How does the Efficiency and Cross-Sectional Area of a River Change Down Stream?
How does the Efficiency and Cross-Sectional Area of a River Change Down Stream?
For my investigation I will be looking at the cross-sectional area of a river and how this affects efficiency in the river of Little Beck as the stream moves downstream from the source to nearer the mouth. To do this I will be taking measurements of the cross sectional area and the efficiency at 12 different sites along the river (See fig. 2 for map of river and 12 sites).
The source of May Beck is approximately at a height of 280m on Fylingdales Moor in the North Yorkshire Moor National Park. The river flows south to join the Parsley beck, and the combined flow of these two rivers, now called the Little Beck, becomes tributary of the much larger river, River Esk, which reaches the sea at Whitby.
During my investigation, I will be comparing my results to that of the Bradshaw Model. The Bradshaw Model is a model of an ideal stream (see fig. 1). By using this I can compare my results with the model to see how ideal Little Beck is. According to the Bradshaw model, at the source of the stream the cross-sectional area and efficiency are low at the source but as you move down stream they increase.
Three key questions I have constructed to help me with my investigation are:
- How does efficiency change as you go down stream compared to the predictions of the Bradshaw Model?
- How does cross-sectional area vary from sites 1 to 12 compared to the predictions of the Bradshaw Model?
- Is there a pattern in the data of the efficiency and cross sectional-area?
Methodology Table
This methodology table shows all the types of data I collected, why and how they were collected. In order to investigate into cross-sectional area and efficiency I will not need all this data, so I have highlighted the methods in green which I will need to use to carry out my investigation.
Data Representation
...This is a preview of the whole essay
Data Representation
To continue my investigation I will need to look at the data I have collected and put it into graphs and tables so I can analysis the data efficiently. I have only used the data that I will need to answer my investigation question.
In the twelve graphs above I have shown a cross-section of each of the 12 sites down stream. According to the Bradshaw model, the depth and width should increase as you go down stream. In figure 5 seen below, it can be seen that the average width and depth increases the closer to the mouth we investigated, ranging from a width of 1.9m at site one to 10.1m at site twelve, and an average depth of 8.1cm at site one to 25.7cm at site 12.
An accurate cross-section of the river allows us to calculate the wetted perimeter of the river at each site, but first I need to calculate the cross-sectional area.
In figure 6, I have shown the cross-sectional area of the river at all twelve sites. By looking at the cross-sectional area I can see how large the river becomes further down stream; however in some areas like at site 4, there is a sudden change. This could be due to human impact changing the river shape. To work out the cross-sectional area I needed to times the depth by the width. For site one, the equation can be seen below:
dw = csa
8.1 x 1.9 = csa
Looking at the Bradshaw model, the cross-sectional area of the stream increases the closer it gets to the mouth. By looking at the data above, it can be seen that this is what the Little Beck river does, starting with 15.39m² at site one and 259.57m² at site 12.
As can be seen in the graph there is a positive correlation, showing that down stream the cross-sectional area increases.
I will now work out the wetted perimeter and use both pieces of data to work out the hydraulic radius.
In figure 7, I have shown the wetted perimeter of all the twelve sites. The wetted perimeter is the line of contact between the water and the channel. To the work this out I need to times the depth by two and then times that by the width. For site one, the equation can be seen below:
2d + w = wp
(2 x 8.1) + 1.9 = wp
16.2 + 1.9 = wp
wp = 18.1
By working out the wetted perimeter I can use the data with the cross-sectional area to work out the hydraulic radius. The hydraulic radius expresses the losses of energy in overcoming friction with a stream’s bed and banks. High values indicate high efficiency, with a channel approaching a semi-circular shape, like site 3, whereas low values indicate low efficiency where a channel will often be wide but shallow like site 4. According to the bradshaw model, the hydraulic radius tends to increase with distance down stream, primarily due to the increasing discharge.
