(2.1)
where QGAS,WH is the natural gas energy input to the water heater per day, hWH is the efficiency of the water heater, and rW, VW, CP,W and DTW are respectively the density, volume, constant-pressure specific heat and temperature change of the hot water produced. The amount of natural gas necessary to run the water heater is related to the necessary gas energy input by the heating value of natural gas6, HVGAS:
(2.2)
where VGAS,WH is the volume of natural gas the water heater uses per day.
Looking only at the turbine generator (see Figure 2-2), the first law states6:
(2.3)
where QGAS,COG is the energy input to the system by means of natural gas, QEXH is the thermal energy released by exhaust gases of the system and ECOG is the electrical energy produced by the system, all on a daily basis. Again, the amount of natural gas needed to
run the system is related by its heating value and the gas energy input:
(2.4)
where VGAS,COG is the volume of natural gas used by the cogenerator on a daily basis. Defining hE as the gas-to-electric conversion efficiency of the cogeneration system,
(2.5)
then
(2.6)
For the hot exhaust gas side of the heat exchanger, which recovers thermal energy from the cogeneration system exhaust (see Figure 2-2), and having an exhaust heat recovery efficiency hR:
(2.7)
where QR is the amount of thermal energy that gets transmitted to the water which is being heated. Substituting equation (2.6) into equation (2.7) yields the following result:
(2.8)
For the fluid in the cool side of the heat exchanger (see Figure 2-2), which is taken as water, the first law of thermodynamics dictates:
(2.9)
Note that all of these equations are still being derived on a daily basis. Hence, VW is the volume of hot water a facility uses per day. Substituting equation (2.7) into equation (2.9) and solving for QEXH gives the following result:
(2.10)
Substituting equation (2.6) into equation (2.10) and solving for QGAS,COG gives the following:
(2.11)
Solving equation (2.5) for QGAS,COG and substituting it into equation (2.11) and solving for ECOG gives this result:
(2.12)
To find the maximum power rating of the cogenerator necessary to meet a given hot water demand of a facility, again on a daily basis, it is necessary to define a cogenerator load factor and relate it to the electrical energy output, ECOG, corresponding to the daily hot water demand, VW:
(2.13)
where 
is the maximum power rating of the cogenerator and tperiod is the time period of interest. The load factor, LF, is the product of two factors: the percentage of time that the cogenerator runs, and the percentage of full load at which it operates:
LF = (% time) (% full load)
(2.14)
If the cogenerator ran at 50% of full load 100% of the time, it would have the same load factor as if it ran at 100% of full load 50% of the time. It is possible that the cogenerator may operate at a different gas-to-electric conversion efficiency depending on what percentage of full load it runs, but in this study the gas-to-electric conversion efficiency was assumed to be constant. Substituting equation (2.13) into equation (2.12) and solving for results in the following equation:
(2.15)
The above equation is the relationship between the size of the cogenerator and the daily hot water produced by that cogeneration energy system.
(2.16a)
or
QGAS,COG - QGAS,WH = QGAS,COG - QGAS,COG(1 - hE)
(2.16b)
Similarly, substituting equations (2.2) and (2.4) into equation (2.16a) results in the following:
(2.17a)
or
(2.17b)
The energy analysis below demonstrates in thermodynamic terminology how and why cogeneration energy systems are beneficial in the area of conserving energy and reducing the energy costs of a facility. An equation is derived to describe how much of the extra gas energy used by a cogenerator is converted to electricity. The ratio of the amount of gas energy used by the water heater to the amount of electrical energy the cogenerator would produce if it were used to produce a corresponding amount of hot water is found by dividing equation (2.1) by equation (2.12):
(2.18)
The ratio of the amount of gas energy the cogenerator uses to the amount of electricity it produces is found by rearranging equation (2.5):
(2.19)
Subtracting equation (2.18) from equation (2.19) and substituting equations (2.1), (2.11) and (2.12) gives the ratio of the excess gas energy consumed to the electrical energy produced by the cogeneration energy system:
(2.20)
For the case where the efficiency of heat recovery, hR, is equal to the efficiency of the water heater, hWH, equation (2.20) simplifies as follows:
(2.21)
This means that 100% of the extra gas energy consumed by the cogenerator is converted to electricity. Contrary to a possible initial reaction, equation (2.21) does not violate the Carnot principle6. The equation does not state that the operating efficiency of the cogenerator is 100%. It states that exhaust heat is being recovered, not wasted. For cases where hR does not equal hWH, e.g., if hR = 70%, hWH = 80%, and hE = 26%, then equation (2.20) states that 61% of the incremental gas energy used is converted to electricity. If hR = 80% and hWH = 70%, again with hE = 26%, equation (2.20) states that 59% of that energy is converted to electricity.
The water temperature rise and the amount of hot water consumed are closely linked. The two terms must be defined in harmony with each other, but this can be done one of two ways. The health facility can be treated as a common mixing junction (see Figure 2-3). The outlet temperature TT is the average temperature of all the water consumed by a facility. The inlet temperature TC is the temperature of the cold water that comes into the club without being heated and is typically equal to the ambient temperature. The inlet temperature TH is the temperature to which water is heated in the water heater, or in a heat exchanger. If the volume of hot water consumed is defined as only and exactly the hot water delivered by the water heater, then the temperature increase would be equal to TH - TC. Alternatively, if the hot water consumed is defined as the total amount of water consumed, i.e., the mixed water leaving the facility, then the temperature increase would be equal to TT - TC. Since the amount of cold water used at a health club facility is negligible, it was more convenient to use the latter definitions of temperature rise and water volume consumed. Although all three of the above temperatures were known, it was impossible to separate the amount of hot water consumed from the amount of cold water consumed, i.e., mT was measured, but mC and mH could not be measured. If mC and mH had been measured, TT could be found in the following manner6:
(2.22a)
and 
(2.22b)
(2.22c)
(2.23a)
(2.23b)
(2.23c)
(2.24a)
(2.24b)
or,
(2.24c)
The mass fractions xC and xH were unknown. Since most of the hot water used at the athletic club facility is shower water, TT was assumed to be approximately 105 °F. This estimate also includes consideration for the water consumed by the clothes washers when towels are washed. The towels are washed in water heated to temperatures greater than 105 °F. The rinse cycle uses cold water. The symbol TC represents the average ambient temperature in Atlanta.