The calculated quantity of gas necessary to heat water to 105 °F, daily, was then subtracted from the total daily gas usage and plotted with the average daily temperature10 in Figure 6-3. The trendline in Figure 6-3 demonstrates that the amount of gas used besides that for heating water, i.e., for space and pool heating, decreases as temperature increases. The reason it does not approach zero at temperatures where no space heating is used is because even during warm weather, the spas and the outdoor pool continue to be heated. Some heat is also used for the dryer, although it is small compared with the space and water heating requirements, as was shown in Figures 5-3.
The economic characteristics of the cogeneration system also depend partly on the size the cogenerator needs to be, and therefore the amount of hot water it can produce. A statistical distribution of the daily water usage data was applied to the probability density function in order to demonstrate the facility’s water usage pattern. This analysis was shown in equations (3.7), (3.8) and (3.9). The probability distribution8 of the daily water usage data is shown in Figure 6-4a. A histogram of the daily water usage data provided in Figure 5-5b was compared with the fitted probability density function in Figure 6-4b. The discrepancies in this comparison can be explained by several possibilities, e.g., some scatter in the data and lack of resolution in the meter. If more daily data was available, the validity of the assumed lognormal distribution could have been better determined.
The inverted cumulative distribution8 of the water usage is shown in Figure 6-5. This figure was created by means of equations (3.10) and (3.11). Figure 6-5 shows that in order to meet 95% (347 days out of the year) of the service hot water thermal load at Athletic Club Northeast, they need 80 kW of turbine power output. Three 28-kW micro-turbines are necessary to meet this requirement, which is actually 84 kW of power. This in turn actually means 98% of their annual hot water needs would be satisfied. The rest of their thermal hot water load can be met as it is now, by the gas-fired water heater, but this would be quite rare.
As discussed earlier, the health club does not consume any hot water when it is closed. If the cogenerator were run when the club is closed, the hot water it produced would have to be stored and the facility would likely have to purchase a large tank for that purpose. Further, the cogenerator would be producing electricity at a time when electricity is not being used. This would be wasteful. It is really only practical to run the cogeneration system during business hours, because that is when the facility experiences thermal and electrical demands. Knowing the business hours of the facility, and assuming 100% of full load, the maximum hours of daily runtime of the cogenerator can be found for the situation where the cogenerator never runs while the club is closed, but only during business hours:
(6.1a)
(6.1b)
where RTMAX is the maximum daily runtime of the cogenerator. This equals the average number of business hours in a day. The corresponding maximum load factor is equivalent to that number non-dimensionalized:
LFMAX = 0.625 or 62.5%
(6.1c)
where LFMAX is the maximum load factor.
The first law of thermodynamics relates the power rating of the cogenerator to the amount of hot water it can produce. This relationship was demonstrated in equation (2.15). Assuming the maximum load factor of 0.625, Figure 6-6 shows what size cogenerator is needed to produce a given amount of hot water.
The general economic characteristics of the cogeneration system can now be plotted with the power rating of the cogeneration system. A fixed cost of electricity was initially assumed for this calculation in order to obtain preliminary estimates of the net savings and simple pay back period of the cogenerator. That fixed rate was taken as $0.062/kW-hr. Knowing how much hot water a certain size cogenerator can produce, and knowing the cost of gas, equation (3.4b) was then used to create Figures 6-7, the Estimated Net Annual Savings and the Estimated Simple Pay Back Period of the cogeneration energy system. These calculations also included sales tax, which is 7% in Georgia. Note how the pay back period increases as the cogenerator equipment cost, CCOG, increases. The impact diversified data has on the economics of cogeneration are also demonstrated in Figures 6-7. Multiplying the standard deviation of the data, in this case by a factor of ten because sLN was very small, shows that the more scatter there is in the hot water usage profile, the lower the savings of the cogeneration system can be. This results in a longer pay back period as well. The economic characteristics can be better, but only for systems that are too large to be practical for this facility’s needs, as will be discussed later.
