PowerPlantCoolingTowerIntegral CalculusProject 3B :
PowerPlantCoolingTowerIntegral CalculusProject 3B Gerardo Rodríguez
Guillermo Punzo
Sofía Cardiel
David Manríquez
Betzy Aceves Introduction… :
Introduction… A nuclear power plant generates energy in the form of heat which is converted to electricity.
The second law of thermodynamics tells us that not all of the heat can be converted to electricity (thermodynamic efficiency cannot be 100%). The unused "waste heat" must be exhausted to the environment. If the plant is near a river the waste heat can be released into the water. Waste heat can also be released into the atmosphere using cooling towers. Slide 3:
Topromote natural circulationofair, theshapeof a coolingtower at a nuclear powerplantis a“hyperbola” Thisisourtower :
Thisisourtower NOTE: For this project, the tower top diameter equals the base diameter. Slide 5:
Our tower is located just outside a small town on the East Coast of the US. We are in charge of the security system, and just before our shift ends, all the alarms start to go crazy. We see people running and screaming, and zombies behind them.
This is our chance to kick some ass! What’s going on!!!?? Slide 6:
We were the first to be captured…
We gave our lives to save our girls! : )
Gino, David and Memo. What can we do? :
What can we do? Thefirstthingthat comes toourmindsistohidepeopleinsidethecoolingtower. How many people can we hide without running out of air? For how long can they stay inside the tower? Calculus has the answer :P Slide 8:
Slide 9:
Now, we need to know how many people can fit into the tower. This can be done by using a volume of revolution. And, weobtainour formula forthecoolingtower Obtainingthevolume. :
Obtainingthevolume. Thisisthe formula wewill use toobtainhalfthevolume: We know that the radius = x. Then, we need to solve for x in the hyperbola formula: Now, we substitute this in our original formula, and solve. How many people can fit inside the tower? :
How many people can fit inside the tower? First, we get the area of the base of the tower (where the people will be). Volumeofhyperbola272π u3. Now, we need to know how many people can fit inside the tower. How manypeople can fitinsidethetower? :
How manypeople can fitinsidethetower? Now, weneedtofind out thevolumethatoccupiesoneperson. Weimagine thatwe are cylinderstokeepit simple…. Slide 13:
Weneedto know how manypeople can be in the base. 130 persons ! Slide 14:
First, wemultiplythevolumeofoneperson times thenumberofpeoplethat can fit. This is the volume that people use Thequestionis…
CAN THEY SURVIVE? FOR HOW LONG? Slide 15:
Now, wesubstractthevolumethatpeopleoccupyfromthe total volumeofthetower (thisistheairavailable).
Now, weassumethatpeopleneed 20m3ofairperdaytosurvive. Slide 16:
Wedivide the 766.96m3 availableandthe 108.33m3 ofairthatpeople use perhour. Butweneedto know how many time we’llhavebeforerunning out ofair… Volumeavailable:
766m3 Volumeneeded:
108.33 m3/hr VS. After 7.07 hourseverybodywillrun out ofair… : ( What’s next? :
What’s next? People outside the tower need to know we are inside. We need to paint the tower.
With 1l of paint we can cover 12m2.
Now, we need to know the area that we will have to paint.
If we extend the 3D figure we have, we obtain a circle.
We need to integrate the formula. Obtaining the exterior surface :
Obtaining the exterior surface We know the formula, and we solve.
This can’t be done by normal means, so we use the Simpson’s Method. Paint needed for the surface :
Paint needed for the surface If 1liter = 12m
42.13 liter= 505.58m2 Weneedtopaint :
505.58m2. The paint didn’t work :
The paint didn’t work However, after 3 days and several dead people, no one noticed the paint.
We decide to put lights to the tower. We will put lights here. Length of the cable :
Length of the cable The cable of the top of the tower will be the perimeter of the circle.
The cable that is needed on the side of the tower will be the arc length of the function.
Now, we substitute and solve. Arc length formula Arc length :
Arc length First, we derivate our original function:
Later, we substitute the derivative on the original function, and solve. Arc length :
Arc length The integration is done by Simpson’s Method.
By solving, we find that we need 24.104m of cable per side of the tower.
Now, it’s a simple sum:
We need 121.54m of cable to light the tower. The thing here is.. :
The thing here is.. No matter what we do, we will become zombies. References :
References www.lp.edu.pe/ecologia/aire/aires1c3elaire.doc
www.ecoportal.net/content/view/full/80579 Co-evaluation :
Co-evaluation Guillermo Punzo Suazo 1
Sofia Cardiel Bernal 1
Gerardo Rodriguez Monroy 1
David ManriquezBuendia 1
Betzy Aceves Muñoz 1