Contents Definition 6. Context 7. Problem Definition and Purpose 8. Sub­questions 9. Conceptualization Entities Relations Formalization 10. Quantities and their Relationships 11. Approximations and Assumptions Approximations Assumptions New assumptions & estimations 12.1 Initial derivations 12.2 Second derivations 13. Special Cases 14. Estimates Execution 15. Rephrase the problem statement in formal terms 16. Calculations / Implementation / Simulation 17. Validation and Verification; Accuracy and Precision Conclusion 18. Presentation and Interpretation Reflections and Discussions 19. Discussion after the Conceptual Model Fountain Wind 20. Discussion after the Formal Model New Assumptions & estimations 21. Discussion after the Result 22. Discussion after the Solution of the Initial Problem Reflection on the Assignment 23. Extension 24. Necessity for Improvement 25. Possibilities for improvement 26. What aspects of your work are you proud of? 27. What have you learned? Appendices 28. Used Literature 29. List of Definitions 30. List of Illustrations Definition 6. Context A decorative fountain is located in the middle of a open square, which is surrounded by terraces, restaurants and cafeteria. The fountain attracts a lot of tourists, because of the impressive height of jet and the beautiful sight of lights involved in the fountain. The tourists have a large contribution to the income of the restaurants and cafeteria. So the more tourists come sightseeing the fountain, the more the restaurants and cafeteria profit. However, when there are windy days, the waterjet of the fountain gets influenced by the wind. Because of this, the wind blows water drops upon the tourists passing by, the visitors of the fountain and terraces. The water jet goes straight upwards and at the top it falls apart in droplets falling towards the ground. One droplet follows a second half of a parabolic path. See figure 1. Figure 1: The route of a water droplet. Source: ​
http://physics.tutorvista.com/motion/projectile­motion.html 7. Problem Definition and Purpose As mentioned earlier, the tourists have a large contribution to the income of the restaurants and cafeteria. The fountain is the main reason for the appearing of tourists. To not disadvantage the restaurants and cafeteria, the fountain has to be attractive, though not spraying water upon tourists because of the wind.​
​
To achieve this, the jet of the fountain needs to be as high as possible. If there is no wind at all, the tourists of the fountain will not get wet. If there is wind, the jet will deviate and the people will get wet. Concluding, the higher the fountain, the more attractive the fountain will be. Also, the higher the fountain, the more the wind can affect the deviation of the jet, so the more tourists will get wet. Our purpose therefore is to optimize the height of fountain without wetting the tourists, given the wind speed and direction. The main questions that needs to be answered is as follows: What can the height of the fountain be, keeping it as attractive as possible, but without wetting the tourists in any wind condition? 8. Sub­questions The following questions have to be answered, in purpose of our model: 1. Should we consider the whole jet or individual droplets? 2. What forces should be taken in consideration? 3. Where should we put the sensor to measure the wind? 4. What is the maximum height of the jet (without wind)? 5. What is the maximum water pressure? 6. What is the shape of the fountain? 7. What are the dimensions of the fountain? 8. What is the maximum distance the water can get from the fountain? 9. How far away are the restaurants and the terraces? 9. Conceptualization Entities Concepts Properties Wind Wind speed: varies Wind force: varies Fountain Width reservoir Height reservoir Shape: circle Water pressure: varies Water Mass Volume water in reservoir People Distance to fountain: varies Water drops Reach: Maximum reach should not be larger than the radius of the fountain. Restaurants Profit: varies Relations This square consists of the fountain in the middle, the restaurants at the edges of the square and the tourists who are walking around. The fountain consists of one single water jet, which creates an attraction for the tourists. The tourists contribute to income of the restaurants and cafeteria that are located around the fountain. The tourists will only come visit the fountain if the fountain height is impressive enough. At the top of the water jet, the jet falls apart into droplets falling towards the ground. The reach and the height of one water droplet is influenced by the wind speed and the wind force. The stronger the wind, the smaller the height and the bigger the reach will be. The water pressure and the gravitational force influence the height of the droplet. The bigger the water pressure, the higher the droplet will get. When the height of the droplet is too high, the tourists will get wet. So the water drop will make the tourists wet. Concluding, the water pressure cannot be too high, because otherwise the tourists will get wet. Also, the water pressure may not be too low, because the water jet will not be high enough and attract enough people. When the fountain attracts less people, the restaurants will have fewer profit. Formalization 10. Quantities and their Relationships Which quantities occur in your model (presumably, these are properties of your concepts)? You will base yourself on the conceptual model, the sub­questions, and the theory you use. For each quantity, apart from its type and its unit, give its role: is the quantity for you to decide, is it asked, is it a constant, or is it an intermediate quantity? (the 4 categories, [1]​
