = 54 belongs to both functions.) This technique enables us to use different types of functions as best fit lines. The first hypothesis was that each trend followed an exponential pattern as shown in Model 2

Model 2

Increasing trend Decreasing trend

2500

2000

1500 y = 370.33e

y = 3163.6e -

0.0279x

1000(cfs) Flow

500

0 0

20

40

60

80 Time (hours)

100

120

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160

It is clear that an exponential correlation is not the best model for the river flow. Indeed, the model’s greatest rate of flow is significantly bellow the data’s maximum value ( t = 54,  flow = 1920 ) while its shape does not closely follow the pattern. The “increasing

trend” best-fit is too steep at first and does not reflect the flow’s initial steady behaviour. Moreover, its y-intercept is below the data’s lowest point. The “decreasing trend” line is also inadequate since it shows a continuous decrease throughout the graph whereas the original data decreases rapidly at first and then slower after a few hours. In addition to the straightforward graphical analysis, it is possible to calculate the average distance between the original data and the model. Let us consider the first point of the data, which is situated on the y-axis. ( t = 0,  flow = 440 ) If the model function is defined as

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f (x) = 370.33e 0.0279x , then its flow value is f (0) = 370.33 cfs. The distance between the model

and the actual data is f (x) - f (0) = 370.33 - 440 = 70 when rounding off adequately. Calculating the distance for all points is long and tedious however, computer software such as Excel make the task much more efficient. Fig 4 shows the tables used to compute the average distance for both increasing and decreasing patterns in the first model. Fig.4

Time (hours) 0 6 12 18 24 30 36 Time (hours) 42 54 48 60 54 66 72 78 84 90 96 102 108 114 120 126 132 138 144

Flow (cfs) F(t) 440 370 450 438 480 518 570 612 680 723 800 855 980 Flow (cfs) F(t)1011 1090 1195 1920 1691 1520 1413 1670 1577 Increasing trend 1920 1671 1440 1471 1380 1300 1150 1060 970 900 850 800 780 740 710 680 660

1372 1280 1194 1114 1039 969 904 843 786 734 684 638 595

F(t)-Flow Av. distance 70 75 12 38 42 43 55 F(t)-Flow31 Av. distance 105 229 54 107 93 249 31 8 20 44 54 69 69 54 43 6 6 26 42 65

Decreasing trend

The average distance between the points and the model is 75 units for the increasing function and 54 units for the decreasing function. The average distance for the entire model is 65 units.

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The second model involves two distinct quadratic functions. The local minima should follow more accurately the original data because of the initial flatness and the rapidly increasing gradient afterwards. This is shown in Model 3 Model 3

2

y = 0.6271x - 8.399x +

y = 0.1622x 2 - 44.772x +

2500

2000 Increasing trend Decreasing trend 1500

1000(cfs) Flow

500

0 0

20

40

60

80 Time (hours)

100

120

140

160

Model 3 illustrates the graph of y = 0.6271x 2 - 8.399x + 476.36 as the increasing function and y = 0.1622x 2 - 44.772x + 3781.9 as the decreasing function.

This model is much better than the previous one. The increasing function cuts the y-axis very close to the first point and remains flattened for about 12 hours. Its gradient then increases rapidly, following the pattern of the river’s flow rate. The model’s maximum flow rate occurs at the intersection of both functions which is just slightly bellow the point where t = 54 . The coordinates of the maximum are found by equating the two functions and solving for x using a Graphic Display Calculator. 0.6271x 2 - 8.399x + 476.36 = 0.1622x 2 - 44.772x + 3781.9 x = 53.835266  and  y = 1841.6813

For simplicity, it is assumed that the actual maximum point has coordinates (54,1920) hence the distance between the two points is found using Pythagoras Theorem: d = (x1 - x 2 ) 2 + (y1 - y 2 ) 2 d = (54 - 53.835266) 2 + (1920 - 1841.6813) 2 d = 78 units

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It is also useful to find the average distance between the model and the data so as to further optimize the analysis. Once again, this can been done using Excel tables as shown in Fig.5 Fig.5 Increasing trend Time (hours) 0 6 12 18 24 30 Time (hours) 36 54 42 60 48 66 54 72 78 84 90 96 102 108 114 120 126 132 138 144

Flow (cfs) 440 450 480 570 680 800 Flow (cfs) 980 1920 1090 1670 1520 1440 1920 1380 1300 1150 1060 970 900 850 800 780 740 710 680 660

F(t) 476 449 466 528 636 789 F(t) 987 1837 1230 1680 1518 1533 1851 1399 1277 1166 1066 979 903 838 786 745 716 698 692 698

F(t)-Flow Av. distance 36 37 1 14 42 44 11 F(t)-Flow 7 Av. distance 83 26 140 10 2 93 69 19 23 16 6 9 3 12 14 35 24 12 12 38

