Mathematics AA SL

Sample Internal Assessment

Table of content

Rationale

Aim

Introduction

Process of calculation

Comparative analysis

Conclusion

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Last year we visited Dubai. I was so excited to visit a place which has already set a benchmark for splendid engineering edifice. One of such engineering marvels is the fountain in Burj Al Arab. Reading articles and watching videos about it, gave me an apparent idea about the design as well as it’s working principle. But watching it right in front, was an outstanding experience. Out of every other magnificent structure, the rhythmic fountain caught my attention the most. Besides its relations with Physics and Mathematics made it more interesting.

Every day whenever I turn on any tap in washroom or basin, the way water flows is quite same as that the Dubai fountain. I have seen fountains with similar flow of water in my hometown as well. However, the water flow in Burj Al Arab was completely different. The water in the fountain seems like rods made up of plastic. There was not a single drop going out of the flow. This let out my inquisitive self and made me question about it’s working fundamentals.

Getting back to my research on flow of water, I came to know about the different types of water flows and their factors.

I was intrigued. I read multiple research papers, articles and books to know more about it. I have done my research on fluid mechanics from different books and journals. I have gone through a few research journals on the construction of Burj Al Arab and the fountain as well. Though I have gone through a large number of articles in the internet, I was unable to find the answer of the exact values of different parameters required for such flow of water in fountain. Another motive of this study is to derive the values and try to recreate a prototype of the fountain in my hometown. I did not get answers to all my questions in any of the writings and I am very interested to know about its scientific principles.

This is the reason I am doing this IA on the same.

The main motive of this IA is to find the range of terminal velocity of water so as to ensure the streamlined flow without disruption. The velocity of water is the key factor which guarantees the rod-like shape and polished texture. Based on this velocity, minimum and maximum range of this streamlined flow of water must be determined.

Burj Al Arab is Dubai’s one of the symbolic hotels. It has got astonishing features and design. One of such wonders is the water fountain. The graceful undisrupted flow makes it one of a kind. It’s a cascade fountain which looks even more astonishing because of the factors around it. One such factor is the pitch perfect integration of the nature of architecture with the fountain. Another key factor is the lighting. But the most important department is the control unit which regulates the timing and synchronization of the water flow. This department is also responsible for controlling the lights and any other visual effects. Although the velocity of water is held as the key factor, but the contribution of proper setup and additional factors around this engineering splendor is undeniable. A few other important attributes are high pressure water pumps, proper waterproof tiles and good software support.

Now, coming to the science behind this structure, we often see that the water which flows out of a pipe is amorphous. But in case of the fountain of Burj Al Arab, it’s streamlined or laminar. This is due to a few important factors, i.e. velocity and pressure. The velocity and pressure of the water helps it to retain its tube-like shape even at its flight after the point of release. Thus, it’s quite obvious that these parameters are not guessed to be of a certain value. The referential value for judgement is Reynolds number. Reynolds number is the ratio of the force of momentum of the mass of a flowing fluid with its viscous force. The range asserted in Reynolds number helps in the judgement of velocity to maintain the streamlined flow of water. But in this case, the velocity as well as the pressure values are unknown.

According to Reynold, critical velocity can be calculated by –

\(R_{e} = \frac{inertial \,force}{viscous \,forces}\)

It can also be written as,

\(R_{e} = \frac{ρuL}{µ}\)

where, R_{e} is Reynolds number

ρ is the density of fluid

u is the flow speed

L is the characteristic linear dimension

µ is the dynamic viscosity of the fluid

To calculate those values, Bernoulli’s principle is extremely helpful. It is based on the law of conservation of energy. It states that, “Within a horizontal flow of fluid, points of higher fluid speed will have less pressure than points of slower fluid speed.” Using Bernoulli’s equation, we can calculate the velocity.

Another important concept is projectile motion. The water constantly flows in projectile motion and thus knowing the angle of elevation and range of the path is very important. Understanding the factors behind projectile path may help us to construct such engineering prototypes with surer chances of success.