discussed in week 5). Which relations among quantities do you assume​ ? Name these relations (equation, inequality, function, constraint, definition, interpolation, educated [2]​
guess,​ , …) For each relation, where meaningful, give the assumptions that need to hold for it to apply, and indicate if the assumptions indeed do hold. ↑Theory: Video lectures 10, 21 & 26; Sections 2.6, 4.2, 5.2 lecture notes (for models involving time also: Video lectures 15­17; Sections 3.1, 3.2 lecture notes) Concept Quantity Role Relation Square width [m] length [m] context maximal area we take in consideration Constant Wind speed [m/s] direction [°] context context influences the horizontal distance of the water drops Equation ­ Fountain diameter [m] nozzle diameter [m] choose choose limits the maximal horizontal distance of the water drops influences nozzle velocity Constraint Equation Water pressure [kPa] nozzle velocity intermediate intermediate influences height and reach of the water drops influenced by water pressure and nozzle size Equation Equation People location [x,y­coördinates] context fountain­enthusiasts within 2 meter of the fountain, other people outside of that area, limits water reach Constraint Water drops height [m] vertical speed [m/s] horizontal speed [m/s] distance (or reach) [m] maximum height of the water influences the horizontal distance influenced by water pressure, nozzle velocity and gravitational force influenced by the wind speed if it exceeds the boundaries, then it should limit the water height Equation Equation Equation Equation Terraces & Restaura
nts location [x,y­coördinates] context limit water reach Constraint intermediate intermediate intermediate asked 11. Approximations and Assumptions Most often, you don’t know exactly how things relate, you only have ideas about approximate relationships. In this item, you list the most important estimates in your model. The choice to leave out a particular entity from the modeled system is also an example of an estimation. For instance: we decide to leave out the effect of moonlight in the street illumination model. Approximations Fountain The dimensions: o The radius is 3 m. This is because 6 meters as a diameter is an impressive enough size for a fountain and to catch falling droplets o The height of the water reservoir is 0,6 m. When it is 0,6 meters it i just high enough for children to watch but not fall in and it looks relatively beautiful with the fountain jet. The jet looks even more high. o
When the fountain reaches it’s highest point, the fountain is 10 meters high. Assumptions Fountain The fountain has a circular shape. To simplify the model, with a circular shape we don’t have to take into account the wind direction. The route of the droplet: o First straight up because of the water pressure, perpendicular to the ground. o When on top of the jet, the droplet tends to make a half parabolic route. o When the droplet is on the end of the parabolic route, it has a downfall till it is on the ground, with a (route)line dependent of the wind force and direction. Due to the distant that the restaurants have from the fountain, we do not take them in consideration for getting wetted by the fountain. The height of the pump of the water jet is equal to the ground level. Wind ●
●
●
The wind only blows in a horizontal direction. o The wind blows in a horizontal direction, because this simplifies the model and this could be similar to a realistic situation. It does not matter in what direction the wind blows, because the fountain is a circle, so the wind has in every direction the same influence. The wind speed is measured by an anemometer that is placed on the roof of a restaurant. With that, we assume that the wind speed on the roof is the same as the wind speed at the fountain. The anemometer is placed on the roof, because it is not visible for the tourists there. Therefore, the fountain is more attractive. New assumptions & estimations ●
Route of the droplet: the droplet leaves the nozzle of the fountain in a completely vertical direction. As soon as the droplet leaves the nozzle its horizontal velocity starts to increase due to acceleration caused by the wind force acting on the droplet. ● We estimate that the average size of a water droplet is around 3 mm in diameter. ● We assume that the droplet is spherical. This is very close to the reality and makes the calculations a lot easier. In reality: Small droplets up to 1 mm are completely spherical. As they become larger (2­3 mm), droplets become more bun­shaped until they fall apart at around 4.5­5 mm in diameter. [1] ● g = 9.81 m/s2 ● The temperature is 20 degrees celsius, because this is a pleasant temperature for watching a fountain and being outside. ● The air drag forces acting on the droplets when they travel vertically up or down can be neglected because are very small (compared to the gravity) due to their small size and low speed . This way the formula does not become o​