Decreasing trend

The average distance is 37 units for the increasing function and 26 units for the decreasing function proving that this model is better than the previous one. Although this third model seems quite reliable, it is possible to optimize it even more by changing the nature of the decreasing best-fit line. Indeed, it was found that a power correlation is closer to reality for 54 £ t £ 144

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Model 4

y = 0.6271x 2 - 8.399x +

y = 147191x

-1.095

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Increasing trend Decreasing trend

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1000(cfs) Flow

500

0 0

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80 Time (hours)

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Model 4 includes the same quadratic function for 0 £ t £ 54 but also a power function y =147191x - 1.095 for 54 £ t £ 144

This fourth model appears to be the most accurate of all. It contains both a quadratic and a power function and its shape is very close to the original data. The maximum occurs when 0.6271x 2 - 8.399x + 476.36 = 147191x - 1.095 x = 54.149827  and  y = 1860.34

hence the distance between the data and the model at the maximum is d = (54 - 54.149827) 2 + (1920 - 1860.34) 2 d = 60 units The average distance between the power function and the data is illustrated in fig.6 and equals 22units.

Fig.6 Time (hours) 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144

Flow (cfs) 1920 1670 1440 1380 1300 1150 1060 970 900 850 800 780 740 710 680 660

F(t) 1866 1663 1498 1362 1247 1150 1067 994 930 874 823 778 738 701 668 637

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F(t)-Flow Av. distance 54 22 7 58 18 53 0 7 24 30 24 23 2 2 9 12 23

The fourth and last model is therefore the most reliable since the maximum is closer to reality and the average distance between the function and the original data smaller. It is now possible to further investigate on the river’s flow rate using this mathematical model. Let the model be defined by the function ì 0.6271t 2 - 8.399t + 476.36             for  0 £  t £ 54 ü Flow = í ý - 1.095                                    for  144 ³ t ³ 54 þ î 147191t

Its derivative is d( flow) ì 1.2542t - 8.399     for  0 £  t £ 54 ü =í ý î - 161174.145t - 2.095   for  144 ³ t ³ 54 þ dt The amount of flowing water therefore increases when the gradient of the graph is positive, i.e. when the first derivative has positive values. Plotting the graph of the derivative helps visualising the situation. Fig.7 y = 1.2542x - 8.399 y = - 161174 x - 2.095

Fig.7 illustrates the first derivative of both function. Note that the blue dotted line occurs for  0 £  x £ 54 and the full red line for  54 £  x £ 144

The flow rate therefore increases when the first derivative y = 1.2542t - 8.399 is positive in

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the given domain. ( between the point where the derivative crosses the x-axis and the model’s maximum value t = 54.1 ) 0 = 1.2542t - 8.399 t » 6.70 hours Rounding off to the nearest hour for consistency, the amount of flowing water increased from 07:00 on 27 October to 06:00 on 29 October. Since the river is controlled by a series of dams, it is also useful to estimate the amount of water that flowed in a given period of time. This could help understanding the river better and allow a more efficient use of resources. The volume of water that flowed past the measuring station between 00:00 on 28 October and 00:00 on 29 October is found by integrating the function between t = 24 and t = 48 . This is é0.6271t 3 8.399t 2 ù48 ò(0.6271t - 8.399t + 476.36)dt = êë 3 - 2 + 476.36túû = 24 404 cubic feet 24 24 48

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Fig.8

y = 0.6271t 2 - 8.399t + 476.36 

y = 147191t - 1.095

Water flowed past

Fig 8 shows the volume of water that flowed between 00:00 on 28 October and 00:00 on 29 October

Another important feature of the original data that can be found using the model is the average flow rate. At first glance and without any calculation, one would estimate the average 10

flow rate to be around 1000 cfs. This exact rate would also occur twice, first during the increase of flow at t » 36 hours and then during the decrease at t » 95. The most reliable way of finding the average of a function in a given domain is to use the Mean Value Theorem that takes the form: mean =

f (b) - f (a) . Because the model is defined b- a

by two distinct functions, it is necessary to find the mean for each function separately and then the average of the results. The average flow rate from 00:00 on 28 October to 00:00 on 2 November is found as an example. Let “00:00 on 28 October” be defined as t = 24 , “00:00 on 2 November” as t = 144 and the “maximum value” of the model occur at t = 54 , then the mean value of the increasing function is: 2 2 f (54) - f (24) [0.6271(54) - 8.399(54) + 476.36] - [0.6271(24) - 8.399(24) + 476.36] = 54 - 24 30 1851.438 - 635.9936                        = 30                        = 1215.4444                        » 1215 cfs

mean( increa sin g) =

and the mean of the decreasing function is: - 1.095 - 1.095 ] - [147191(144) ] f (144) - f (54) [147191(54) = 144 - 54 90                        =1228.503 cfs

mean( decrea sin g ) =

                       » 1228

Hence the average flow rate from 00:00 on 28 October to 00:00 on 2 November is 1215.444 +1228.503 » 1222 cfs 2

