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Design for the Environment/Institutional Cogeneration

This page is part of the Design for the Environment course

As Toronto Medical Officer of Health Dr. David McKeown said “Air pollution is responsible for 1,700 premature deaths and 6,000 hospital admissions in Toronto each year”. Moreover, a significant portion of these impacts is due merely to the production of electricity and heat. Therefore, it is vital to understand these environmental issues and to develop more proactive designs in order to solve them. This article analyzes the environmental, societal, functional and economic impacts of three alternatives for providing heat and electricity to the University of Toronto (institutional cogeneration). The baseline alternative is for the university to purchase electricity from Toronto Hydro and produce heat by means of the already existing natural gas boilers. Additionally, two more environmentally friendly alternatives are considered, they are: the utilization of Solid Oxide Fuel Cells (SOFC) or to retrofit the existing boiler system for cogeneration by adding a steam turbine and generator to produce electricity. As a common function, each of the alternatives considered must provide 985 000 MMBTU of steam and 188 000 000 kWh of electricity per year to the university. This analysis considers the impacts of the various processes along with the acquisition of the fuels and utilities. Furthermore, it looks at the impacts of transportation of the fuels; as well as recycling of the utilities. The optimum goal of this analysis is to recommend to the client, the University of Toronto, the best option to implement among the ones analyzed.

Painter: Sir Edmund Walker Year: 1859 Source URL: http://www.fineart.utoronto.ca/canarch/ontario/toronto/uc.html]]

Project Information
Section 1 Group A2 Maby Boado Amador (boadoama) Michael Bosompra (bosompra) Joshua Sissons (sissonsj) Michael Wong (wongmi18)

Highlights and Recomendations
The main functional similarity between the three alternatives is the use of natural gas as the fuel for the generation of heat and electricity. The main difference is the process by which each of the alternatives accomplishes the required functions. Comparing the performance of the three alternatives, it is apparent that Alternative 1 has an advantage over the Baseline and Alternative 2. The whole plant efficiency of Alternative 2 is approximately 60%, whereas with just boilers (Baseline alternative), the efficiency would be around 40%. Furthermore, the efficiency of the fuel within the solid oxide cell when only generating electricity is 50%. A thermal to electric ratio of one was assumed, implying 100% efficiency. However, this is not a valid assumption, since heat will inevitably be lost in the process. Therefore, an estimated efficiency of this process would be 70%. Another important issue considered is the space required for the power plants. Alternative 1 would require the construction of a large facility since it needs 44000m2 of land for the placement of the stacks. See details below.

Figure 1 serves as a quick visual aid in comparing the three alternatives based on their Streamlined LCA. As it can be seen, the environmental impacts of the three alternatives are similar in the resource provisioning stage. The reason for this similarity is that they all use natural gas as the primary fuel. During the process implementation stage Alternative 1 ranked the highest, meanwhile the Baseline and Alternative 2 were given similar, lower scorings. The main reason for this is that the production and installation of the fuel cells require simple manufacturing techniques using fairly inexpensive materials, while the other two require the extensive use of high strength steel and other virgin metals. Furthermore, Figure 1 shows that Alternative 2 scored close to Alternative 1 for the primary process operation. The reason for this is intrinsic of an ideal cogeneration system. By using the byproduct of one process, in this case heat, as the input product of a subsequent process the total energy used can be reduced considerably. Alternative 1 has the advantage over the Baseline and Alternative 2 in terms of gaseous emissions. Carbon dioxide and methane are reduced by approximately 75%, and the NOx, SOx, CO, and particulate matter emissions are almost negligible. Complementary processes operation follows the same trend that the primary process one and as so, Alternative 1 has the highest score of the three. The three alternatives scored similarly for the end-of-life stage. The reason for this is the similarity in the materials used in the systems, mainly metals, which are mostly recyclable. In addition, since the fuel burned is natural gas, all pipes, vessels, tanks, etc are fairly clean. Based on the Streamlined Life Cycle Analysis the usage of Solid Oxide Fuel Cells has the least environmental impacts with a matrix total of 70, while the Baseline and Alternative 2 had totals of 53.5 and 62, respectively. See details below.

The EIO-LCA model is another means of estimating environmental impacts by dividing the process into different sectors and determining the burden releases per unit of economic activity. Figure 2 and Figure 3 serve as quick visual aids in comparing the three alternatives based on their EIO-LCA model results. Totals regarding Global Warming Potential(GWP), SO2, CO, NOx, VOC, PM10 and toxic releases are compared among the alternatives. Alternative 2 seems to be the worst polluter regarding Global Warming Potential emissions (7,240,884 MTCO2E). The largest portion of this total comes from the process of burning and distributing natural gas, which releases large amounts of CO2 into the environment. On the other hand, Alternative 1 produces far less GWP than the other two alternatives, which explains why Solid Oxide Fuel Cells are being researched so thoroughly, but releases the most toxic releases out of the three alternatives. The extraction of raw materials accounts for almost all of the toxic releases, due to the large amounts of residue from mining the metals the Solid Oxide Cells are made of. The Baseline Alternative releases the most conventional pollutants. The largest contributor to this result is the generation and distribution of electricity. Once again, Alternative 1 has the least emissions. For this alternative, most of the pollutants come from the natural gas processes; however, the PM10 are split between the natural gas processes and the extraction of raw materials of the system. Overall Alternative 1 seems to be the more environmentally friendly and this agrees with the Streamlined Life Cycle Analysis. However, the Baseline and Alternative 2 seem to be similar, since Alternative 2 has higher GWP and the Baseline alternative has higher toxic releases and conventional pollutants. See details below.

Figure 4 summarizes the economic analysis of the three alternatives and it divides the total cost into its respective areas of occurrence. Reoccurring costs for the Baseline Alternative are the purchasing of the electricity from Ontario Power Generatin(OPG), for Alternative 1 the acquisition of the new fuel cells every three years and for Alternative 2 are minor things such as control system upgrades. This reoccurring cost is where most of the cost of the Baseline alternative comes from (approximately $ 171 million), since this alternative is already implemented and has no capital cost. Natural gas needs to be purchased by every alternative but by different amounts. The Baseline alternative uses much less fuel than the other two, approximately 40% less than Alternative 1, since it only uses it to produce heat. Furthermore, Alternative 1 consumes much less fuel than Alternative 2 in the use phase, approximately 60% less, since it does not have a turbine. It is evident now that the bulk of the cost of each alternative come from different sources, the Baseline’s comes from the purchase of electricity ($ 171.91 million), Alternative 1’s comes from the implementation of the process ($ 227.88 million) and Alternative 2’s comes from the larger amount of fuel consumption ($ 246.82 million). Overall Alternative 1 is the pricest. See details below.

Alternative 1 is the one that has the least negative effect to the community surrounding the University, while the Baseline and Alternative 2 would be ranked equally. The main advantages of the Solid Oxide Fuel Cell cogeneration system are the low noise and emissions levels of the process. The lower greenhouse gas and toxic release emissions would improve the air quality of the University’s community and would contribute to maintain the aesthetics of the city of Toronto by reducing smog and odours. These lower emissions would also improve the health of the population since the prolonged exposure to increased NOx levels could affect people with asthma or bronchitis. See details below.

Based on the above shown decision-making matrix (Table 1) a Solid Oxide Fuel Cell power plant would be the best option for the University of Toronto. However, other issues, primarily the availability and high cost of this alternative, must be considered. Nowadays, this technology is only generating 250 MW per year and U of T requires 476 602 MW. Furthermore, the cost for producing energy with this technology is approximately $850/kW, which results in around $ 440.18 million over its entire life. Just over half of this total amount would be spent during the implementation of the process ($ 227.88 million). However, what make this alternative feasible are the fast improvements on its technology and efficiency. It is expected that by 2010 it will break the barrier of $400/kW. On the other hand, the University of Toronto’s annual budget shows an operating surplus average of $20 million per year. This outstanding total could be used to implement this technology in a span of 10 years after this alternative becomes economically viable. It is important to highlight that Alternative 2 is better than the Baseline Alternative since it has higher efficiency (60% and 40% respectively) and according to the SLCA less environmental impacts. Furthermore, the University would not be dependent on external parties for the acquisition of electricity and would not be influenced by market price fluctuations. Alternative 2 also has some advantages with respect to Alternative 1. For instance, Alternative 2 could be implemented in the same boiler house that the University already owns, rather than needing another building, which saves money and time. However, it is not advisable to switch to Alternative 2 just yet since Alternative 1 is predicted to be commercially viable in less than 5 years.

After careful analysis it was decided that the final recommendation for the power generation of the University of Toronto is to maintain its current technology (Baseline Alternative) and to implement gradually the Solid Oxide Fuel Cell power technology (Alternative 1) as it becomes commercially available and economically suitable.

Baseline Alternative
Functionally, the baseline consists of two parts; firstly, purchasing electricity from Toronto Hydro, and secondly heating the university via a boiler system like the one shown in the adjacent diagram.

Alternative 1
The Solid Oxide Fuel Cell (SOFC) utilizes cathodes, electrolytes, anodes, and interconnects to electrons. The Solid Oxide Fuel Cell system consists of three main sectors, the balance of plant (heater, control system, etc), the balance of stack (seals, electrical, pressure vessel, etc), and the stack structure (fuel cell). The advantages of the system include waste heat, various fuel compatibility, no moving parts, and lowered emissions. However, the system requires a slow start up time and is not yet commercially viable. The fuel cells are to be replaced every three years due to degradation. The thermal to electric ratio of the system is assumed to be one, so the heat energy generated will be equal to the electric energy. Natural gas will be used in the analysis to generate 813 322 MMBTU of electricity at 50% efficiency.

Alternative 2
Alternative 2 involves retrofitting the existing boiler system to include a 35 MW gas turbine-generator linked to four boilers. AC power is generated simultaneously with steam as hot exhaust gases from the gas turbine are collected in a duct burner attached to a boiler. Oxygen-rich gas allows additional fuel gas to be burnt in the boilers, thereby increasing the steam capacity and increasing the overall efficiency of the system. To further increase the plant’s efficiency, heat exchangers, economizers and a percotherm are used in the heat recovery system. The steam created flows to most of the buildings on the St. George Campus where it is used in radiators, chillers, etc. The main advantage of this system is that it utilizes the existing boiler house on campus and is able to contribute to the full electricity load. Disadvantages of the system are that it requires a lot of natural gas and contributes to direct emissions on campus. The following figures outline the overall process.

Baseline Alternative
The lowest scores over all were during the primary process life stage and specifically due to the energy use during this stage. The reason for this is two fold, firstly, the baseline boilers do not have any cogeneration implemented. Thus the heat from the boilers escapes unused. Secondly, the relative size of the primary process equipment used to generate electricity necessitated the low score. The fact that energy use turned out to be the most critical environmental stressor is not surprising considering that the fuels are all virgin materials, and that the generation processes and equipment are large in scale.

Alternative 1
The resource provisioning involves the extraction of natural gas, metals, plastics, and other materials required for the system. The extraction of the metals will result in solid residues, acid mine drainage and other toxic liquid residues, and greenhouse gases. Energy is also required for all resource provisioning processes. The implementation of the system will require lots of energy for manufacturing. The carbon dioxide and methane emissions from the primary process are reduced to 75% of emissions from the national grid and heat from a natural gas turbine, while NOx, SOx, CO, and PM10 are almost negligible. The majority of the machinery is made from steel, and therefore can be recycled, but the fuel cells are made of a complex composition of metals, which cannot be reused. The plastics and other instruments that cannot be reused or recycled will be sent to landfills.

Alternative 2

Natural gas is the resource used in alternative 2, therefore the resource provisioning stage is the same as the baseline. Implementing the large gas turbine on the fourth floor of the boiler house is an energy intensive task; high strength steel in the gear box and the turbine are needed for reliable operation which means virgin metals are needed. The primary and complementary process operations have relatively smaller environmental impacts than the other life stages due to the use of reheatable water. The process is designed to recover the water sent out to buildings around campus by re-circulating the return water through the heat recovery system or boilers. The plant uses minimal energy because the hot exhaust gas from the turbine is high in oxygen content and allows additional fuel gas to be burnt by using duct burners. At the end-of-life of the boilers and turbine, dismantling is an energy intensive process since the boilers are approximately 3 ½ stories high and are not modular. However, most of the steel is salvageable since natural gas is clean burning unlike liquid fuel.

Baseline Alternative
The flow charts illustrate the various stages and how they fit together to form the over all process. At each stage there were associated environmental impacts which are outlined in detail in the table below. It is important to note that the rows titled “Total” are the total emissions at the end of the 30 year life, and not simply the addition of all emissions.