verly complex and still remains accurate. 12.1 Initial derivations We first need to determine the time (t) a droplet needs to travel from the maximum height (h) to the ground. This time is the same as the time a droplet needs to travel from the maximum height to the ground when the wind influences the droplet. This is because the time only depends on the gravitational force. t = √2h/g The distance (d) of the droplet to the middle of the fountain cannot be larger than the radius of the fountain. Otherwise, the tourists will get wet. The distance d depends on the speed of the wind (v​
) and the time t. The wind speed is variable and is measured by the w​
anemometer. d = vw * t d = r We need to find the relation between the maximum height and the radius. Through substitution, we find the following formula: r = vwind * √2h/g We can rewrite this function to a function where h depends on v​
w h=
g*r2
2*vwind2
With this formula we can determine what the maximum height of the fountain can be with different wind speeds. 12.2
Second derivations Since the results obtained using our first derivations were very unrealistic, we revised the original model. We found that the biggest flaw was that the droplet would immediately get the speed of the wind. In this model this is solved by calculating the acceleration of the droplet cause by the wind. So now the acceleration is gradual and realistic instead of instant. The radius of the fountain is the maximal horizontal distance that a droplet of water can travel. So rfountain
=d=
where t
and 1
2
2
*a*t √
= time to go down f rom hmax =
a = adroplet =
F wind
mdroplet
2*hmax
g
h
This can be rewritten to:​ max
=
mdroplet*g*rfountain
F wind
where mdroplet = rdroplet2* ρwater * 43 π and F wind =
1
2
* ρair * π * rdroplet2 * C * vwind2 When we now fill in the following values → g = 9.81 m/s2 rfountain = 3.0 m rdroplet = 15 * 10−4m C = 0.47 ρair = 1204 kg/m3 (at 20°C) ρwater = 1000 kg/m3 We find And mdroplet = 9.42 * 10−3 kg −3
F wind = 2.0 * 10
* vwind2 So
hmax =
9.42*10−3*9.81*3
2.0*10−3*vwind2
New formula: hmax
=
2.77*10−1
2.0*10−3*vwind2
13. Special Cases We found it interesting to look into special cases regarding the wind speed. A special case that caught our attention, is when the wind speed goes up to infinity. Wind speed = vw ​
= ∞ → h = 0 We see that the maximum height of the fountain goes up to zero. 14. Estimates The width of the fountain is 6 meters, so the radius is 3 meters. A droplet has a maximum reach of the radius of the fountain, so the droplet has a maximum reach of 3 meters. The maximum distance of the droplet is then 3 meters. To derive the maximum height of the jet, the wind speed should be close to zero. So maximum height ​
hmax
​​
when the wind speed is close to 0. To derive the maximum height we can use a limit that gets close to zero. hmax =
lim =
x→0
mdroplet*g*rfountain
F wind
9.42*10−3*9.81*3
x
= ∞ When the wind speed is close to 0, the maximum height of the jet is infinitely large. This is not a realistic estimate, so instead of using the value 0 as a wind speed, we can use 1.0 m/s as wind speed. In the maximum height formula this yields: hmax =
mdroplet*g*rfountain
F wind
= 138.6 m So the maximum height of the jet with a wind speed of 1.0 m/s is 138.6 m. We did some research on the average wind speed in The Netherlands and we found a chart of the average yearly wind speed in the month March of all different places in The Netherlands. [2] The results are: A minimum average wind speed of 3.5 m/s and a maximum average wind speed of 6.0 m/s. To take the average wind speed of the whole country, the average wind speed is (6.0+3.5)/2 = 4.75 m/s. The maximum height of the jet with a wind speed of 4.75 m/s is: −3
2
*9.81*3
4.51*10−2
hmax = 9.42*10
= 6.14 m To indicate a range of wind speeds from soft winds to harsh winds we use the scale of Beaufort. [3] Number 6 on the scale of Beaufort is a condition of a strong breeze where large branches are in motion, use of umbrellas becomes difficult and empty plastic bins tip over. We take this number 6 scale as the maximum wind speed, which is a wind speed of 13.8 m/s. So we use a wind speed range of 1.0 m/s to 13.8 m/s, or you can also say, a wind speed range of scale 1 to 6 of the scale of Beaufort. Execution 15. Rephrase the problem statement in formal terms v ​
h ​
is as high as possible when r ​
is 3. When h i​
s high the fountain is the most attractive to the tourist. Quantity w should be as small as possible, so 16. Calculations / Implementation / Simulation v​
w [m/s]
​