At certain times of the year the cogenerator is oversized to meet its probable daily requirement. The maximum amount of hot water that the cogeneration system could produce in a year, VW,Y, was found by using equation (3.12). Dividing this number by the maximum daily hot water production of the cogenerator, which is equal to x, gives the number of hours the cogenerator would actually be run throughout a year:
(6.2)
where ty is the annual runtime of the cogenerator. The calculation of ty is always less than or equal to 62.5% of the number of hours in a year, 5475. This is how many business hours there are in a year. Knowing the power rating of the cogenerator and the number of hours per year that it runs, the amount of electrical energy the cogenerator would actually produce in a year, ECOG,Y, is found as follows:
(6.3)
Georgia Power’s electrical billing structure7 was used with ACN’s electrical usage history to determine what the exact savings and simple pay back period of implementing a cogeneration energy system would be if ACN’s electrical usage continued as it did in the past year (see Appendix C). Therefore, to find the value of the electrical energy produced, again on an annual basis, multiply equation (6.3) by the exact cost of electricity noted in the electric bill history7:
(6.4a)
where CE,DAT is the cost of electricity based on electric bill data. Substituting equation (6.3) into equation (6.4a) yields:
(6.4b)
Since ACN is billed for electricity on a monthly basis, it is necessary to calculate how many hours per month the cogenerator would run. Since the cogenerator runs to satisfy the hot water thermal load, and this load is fairly constant, it is reasonable to assume that the cogenerator runs the same amount of time every month:
where tm is the number of hours the cogenerator would run every month. Equation (6.4b) is then adjusted for monthly terminology:
(6.4c)
where $E,COG,M is the value of the electricity the cogenerator would produce every month. Twelve of these are summed together to find the value of the electricity the cogenerator would produce over a year:
(6.4d)
It is necessary to use a full year of electric bill data because the amount of electricity used for space cooling at the athletic club is very different during winter months than it is during summer months, as demonstrated in Figure 5-2c.
In order to find the total annual savings of running the cogenerator, the added cost of gas to run it must be accounted for, as stated earlier in equation (3.1). Since this calculation is now based on electric bill data7, the new notation, $SAV,DAT, is used to describe the savings:
(6.5)
Modifying equations (3.3b) and (3.3c) to use the annual hot water production of the cogenerator, VW,Y, instead of the daily production, VW, results in the following equations:
(6.6)
(6.7)
where the subscript "Y" merely denotes that these values are annualized. Here again, the current cost of gas, CG, is $3.40/MBtu. The change in temperature is still based on the assumption that the average temperature of the water consumed, TT, is 105 °F. The average ambient temperature in Atlanta over the past year10, 63.8 °F, was used for TC. Subtracting equation (6.7) from (6.6) gives:
(6.8a)
where D$GAS,Y is the increase in the amount of money that is spent on gas throughout a year due to replacing the water heater with the cogenerator. The term in parentheses can be simplified to obtain:
(6.8b)
Having derived equations (6.4d) and (6.8b), equation (6.5) can be re-written:
$SAV,DAT = $E,COG,Y - D$GAS,Y
(6.9)
where all three terms are in units of $/yr. The simple pay back period15 is then found in the same manner as described in equation (3.6), only now with nomenclature expressing that the terms are based on the electric bill data:
(6.10)
Because the equations derived in this section are based on the exact cost of electricity according to ACN’s billing history, they were used to create Figures 6-8. Figure 6-8a demonstrated the Net Annual Savings and Figure 6-8b, the corresponding Simple Pay Back Period. Here again, a sales tax of 7% was accounted for. These figures reemphasize the effect a more diverse water usage profile has on the economic characteristics of the cogeneration energy system by showing curves in which the standard deviation was multiplied by ten.
The 28-kW Capstone stand-alone low-pressure micro-turbine generators cost $36,000. A reasonable maintenance cost for such a system would be around 3% of its purchase cost per year1. Therefore, assuming $1080 per year (for each turbine) in maintenance costs throughout a ten-year life cycle, and compatible heat exchangers for about $400 each, the cost of the cogeneration system, CCOG, would be equivalent to $1685.7/kW (see Appendix B).
The simple pay back period is found by linear interpolation in Figure 6-8b to be 6.4 years both for a system comprised only of one turbine and for a system comprised of two turbines, 6.8 years for three turbines and 7.4 years for four turbines. Although the simple pay back period is shorter for smaller cogenerator systems, it is probably best in this case that a three-turbine system be implemented at the facility. The difference between the pay back period for a one or two-turbine system and a three-turbine system is less than five months. Figures 6-5 and 6-6 show that only having two cogenerators would not satisfy a significant amount of the health club’s hot water needs.