Knowing the range of the flow of water is extremely important to ensure that the control unit always keeps the flow well within limit so as to avoid any mishap under any circumstances. To calculate the range of the flow, knowledge of projectile motion is very important. Projectile is only subject to acceleration due to gravity and follows a curved path from its point of release.

By law of conservation of energy,

W1 = P1A1 (v1∆t) = P1∆V

where, W_{1} is the initial work done

P_{1} is the initial pressure

A_{1} is the initial cross-sectional area

v1∆t is the initial distance covered from B to C

Also, considering the exit,

W2 = P2A2 (v2∆t) = P2∆V

where, W_{2} is the final work done

P_{2} is the final pressure

A_{2} is the final cross-sectional area

v_{2}∆t is the final distance covered from D to E

Therefore, to find the total work done,

W1 – W2 = (P1 - P2) ∆V

Taking into consideration, the fluid density as ρ, mass in the pipe as ∆m and time interval ∆t-

∆m = ρA1 v1∆t = ρ∆V

Now calculating change in the gravitational potential energy ∆U,

∆U = ρg∆V(h2 - h1)

where, g is the acceleration due to gravity

h_{1} is the initial height

h_{2} is the final height

Similarly, by calculating the change in kinetic energy ∆K,

∆K = \(\frac{1}{2}ρ∆V(v\ ^2_2-v\,^{2}_{1})\)

Since the prime objective is to find the velocity of water in the fountain. In order to achieve it by using the known parameters, Bernoulli’s equation can be used.

Applying work-energy theorem,

(P1 - P2)∆V = \(\frac{1}{2}\)ρ∆V\(\biggl{(}v\,^{2}_{2}-v\,^{2}_{1}\biggl)+\) ρg∆V(h2 - h1)

Rearranging the equation,

P1 + \(\frac{1}{2}\)ρv \(^{2}_{1}\) + ρgh1 = P2 + \(\frac{1}{2}\)ρv \(^{2}_{2}\) + ρgh2

Thus, the general equation can be written as,

P + \(\frac{1}{2}\)ρv^{2} + ρgh = constant

Now putting the values,

P + \(\frac{1}{2}\)v^{2} + 1 × 980 = constant

As pressure and velocity both are variables and both can be changed independently. Thus, in this case, both are significant driving factor for the streamline flow.

Now, after calculating the value of velocity, it must be ensured that whether it satisfies the condition for being streamlined. To check that, one must make use of Reynolds number. Reynolds number helps to predict the flow of fluid with a set of parameters. There is a fixed threshold which marks the zone for the flow of a fluid to be either streamlined or turbulent or in some cases a mixture of both.

On putting the value of the velocity of water, Reynolds number can be calculated. The threshold for the flow of water to be streamlined is 2000. Thus, velocity and pressure must be at par to maintain the undisrupted flow of water.

\(R_{e}=\frac{ρuL}{μ}\)

For the case of the fountain in Burj Al Arab, the values of the above parameters are mentioned below:

ρ = 1 gm/cm^{3}, L = 1 cm, μ = 1 dyne. cm^{-2}.sec^{6}

Putting the values-

2000 = \(\frac{1×u×10}{1}\)

So, velocity is-

u = 200 cm/s

**Derivation using concepts of solving simultaneous equation:**

Using displacement-time equation and the parametric values, we can write,

S_{y} = u_{y} t + \(\frac{1}{2}a\,_yt\,^{2}\)

=> 0 = u × sin θ × t - \(\frac{1}{2}gt\,^{2}\)

=> (2u × sin sin θ × t - gt^{2}) = 0

=> (2u × sin sin θ - gt) × t = 0

∴t = 0

∴2u × sin sin θ - gt = 0

=> t = \(\frac{2u\,×\,sin\,sin\, \theta }{g}\)

This implies, at these two-time instances, the vertical displacement will be zero. Thus, the total time of flight will be:

\(T=\frac{2u\,×\,sin\,sin\, \theta }{g}\)

Now using equation (2) and the parametric values, we can write:

s_{y }= u_{y }t + \(\frac{1}{2}a\,_{y}t\,^{2}\)

=> R = u_{x} T + \(\frac{1}{2}×(0)×T\,^{2}\)

Here, R is the range and T is the total time period or total time of flight.