hmax NEW [m] hmax OLD [m] 1 138.6 44.145 2 34.64 11.036 3 15.4 4.905 4 8.663 2.759 5 5.545 1.765 6 3.85 1.226 7 2.829 0.9 8 2.166 0.689 With the new formula, the results of the model are higher. This means that the height of the water jet can be much higher than in the first model. The air drag has a big influence on the height of the water jet. This item contains the required calculations to achieve your model purposes. This may involve implementing one or more formulas in a computer program, or filling in expressions in a spreadsheet. It also may mean that you will be ‘playing’ with a simulation – where, of course, playing is directed at finding an answer to je modeling purpose. Document all non­trivial choices (for instance: ‘since we cannot solve f(x)=0 in closed form, we use the bisection method [4], see module SOLVE_IT, lines 23­28’). 17. Validation and Verification; Accuracy and Precision As stated before, we assume that ​
when the droplet is influenced by the wind, the droplet immediately goes horizontal and that it makes a parabolic route. ​
However, in reality the droplet will first be blown obliquely upwards by the water pressure and obliquely by the wind. The droplet makes a longer parabolic route than we assume in our model and it therefore will hit the ground further away than we can calculate. The change of the height of the fountain with different wind speeds seems a bit strange in our opinion, because it varies widely. The height of the fountain at for example a wind speed of 1 m/s is 138.6 meters and the height of the fountain at a wind speed of 2 m/s is 34.64 meters. The difference between these heights is very big and therefore we do not necessarily believe that this is true. Of course, the higher the wind speed, the lower the height of the fountain, but we expected it not to be this kind of difference. Conclusion 18. Presentation and Interpretation Data Visualization is important for communicating our model. In order to create a clear presentation of our model, we chose to make a graph and a table. The table consists of the wind speed and the exact numbers for the height of our fountain regarding the wind speed. We also put those outcomes of our calculations in a graph, which shows the relation between the wind speed and the fountain height. We chose this graph, because you can immediately see how the wind speed influences the height of the fountain. The higher the wind speed, the lower the height of the fountain will be. In our daily life, we determine the wind scale with the scale of Beaufort. [3] Bft Designation Wind speed in m/s 0 Wind still < 0.2 1 Weak 0.3 – 1.5 2 Weak 1.6 – 3.3 3 Average 3.4 – 5.4 4 Average 5.5 – 7.9 5 Quite powerful 8.0 – 10.7 6 Powerful 10.8 – 13.8 7 Strong 13.9 – 17.1 8 Stormy 17.2 – 20.7 9 Storm 20.8 – 24.4 10 Heavy storm 24.5 – 28.4 11 Severe storm 28.5 – 32.6 12 Hurricane > 32.6 With our new formula, we always maintain a spectacular height of our fountain for the tourists. On a day with a scale of Beaufort rate of 3, wind speeds up until 5.4 m/s, the fountain will have a maximum of 4.75 meter, which is just spectacular enough to look at In our previous calculations, the height of the fountain would have a maximum of 1.51 meters with a scale of Beaufort rate of 3. It’s clear that in our new calculations, where the air drag was taken into consideration, has a more realistic outcome. Reflections and Discussions 19. Discussion after the Conceptual Model Fountain The fountain has a circular shape. This assumption simplifies our model and is also realistic because most fountains are circular. the dimensions: o The radius is 3 m. This assumption is realistic. o The height of the water reservoir is 0,6 m. This assumption is realistic, but irrelevant for our model. The route of the droplet: o First straight up because of the water pressure, perpendicular to the ground. true o When at the top of the jet, the droplet tends to make a half parabolic route. Not entirely realistic, because when coming closer to the top of the jet, the wind starts to have an increasing influence on the droplet. The semi­parabolic route starts sooner than in our model. In our model, we assume that when the droplet is influenced by the wind, the droplet immediately goes horizontal and that it makes a parabolic route. However, in reality the droplet will first be blown obliquely upwards by the water pressure and obliquely by the wind. Therefore, the droplet makes a longer parabolic route than we assume in our model. You can also see this in figure 4 and 5. Because of this, the droplet in our model will come less far than it would go in reality. A real fountain can therefore not be as high as the fountain in our model. We assume that the difference of the distance that the droplets will travel is negligible, in order to simplify our model. Figure 4: The route of the droplet in reality Figure 5: The route of the droplet in our previous model o When the droplet is on the end of the parabolic route, it has a downfall till it is on the ground, with a (route)line dependent of the wind force and direction. This slightly simplifies the model whilst still being realistic, so that we can work with it. Due to the distance that the restaurants have from the fountain, we do not take them in consideration for getting wet by the fountain. This simplifies our model and it is a realistic assumption, because otherwise everything would get wet. and the height of the fountain would not be attractive anymore The height of the pump of the water jet is equal to the ground level. The pump height of the water jet is equal to the level of the ground. This is a realistic assumption. Wind ●