Ontario Power Generation spends $112 million per month on fuels for their generation stations; the three types include hydro, fossil fuel, and nuclear powered. By knowing how much electricity they produce each month and how much the university consumes it is estimated that the university is responsible for $1.9 million per year. Therefore, to model the electricity acquisition process $1.9 million was split up into various mining and refining sectors to simulate the acquisition and processing of fossil fuels and uranium. Following this, the money was inputted into appropriate transportation sectors to simulate transporting the fuels from the mining sites to the power plants. It should be noted that in the case of hydro power plants the water does not need to be transported from a “mining site” but it still needs to be transported from a holding area through the plant. After transportation, the total sum was inputted into the power generation and distribution sector to finish off the process. Lastly, it is important to note that there is no “end of life” stage for this process. Based upon the amount of steam the univeristy uses each year and the price of natural gas in January of 2008, it is estimated that the university is responsible for the equivalent of $5.1 million in 1997. This dollar amount was inputted into oil and gas extraction sectors then into various distribution sectors. Following that a process model had to be constructed to accurately model the effects of burning natural gas; and lastly, using the 1-3-9 manufacturing rule recycling values were estimated.

Alternative 1
The amount of natural gas required is equivalent to 1.63 million MMBTU, and the extraction and distribution of the natural gas have to be considered. The material extraction and manufacturing of the balance of plant (machinery), balance of stack (seals, electrical), and the stack structure (cathodes, electrolytes, anodes, and interconnects). The amount of resources required was estimated using the 1-3-9 method. The global warming percentage during the use phase compared to electrical energy obtained from the grid and heat from a gas boiler is approximately 75%, and the NOx, SOx, CO, and PM10 emissions are almost negligible. At the disposal phase, the steel is to be recycled and all other components are put into landfills. Therefore, the emissions related to the mining of iron (which is used to make steel), is subtracted from the total (included as a credit). The table below illustrates the environmental effects of a Solid Oxide Fuel Cell Power Generation system over 30 years. The majority of the greenhouse gas potential comes from the generation of electricity and heat, however, the extraction and distribution of natural gas also accounts for a large portion. The conventional air pollutants, with the exception of PM10, originate mostly from the natural gas processes. The inhalable particles are split between the natural gas processes and the extraction of raw materials of the system. The extraction of raw materials also accounts for almost all of the toxic releases, due to the large amounts of residue from mining metals. The manufacturing phase, in relation to the other phases, does not have a big influence on the environment when compared to the other phases such as natural gas processes and the extraction of raw materials. The disposal phase, which involves the recycling of steel, actually reduces the sulphur dioxide, NOx, and inhalable particles.

Alternative 2
The total required energy per year for new plant is the summation of the turbine process, the boiler process and the fuel gas used in the gas compressor, respectively: 3,012,000 + 819,000 + 25,000 = 3,856,000 MMBTU/year. The cost of natural gas per year in the 1997 amount is then $20.16 million. This amount was put into the EIO-LCA model under oil and gas extraction sector to account for the process of acquiring natural gas from the ground, as well as natural gas distribution sector, which includes the transmitting and distribution of natural gas to final customers. The use phase was done conventionally with the help of an emissions table from Natural Gas Issues and Trends which gave pollutants in lb/billion Btu which was converted to MT/MMBtu and then multiplied by the amount of required energy. Emissions from the manufacturing of the primary process (turbine and generator) and complementary (heat exchangers and percotherm) were estimated using a cited $50 million capital cost for the whole system implementation. 75% of that cost was assumed to be from the primary process equipment manufacturing, 23% from the complementary and 2% from shipping, packaging and labour. The EIO-LCA sectors used for these percentages were: Turbine and turbine generator set units manufacturing, Wiring Device Manufacturing, Air and Gas Compressor Manufacturing and Heating Equipment, Except Warm Air Furnaces. End-of-life emissions are minimal: wiring and circuitry would be discarded to landfill but a credit would be afforded for recycled steel of about $16.38 million meaning the amount of pollutants correlating to this amount in the iron ore mining sector can be subtracted from the total pollutant tally.

Baseline Alternative
Additionally to the direct costs outlined in the table, there would be indirect costs which would include taxes, depreciation of equipment, office supplies and insurance. Basically, indirect costs encompass espenses that are very necessary but do not contribute directly to the manufacture of the end product.

Alternative 1
The direct cost includes the facility, fuel, personnel, and maintenance of the system. The capital cost for the Solid Oxide Fuel Cell power generation system is the cost of the Balance of Plant, the Balance of Stack, and the Fuel Cell Stack. The recurring costs consist of the annual cost of natural gas, personnel, and maintenance, and the replacement of the fuel cell stack.

The indirect costs associated include insurance, taxes, depreciation, and utilities. The high operating temperatures and heavy machinery will induce safety hazards, which would increase the insurance rates. The power generation facility is used for generation within the university and would not result in any significant profits, however, taxes for the purchasing of parts and land taxes will have to be paid. All the components of the plant have a finite life span, so there will depreciation of the value of the parts. Lastly, because the system generates its own power and heat, the only utility fee that will have to be considered is water consumption.

Alternative 2

An advantage to this system is that the boilers are considered a sunken cost at this point as they currently exist and a decision on implementation no longer requires analyzing them except when recycling. Fuel gas is the cost of natural gas per year as calculated in the EIO-LCA section. Labour costs include the operator and engineer salaries with an estimated 3 operators required at $45,000/year and 2 engineers at $90,000/year. Lastly, a recycling deposit was subtracted from the total cost as credit to the plant. For scrap steel there is $49.4 per tonne recoverable; in 2008 this value is $64.17/tonne and with 348 m3 estimated volume of recoverable steel a recycling deposit of $175,505 results. The resulting total cost of operating the cogen system over the next 30 years is equivalent to paying $303,467,000 today.

The indirect costs associated with this alternative also include insurance, taxes, depreciation, and utilities as the previous alternatives. Insurance would be slightly higher for this alternative than the baseline because of the turbine-generator. It is known that no taxes are paid since the plant is on university land. After 30 years, the depreciation would take the salvage value to zero if it followed the linear curve. The same held true for boilers. Therefore, salvage value did not play a role in the total cost. Utilities such as electricity for lighting and water for boilers report an annual amount of $ 7.2 millions, which in present worth would be $ 0.72 million.

Baseline Alternative
The major societal concern related to the baseline is the emissions out of the stack. The clouds of grey entering into the atmosphere are not aesthetically pleasing and more importantly could damage the university’s reputation. The university is perceived to be home to some of the brightest minds, and newest technologies in the country; with such a large emphasis being placed on “green” alternatives and smaller ecological footprints the billowing stacks could symbolize apathy on the part of the university. Without considering the entire life cycle it would appear that the university elected for the cheapest alternative without consideration for their neighbours or the environment; a conclusion that could have negative long term effects.

Alternative 1
The size of the facility would be quite large, so an appropriate land space would have to be found. There is not much land around the campus that could be used for construction, and existing buildings all serve their own purpose, so to sacrifice their land space would not be a solution. However, because the noise and vibrations emitted from the system are minimized due to lack of moving parts, such as turbines, the facility can be placed in any area without causing too much disturbance. The lower greenhouse gas and toxic release emissions, as discussed in previous sections, would improve the air quality around the campus, and improve aesthetics by reducing smog and odours.

Alternative 2
Noise and pollution are the main societal effects this cogeneration process will have on the people around the university. Given the right conditions (wind direction and turbine rpm) noise generated from the boiler house assuming the windows are open can reach 95 dB maximum. Constant exposure to this amount can damage hearing. For pollution, direct burning of natural gas emits 441 kg of NOx and 192 kg of carbon monoxide per day based on the values from the EIO-LCA. NOx is what creates the smog in the atmosphere and people with asthma or bronchitis have increased sensitivity when exposed to it. Increased CO means decrease of oxygenated blood, which impairs learning ability and visual perception.


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User talk:Miroslaw.Makowiecki

thumb|350px|gęstość rozkładu normalnego Artykuł zawiera wszystkie informacje dotyczące rozkładu normalnego, które są potrzebne do jego wyprowadzenia i prostą jego interpretację, a w szczególności interpretację $$\sigma$$ oraz punktu przegięcia, kiedy wyniki pomiaru należy odrzucić, a kiedy przyjąć.

Niżej znajdują się wszystkie momenty statystyczne rozkładu normalnego. Należy zaznaczyć, że momenty statystyczne o nieparzystym stopnia są równe zeru, a pozostałe są od niego różne. Można tak interpretować, że rozrzut wyników doświadczalnych wokół wartości najprawdopodobnej $$x_0$$ jest symetryczny, tzn. ilość wyników pomiarowych przed i za $$x_0$$ jest taka sama, czyli funkcja gęstości prawdopodobieństwa jest funkcją parzystą, z tego powodu momenty statystyczne nieparzystego stopnia są równe zero, gdyż funkcja podcałkowa jest funkcją nieparzystą w całce symbolizującej moment statystyczny.

Twierdzenie o rozkładzie normalnym – szczególny przypadek twierdzenia o rozkładzie dwumianowym w szczególności Bernoulliego dający odpowiedź na pytanie jak postępować z danymi doświadczalnymi: czy je odrzucić jako mało prawdopodobne, czy je przyjąć w zależności od danych doświadczalnym uzyskanych z doświadczenia poprzez uprzednie obliczenie $$x_{srednie}$$ i $$\sigma$$, czyli wartość średnią (arytmetyczną) i średnie odchylenie standardowe (zob. reguła trzech sigm). Zasady obliczania tych wskaźników $$x_{srednie}$$ czy $$\sigma$$ będziemy tu przyjmować za znane i nie będziemy do tego problemu podchodzić, tylko wyprowadzimy wzory na $$\sigma_p$$- to jest niepewność standardowa pojedyńczego pomiaru, a $$\sigma_{srednie}$$ to jest niepewność standardowa średniej arytmetycznej uzyskanych z doświadczenia wszystkich nie odrzuconych pomiarów, oraz $$x_{srednie}$$ określa wartość, której prawdopodobieństwo uzyskanie w naszym doświadczeniu jest największe, jest to wartość najdokładniejsza ze wszystkich uzyskanych  $$x_i$$ względem wartości dokładnej $$x_0$$, i celem doświadczenia jest uzyskanie wartości $$x_0$$, ponieważ nie jest możliwe uzyskanie tej wartości,lub dokonania nieskończenie wielu pomiarów, bo wtedy $$x_{srednie}=x_0$$, to zamiast niej uzyskujemy $$x_{srednie}$$ (średniej arytmetycznej), która jest liczona dla każdej serii danych doświadczalnych wykorzystując do tego celu rozkład normalny, w której w miejsce wartości dokładnej wsadzamy $$x_{srednie}$$.

Zwykle tak się robi w fizyce doświadczalnej, nie patrząc na opinie matematyków, w której matematycy mówią że wszystko musi być dokładne, nie ma wartości przybliżonej a jeśli są to trzeba określić je określić w sposób ściśle określony, a według fizyków nie ma wartości dokładnej, są tylko wartości przybliżone. W doświadczeniach fizycznych dane doświadczalne układają się wokół punkt x_0 (wartości dokładnej}, i bardzo rzadko się zdarza, że wyjdzie jakiś wynik(pomiar), który jest odległy od $$x_0$$ nawet $$o 2\sigma$$, i jest to raczej błąd systematyczny, które może mówić o błędzie eksperymentatora, niedostosowanej np. temperatory do warunków doświadczalnych itp. dlatego odrzucanie wyników doświadczalnych bardzo oddalonych od $$x_0$$, należy przyjąć jako odrzucone Gdy $$|x_{i}-x_{srednie}|>2\sigma$$ odrzuca się te $$x_i$$ z poszczególnych prób i serii. Liczba wyników doświadczalnych powinna być na tyle duża, nawet po odrzuceniu pewnych danych doświadczalnych szczególnie zawężonym do $$|x-x_i|>\sigma$$. Mówi się, że przy typ, że przybliżeniu $$n \to \infty$$ jest spełniony tutaj rozkład normalny, a przy małym n nie jest, ponieważ w doświadczeniu powinno się robić co najmniej 30 doświadczeń w poszczególnych próbach, by rozkład w miarę był spełniony. Rozrzut wyników wokół wartości dokładnej jest porozrzucany po obu stronach $$x_0$$ i tak się dzieje, że wyniki doświadczalne znajdują się bardzo blisko wartości dokładnej. Prawdopodobieństwo uzyskania wyników o błędzie większym od $$\sigma$$ a nawet $$2\sigma$$ jest bardzo mało prawdopodobne i odrzucanie tychże wyników doświadczalnych nie wnosi prawie żadnego wkładu do obliczania $$\sigma$$ i $$x_0$$. Rozkład normalny jest spełniony dla nieskończonej ilości stopni swobody (liczby uzyskanych danych doświadczalnych), oczywiście minus jeden, bo mamy wzór na średnią arytmetyczną, licząc je dla poszczególnych prób.

Niepewność pomiarowa w doświadczeniu, określmy jako średni błąd, który można popełnić, dla poszczególnych doświadczeń w próbach. A jeśli niepewności są większe niż $$\sigma$$, to taki wynik częściowy doświadczenia należy odrzucić, toteż $$x_{srednie}$$ i $$\sigma$$ należy przeliczyć od nowa, czy nie ma dalszych wyników doświadczenia tzn.: $$x_i$$ określić jako odrzucone.Jeśli są to tą procedurę powtarzamy aż do skutku. Po tej czynności należy przejść do dalszego etapu badania jakim są zależności w naszym doświadczeniu w poszczególnych próbach.