If the system were only comprised of one or two cogenerators, it would be run at the maximum load factor of 0.625, without back-up. Figure 6-9 demonstrates this by showing the hours of runtime per year and the equivalent load factor plotted with the power rating of the cogeneration system. The equivalent load factor is an annual average, based on the annual runtime calculated in equation (6.2). The pay back periods for the one and two-turbine systems are equal because their annual runtimes are equal. If the thermal demand of the facility was so large that a third cogenerator would also be run at the maximum load factor, then the three-turbine system would also have a simple pay back period of 6.4 years. As it is, a third cogenerator would not need to be run during all business hours and hence could also serve as a back-up for the other two. A three-turbine system could satisfy nearly all of the facility’s hot water needs. A fourth cogenerator would probably be excessive. It would rarely be run, and it adds over seven months to the pay back period over a three-turbine system.
The 84-kW micro-cogeneration system runs 4560 hours per year (see Figure 6-9). The amount of electricity produced by this system is found by means of equation (6.3) to be 383,000 kW-hr. The net annual savings of the cogeneration system is $20,800 (see Figure 6-8a). Dividing the net annual savings, $20,800, by the amount of electricity produced, 383,000 kW-hr, gives the net average savings from the electricity that was generated from natural gas instead of having been purchased from the utility. That net savings is $0.0543/kW-hr that was produced by the cogenerator, instead of being purchased. The value of the electricity produced equals the sum of the net savings and the cost of the natural gas purchased to generate that electricity, which conceptually can be seen by solving equation (6.9) for $E,COG,Y. The cost of natural gas is $3.40/MBtu. Converting units and adding 7% sales tax, the cost of natural gas is $0.0124/kW-hr of electricity produced by the cogeneration system. Adding that to $0.0543/kW-hr gives the average value of the electricity produced by the cogenerator, instead of being purchased from the utility, which comes to $0.0667/kW-hr.
It was noted earlier that the club would save $20,800 per year in utility costs throughout the three-turbine cogeneration system’s life. This is approximately 9.5% of their current utility costs and corresponds to an internal rate of return on the investment15 (IRRI) of 9.73%, assuming the cogeneration system lasts 10 years (see Appendix B). The IRRI is for systems that have longer lives. Further, 10 years is a very low estimate for the life of a turbine. These types of system components can last anywhere from 10 to 34 years1. If the turbine’s life is assumed to be 15 years instead of 10, the same calculation yields an IRRI of 13.8%. If the life of the system is 20 years, the IRRI becomes 15.1%.
Another way to increase the IRRI is to finance some of the capital investment at a lower APR (Annual Percentage Rate) than the original IRRI15. For example, if 80% of the capital investment were financed over a 10 year period at an interest rate of 7% APR, the IRRI for the system that lasts 10 years would be 19.5% (see Appendix B). Again, the longer the system lasts, the higher the IRRI.
The profit of each ton of carbon reduced, like the IRRI, is larger for longer-lasting systems. Using an interest rate of 7.5% to represent the time value of money15, it was found that for a system that lasts 10 years, the profitability of emissions reduction is $35.32 per ton of carbon. It is $111.08 per ton for a 15-year system and $146.51 per ton for a 20-year system (see Appendix B). Note that these are positive numbers. Studies have calculated that carbon emissions reductions cost $100-200 per ton13. These studies did not consider the recent developments in technology which make cogeneration systems possible for commercial, as well as residential, applications. Instead of costing $100-$200 per ton of carbon emissions reduction, a profit in a similar range is made.
It may seem on the surface that the power industry would suffer if a cogeneration revolution occurred. In fact, this is not the case. Energy demands increase every year because of natural economic and population growth. If some of the increasing power demand were met by dispersed power generation, central utility power providers would not have to invest in new capacity quite as quickly.
Government provides incentives for certain industries that benefit national interests. The environment has become a high international priority. Energy consumption can be lowered as a result of these incentive programs. Incentives such as tax credits could motivate more business and home owners to install cogeneration energy systems. Deregulation also may have a significant impact as more states begin to permit their residents and businesses to generate and sell power, rather than being required to purchase all of their electricity from regulated power utilities.
The problem remains however of informing energy consumers of cogeneration technology. Infomercials, newsletters, radio announcements and any other form of communication that people would not have to purchase or subscribe to can accomplish this. Once informed, some persons would take some time to investigate and decide to take advantage of this new micro-turbine technology that benefits both the national economy and the environment.