This exact rate also occurs twice, once when the quadratic and the power functions equals the mean value: 1222 = 0.6271t 2 - 8.399t + 476.36 t = 41.82 t » 42

1222 = 147191t - 1.095 t = 79.48 t » 80

Hence the average flow between those two dates occurs at 18:00 on 28 October and 08:00 on 29 October. Using the information extracted from the mathematical model of the data, we can suggest a possible weather patter that would account for the shape of the graph. Because the river’s

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average flow rate between the 28th and the 2nd was significantly higher than the initial and final rates, the total volume of water that flowed seems to be greater than usual. This could be caused by rainfall in the area, which would then be collected in the river. Moreover, the model shows a steep increase in flow rate that reaches its maximum value within a few hours. A strong and sudden precipitation is therefore likely to have caused the shape of the graph. When heavy rainfall occurs on land that cannot absorb and store the water because it is urbanised, saturated or too compact, it flows on the surface and reaches the river within a short time3. This will then lead to a temporary but rapid increase in flow rate and sometimes floods. If the rain comes to an end, the surface runoff disappears and the flow of the river returns to its normal level. This “falling limb” is always more gentle than the rising limb because the water infiltrated in the ground starts reaching the river and continues contributing to its flow. The graph of the original data is therefore an example of a “storm hydrograph” and can be compared to the typical one in Fig.94

Fig.9 shows the sketch of a typical storm hydrograph. The increase in flow is the rising limb, the peak is the maximum and the decrease is the falling limb, or flood recession. Note that subsurface flow arises after the peak because of water infiltration and the normal flow rate is the baseflow.

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http://www.uwsp.edu/geo/faculty/ritter/geog101/textbook/hydrosphere/surface_water.html

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Ecole Polytechnique Federale de Lausanne: http://hydram.epfl.ch

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Model 4

2

y = 0.6271x - 8.399x +

y = 147191x

-1.095

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Increasing trend Decreasing trend

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Model 4 shows the graph and model of the original data. The model appears to be close to reality confirming our hypothesis. Nevertheless, there are still several limitations that will affect the results. The first and most important limitation is the transition between the rising and falling limb. The model shows it as an abrupt change in gradient meaning the flow rises rapidly and then suddenly falls with not intermediate stage in between. Most models, however, contain a somewhat flattened peak since the runoff water may flow through different path and takes some time to disappear completely. One way to solve this problem is to divide the model’s composite function further and use another quadratic polynomial for the 3 highest points. On the other hand, transitions between the various new functions could be even more ambiguous and inaccurate hence this is a possible limit of technology. Secondly, the model does not take into account the slight change in direction of the rising limb at the 8th point of the graph. This point could have a great importance in a complex and technical analysis as it could show, for example, that subsurface flow takes less time than expected to reach the river and therefore adds to the surface runoff. Taking measurements of the flow at smaller time intervals is a possible solution to this limitation and it would greatly improve the accuracy and reliability of the model.

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Finally, we may attempt to apply a similar model and reasoning to a different set of data in order to verify the hypothesis and analysis made in the discussion. The new data originates from the Nolichucky River in Tennessee between 1st May 2007 and 20th May 2007. The flow rate values are the averages of each day and are shown below5. Day Flow

1 756

2 729

3 766

4 1780

5 2000

6 1670

7 1370

8 1170

9 1060

10 984

Day Flow

11 921

12 870

13 842

14 783

15 726

16 707

17 715

18 676

19 639

20 613

Note that some parameters of the data are changed. Here, 24 hours pass between each measurement. The rise and fall of the flow rate also takes place over a much longer period of time. (20 days) Applying the same model to this set of data produces the graph in Model 5.

Model 5

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y = 105.07x - 276.53x + 880

y = 6698.6x

-

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0 0

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Day, May 2007

It appears that the power function for the falling limb fits reality closely. The rising trend, 5

http://nwis.waterdata.usgs.gov/tn/nwis/dv?cb_00060=on&format=html&begin_date=2007-0427&end_date=2007-05-20&site_no=03465500&referred_module=sw

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however, is visibly inaccurate since the data shows a smooth peak whose gradient changes slowly. Our initial mathematical model therefore cannot be effectively applied to this situation. Surprisingly, the fact that the model did not fit the new data reinforces our confidence in its reliability. Indeed, the new data was taken over a much longer period of time meaning that the weather pattern was different. The new data might have occurred in a month where it rained often but less strongly. The water could then infiltrate more into the ground leading to less surface runoff but more subsurface flow, explaining why it took about 4 days for the flow rate to reach its maximum level. In this case, we our not dealing with a short and violent storm so we cannot apply the same model. We may therefore contest the first limitation of the model that said that the maximum flow rate should show a smoother pattern. Indeed, it appears that such behaviour is likely to be caused by underground flow, not surface runoff.

Sources and references www.water.usgs.gov, last visited 2007-12-11 http://hydram.epfl.ch, last visited 2007-12-11

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