Dlaczego trzeba odrzucać wyniki dla której $$(x_i-x_{srednie})>2 \sigma$$, dlatego jak zapoznamy się z wyprowadzeniem wzoru na rozkład normalny zastosowano szereg Taylora zastosowanej wokół wartości $$x_0$$,i jak znamy z analizy matematycznej gdy większa różnica między $$x_i$$- wynikiem pomiaru a $$x_0$$- wartością dokładną to szereg Taylora przestaję być spełniona poza promieniem promieniem zbieżności i dlatego odchyły od $$x_0$$ nie powinny być za duże.

Wzór na rozkład Bernoulliego
Rozkład Bernoulliego jest szczególnym przypakiem twierdzenia rozkładu Bernoulliego, a jeszcze dokładniej dwumianowego. Aby wyprowadzić twierdzenie na rozkład normalny, należy skorzystać z twierdzenia na rozkład Bernoulliego. Jest to twierdzenie, które zależy od n,k, czyli ilości przeprowadzonej doświadczeń na obiekcje S i uzyskanych wyników poprawnych na S czyli k razy. Na sam początek: twierdzenie Bernoulliego, znane ze szkoły średniej, ze statystyki matematycznej:

 * $$P_{nk}={n \choose k} p^k q^{n-k}$$

gdzie:

 * $$P_{nk}$$-to prawdopodobieństwo uzyskania k razy elementu z n losowań,

ale
 * $$q=1-p$$

 * n- ilość doświadczeń przeprowadzonej na obiekcie S

 * k - ilość doświadczeń na S, w której uzyskaliśmy zdarzenie K, którego wartość uzyskania wynosi p przy pojedyńczym pomiarze. Ale gdy $$k=n$$, ze wzoru na rozkład normalny, uzyskujemy $$P_{nn}=p$$, oraz gdy $$k=0$$, to $$P_{n0}=q=1-p$$, czyli zdarzenie całkowicie przeciwne.

Mamy n doświadczeń uzyskanych,w których dana wartość(właściwość K jest k razy wylosowana), trzeba pamiętać że p jest to prawdopodobieństwo uzyskania pojedyńczego zdarzenia oczekiwanego, w pojedyńczym losowaniu, ale też jest q=p-1 dla pojedyńczego zdarzenia nieoczekiwanego uzyskania wyników przeciwnych niż oczekiwane.

Wyznaczanie wzoru przybliżonego na n!
Obliczająć n! w sposób przybliżony, korzystam z logarytmów naturalnych, by udało się ustalić przybliżoną zależność na n!. W rozkładzie dwumianowym bardzo często występuje silnia, korzystanie z niej szczególnie jest nie wygodne gdy $$n\rightarrow \infty$$ Silnia jest to iloczyn kolejnych liczb naturalnych licząc od 1 do n
 * $$n!=1\cdot2\cdot...\cdot n\;$$

Logarytmując silnia n-tego stopnia mamy:
 * $$ \ln n! =\ln (1\cdot2\cdot ...\cdot n)\;$$

Korzystając, że logarytm (w szczególności naturalny) iloczynu dwóch czynników, jest to sumą logarytmów naturalnych, czyli według wzoru:
 * $$\ln {xy} =\ln x+\ln y\;$$

Wykorzystując tą prawdę mamy:
 * $$ \ln n!=\ln 1+\ln 2+..+\ln n\;$$

Stąd jego dyskretna suma jest równa:
 * $$\ln n!=\sum\limits_{i=1}^n \ln i\;$$

Udowodnijmy do zapomocą indukcji matematycznej. Dla n=1 wiadomo że $$\ln 1!=\ln 1=0\;$$ Załóżmy że twierdzenie jest spełnione dla n i udowodnijmy to dla n+1, otrzymujemy:

$$ \ln (n+1)!=\ln (n! (n+1))=ln n!+ln (n+1)=\ln 1+\ln 2+..+\ln n +\ln (n+1)\;$$

Czyli twierdzenie na $$ \ln(n!)$$ zostało udowodnione. Można dokonać małego przybliżenia zastępując sumę logarytmów na całkę, czyli zamienić na całkę, która jest w pewnym sensie sumą ciągłą, z definicji całki, a jeszcze poza tym zamieńmy i na x. Zastosowanie takiego przybliżenia dla dużych n nie powoduje dużego błędu, czyli np. pole pod krzywą powstałych z połączeń punktów $$\ln(i)$$ odcinkami. gdzie i to numer argumentu dla której liczymy wartość. Udowodnijmy twierdzenie:

 * $$\int\limits^n_1 ln(x)=\sum_{i=1}^{n-1} \int\limits^{i+1}_{i} ln(x)dx$$

Korzystając z twierdzenia Langrange'a, tzn.:$${{f(b)-f(a)}\over{b-a}}=f^'(c)$$ a więc
 * $${\int\limits^n_1 \ln n}=\sum_{i=1}^{n-1} {\ln(i+\theta)\int\limits^{i+1}_i dx}=\sum_{i=1}^{n-1}\ln(i+\theta)$$

Trzeba pamiętać $$\theta\in(0,1)$$

W porównaniu z dwoma sumami skrajnymi mamy:

 * $$\sum_{i=1}^{n-1}\ln(i+0)\leq\int\limits^n_1 \ln i\leq\sum_{i=1}^{n-1}\ln(i+1)$$

Obliczając różnice prawej i lewej strony mamy:

 * $$\sum_{i=1}^{n-1} \ln(i+1)-\sum_{i=1}^{n-1}\ln(i)=\sum_{i=2}^{n}\ln i-\sum_{i=1}^{n-1} \ln i=\ln n-\ln 1=\ln n$$

I obliczmy całkę $$\int\limits^n_1 \ln x$$ Dokonując całkowania przez części otrzymuję:

 * $$\ln n!\simeq[{x} \ln {x}]_{1}^{n}- \int\limits^n_1 {x} {1 \over {x}} d{x}=n \ln n-n-1\simeq n\ln n-n=\ln n^n-\ln e^n=\ln {{n^n} \over {e^n}}\;$$

A więc mamy:

 * $$ \ln n!\simeq\ln {{n^n}\over{e^n}}\;$$

A dokładniej:

$$ \ln n!= ln{{n^n}\over{e^n}}\pm \ln n\;$$

Błąd bezwzględny jest równy $$\ln n\;$$

Obliczmy błąd względny, który tak naprawdę mówi, jak nasza wartość przybliżona $$\ln n!\;$$ jest poprawna. Otóż mamy:

$$\lim_{n \rightarrow \infty}{{\ln n}\over{ln{{n^n}\over{e^n}}}}=\lim_{n \rightarrow \infty}{{\ln n}\over {n ln n-n}}=\lim_{n \rightarrow \infty}{1\over{n-{n\over \ln n}}}=\lim_{n \rightarrow \infty}{1\over{n(1-{1\over \ln n})}}=(\lim_{n \rightarrow \infty} {1\over n}){1\over{1-\lim_{n \rightarrow \infty} {1 \over ln n}}}=0 \cdot 1=0$$

Czyli wyznaczone wartości przybliżone tzn. :$$n!\simeq={{n^n}\over{e^n}}$$ jest spełnione dobrze dla dużych n,a dla $$ n \rightarrow \infty$$ to przybliżenie staje się dokładne. W poniższym dowodzie będziemy operować na n nieskończenie dużym, i małych odchyleń od $$x_0$$ w porównaniu z $$\sigma$$, bo tylko wtedy nasze twierdzenie jest twierdzeniem względnie dokładnym. Należy pamiętać, mimo że błąd bezwzględny dąży do nieskończoności, to w tym przypadku dla $$n\rightarrow \infty\;$$, błąd względny jest wprost równy zero, i dlatego ten rodzaj błędu jest dla nas ważny.

Etap obliczeń korzystając z przybliżonego wzoru na n!
Mając uzyskany wzór na n! mogę wykorzystać go w dalszej części rozumowania. Przekształcając wzór z twierdzenia Bernoulliego za pomocą wzoru ostatnio wyprowadzonego na n! uzyskujemy:
 * $$P_{nk}= {{n!} \over {k! (n-k)!}} p^k (1-p)^{n-k} \simeq {{{n^n} \over {e^n}} \over }$$

Grupujemy wartości względem $$e^{cos}$$od $${cos}^{cos}$$ w osobne czynniki i mamy:
 * $$P_{nk}={n^n \over {{(n-k)^{n-k}}k^k}}{{e^k e^{n-k}} \over e^n}p^k (1-p)^{n-k}$$

Wykonując prostych obliczeń na $$e^{cos}$$ i skróceniu tychże czynników w sposób całkowity zależnych od liczby e mamy:
 * $$P_{nk}={{n^n}\over{ {(n-k)^{n-k}}k^k}}{{e^n \over e^n}p^k (1-p)^{n-k}}={{n^n} \over }p^k(1-p)^{n-k}$$

rozkładając na czynniki wyrażenie $$(n-k)^{(n-k)}\;$$ na $${{(n-k)}^n}\over{{(n-k)}^k}\;$$ mamy: $$P_{nk}={{n^n} \over {{(n-k)^{n}} \over {(n-k)^k}}k^k}p^k(1-p)^{n-k}$$

Zaczynamy grupowania potęg względem wykładników k oraz n, mamy:

 * $$P_{nk}={{n^n} \over {(n-k)^n}} {{(n-k)^k} \over {k^k}}p^k(1-p)^{n-k}$$

Dokonując przekształceń na ilorazach typu $${{x \pm 1}\over{x}}=1 \pm {1\over x}$$, mamy:

$$P_{nk}={1 \over {(1- {k \over n})^n}}{({n \over k} - 1)^k}p^k(1-p)^{n-k}$$

Następne warunki przybliżające lub wzory
Wyprowadzenie rozkładu normalnego z obfituje w obliczeniami z przybliżonymi wartościami, $$k,n \rightarrow \infty$$ a także $$k<<n$$ Te warunki jak wiadomo są w naszych doświadczeniach całkowicie są spełnione, w przeprowadzonym, tzn.mamy nieskończenie wiele liczb z przedziału $$x\in(-\infty,\infty)$$, a liczba trafień np. w wartość $$x_i$$ jest mała w porównaniu z liczbą wszystkich możliwych trafień jakie można dokonać w doświadczeniu.

Trzeba pamiętać, że obliczenia są przeprowadzone w przybliżeniu bo n skończone, ale gdy przejdziemy do $$n \rightarrow \infty$$, to wzory są całkowiecie względnie dokładne, ale przy pewnych warunkach przeprowadźmy nieskończoną ilość doświadczeń, by uzyskać $$x_0=x_{srednie}$$,tzn.$$x_0$$ dokładnie i $$\sigma$$ też dokładnie. Określmy przedział $$\Delta x<\infty$$, ale na tyle małe, by doświadczenia fizyczne miały sens, i mówiąc gęstość prawdopodobieństwa, w tymże przedziale jest jednakowe, ale można powiedzieć, że jeszcze $$Delta x$$ jeszcze nie 0, ale dostatecznie małe uzyskania x'a, jest z małym w przybliżeniem dostatecznie małe tzn. $$\rho_{max(x)}-\rho(x)_{min(x)}\simeq 0$$. Liczby trafione w przedział $$x\in(x-{\Delta x},x+{\Delta x})$$ mają takie same prawdopodobieństwo trafienia w ten obszar podczas eksperymentu fizycznego.