In figure 8, I have shown the hydraulic the radius for all the twelve sites. To work out this data, I had to use the data I calculated previously (the wetted perimeter and cross-sectional area). I needed to take the cross-sectional area and divide it by the wetted perimeter. For site one, the equation can be seen below:
CSA ÷ WP = HR
15.39 ÷ 18.1 = HR
HR = 0.850276243
By looking at fig. 8, I can see that the hydraulic radius increases the further you get down stream. This indicates high efficiency. This means that the hydraulic radius follows the trend of the bradshaw model, which shows that the river’s efficiency increases down stream.
By looking at my data so far, this should mean that as the cross-sectional area increases, as does the efficiency of the river. To show this, I will produce a scatter graph.
As can be seen in the scatter graph (graph 4) above, overall there is a positive correlation. With increasing cross-sectional area the hydraulic radius does increase. This is shown by the line of best fit. All points follow an obvious trend, showing no anomalies in the graph.
We can see from the data above that the efficiency and cross-sectional area increases, but this should also mean that the velocity of the river increases also.
In graph 5 we can see that there is a negative correlation from the line of best fit. The average speed at site 7 and 8 is higher than expected, making it a positive anomaly. The average speed at site 11 is lower than expected, making it a negative anomaly. This could be due to foliage catching the biscuit used, stopping it from flowing down stream efficiently, or the biscuit may have been lost under the current of the river. These results aren’t very reliable due to the method we used to collect the data.
To show that velocity increases with hydraulic radius I have produced a scatter graph, which can be seen to the right (graph 6). As can be seen there is a positive correlation, but several positive and negative anomalies are present. These anomalies are mainly results with a hydraulic radius between 2 and 3, and a low velocity between 10 and 30 seconds. This will have been due to the same reason the velocity graph is inaccurate.
My graphs show quite good correlations between the data, but there are some strong positive and negative anomalies. To get a clearer source of correlation data, I will produce a Spearman’s rank correlation coefficient. If two sets of data correlate exactly, they are said to have a perfect correlation. If one set of values goes up as the other set goes up, a perfect correlation is found, which would have the Rs value of 1.0. A perfect negative correlation (as one set of data goes up, the other goes down) would have a Rs value of -1.0. No correlation would be mid way between these values, or 0. This can be seen in the figure 3.
I will firstly produce a Spearman’s rank correlation coefficient between the cross-sectional area and the hydraulic radius with 99% certainty.
I now need to use the data from figure 10 to calculate the correlation coefficient, which can be seen below:
Rs = 1 - 6 x ∑d²
n (n² - 1)
Rs = 1 - 6 x 18
12 (12² - 1)
Rs = 1 - 108
1716
Rs = 1 - 0.062937
Rs = 0.937063
The correlation between the cross-sectional area and hydraulic radius is 0.937063. This is a near perfect positive correlation, showing graph 4 (see page 10) to be correct. From graph 6, showing a correlation between velocity and hydraulic radius (see page 11), there is a positive correlation, but it is a weak one with a few anomalous results. I will repeat the spearman’s rank correlation process to give me a more reliable correlation. According to the Bradshaw model, there should be a positive correlation. The table I will use to show this can be seen below.
I now need to use the data from the table above to calculate the correlation coefficient, which can be seen below:
Rs = 1 - 6 x ∑d²
n (n² - 1)
Rs = 1 - 6 x 341
12 (12² - 1)
Rs = 1 - 2046
1716
Rs = 1 - 1.192308
Rs = -0.19231
The correlation between velocity and hydraulic radius is -0.19231. This is a weak, negative correlation. As there should be a strong positive correlation between the two, a problem must have occurred. This is most likely due to the method in which the data was collected, as mentioned earlier (see page 10/11).