∴R = u_{x}T

=> R = u × cos cos θ × \(\frac{2u×sin\,sin\,\theta}{g}\)

=> R = \(\frac{u^{2}×2sin\,sin\,\theta\,cos\,cos\,\theta}{g}\)

=> R = \(\frac{u^{2}×sin\,sin\,2\theta}{g}\).......(3)

Now on differentiating the equation no (3) with respect to ,the value of angle of projection for which the range will be maximum can be calculated.

\(\frac{dR}{d\theta}=\frac{d}{d\theta}(u\,^{2}×sin\,sin\,2\theta)\)

=> \(\frac{dR}{d\theta}=u\,^{2}×\frac{d}{d\theta}(sin\,sin\,2\theta)\)

=> \(\frac{dR}{d\theta} = u\,^{2}×2×\, cos\,cos\,2\theta\)

=> 0 = 2u^{2 }× cos cos 2θ

=> cos cos 2θ = 0

=> 2θ = \(\frac{\pi}{2}\)

=> θ = \(\frac{\pi}{4}\)

Therefore, for maximum range, the angle of projection should be \(\frac{\pi}{4}\)or 45°.

So, by putting the value,

R = \(\frac{200\,^{2}×sin\,sin\,(2×45°)}{980}\)

R = 40.81 cm

Thus, the maximum range for velocity 200 cm/s is 40.81 cm.

The angle of elevation is derived to be 45° for having maximum range.

By keeping the angle fixed at 45° maximum range can be attained. This gives better control on the streamlined flow because the control unit needs to keep only pressure and velocity at parity to restrict it within 2000 and need not concentrate on the angle once the structure elevation is locked. According to the concept of Reynolds number, the range from 0 to 2000 is the laminar zone. In this range, the flow is streamlined. The region between 2000 and 3000 is an unstable zone in which the state constantly changes from laminar to turbulent and vice versa. And the flow beyond 4000 is turbulent.

In this IA we have studied about the mathematical and scientific backbone of a praiseworthy engineering design. Most of the time we admire the beauty of a magnificent structure but fail to look into the scientific pillars behind such wonders. The fountain of Burj Al Arab continues to be awestruck me but after completion of this IA, I can understand the principles and laws behind the rhythmic ejection of tubular water.

Reynolds number is the most important factor to ensure streamlined flow. Another key factor that we came across was the angle of elevation. It plays a major role in the range if water flow. The threshold provided by Reynolds number has made it possible to get an idea about the velocity and pressure of the water. Since, a limit of 2000 must be maintained, the velocity and pressure should be at parity. Both are together responsible for maintaining the stability of the shape of water at flight and also maintaining the distance at within range. We have calculated the angle of projection to be 45°, these leaves us with the rest of the values to be controlled by the software and control unit.

A few other factors which must be taken into account is the moisture control facility for the entire system. The accuracy of the water pumps to generate feedback after receiving instructions from the software unit should be up to the mark. Also, if we closely observe the fountains, we can conclude that it runs on a definite pattern. These patterns are generally set by loops in programs. Thus, proper care must be taken to resolve latency issues in case of complex manipulations of the flowing water. This will also ensure that irrespective of the complexity of the behavior of the fountain, when integrated with light and other visual effects, the laminar flow remains flawless.

Thus, in a nutshell, we can infer that although there are a lot of factors for the effortless motion of the water flow in the fountain of Burj Al Arab, the velocity and pressure parameters to keep Reynolds number within limit is the most important factor for the streamlined flow of the fountain.

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- Mathematics by RS Aggarwal – Class 11