Następne zastosowanie przybliżeń lub warunków przybliżających
Ale liczba wyników liczby x uzyskanych wyników ściśle określonych k razy, jest o wiele mniejsza od liczby n, czyli od liczby doświadczeń wykonanym na obiekcie powiedźmy S. W obliczeniach będziemy przyjmować, że $$k\rightarrow \infty$$ oraz $$n \rightarrow \infty$$ Ale tak że spełniony jest warunek$${{k}\over{n}}<<1$$

Wzory przybliżone i zastosowania w wyprowadzeniu twierdzenia
Korzystająć z ostatniego warunku często należy korzystać ze wzoru przybliżonego:

 * $$ {1 \over {1-{x}}} \simeq {1+{x}}$$

Ostatni wzór można wyprowadzić z twierdzenia Taylora czyli:

$${1\over {1-{x}}}={1+\sum^{\infty}_{i=1}}{{x^i}\over{n}}\simeq 1+x$$, Stosując to przybliżenie dla pierwszego czynnika w ilorazie mamy:

 * $$P_{nk}\simeq{(1+ {k \over n})^n}{({n \over k}-1)^k}p^k(1-p)^{n-k}$$

Zastosowanie wzorów przybliżonych i warunków przybliżających
Stosując $$\lim_{n\rightarrow \infty}{(1+{k\over n})}^n=e^k$$ otrzymujemy:
 * $$P_{nk}\simeq{(1+ {k \over n})^n}{({n \over k}-1)^k}p^k(1-p)^{n-k}=(1+ {1 \over {n \over k}})^{{n \over k}k}({n \over k}-1)^k p^k(1-p)^{n-k}\rightarrow{e^k}({n \over k}-1)^k p^k(1-p)^{n-k}$$

Rozszerzenie przedziałów uzyskania wyników na R (liczby rzeczywiste)
Dla doświadczenia spełnione są warunki, wiedząc jednak ,że $$\Delta x \rightarrow 0$$, gdzie jest to malutki przedział dla którego wartości naszego pomiaru są nierozróżnialne względem niepewności pomiarowej, ale i też $$d \rightarrow \infty$$, co przedstawia rozległość pomiarów co spełnia warunki dla rozkładu normalnego, a więc z użyciem tych warunków wzór na $$p= {{\Delta x} \over d}$$,czyli $$p \rightarrow 0$$, oraz następny wzór $$n={d \over { \Delta x}}$$, który przedstawia liczbę punktów pomiarowych w przedziale d, tzn. punktów $$x\in(x_0-{d\over 2},x_0+{d\over 2})$$czyli $$n \rightarrow \infty$$. Zwykle w doświadczeniach w fizyce mamy, że d jest bardzo duże w porównaniu $$\Delta x$$ bo $$d>> {\Delta x}$$ i jest conaj wyżej skończone,bo nie możemy badać wyników, które uzyskamy w przedziale $$x\in{(-\infty,\infty)}$$,nie dość że długość przedmiotów miała by wiele do życzenia, jak i prawa fizycznego, które badamy może być nie spełnione dla aż tak dużego, np. prędkość świata dla d względnie miarę małego w porównywana tysiącami kilometrów, prędkość światła można zaniedbać, a dla dużych odległości, raczej już nie. Również $$\Delta x$$ nie może być nieskończenie małe ponieważ niepewność pomiarowa przedmiotów jest jakaś tam, tak że rozróżnienie wyników w $$x\in(x-\chi,x+\chi)$$ graniczy z cudem. Czyli pomiary jakieś wielkości fizycznej nie mogą być pomiarami zbyt rozległymi względem naszego doświadczenia, jak i pomiarami bardzo małymi względnie wielkości które mierzymy.

Zastosowanie ostatnich warunków
Podstawiając za p,mamy:
 * $$P_{nk}\rightarrow e^k({1 \over {k \over n}}-1)^k p^k(1-p)^{n-k}={e^k ({1 \over {k \over n}}-1)^k} {({{\Delta x} \over d})^k}{(1-{{\Delta x} \over d})^{n-k}}$$

stosujemy, że $$k<<n\;$$ otrzymujemu: $$P_{nk}\simeq {e^k ({1 \over {k\over n}}-1)^k} {({{\Delta x} \over d})^k}{1 \over{(1+{1 \over {d \over {\Delta x}}})^n}}$$

Korzystając ze wzoru przybliżonego na odwrotność różnicy ilorazu
Stosując: $$({1+{1\over{{d}\over{\Delta x}}}})^{d \over {\Delta x}}\rightarrow e$$ mamy

 * $$P_{nk}\rightarrow {e^k {({1 \over {k \over n}}-1)^k}({{\Delta x} \over d})^k}{1 \over{(1+{1 \over {d \over {\Delta x}}})^}}\rightarrow e^k ({{1 \over {k \over n }}-1})^k ({{\Delta x} \over d})^k {1 \over {e^{n {{\Delta x} \over d}}}}={e^k e^{-n{{\Delta x} \over d}}}({{1 \over {k \over n}}-1)^k}({{ \Delta x} \over d})^k=$$

Ogólne stosowane twierdzenia

 * $$e^x\simeq 1+x$$,  oraz     $$ x\simeq e^x-1$$

Zastosowanie ostatnich przybliżeń
Wykorzystując wzór $${k \over n}\simeq e^{k \over n}-1$$,oraz $${{\Delta x}\over{d}}=e^{{\Delta x}\over{d}}-1$$
 * $$P_{nk}\rightarrow { {e^k e^{-n {{\Delta x} \over d}}} ({{1 \over{e^{k \over n}-1}}-1})^k({e^{{\Delta x} \over d}}-1)^k}=e^k e^{-n{{-\Delta x}\over{d}}}({{1-e^{k \over n}+1}\over{e^{k \over n}-1}})^k({e^{{\Delta x} \over d}-1})^k$$

Następnie stosując twierdzenie $$e^{k \over n}={k \over n}+1$$

 * $$P_{nk}=e^k e^{-n{{-\Delta x}\over{d}}}({{2-e^{k \over n}}\over{e^{k \over n}-1}})^k({e^{{\Delta x} \over d}-1})^k={e^k e^{-n{{\Delta x} \over d}}}({{2-(1+{k \over n})}\over{e^{k \over n}-1}})^k(e^{{\Delta x} \over d}-1)^k$$

Tutaj stosujemy twierdzenie:$${1-{k \over n}}\simeq{{1}\over{1+{k\over n}}}$$
 * $$P_{nk}={e^k e^{-n{{\Delta x} \over d}}}{{({1-{k \over n}})^k}\over {({e^{k \over n}-1})^k}}({e^{{\Delta x}\over d}-1})={1 \over{(1+ {k \over n})^k}}({e^{{\Delta x} \over d}-1})^k$$

Tutaj możemy zastosować troszkę zmodyfikowaną wersję:$$\lim_{n \rightarrow \infty}{(1+{k\over n})}^n=e^k$$
 * $$P_{nk}\rightarrow{1 \over{({(1+{1 \over{n \over k}})^)^{{k \over n}k}}}}({e^{{\Delta x} \over d}-1})^k\rightarrow {{e^k e^{-n {{\Delta x} \over d}}}\over}{e^{-{{k^2} \over n}}}{(e^{{\Delta x} \over d}-1)^k}$$

Odpowiednio grupując wyrazy:
 * $$P_{nk}\rightarrow{{e^k e^{-n{{\Delta x} \over d}}e^{-{{k^2}\over n}}}\over{(e^{k \over n}-1)^k}}({e^{{\Delta x} \over d}-1})^k$$

Do czego dąży $$P_{nk}$$ gdy $$ n\rightarrow \infty$$ i gdy $$k<<n$$
Ale $$n\Delta x=d\;$$ $$\lim_{n \rightarrow \infty} P_{nk}={{ { {e^k e^{-1} e^0} } {({\Delta x \over d})}^k}\over {k \over n}}=e^{k-1}{ { {\Delta x}\over{n \Delta x} }^k \over{k \over n}}\simeq {{e^k n^{-k}} \over{k n^{-1}}}={e^k\over k}n^{-n-1}={e^k \over{n^{k+1}}}=\simeq e^k\cdot 0=0$$ A więc niema sensu stosowania twierdzenia że $$P_{nk}!=0\;$$ tylko $$\lim_{n \rightarrow \infty}P_{nk}=0$$

Ilość pomiarów dąży do nieskończoności
Jeśli $$k<<n$$, to mamy$$P_{nk}\rightarrow 0$$ A więc należy zastosować definicję gęstości prawdopodobieństwa, bo posługiwanie się samym $$P_{nk}$$ nie ma wielkiego sensu bo:

 * $$\lim_{n \to \infty} P_{nk}=0$$

Przejście od prawdopodobieństwa do gęstości prawdopodobieństwa

 * $$\rho_{nk}\rightarrow{{e^ke^{-n{{ x} \over d}}e^{-{k^2 \over n}}}\over{({e^{k \over n}}-1)^k}}{({e^{{\Delta x} \over d}-1})^k}{1 \over }$$

bo
 * $$\rho_{nk}={{P_{nk}}\over {\Delta x}}$$

gdzie:
 * $$\rho_{nk}$$ jest to gęstość prawdopodobieństwa uzyskania k razy jakiegoś wyniku w n losowaniach

Uzyskanie wzoru na gęstość prawdopodobieństwa

 * $$\rho_{nk}\rightarrow {{e^ke^{-{k^2}\over n}}\over{e^{k \over n}-1}}{{e^{-n{{\Delta x} \over d}}({e^{{ \Delta x}\over d}-1})^k}\over}$$

Względem dwóch ostatnich czynników w liczniku przez $$\Delta x$$ stosując regułę de'Hospitala uzyskując:

 * $$\rho_{nk}\rightarrow {{e^ke^{-{{k^2}\over n}}}\over{(e^{k \over n}-1)^k}}{{e^{-n{{\Delta x}\over d}}{(-{n \over d})}({e^{{\Delta x}\over d}-1})^k+e^{-n{{\Delta x}\over d}}k({e^{{\Delta x}\over d}-1})^{k-1}e^{{\Delta x}\over d}{1\over d}}\over{1}}$$

\rho_{nk}\rightarrow e^{-n{{\Delta x}\over d}}[-{n \over d}(e^{{ \Delta x}\over d}-1)^k+k(e^{{\Delta x} \over d}-1)^{k-1}{{e^{{\Delta x}\over d}}\over d}]$$$$\rho_{nk}\rightarrow {{e^k e^{-{{k^2}\over n}}}\over{(e^{k\over n}-1)^k}}e^{-n{{\Delta x}\over d}}({e^{{\Delta x}\over d}-1})^{k-1}[{-{n}\over{d}}({e^{{\Delta x}\over d}-1})+k{{e^{{\Delta x}\over d}}\over {d}}]$$

 * $$\rho_{nk}\rightarrow {{e^k e^{-{k^2}\over n}}\over{(e^{k \over n}-1)^k}}e^{-n{{ \Delta x}\over d}}({e^{{\Delta x}\over d}-1})^{k-1}[-{{n}\over{d}}{\over{d}}+{{k}\over{d}}({{\Delta x}\over d}+1)]=$$
 * $$\rho_{nk} \rightarrow {{}e^ke^{-{{k^2}\over n}}\over{({e^{k\over n}-1})^k}}e^{-n{{\Delta x}\over d}}({e^{{\Delta x}\over d}-1})^{k-1}[-{{{\Delta x}n}\over{d^2}}+{{k{\Delta x}}\over{d^2}}+{k\over d}]$$

Upraszczanie ostatniego wzoru względem ostatniego czynnika korzystając $$ d = {\Delta x} n$$
Ale dla czynnika w nawiasie kwadratowym zachodzi:
 * $$-{{{\Delta x}n}\over{d^2}}+{{k{\Delta x}}\over{d^2}}+{{k}\over{d}}=-{{{\Delta x}n}\over{n^2{\Delta x}^2}}+{{k{ \Delta x}}\over{n^2{\Delta x}^2}}=-{{1}\over{n{\Delta x}}}+{{k}\over{n^2{\Delta x}}}+{{k}\over{n{\Delta x}}}\simeq{{k-1}\over{n{\Delta x}}}$$

Stosując $$e^{k \over n}-1={k\over n}$$ oraz $$d=\Delta x n\;$$

A więc:
 * $$ \rho_{nk}\rightarrow {{e^ke^{-n{{k^2}\over n}}}\over{({{k}\over{n}}})^k}e^{-n{{{\Delta x}}\over{n{\Delta x}}}}({e^{1\over n}-1})^{k-1}{{k-1}\over n{\Delta x}}\rightarrow {{e^ke^{-{k^2}\over{n}}}\over{e{({k\over n})^k}}}{({{1}\over n})}^{k-1}{{k-1}\over{n{\Delta x}}}=$$

Odpowiednio przekształcając manewrując czynnikami

\rho_{nk}\rightarrow e^{k-{{k^2}\over n}-1}{{n^k}\over{k^k}}{{n}\over{n^k}}{{k-1}\over{n{\Delta x}}}$$ Stąd ostateczny wzór:
 * $$\rho_{nk}\rightarrow e^{k-{{k^2}\over n}-1} {{k}\over{k^{k-1}n{\Delta x}}}$$

Następne przybliżenia
Ale wiadomo $$k<<n\;$$,a także $$k^2<<n\;$$, a uzyskując:$$k-{{k^2}\over n}\simeq k\;$$,i jeszcze $$k-1\simeq k\;$$ Korzystając z ostatniego wyprowadzonego wzoru oczywiście mamy:

 * $$\rho_{nk}\simeq {{e^{k-1}}\over{k^{k-1}}}{{1}\over{n{\Delta x}}}\simeq {{k^{-k+1}}\over{e^{-k+1}}}{{1}\over{n{\Delta x}}}\simeq {{k^{-k}}\over{e^{-k}}}{{1}\over{n{\Delta x}}}\rightarrow {{e^{ln k^{-k}}}\over{e^{-k}}}{{1}\over{n{ \Delta x}}}$$

 * $$\rho_{nk}\rightarrow {{e^{-k ln k}}\over{e^{-k}}}{{1}\over{n{ \Delta x}}}\rightarrow {{e^{-k ln k +k}}\over {n{\Delta x}}}={{e^{-k[ln k-1]}}\over {n{\Delta x}}}$$ale$$ k>>1\;$$ stąd $$\rho_{nk}\rightarrow {{e^{-k ln k}}\over{n{\Delta x}}}$$