The wind only blows in a horizontal direction. This assumption simplifies the model and it is a realistic assumption, because the square is small enough to consider the wind direction as horizontal. ● It does not matter in what direction the wind blows. This assumption simplifies the model. It is also realistic because the fountain is round and the pump is in the middle so it does not matter in which direction the wind goes, the droplets will always travel the same distance relative to the pump. ● The wind speed is measured by an anemometer that is placed on the roof of a restaurant. With that, we assume that the wind speed on the roof is the same as the wind speed at the fountain. Because of the possible height difference between the roof and the fountain, an anemometer placed at the fountain may detect a different wind speed than one on the roof. The anemometer on the roof may detect higher wind speeds than an anemometer placed close to the fountain. It would be more precise to place the anemometer as close as possible to the fountain. If the anemometer would be placed on the roof in the real situation, the maximum height would be lower because the anemometer detects a higher wind speed. In that way, the height of the water jet can be higher, without wetting the tourists and is therefore more attractive. 20. Discussion after the Formal Model Since the results obtained using our first derivations were very unrealistic, we revised the original model. We found that the biggest flaw was that the droplet would immediately get the speed of the wind. In this model this is solved by calculating the acceleration of the droplet caused by the wind. So now the horizontal acceleration of the droplet due to the wind is gradual as in reality instead of instant as in our previous model. A list with the new assumptions along with a short explanation can be found under the original assumptions. 21. Discussion after the Result The results with our initial formula where not realistic, at an average speed the fountain did not even reached a height of 2 meters. That is why we have changed the initial formula into a more realistic formula. We have taken the wind force into account in this second formula. Now the results that are shown are valid and plausible. Because we changed the formula and the uncertainties are taken into account the answer on the main question is valid. 22. Discussion after the Solution of the Initial Problem We asked ourselves what the height of the fountain can be, keeping it as attractive as possible, without wetting the tourists in any wind condition. Out of our calculations we have derived a maximum height with the highest wind speed and a maximum height with the lowest wind speed. Therefore we can conclude that we answered the question of our initial problem. This completes the actual modeling assignment, the report with items 6 – 22 is intended for a hypothetical problem owner. Reflection on the Assignment 23. Extension A less simplified model which is more realistic. We already started working on a new model which also takes air resistance into account when going up and down but we do not have enough time to finish the formula. There are too many unknown variables. Also we would like to extend the model so that we can make a distinction between average water drops and water spray. The water spray gets carried away much further by wind than the normal droplets because of their lower weight. 24. Necessity for Improvement The model can be improved with respect to several criteria. First of all, the criteria specialization could be improved. Now, the model is not very well explained for normal people who are not specialized. So the tourists, who are an important stakeholder group, maybe cannot always understand the way the model is created. The model addresses a big intended audience. The model addresses the administrator of the fountain, because with this model he knows how high the fountain should be with a various of wind speeds, the restaurant owners, because they get an indication of how many tourists will visit the fountain (with a high wind speed, less tourists and with a low windspeed, more tourists). However, there is a negative point of our model in this criteria, because the model addresses a big audience, the specialization should be low, so everyone can understand it, but in our model this should be improved. Also, the criteria convincingness could be improved. Our assumptions are not always logically deducible from other, less problematic assumptions. Also, not all of our assumptions are based on first­principle ‘laws’. We constructed a simplified formal model system. We can improve the model, by making the assumptions more realistic. Finally, the impact of our model is not really big. Tourists now know how high the fountain will be that day, so the model impacts the amount of tourists at the square. So it indirectly impacts the profit of the restaurants. 