Conclusion
The main purpose of my investigation was to find out how the cross-sectional area and efficiency changed down stream, by looking at the velocity of the stream, the inner and outer friction and the cross-sectional area. I also calculated the hydraulic radius so I could see the efficiency of the stream. The table below shows the key questions I constructed at the start to help me with my investigation:
If the efficiency of the river increased down stream, this meant that the velocity of the river should also increase. A graph was produced (see graph 6, page 11) comparing hydraulic radius with the velocity, and a positive correlation is produced but the results didn’t appear very reliable as there were a few anomalies. I produced a spearman’s rank correlation coefficient to see how strong the correlation was, and it shows there is a weak negative correlation. This shows how the spearman’s rank correlation can give a more clear answer. This negative correlation could be due to a few reasons. One of these reasons could be obstacles in the river, like foliage and the current of the river. Because of the method of throwing a dog biscuit into the river, it was easy for it to get caught by rocks and foliage hanging over the stream, and easy for it to be caught by the current. If this test were to be done properly, a flow metre would have been used, which can measure the velocity of the river by being placed in the river and can give an immediate, accurate reading.
My investigation question was “How does the Efficiency and Cross-Sectional of a River Change Down Stream?” To answer this question I first needed to work out the efficiency and cross-sectional area separately by using the key questions on page 1. I worked out the cross-sectional area increased down stream, which can be seen on pages 7-9 in figures 5 and 6, ands graphs 2A-L showing graphs of the cross-sectional area of the river and graph 3. Graph 3 shows that there is a positive correlation, showing that down stream the cross-sectional area increases.
I then calculated the efficiency, which can be seen from page 9 to 10, in figures 7 and 8. By looking at fig. 8, I can see that the hydraulic radius increases the further you get down stream which indicates high efficiency. This means that the river’s efficiency increases down stream.
Now I know that the cross-sectional area and efficiency increases down stream, I needed to see if there was a pattern between the two to answer my question, which can be seen in figure 9 and graph 4 on page 10. These shown a positive correlation of results, but to make sure this was a reliable source I produced a Spearman’s Rank Correlation Coefficient, which can be seen in figure 10 on page 11-12. By doing this I was producing a reliable source of data, and correlation came out to be a near perfect, strong positive correlation, thus proving that as the cross-sectional area increases down stream, the more efficient the river becomes.
Evaluation
For my geographical enquiry, I visited the river Little Beck in North Yorkshire where I intended on doing several tests to acquire the data needed to answer my investigation question. I spent the day of the visit collecting the data using methods that can be seen in the methodology table on page 3, fig. 3. Then after the visit, the data was collected and observed in school to be used in our investigation. I found the day very successful, and found the results interesting to observe, and enjoyed collecting the data. If I were to repeat this investigation, I would perhaps increase the number of sites where data was collected so I could strengthen my finding, perhaps more sites closer to the mouth as we weren’t very close to the mouth, or maybe do the same tests in another river so I could compare the two rivers results.
To collect the data needed for my investigation, there were several methods I needed to carry out (to see what data was needed to be collected and what methods were used, see figure 3 on pages 3). These included measuring the velocity and stream surveys looking at the width and depth of the river.
I feel the method used to measure the width and depth worked well and was done successfully, but perhaps more intervals could have been taken to give a stronger set of results. Graphs were produced to show the cross section of each site we visited, which can be seen on page 7 and 8, graphs 2A to 2L. These shown the shape of the river well, however they aren’t very reliable as the width hasn’t be used on the x axis, not showing an accurate cross-section.
By collecting this data, I was able to calculate the hydraulic radius of the river which would tell me how efficient a river was using a particular method (see pages 8-10, 11-12), which gave an accurate and reliable result.
To measure the velocity, a dog biscuit was thrown into the stream at each site; this method wasn’t very effective and gave inaccurate results. This was due to foliage catching the biscuit thus stopping it from flowing efficiently, or the biscuit could have gotten lost under the currents. If I were to repeat this, I would have employed a better method by using a flow metre and collected more data to give a more reliable set of data.
Although some of the methods used didn’t give accurate results, I was able to answer the question to my investigation, making the trip and methods successful.
Matthew Southward Geography: Personal Investigation