Określanie względem wartości najprawdopodonej
Ale przez $$ \rho_{n}^{s}$$ oznaczając jako zdarzenie najbardziej prawdopodobne, a przez $$\rho_{nk} $$jako zwykłe zdarzenie, stąd możemy powiedzieć:

 * $$ {{\rho_{nk}^{s}}\over{\rho_{nk}}}\rightarrow {{e^{-k_s ln k_s}}\over{e^{-k ln k}}}{{1 \over{n{\Delta x}}}\over{1\over {n{\Delta x}}}}$$

stąd po skróceniu skadników mamy:

 * $$\rho_{nk}\rightarrow {{e^{-k \ln k}}\over{e^{-k^s_n \ln k^s_n}}}\rho_{n}^s$$

A więc określając stał przez $$\beta(k_s)={1 \over{e^{-k_n^s \ln k^s_n}}}$$,i po zastąpieniu otrzumuję:
 * $$\rho_{nk}\rightarrow\beta(k_s) e^{-k \ln k}$$

Zerowy moment statystyczny, czyli normowanie funkcji $$\rho_{nk}$$
Teraz normujemy funkfcję $$\rho_{nk}$$ do jedynki, bo prawdopodobieństwo znalezienia zdarzenia  w $$ x\in(-\infty,\infty)$$ jest pewne czyli jest równe jedynce
 * $$1=\int\limits^{\infty}_{-\infty}{\rho_{nk}dx}=\int\limits^{\infty}_{-\infty}=\beta(k_s) \int\limits^{\infty}_{-\infty}{e^{-k \ln k}dx}$$

stąd:

 * $$\beta(k_s)={1 \over{\int\limits^{\infty}_{-\infty}{e^{-k \ln k}dx}}}$$

A więc: Stosując twierdzenie Tayelora można uzyskać to co poniżej, i dlatego trzeba ucinać pewne wyniki doświadczalne które są daleko od $$x_0$$,tak żeby był spełniony rozkład Tayelora, która jest napewno spełniony dla małych odchyleń x od $$x_0$$ A nawet nieznamy promienia zbieżności takowego szeregu przybliżającego $$k \ln k$$ ale spróbujemy je wyznaczyć później.

 * $$ \rho_{nk}={{e^{-k \ln k}}\over{\int\limits^{\infty}_{-\infty}e^{-k \ln k}}}$$

Mając n pomiarów, w której gęstość prawdopodobieństwo jest równe $$\rho(x_i)$$,a więc łączna gęstość prawdopodobieństwa jest równa:
 * $$\rho^{all}=\rho(x_1)\rho(x_2)...\rho(x_n)\;$$

Zakładam że gestości prawdopodobieństwa w poszczególnych pomiarach są takie same. A więc:
 * $$\rho^{all}_{nk}={{e^{-\sum^n_i(k_i \ln k_i)}}\over{(\int\limits^{\infty}_{-\infty}e^{-k \ln k})^n}}\;$$

-to jest gęstość prawdopodobieństwa uzyskania n wyniku $$x_i$$ $$k_i$$ razy jednocześnie dla $$i\in Z$$. A łączne prawdopodobieństwo uzyskania tych wszystkich wartości jest równe: Rozłóżmy wyrażenie $$k \ln k$$ na szereg Tayelora,wokół punktu $$x_0$$czyli wokół wartości dokładnej, a więc mamy:
 * $${k_i\ln k_i}={k_{si} ln k_{si}}+_{k_{si}}{({x_i-x_0})}+{({{\delta x(k_i\ln k_i)}\over{\delta x^2}})}_{k_{si}} {(x_i-x_0)^2\over 2}+...$$

czyli
 * $$k_i \ln k_i= D_i+A_i(x_i-x_0)+{{B_i}\over 2}(x_i-x_0)^2$$

Stąd wszystko podstawiając według wzoru $$\rho^{all}(k)$$ Przyjmujemy,że rozkłady prawdopodobieństw poszczególnych pomiarów są jednakowe,takie same.

 * $$\rho^{all}_{k}={{e^{-\sum^n_{i=1}D_i-\sum^n_{i=1}A_i(x_i-x_0)-\sum^n_{i=1}{{B_i}\over 2}(x_i-x_0)^2}}\over{(\int\limits^{\infty}_{-\infty}\rho(x)})^n}$$

Po przekształceniach:
 * $$\rho^{all}_{k}={{e^{-nD}}\over{(\int\limits^{\infty}_{-\infty}\rho(x))^n}}e^{-A\sum^n_{i=1}(x_i-x_0)}e^{-{{B}\over 2}\sum^n_{i=1}(x_i-x_0)^2}$$

Funkcję którą otrzymaliśmy, nie da się unormować, a nawet mamy wątpliwości co to stosowalności szerego Tayelora dla $$x\rightarrow\pm\infty$$, gdy $$A\not=0\;$$, a więc przyjmnijmy A=0, które na pewno spełnia wszystkie warunki naszego rozkładu. Wiadomo że musi być B\not=0, bo z tym warunkiem rozkład $$\rho(k)$$ daje się w ogólności unormować,i szereg Tayelora jest spełniony Oznaczając przez $$A=A_i$$ i $$B=B_i$$, oraz $$D_i=D$$ ponieważ ma symetryczność względem numeru pomiaru,tzn.: niezależy od wyniku pomiarowego, które uzyskaliśmy, ale istnieje takie same prawdopodobieństwo uzyskania pomiaru $$x_i$$ jak i pomiaru $$x_j$$, jak przy pomiarze i, jak pomiarze j, czyli ten sam wynik można uzyskać z takim samym prawdopodobieństwem niezależnie od kolejności wykonywania doświadczeń. Oznaczając
 * $$C={{e^{-D}}\over{({\int\limits^{\infty}_{-\infty}\rho(x)}})^n}$$

Stąd mamy:
 * $$\rho^{all}(x)=C e^{{-B\over 2}\sum^n_{i=1}{({x_i-x_0})^2}}$$

Gęstość prawdopodobieństwa uzyskania średniej arytmeycznej jest równa:
 * $$\rho^{all}(x^{srednie})=C e^{-{B\over 2}\sum^n_{i=1}{(x^{srednie}-x_0)}}$$

Określimy czemu jest równa gęstość prawdopodobieństwa jeśli znamy tylko gęstość prawdopodobieństwo, a mianowicie:
 * $$\rho^{srednie}={{C_1}\over{c_2}}{{e^{-{B\over 2}\sum^n_{i=1}(x_i-x_0)^2}}\over {e^{-{B\over 2}\sum^n_{i=1}(x^{srednie}_i-x_0)^2}}}$$

A teraz
 * $$\rho^{srednie}=C_3 e^{-{B\over 2}(\sum^n_i((x_i-x_0)^2-(x^{srednie}_i-x_0)^2))}$$

W wykładniku liczby e przekształcając:

 * $$(x_i-x_0)^2=(x_i-{x_{srednie}}+-x_0)^2=(x_i-x_{srednie})^2+({x_{srednie}}-x_0)^2+(x_i-x_{srednie})({x_{srednie}}-x_0)\;$$

Otrzymujemy wzór:
 * $$\rho^{srednie}=C_3 e^{-{B\over 2}\sum^n_{i=1}(x_i-x_{srednie})^2} e^{-{B\over 2}\sum^n_{i=1}({x_{srednie}}-x_0)}e^{-{B\over 2}\sum^n_{i=1}((x_i-x_{srednie})({x_{srednie}}-x_0))}e^{{B\over 2}\sum^n_{i=1}(x_0-x_{srednie})^2}$$

Skracając drugi i czwarty wyraz,i korzystając z wyniki odtrzymane z doświadczenia są nie zeleżne od siebie, mamy:
 * $$\rho^{srednie}=C_3 e^{-{B \over 2}\sum^n_{i=1}(x_i-x_{srednie})^2}$$

Mając wzór na rokład średniej $$x_{srednie}$$wokół wartości dokładnej $$x_0$$
 * $$\rho(x_{szrednie})=C e^{-{B \over 2}\sum ^n_{i=1}(x_{srednie}-x_0)}$$

Korzystająć z definicji wartości dokładnej: tzn:$$x_{srednie}=\sum^n_{i=1}{{x_i}\over{n}}$$
 * $$\rho(x_{srednie})=C e^{-{B\over 2}{\sum^n_{i=1}({{\sum^n_{j=1}{x_j}}\over {n}}-x_0)^2}}$$
 * $$\rho(x_{srednie})=C e^{-{B \over 2}{\sum^n_{i=1,j=1}({{x_j-x_0}\over{n}})^2}}$$$$=C e^{-{B \over 2}n\sum^n_{i=1}({{{x_i-x_0}}\over{n}})^2}$$$$=C e^{-B{\sum^n_{i=1}(x_i-x_0)^2}\over {2n}}$$

Jeszcze raz przejdźmy do $$\rho^{srednie}$$czyli
 * $$\rho^{srednie}=C_3{{e^{-{B\over 2}\sum^n_{i=1}(x_i-x_0)^2}}\over {e^{-{B\over {2n}}\sum^n_{i=1}(x_i-x_0)^2}}}$$$$=C_3 e^{-{B\over 2}(1-{1\over n})\sum^n_{i=1}(x_i-x_0)^2}$$=$$=C_3 e^{-{B\over 2}{{{n-1}\over{n}}\sum^n_{i=1}(x_i-x_0)^2}}$$

Wyznaczamy stałe B i C we wzorze na rozkład normalny
Określmy całki znane analizie matematycznej: Zakładamy,że mamy rozkład:
 * $$\rho=C e^{-{B\over 2}(x-x_0)}$$

gdzie $$x_p$$ jest to dowolna stałą taką, że $$E(x)=x_0$$
 * $$\int\limits^{-\infty}_{\infty} e^{-\alpha x^2}=\sqrt{{\pi}/{\alpha}}$$

oraz całkę

 * $$\int\limits^{-\infty}_{\infty} x^2 e^{-\alpha  x^2}={{1}\over{2}}\sqrt{{\pi}\over{\alpha^{3}}}$$

Funkcję na g(x), możemy unormować,

 * $$1=\int\limits^{\infty}_{-\infty}Ae^{-{B\over 2}( x- x_{0})^2}=A \int\limits^{\infty}_{-\infty}e^{-{B\over 2}( x- x_{0})^2}\Rightarrow 1=A \sqrt\Rightarrow A=\sqrt{{B\over 2}\over{\pi}}\Rightarrow \rho(x)=\sqrt{{B \over 2}\over{\pi}}e^{-{B\over 2}( x- x_{0})^2}$$

Stąd średnie odchylenie kwadratowe wyrażone jest wzorem:

 * $$\sigma^2=\int\limits^{\infty}_{-\infty}{( x- x_0)^2} \sqrt{{B}\over{2\pi}}e^{-B( x- x_0)^2}d x\Rightarrow\sigma^2=\sqrt{{B}\over{2\pi}}\int\limits^{\infty}_{-\infty}{t^2e^{-Bt^2}}$$

 * $$\sigma^2=\sqrt{{B\over 2}\over{\pi}}{{1}\over{2}}\sqrt{{\pi}\over{{B^3}\over 8}}\Rightarrow\sigma^2={{1}\over{B}}\Rightarrow B={{1}\over}$$

Związki w rozkładzie normalnym
Przedstawmy podstawowe wzory, które otrzymaliśmy z powyższych obliczeń,by określić w sposób ścisły stałe B i C
 * $$\rho^{srednie}=C_3 e^{-{B\over 2}{{{n-1}\over{n}}\sum^n_{i=1}(x_i-x_0)^2}}$$
 * $$\rho(x_{srednie})=C e^{-B{\sum^n_{i=1}(x_i-x_0)^2}\over {2n}}$$
 * $$\rho^{srednie}=C_3 e^{-{B \over 2}\sum^n_{i=1}(x_i-x_{srednie_i})^2}$$

Względem ostatniego wzoru mamy:
 * $$ C={{1}\over{\sqrt{2\pi}\sigma}} $$ i
 * $$B={{1}\over{\sigma^2}}$$

Jest to błąd pomiaru pojedyńczej wartości Dla drugiego wzoru mamy:
 * $${B\over n}={1\over \sigma^2}\Rightarrow {1 \over {n{(\sigma^{srednie})}^2}}={{1}\over {\sigma^2}}\Rightarrow {({\sigma^{srednie}}_2)}^2={{\sigma^2} \over n}$$

Wiadomo, że wzory pierwszy i trzeci oznaczają to samo,a więc:
 * $$C e^{-{B\over 2}{{{n-1}\over{n}}(x_i-x_0)^2}}=C e^{-{B \over 2}(x_i-x_{srednie_i})^2}$$

stąd sumując obie strony po 'i' mamy:
 * $$\sum^n_{i=1} {{n-1}\over n}(x_i-x_0)^2=\sum^n_{i=1}(x_i-{x_{srednie_i}})^2$$
 * $$\sum^n_{i=1}(x_i-{x_{srednie_i}})^2={n\over {n-1}}\sum^n_{i=1}(x_i-x_0)^2$$