25. Possibilities for improvement The criteria specialization can be improved. First, several (randomly chosen) tourists should read the model. Everything they do not understand should be formulated and presented in a different way. We think that we will not have troubles with biased users. Tourists (and with that also restaurant owners) want to see a fountain that is as high as possible, but they also want to stay dry, so the purpose of the model is exactly what the people need. The criteria convincingness could be improved by means of the assumptions. If the assumptions were based on first principle laws or logically deducible from other, less problematic assumptions. For example, we assumed that all of the droplets are perfectly round and have the same proportions. However it should be more realistic if the droplets vary in shape and proportions. Also, the assumption about the measuring of the wind speed. We assume that the wind speed on the roof of the restaurants (where the anemometer is placed) is the same is the wind speed at the fountain. However, the wind speed at the rooftop is higher than the wind speed at the fountain, so the air drag is also different, so the result of the model will be different.[5] If we do not neglect the difference in wind speed at the rooftop and the fountain the model would be a lot more realistic and thus a lot more convincing. We already added the air drag. We first made a model where we assumed that the air drag was negligible. After that we discussed that the model was not convincing enough, so we added the air drag, because of that the model already a lot more convincing. 26. What aspects of your work are you proud of? We are proud that we managed to incorporate the air drag in our new formula. We were not satisfied with the unrealistic outcomes of the first model. When we made the model with the air drag, it had much more realistic outcomes. When we made the model wíth the air drag and it had much more realistic outcomes, we were satisfied that the work we did resulted in a better working and more realistic model. That made us quite proud of ourselves. Next to this, it was not a small effort to change almost every calculation with the previous formula with the new formula, incorporated with air drag. Even though we had not much time to adjust all this in our report, we still managed to do this before the deadline. We are also proud of the fact that we made a model with our group, while none of us had any experience with modeling. 27. What have you learned? We have learned how to make a model, set up formulas, through observation of the problem statement/ realistic situation and knowledge of the course Physics. We have learned to look at a problem in a different perspective. We looked at all the parts of the problem, took everything in consideration and decided how to handle the different components. By means of almost all the lectures we learned that we can tackle problems in a better and more convincing way. Appendices 28. Used Literature [1] Bad Meteorology: Raindrops are shaped like teardrops. (date unknown). Retrieved from: http://www.ems.psu.edu/~fraser/Bad/BadRain.html [2] Koninklijk Nederlands Meteorologisch Instituut. (date unknown). Klimaatatlas; Langjarig gemiddelde 1981­2010. . Retrieved from: http://www.klimaatatlas.nl/klimaatatlas.php?wel=wind&ws=kaart&wom=Gemiddelde%20wind
snelheid [3] Beaufort Scale. (2016, April 1). Retrieved from: https://en.wikipedia.org/wiki/Beaufort_scale​
. [4] Bisection Method. (2016, January 8). Retrieved from: http://en.wikipedia.org/wiki/Bisection_method​
. [5] Swiss Federal Office of Energy. (date unknown). Wind Profile Calculator. Retrieved from: http://wind­data.ch/tools/profile.php?lng=en 29. List of Definitions t
h​
max g
d
v​
wind
r​
fountain
r​
droplet
m​
droplet
F​
wind
ρ​
water
Time Maximum height of the droplet in meters 2 Gravity in m/s​
The distance of the droplet to the middle of the fountain in meters The wind speed in m/s Radius of the fountain in meters Radius of the droplet in meters Mass of a droplet The force of the wind on the droplet in Newton Density of water ρ​
air C​
air
a​
droplet
Density of air The drag coefficient of air Acceleration of the droplet 30. List of Illustrations [1]​
Here, we mean: relations following from the correspondence between the model and the modeled system, or relations taken from the theory as used in the derivation of your model. We don’t mean the mathematical derivations you do yourself, in order to fulfil the purpose; these occur in item 12. [2]​
An example of a ‘guess’ is: ‘I assume that the relation between the price of this product and the number of sold copies is given by the logistic function (​
http://en.wikipedia.org/wiki/Logistic_function​
) with quantities … because ... . For these guesses, you may want to use the Relation Wizard or the Function Selector.