Biorąc średnią arytmetyczną prawej i lewej strony otrzymujemy:
 * $${(n-1)\over n}\sum^n_{i=1}(x_i-x_0)^2=\sum^n_{i=1}(x_i-x_{srednie})^2$$

stąd otrzymuję:
 * $${\sum^n_{i=1}(x_i-x_0)^2}={n\over {n-1}}\sum^n_{i=1}(x_i-x_{srednie})^2$$

Biorąc srwdnią prawej i lewej strony otrzymuję błąd pojedyńczego pomiaru:
 * $$\sigma=\sqrt{\sum^n_{i=1}({x_i-x_{srednie})^2}\over{n-1}}$$

A stąd błąd średniej arytmetycznej:
 * $$\sigma_{srednie}=\sqrt{\sum^n_{i=1}({x_i-x_{srednie})^2}\over{n(n-1)}}$$

Rozkład normalny
Stąd po krótkich przekształceniach mamy:

 * $$\rho^{all}(\vec{x})={{1}\over{\sqrt{2\pi}\sigma}}e^{-{{\sum^n_{i=1}(x_i- x_0)^2}\over{2\sigma^2}}}$$

W zastosowaniach fizycznych stosowany jest przede wszystkim:
 * $$\rho(x_{srednie},\vec{x})={{1}\over{\sqrt{2\pi}\sigma}}e^{-{{\sum^n_{i=1}(x_i- x_{srednie})^2}\over{2\sigma^2}}}$$

Zwykle pomija się znaki sumy w wykładników potęgi, i piszemy:
 * $$\rho^{all}(x_i)={{1}\over{\sqrt{2\pi}\sigma}}e^{-{{(x_i- x_0)^2}\over{2\sigma^2}}}$$

W zastosowaniach fizycznych stosowany jest przede wszystkim:
 * $$\rho(x_i,x_{srednie})={{1}\over{\sqrt{2\pi}\sigma}}e^{-{{(x_i- x_{srednie})^2}\over{2\sigma^2}}}$$

Z obliczeń otrzymaliśmy też:

 * $$A=({{\delta k_i \ln k_i}\over{\delta x}})_{k_s}=0$$ oraz: $$B={({{\delta^2 k_i \ln k_i}\over {\delta x^2}})_{k_s}}={1 \over {\sigma^2}}$$ oraz $$C={{e^{-nD}}\over{({\int\limits^{\infty}_{-\infty}\rho(x)}})^n}={{1}\over{\sqrt{2\pi}\sigma}}$$

oraz $$D=k_i \ln k_i\;$$

Obliczanie poszczególnych momentów statystycznych dla rozkładu normalnego
Momenty ogólnie definiują się:
 * $$\gamma_n=\int\limits^{\infty}_{-\infty}(x-x_0)^{n}\rho(x)dx$$

Dla momentów statystycznych o stopniu nieparzystym:

 * $$\gamma_{2n+1}=\int\limits^{\infty}_{-\infty}(x-x_0)^{2n+1}\rho(x)dx$$$$=({{\int\limits^{x_0}_{-\infty}}+{\int\limits^{\infty}_{x_0}}})(x-x_0)^{2n+1}\rho(x)dx$$

Obliczamy tą całkę uwzględniając $$ t=x-x_{0}$$, toteż mamy:

 * $$\gamma_{2n+1}=({\int\limits^{0}_{-\infty}+\int\limits^{\infty}_{-0}})t^{2n+1}\rho(t)dt$$

A następnie biorąc podstawienie dla pierwszej całki$$t \rightarrow -t$$ mamy

 * $$\gamma_{2n+1}=\gamma_{2n+1}=({\int\limits^{\infty}_0-\int\limits^{\infty}_0})t^{2n+1}\rho(t)dt=0$$

Czyli momenty statystyczne o nieparzystym stopniu są równe zero.

Obliczenie momentów statystycznych o stopniu parzystym:

Wiadomo, że:

 * $$\sqrt=\int\limits^{\infty}_{-\infty}e^{-\alpha(x-x_0)^2}$$

Obliczając kolejne pochodne z lewej i z prawej strony ostatniego wzoru otrzymujemy dla pierwszej pochodnej::

 * $$\sqrt{\pi} {1\over 2}\alpha^{-{{3}\over{2}}}=\int\limits^{\infty}_{-\infty}(x-x_0)^2 e^{-\alpha(x-x_0)^2}$$

biorąc jeszcze raz pochodną ostatniej obliczonej obustronnie pochodną mamy:

 * $$\sqrt{\pi}{1 \over 2}{3 \over 2}\alpha^{-5 \over 2}=\int\limits^{\infty}_{-\infty}(x-x_0)^4 e^{-\alpha(x-x_0)^2}$$

Stąd ogólnie:

 * $$ \sqrt{\pi}{{(2n-1)!!}\over{2^n}}\alpha^{-{2n-1}\over{2}}=\int\limits^{\infty}_{-\infty}(x-x_0)^{2n}e^{-\alpha(x-x_0)^2}$$

Momenty statystyczne dla rozkładu normalnego dla potęg parzystych po podstawieniu $$\alpha \rightarrow {{1}\over{2\sigma^2}}$$ wynosi:

 * $$\sqrt{\pi}{{(2n-1)!!}\over{2^n}}2^{n+{{1}\over{2}}}\sigma^{2n+1}=\int\limits^{\infty}_{-\infty}(x-x_0)^{2n}e^{-{(x-x_0)^2}\over{2\sigma^2}}$$

Mnożąc obustronnie przez:

 * $${1\over{\sqrt{2\pi}\sigma}}$$

Stąd 2n-ty moment statystyczny wynosi:

 * $$\gamma_{2n}={(2n-1)!!}\sigma^{2n}\;$$

przyjmując jednocześnie $$(-1)!!=1\;$$, to obejmiemy również szczegół normowania gęstości prawdopodobieństwa dla $$n=0$$;

Punkt przegięcia w rozkładzie normalnym
Mamy oto rozkład normalny:

 * $$\rho(x)={{1}\over{\sqrt{2\pi}\sigma}}e^{-{{(x-x_0)}^2}\over{2\sigma^2}}$$

Wykonując pochodną względem x otrzymujemy:

 * $${{d\rho(x)}\over{dx}}={{1}\over{\sqrt{2\pi}\sigma}}e^{-{{(x-x_0)}^2}\over{2\sigma^2}}(-2){{x-x_0}\over{2\sigma^2}}$$

Oznaczając za stałe przez C, mamy:

 * $${{d^2\rho}\over{dx^2}}=C (e^{-{{(x-x_0)}^2}\over{2\sigma^2}}(-2){{(x-x_0)^2}\over{4\sigma^4}}+ e^{-{{(x-x_0)}^2}\over{2\sigma^2}}{{1}\over{2\sigma^2}})$$

 * $${{d^2\rho}\over{dx^2}}=Ce^{-{{(x-x_0)}^2}\over{2\sigma^2}}{{1}\over{2\sigma^2}}(1-{{(x-x_0)^2}\over{\sigma^2}})$$

Ponieważ liczymy punkty przegięcia funkcji Gausa rozkładu normalnego, to mamy:

 * $${{d^2\rho(x)}\over{dx^2}}=0$$

Stąd mamy:

 * $$1-{{(x-x_0)^2}\over{\sigma^2}}=0$$

W ostatecznych obliczeniach mamy:

 * $$x=x_0 \pm \sigma$$

Stąd można z interpretować jako że otrzymane $$x_i$$ w obliczeniach powinno się mieścić w granicach $$x_{i} \in (x_0-\sigma,x_0+\sigma)$$, a te wyniki, które nie spełniają tego warunku, powinny być odrzucone z danych doświadczalnych, i odnowa trzeba wyznaczać $$x_{srednie}$$ i $$\sigma$$ Przypominając przy liczeniu $$\rho(x)$$ trzeba wstawić za $$x_0$$ $$x_{srednie}$$ co dla dużej ilości wyników doświadczalnych jest w zupełności spełnione, ponieważ średnia arytmetyczna ma najmniejszą niepewność pomiarową z tych wszystkich otrzymanych $$x_i$$ z danych doświadczalnych. Dla tych wyników doświadczenia, które nie powinny być odrzucone funkcja z matematycznego punktu widzenia dla tego punktu powinna być wklęsła. Funkcja Gausa dla $$x\in (x_0-\sigma,x_0+\sigma)$$ jest funkcja wklęsła,a dla dopełnienia wcześniej przedstawionego wzoru jest wypukła.

Warto zauważyć, że definicja $$\sigma$$ dana za pomocą momentu statystycznego o stopniu dwa czyli średnie odchylenie standardowe jest poprawna.

Bibliografia
Siegmund Brandt: Analiza danych, metody statystyczne i obliczeniowe - PWN Warszawa 1999

B.M. Jaworski: A.A. Dietław Fizyka Poradnik encyklopedyczny - PWN Warszawa 1996

I.N. Bronsztejn: K.A.Siemiendiawjew Matematyka Poradnik encyklopedyczny - PWN Warszawa 1996

G.M.Fichtenholz: Rarunek różniczkowy i całkowy,tom 1 PWN Warszawa 1999

Zobacz też

 * rozkład normalny wielowymiarowy,
 * centralne twierdzenie graniczne,
 * przegląd zagadnień z zakresu matematyki,
 * przegląd zagadnień z zakresu statystyki

Twierdzenia matematyczne Rozkłady prawdopodobieństwa Dowody matematyczne Fizyka_matematyczna

title: User talk:Miroslaw.Makowiecki


text:
Design for the Environment/Frozen Yogurt Cups

This page is part of the Design for the Environment course

The following is a comparative Life Cycle Analysis (LCA) of three alternative materials for  frozen yogurt cups. The client is Yogen Fruz, and the current baseline is Expanded Polystyrene (EPS). EPS sets the standard for the functional analysis. The cups must be able to hold 125 ml frozen yogurt, be less than 10 grams, not easily crushed, and should ideally be a thermal insulator. The first alternative is polylactic acid (PLA), a derivative of sweet corn glucose. The second alternative is bagasse, a by-product of  sugar cane crushing. The functional unit for analysis is one million cups from preproduction through to the end of life. The motivation for this analysis was to determine the most environmentally friendly alternative, and to investigate the profitability and feasibility of investing in renewable products for cup production.

Highlights and Recommendations
From the various analysis tools used to look at the alternatives a rough ranking scheme is produced and tabulated below.

Summarised Results

This table does not provide a clear leader and so a more in depth discussion is required. Looking at the Economic Input Output Life Cycle Analysis (EIOLCA) results, it is shown that PLA and Bagasse have a similar environmental impact. However, it is not known whether the EIOLCA incorporates photosynthesis and the absorbtion of CO2 when growing corn in its results. As such the Global Warming Potential (GWP) output for PLA can be viewed as inflated. Further, when analysing the Streamlined Life Cycle Analysis (SLCA), Bagasse and PLA are very close. Since the SLCA is a subjective comparison tool, it is not reasonable to assume either product is superior from an environmental perspective based solely on these results.

The economic and societal impacts are leftover to judge these products. From an economic perspective, EPS cups are the clear winner. They are cheap and effective. Any client interested solely in direct economic costs would choose EPS cups, a choice which is reflected in day to day life by the ubiquity of EPS. However, from a societal perspective, PLA emerges as a highly effective product. It is deemed that the direct costs of PLA are likely to be offset by the positive brand image they represent. The 'green' factor of PLA cups is an obvious marketing tool. From these results, PLA is recommended as the most environmentally friendly and economically viable alternative.

Functional Analysis
EPS frozen yogurt cups are the baseline alternative, as such they define the functions that must be performed by the other alternatives in order to make a reasonable comparison. ‘Yogen Fruz’ frozen yogurt products are used to classify typical frozen yogurt cup requirements. A ‘Yogen Fruz’ serving size contains 125 ml and is solid enough to be held comfortably by a person without crushing it. An EPS cup weighs approximately 2 g, for functional purposes however, any alternative cup should weigh less than 10 g, in order to be held comfortably by a person. The cups must be produced so they are stackable and wide enough for a spoon. Additionally, EPS has an R-value of 1.2 m² °C/W for a 51 mm thick insulation board and is therefore a good thermal insulator. The functional unit analysed for all alternatives is one million cups.

Alternative 1: Bagasse Cups

Bagasse cups weigh 5 g, are soak proof and require no lining. Furthermore, bagasse is often used in composite materials to minimize heat conduction and increase tensile strength and is therefore both sturdy and functional as an insulator. Bagasse cups may be manufactured in 125 ml sizes. It is assumed from this that a bagasse cup will function as well as an EPS cup.

Alternative 2: Polylactic Acid Cups

The mechanical properties of PLA are comparable to those of polyethylene-terephthalate (PET). It can be assumed that the mass of one PLA cup is approximately the same as PET cup. Thus, the mass of one PLA cup is assumed as 2.5 grams. Given the small thickness of the cups, there is very little discrepancy between the insulation coefficient value of the PET cups and the PLA cups. PET cups are commonly used by large retailers such as Tim Hortons and  Starbucks to sell frozen drinks. It is assumed that it will not be uncomfortable to hold frozen yogurt in a cup made of PLA and that a PLA cup will function otherwise as well as an EPS cup.

Economic Input-Output Life Cycle Analysis
An Economic Input-Output Life Cycle Analysis (EIOLCA) is a publicly available tool for gauging the environmental impact of a product or service based on a dollar amount of production or use.

Methodology

The EIOLCA is governed by the 1997 U.S. Dept. of Commerce Industry Benchmark. As such, any dollar values input must be in 1997 USD. For this study the U.S. Consumer Price Index is used for all time conversions; The conversion factor is 0.758173333 from 2008 to 1997.

The packaging and shipping section of the EIOLCA is assumed only to be that required for the various internal processes. It is unnecessary to further complicate the EIOLCA with the addition of a shipping sector because the final comparison of packaging and shipping will be very similar for all three alternatives. The use phase is also ignored, because it is a disposable product with negligible environmental impact during its short use phase. Additionally, a mark up of 10% is assumed for the retail prices of all cup alternatives.

Expanded Polystyrene

For the purpose of this study, the premanufacture and manufacturing phases of the life-cycle are organized as shown in the flow chart at right. End-of-life assumes that 100% of the EPS cups end up in land fills. While EPS is theoretically recyclable, in practice it is rarely accomplished.

Included in the available EIOLCA sectors is ‘Foam Product Manufacturing’. This sector includes the manufacturing of polystyrene cups as an activity. It is assumed that this sector does not incorporate the premanufacture phase of the life-cycle. According to Bishop, the cost of the styrene monomer is roughly 40% of the total manufacturing cost of an EPS product. This 40% cost is input through the economic sector of ‘Petrochemical Manufacturing’, which is responsible for styrene production under the EIOLCA documentation.

A rough estimate for cost was gleaned from web searches for bulk cup sales. The cost used is averaged over three suppliers and is $0.025 per unit. Including the markup, the cost to the manufacturer is $0.0225. Therefore, 1,000,000 cups create $22500 in direct economic activity for the ‘Foam Product Manufacturing’ sector. Converted to 1997 seven dollars for input this is equal to $17058.90. Following the life-cycle breakdown 40% of this value is input into the ‘Petrochemical Manufacturing’ sector. To account for end-of-life land filling a weight of 2 grams is assumed per cup. The land filling cost for 2 tonnes of waste is $128.89 in 1997 USD. This value is added to the ‘Waste and Remediation Services’ sector. A breakdown of the economic inputs is given in the table below.

EIOLCA inputs for EPS

Results

The pertinent results are given in the figure at right organised in descending order by the top 5 contributors to GWP. The results seem reasonable on an intuitive level; based on the flow chart given above it is logical that petroleum related industries would be major contributors. Additionally, the ‘Power Generation and Supply’ sector pollutes very heavily for little economic activity, its position is not surprising.

Sensitivity Analysis

By assuming that the premanufacture process is not incorporated in the ‘Foam Products Manufacture’ sector, more than $6823.56 of economic activity has been added to the life cycle. With a total of only $62000 of economic activity, this amounts to a 12% increase. In Table 2 it is clear that the relationship between GWP and the premanufacture input is linear. The $6823.56 input to the ‘Petrochemical Manufacturing’ sector contributes roughly 50% to the GWP of the EIOLCA. It is interesting to note that when the premanufacture input is set to $0 then all petrochemical contributions drop out of the top 5 contributors to GWP in the EIOLCA. The assumption is therefore justifiable based on the definition for premanufacture given in Appendix A – Fig. A2.

Polylactic Acid
In order to create an EIOLCA model of PLA frozen yogurt cups, it was necessary to create a custom sector in order to obtain meaningful data. Two sectors, wet corn milling and power generation and supply, were combined to create an approximate equivalent sector for the production of PLA cups. These two sectors were used because the material output of wet corn milling is dextrose, and after that, the only major input to manufacture the final product is energy (See right).

To find the required input value for wet corn milling, values were used from the LCA of Natureworks PLA written by Vink et al, the Wall Street Journal , and three internet retailers. From these values, the cost of wet milled corn used to make one million cups was found to be $9700 in 2007. Converted to 1997 US dollars, this value is $7,360. This was the input for wet corn milling used in the final EIOLCA model.

The cost of energy required to make one million cups was found using the total energy used to create 1 kg PLA (54.1 MJ/kg), the mass of one cup (2.5 g), and the cost of 1 TJ found from the linearity of the EIOLCA model ($8559/TJ). After calculations based on these values, the dollar input to the power generation and supply sector was $1157 for one million cups.

It was found in the EIOLCA of expanded polystyrene that the end of life phase has minimal influence on the overall GWP or total releases. Thus, it was not included in the model used for PLA.

EIOLCA inputs for PLA

Results

The final EIOLCA table can be found in the image to the right. When compared to the literature, the results are quite different. Vink et al records a GWP of about 1.9 kg CO2 eq./kg PLA. When converted to one kg PLA, the EIOLCA gives a GWP of 19. From this alone, it is very clear that the EIOLCA is only a very approximate tool and cannot be relied on for very exact results, especially given the need to create a custom sector for PLA cups. One significant factor in this error is the lack of GWP offset in the EIOLCA model from the growth of the corn used to make PLA. In growing corn, significant amounts of CO2 are absorbed in photosynthesis. This would decrease the GWP of PLA substantially.

Sensitivity Analysis

A sensitivity analysis was conducted on the EIOLCA to find how much influence the source of energy has on the final GWP. This was done by varying the economic input to power generation and supply. Although the alternative sources of power would actually cost more, the GWP of these sources is minimal, so simply varying the economic input is a valid method of finding the sensitivity. The results of this analysis can be found right:

Bagasse
The resource extraction life stage for this analysis is considered to be an input of coal burning energy to displace the energy produced by burning bagasse, and a resulting output of bagasse. The EIOLCA sector for the bagasse premanufacture phase is the ‘Power Generation and Supply’. According to a study performed on a sugar cane mill in Catanduva,  Brazil, every season 374400 tonnes of bagasse is burned to generate 119700 MWh of energy. For the purpose of this study, this translates to 0.8688 kg Bagasse displaced per MJ. For the functional unit of 1 million cups, 5000 kg of bagasse is required (at 5 g/cup ) and therefore a total of 5755 MJ is required. It was calculated that $48.77 (1997 $USD) produces the appropriate energy (5755 MJ). According to the EPA, the emissions from burning bagasse in a boiler are summarized in the table below. These pollutants are subtracted from the final EIOLCA model in order to better reflect the displacement of bagasse burning.

Pollutants produced per 5000 kg bagasse burned

The EIOLCA sector used for the manufacturing life stage is ‘All other converted paper product manufacturing'. Outlined in the figure to the right is detailed information regarding the inputs to a bagasse cup manufacturing facility in China. Scaled to the production of one million cups at 4 g/unit, the required electricity in the factory is 38764.84 kWh. By using the linearity of the EIOLCA model to set the electricity used, it was clear the sector does not represent the manufacturing of bagasse cups from pulp very well. The Motor Gas used in the sector was lower by a factor of 7.87 and the Natural gas was higher by a factor of 1.45 compared to the data given by Worldcentric. The following Hybrid-LCA was performed: The cost of one 4 oz. bagasse cup was averaged from sources to be $0.04/cup. With an estimated 10% mark-up value for 1 million cups, the total cost in 1997 $USD is $27294.24. This total cost was used as an input into a hybrid version of the ‘All other converted paper product manufacturing’ sector which eliminates paper-specific and harvesting processes such as Logging and Sawmills in order to simulate the conditions for resource extraction. The following table provides a summary of inputs for the EIOLCA:

EIOLCA input for Bagasse

From the output of this EIOLCA the pollutants from burning 5000 kg of bagasse are subtracted and the final results are given in the figure to the right.

Results

The displacement of bagasse as an energy source with coal burning is actually beneficial to the environment with a GWP of -2.846 Metric Tonnes CO2 Equivalent (MTCO2E). This is justified by the energy density, and emission factors of each. The results obtained in the EIOLCA is compared to a process model (Bagasse Flow Chart, right) based on inputs obtained from Worldcentric and outputs calculated based on the work of Das and Houtman. The results of the two methods are markedly different, as expected. The EIOLCA is very general and not specific to bagasse or specific to different pulping processes. It is noted that the GWP emissions for the EIOLCA are a factor of 22.475/8.28 = 2.7 higher than that of the process model.

Sensitivity Analysis The sensitivity analysis measures the effect of increased transportation costs on the GWP, and to find a maximum economic value to match the GWP of bagasse to the other alternatives. The results are illustrated in the figure to the right. The chart concludes that for a total economic value less than roughly $6000 into the transportation sector, bagasse will still have less GWP emissions than the baseline polystyrene. In order to transport 1 million cups by cargo ship from China to the United States, the price of a 8 ft x 8 ft x 20 ft container costs on average $1856 ($2005). Using the dimensions of the 4 oz cups at Worldcentric, roughly 430000 cups can fit in one bin.

Converting the cost per bin to $1997, and assuming 2 bins will ship 1 million cups. The cost is as follows:

1856($2005) = 1525.28 ($1997)

2*$1525.28 = $3050.56

Streamlined LCA
A Streamlined Life Cycle Analysis (SLCA) is a tool to rapidly assign quantitative values to a product or service. The process is outlined by T. Graedel in the work ‘Streamlined Life Cycle Analysis’. This method allows for a quick comparison of different products or services to provide direction for a more in depth life cycle analysis.

Methodology

In the case of frozen yogurt cups, the use phase is ignored for environmental impact, as it is effectively negligible. The major areas of concern are premanufacture, manufacture and end-of-life. Premanufacture and manufacturing each receive a weighting of 1.75 and end-of-life receives a weight of 1. The remaining 0.5 is assigned to shipping.

Expanded Polystyrene
The streamlined LCA (shown below) for EPS indicates that it is extremely hazardous to the environment. The low scores are mainly attributable to the premanufacture and end of life phases. This is expected as petrochemical processes are defined as the some of the most energy intensive processes by T. Graedel. Additionally, EPS is not biodegradable and scores poorly in the solid-residues stressor of the end-of-life phase. These results correlate well with the EIOLCA results.

EPS Streamlined Life Cycle Analysis

Polylactic Acid
In the both the SLCA and the EIOLCA the phase with the largest environmental impact is premanufacturing (shown below). This is because the processing of the corn into dextrose and the dextrose into PLA is energy intensive compared to the rest of the phases. The manufacturing only includes the moulding of the PLA into cups which requires energy, but not significant amounts compared to premanufacturing. Although the SLCA would suggest that PLA is beneficial compared to the alternatives, there is a large discrepancy between the results of the SLCA and the EIOLCA. This is likely because the EIOLCA was not capable of accurately representing the environmental impact of PLA. However, the results of the SLCA are more consistent with the results of two independent LCA papers which utilize different tools,. Thus, the results of the SLCA, despite being less quantitative, may be interpreted as more meaningful than the EIOLCA.

PLA Streamlined Life Cycle Analysis

Bagasse
It is clear in the streamlined LCA (shown below)that the largest source of environmental impact over the Life Cycle of bagasse cup production is the manufacturing phase. This is due to the energy used and wastes and emissions generated from the chemical pulping process. This correlates well with the results generated from the EIOLCA model, as “Power Generation and Supply” and “Paper and Paperboard Mills” (which includes pulping) are the largest sources of GWP. In addition, there is a large amount of emissions associated with the shipping of bagasse cups overseas, resulting in poor grading for the energy use and gaseous emissions in the transportation life stage. However, bagasse cups performed well in the resource extraction and disposal life stages – an expected result for a renewable, biodegradable material.

Bagasse Streamlined Life Cycle Analysis

Cost Analysis
When designing a product for the environment it is difficult to assign economic value to many aspects of the life cycle. However, there are direct costs to a buyer that can be quantified when a purchase is made. These costs are summarised in the table below. The important costs are the initial product costs and the end of life costs. The end of life costs are assumed on a conservative basis, in that the end user will be ignorant of recycling facilities and an equal number of cups will end up in a land fill for each alternative.

Direct Costs Summary

The carbon credit cost is a method of quantifying GWP with a dollar value. One carbon credit is worth one MTCO2E and these credits are traded online on the Chicago Climate Exchange. The market value of one credit at the time of writing is $5.50. This market value is multiplied by the GWP given by the EIOLCA for each alternative.

The total costs to the buyer are lowest for the baseline polystyrene cups. This is a reasonable result based on the ubiquity of polystyrene cups in daily life. The important aspect of this data is that the direct costs of purchasing the cup are by far the most significant in the total costs. This cost outweighs any possible economic benefit derived from the lower environmental impact of the alternatives. Even if we assume that all of the bagasse cups are left to biodegrade and not land-filled, they are still $15208.25 more expensive to the buyer than the polystyrene cups. A similar issue arises with the polylactic acid cups, if they were recycled at 100% efficiency and at no cost (which is idealized and not possible) they are still $25193.50 more expensive than the baseline. It is clear that the unit cost is the most important property of any alternative from a purely economical point of view.

The indirect economic aspects are the long term costs of global warming and land consumption and are not as easy to quantify. Polystyrene is not biodegradable, when it ends up in a land fill it will continue to take up that space indefinitely. This results in a cost of land use to the average citizen through taxes. This space usage does not occur with bagasse or PLA when they are properly recycled or composted. The GWP of the various alternatives results in air pollution, health problems and other environmental issues. The indirect costs associated with each alternative were assumed to be insignificant in determining a final recommendation on the order of one million cups.

Societal Analysis
Although the difference in costs to Yogen Fruz for the various alternatives is significant, the effect on the end cost of the frozen yogurt is very small. PLA cups would cost Yogen Fruz double what EPS cups would cost. However, one small frozen yogurt costs about $4.00. Thus, the final influence of $0.025 on the final product cost is not very significant. This cost can be balanced by the positive brand image resulting from being more environmentally friendly.

Another societal influence on the viability of the different alternatives is aesthetics. While bagasse is in fact cheaper than PLA, it is being used on a smaller scale in North America. This is likely because it does not look as appealing as PLA which is clear and easily printable. Once again, the difference in cost per cup is negligible compared to the increased aesthetic appeal of PLA.


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Design for the Environment/Power Generation

Ontario Power Generation (OPG) is one of the largest power generators in North America. The Crown Corporation is owned by the Government of Ontario and it was established in 1999 under the provincial premier, Mike Harris. OPG currently owns 75 power generation facilities across Ontario to generate electricity at a capacity of 22000 MW. It generated over 105 terawatts-hrs (TWh) of electricity in 2006 to supply Ontario’s needs. Nuclear power generation accounts for 44% of the total power generation mix, hydroelectricity for 32%, and 24% for fossil fuel. One of OPG’s primary focuses is to generate electricity in an environmental friendly approach by reducing CO2 and other Green House Gas (GHG) emissions.

Background

The Nanticoke power plant located in Haldimand County, Ontario is the largest coal fuelled power plant owned by OPG. Currently, Nanticoke is using Pulverized Coal (PC) power generation process to generate electricity. The power plant has a total of eight steam turbines which generate electricity at a maximum capacity of 3920MW.

The use of coal to generate electricity produces Green House Gas(GHG) emissions to the environment hence, a retrofit proposal is provided to improve the current electricity generating method at the Nanticoke power plant. The Integrated Gasification Combined Cycle (IGCC) is a well developed technology known to generate electricity using less coal compared to the PC method. Additionally, with the Carbon Capture and Storage (CCS) technology implemented to the IGCC process it reduces the CO2 emission which aids in decreasing the GHG's released into the atmosphere.

Project Information
Section 1 Group 7A Chiu Yuen Huen (THuen) Michael Khelawan (Michael Khelawan) Kiri Panchalingam (Kiri) Geerthanan Thiviyarajah (thiviya1)

Highlights / Recommendations
The Pulverized Coal (PC) power generation method is a traditional method of generating electricity which has inherent negative impacts on the environmental. PC plants are relatively cheap and easy to construct; however, there are increasing demands to reduce GHG emissions. Therefore, the PC generation method, which generates 5100000 MtCO2e of GWP per year, will need to make some changes to reduce emmissions. To replace this old technology, the Integrated Gasification Combined Cycle (IGCC) or IGCC with Carbon Sequestration (CCS) are two promising technologies capable of delivering cleaner energy. CCS is a costly technology at the present day to construct. Nonetheless, the benefits of this technology over-ride the negative aspects and this method can greatly reduce GHG emissions by 90%. Furthermore, considering it can yield oil production for Canadian oil fields by 100% this alternative is best suited as a retrofit to the existing Nanticoke Power Generation Station.

Pulverized Coal
Using coal to produce electricity was initially developed in the 1880’s; however, using pulverized coal (PC) was introduced later in the 1920’s. The success of this technology (at least from an economic sense) is evident from its wide use even today. For example, according to, in 2001, 51% of the electricity generated in the United States was from coal burning plants. The popularity of using PC in power generation stems from the fact that coal “is the most abundant fossil fuel in the world”. However, it would be insufficient to determine the success of this technology solely on economics.

In the United States, roughly 30% of all solid residue from coal combustion is recycled ; a relatively low percentage. At the Nanticoke plant, the ash residue is recycled and sold for use for making cement. However, it should be noted that flue gas desulphurization (FGD) technology has not been used. This technology essentially reduces the amount of sulphur dioxide emitted, by over 90%. In fact, it is evident that the Nanticoke station is a heavy polluter.

The monetary cost of a process is a key factor in determining its relative success. The direct costs associated with the operation of a plant such as the Nanticoke generating station is summarized below:

The above values are from the International Coal Development Institute (ICDI) for typical coal powered power plants. However, since the size of power plants varies significantly, some of the assumptions made by ICDI may not be accurate for the case of the Nanticoke power plant. The above capital costs are calculated at $900/kW of capacity. Therefore using this estimate for the Nanticoke plant, the capital costs would be 3.5 billion dollars (1986 US $); a significant difference. Indirect costs associated with the amount of pollution caused to the environment by the plant also have monetary value. These damages have been quantified by in terms of $/ton for various gaseous releases. The table below shows the costs for the major gaseous releases at the Nanticoke power plant.

If we compare the values from the above two tables, it is evident that the indirect cost is roughly half the direct cost; a substantial amount. Summing the direct and indirect costs, we get a total of 686.4 million dollars. Since many assumptions were made in the process of determining these costs, the numbers are likely skewed. For example, the cost per ton of gaseous releases stated by had a broad range and it was assumed to be the mean of these values.

Integrated Gasification Combined Cycle

Integrated Gasification Combined Cycle (IGCC) is a fast growing technology in the power generation industry. IGCC is being used to eliminate the oxides of nitrogen, sulfur, and greenhouse gas emissions to improve fuel efficiency of the pulverized coal system. This technology generates power with little use of solid and liquid fuel.

The IGCC power plant burns coal to generate electricity but additionally, it uses a gasification method. The added gasification process allows the coal elements to be differentiated from the synthetic gas, which is often referred to as syngas. Syngas is a mixture of carbon monoxide and hydrogen. Followed by the production of syngas, gasses such as hydrogen sulfide, particulate matter (PM) and many other pollutants are removed from the syngas through the use of filters. The cleaned syngas is then combusted in a gas turbine which produces electricity. The steam from this process drives a steam turbine and generates electricity. The process of generating electricity in an IGCC power plant can be seen below.

There are many advantages in choosing an IGCC system to generate electricity rather then the common Pulverized Coal system. One of the advantage that an IGCC system brings forth is its ability to produce electricity through a combination of gas and steam turbine. Additionally, this clean coal technology is an air pollution control, and an environmentally friendly approach. It also achieves ~50% thermal efficiency, which means less coal is needed to produce the same amount of energy as the coal-based system. Nonetheless, with the combination of the new technology such as, Carbon Capture Sequestration (CCS) emissions can be further reduced as explained in CCS section.

The IGCC power plant is a cleaner technology for the environment; however, the economic analysis must also be conducted in order to see the overall benefit of the technology. There are many costs that are taken into consideration in the analysis such as capital, operation and maintenance cost. Refer to Cost Tables for details. However, an IGCC plant cannot be retrofitted to an existing pulverized coal plant therefore it must be built from scratch. Though, many parts can be used from the pulverized coal plant, there are some extra fitting that must be taken into consideration such as the gasifier, syngas cooler and the air separator. Refer to Cost Tables for details. The additional cost that is involved in generating electricity through the IGCC is the disposal of residue cost. The plant requires about 1.1 metric tonne of coal to generate electricity for a year with 44 rails transporting the coal to the power plant. The cost of low sulphur coal as of today is approximately $55 million and the cost of transportation is ~$9.20 million [34]. On average an IGCC power plant uses about 1.1 metric tonne of coal an year which produce roughly 245,000 mt of coal ash. However, only about 85,000 tonne is the fly ash. The disposal of the fly ash is ~$30 a tonne hence, an IGCC power plant can expect to spend about $2.55 million a year for disposal fees.

Many indirect costs should be also considered in a full economic analysis as it reveals the weakness of the power plant. The IGCC power plants’ main source in generating electricity is coal, as mentioned before, it releases many gases into the atmosphere causing premature deaths and long-term health risks. Under the Canada’s Kyoto agreement, the cost of greenhouse gas (GHG) is ~$15/tonne. On average, health damages range from a minimum of ~0.4 billion yearly. The generation of electricity not only cause health issues but, also causes environmental damages. It is evident from the studies that in general, the environmental damages range from a minimum of ~$48 million. It should be noted that the cost of damages does not include all damages and therefore the figures mentioned above should be interpreted with care.

Integrated Gasification Combined Cycle with Carbon Sequestration
Carbon sequestration can be defined as the capture and storage of carbon that would otherwise be emitted into the atmosphere. The process is otherwise known as carbon capture and storage (CCS) and the purpose of this process is to enable the use of fossil fuels while reducing the emissions of carbon dioxide into the atmosphere therefore alleviating global warming which has become a major source of environmental concern.

When carbon sequestration is applied to an IGCC power plant, it is capable of reducing atmospheric emissions drastically by 80-90%. The process would increase the energy needs by about 10-40% and the costs of storage and other system costs are estimated to increase the costs of energy by 30-60%.

There are three types of CCS technologies that currently exist. The method that will be studied is post-combustion, otherwise known as flue gas separation is a process in which CO2 is captured from flue gases. This is currently the best understood method and most commercially used.

The CO2 product can be stored in deep geological formations, deep oceans, or also in the form of mineral carbonates. The most effective way to store sequestered CO2 is the Enhanced Oil Recovery which is the injection of inert gases into depleted oil fields to increase the yield. Therefore, more oil can be extracted from the fields leading to increased energy supply. For this study the captured CO2 will be transported by pipelines from the power plant to the Weyburn Oil Field in Saskatchewan. This field has been chosen for the study because it is currently being used for enhanced oil recovery. The Weyburn Field allows for permanent CO2 storage and over the anticipated 30-year life of the project, it is expected that the injection of 18 million tons of CO2 injected will produce 130 million barrels of oil. This has been calculated to be equivalent to approximately 14 million tons of CO2 being prevented from reaching the atmosphere, including the CO2 emissions from electricity generation that is required for the whole EOR operation. Transporting the CO2 from the Nanticoke plant in Ontario to the Weyburn field will require about 1500 miles of pipelines. In order to make an existing IGCC plant capable of capturing, it needs to be retrofitted with Carbon Capture equipment including 2 CO- Water Gas Shift Reactors, a CO2 Acid Gas Removal Section (AGRU), and a CO2 compressor.

In the resource provisioning stage of the life cycle analysis, the building materials for the plant and the coal that the plant will use for fuel needs to be supplied. The major building materials needed for the plant include concrete and steel. More coal is needed in comparison to an IGCC plant without CCS. It is estimated that 1.35 Mt of coal is used a year, and in a total life-time of 30 years a total of 40.5 Mt of coal is used. To mine this coal it will cost $67.5 million/year and $2025 million/life-time. It is also estimated that it will cost $20.3 million/year or $609 million/life-time to transport this coal to the plant via railway. The retrofits which are implemented to facilitate carbon capture include 2 CO Water Gas Shift Reactors, a CO2 Acid Gas Removal Units and a CO2 Compressor costing $34, $98 and $40 million respectively. The capital cost is $2600/KW and therefore for a 500MW plant $1300 million.

The cost of electricity for IGCC with capture and Enhanced Oil Recovery is 4-8¢/kWh. For this study the cost of electricity at 6.74¢/kWh will be used. In general, there is a 1-2¢/kWh increase in price for IGCC plant to adopt carbon capture technology. For a 500MW plant, the cost of electricity is $29.5 million/year and $885 million/life time of 30 years .The Operating and Maintenance cost is about $31.2/year or for entire life it is $936 million. If carbon dioxide will be transferred via pipeline onshore from the Nanticoke Power Generation center is southern Ontario to Weyburn oil field in Saskatchewan, which is approximately 1500 miles, it is calculated to cost approximately $25/t CO2.

Indirect costs include environmental, health, and social costs. The major environmental benefit of the Carbon Capture and Storage system is that it decreases atmospheric emissions of carbon dioxide by 90%. This will play a major role in alleviating global warming in the future. If carbon sequestration is made commercially available worldwide, and if governments work together to enforce policies for decreased emissions, this global warming crisis can be averted. This technology can not only be retrofitted to power plants, but all industrial and commercial sites emitting carbon dioxide. This technology will help Ontario become a leader in power generation and it will play a major role in Canada’s plans to meet its Kyoto protocol commitment as Ontario still heavily relies on fossil fuels.