Carnot method

The Carnot method is an allocation procedure for dividing up fuel input (primary energy, end energy) in joint production processes that generate two or more energy products in one process (e.g. cogeneration or trigeneration). It is also suited to allocate other streams such as CO2-emissions or variable costs. The potential to provide physical work (exergy) is used as the distribution key. For heat this potential can be assessed the Carnot efficiency. Thus, the Carnot method is a form of an exergetic allocation method. It uses mean heat grid temperatures at the output of the process as a calculation basis. The Carnot method's advantage is that no external reference values are required to allocate the input to the different output streams; only endogenous process parameters are needed. Thus, the allocation results remain unbiased of assumptions or external reference values that are open for discussion.

Fuel allocation factor
The fuel share ael which is needed to generate the combined product electrical energy W (work) and ath for the thermal energy H (useful heat) respectively, can be calculated accordingly to the first and second laws of thermodynamics as follows:

ael= (1 · ηel) / (ηel + ηc · ηth)

ath= (ηc · ηth) / (ηel + ηc · ηth)

Note: ael + ath = 1 with ael: allocation factor for electrical energy, i.e. the share of the fuel input which is allocated to electricity production ath: allocation factor for thermal energy, i.e. the share of the fuel input which is allocated to heat production

ηel = W/QF ηth = H/QF W: electrical work H: useful heat QF: Total heat, fuel or primary energy input

and ηc: Carnot factor 1-Ti/Ts (Carnot factor for electrical energy is 1) Ti: lower temperature, inferior (ambient) Ts: upper temperature, superior (useful heat)

In heating systems, a good approximation for the upper temperature is the average between forward and return flow on the distribution side of the heat exchanger. Ts = (TFF+TRF) / 2 or - if more thermodynamic precision is needed - the logarithmic mean temperature is used Ts = (TFF-TRF) / ln(TFF/TRF) If process steam is delivered which condenses and evaporates at the same temperature, Ts is the temperature of the saturated steam of a given pressure.

Fuel factor
The fuel intensity or the fuel factor for electrical energy fF,el resp. thermal energy fF,th is the relation of specific input to output.

fF,el= ael / ηel = 1 / (ηel + ηc · ηth)

fF,th= ath / ηth = ηc / (ηel + ηc · ηth)

Primary energy factor
To obtain the primary energy factors of cogenerated heat and electricity, the energy prechain needs to be considered.

fPE,el = fF,el · fPE,F fPE,th = fF,th · fPE,F with fPE,F: primary energy factor of the used fuel

Effective efficiency
The reciprocal value of the fuel factor (f-intensity) describes the effective efficiency of the assumed sub-process, which in case of CHP is only responsible for electrical or thermal energy generation. This equivalent efficiency corresponds to the effective efficiency of a "virtual boiler" or a "virtual generator" within the CHP plant. ηel, eff = ηel / ael = 1 / fF,el ηth, eff = ηth / ath = 1 / fF,th with ηel, eff: effective efficiency of electricity generation within the CHP process ηth, eff: effective efficiency of heat generation within the CHP process

Performance factor of energy conversion
Next to the efficiency factor which describes the quantity of usable end energies, the quality of energy transformation according to the entropy law is also important. With rising entropy, exergy declines. Exergy does not only consider energy but also energy quality. It can be considered a product of both. Therefore any energy transformation should also be assessed according to its exergetic efficiency or loss ratios. The quality of the product "thermal energy" is fundamentally determined by the mean temperature level at which this heat is delivered. Hence, the exergetic efficiency ηx describes how much of the fuel's potential to generate physical work remains in the joint energy products. With cogeneration the result is the following relation:

ηx,total = ηel + ηc · ηth

The allocation with the Carnot method always results in: ηx,total = ηx,el = ηx,th with ηx,total = exergetic efficiency of the combined process ηx,el = exergetic efficiency of the virtual electricity-only process ηx,th = exergetic efficiency of the virtual heat-only process

The main application area of this method is cogeneration, but it can also be applied to other processes generating a joint products, such as a chiller generating cold and producing waste heat which could be used for low temperature heat demand, or a refinery with different liquid fuels plus heat as an output.

Mathematical derivation
Let's assume a joint production with Input I and a first output O1 and a second output O2. f is a factor for rating the relevant product in the domain of primary energy, or fuel costs, or emissions, etc.

evaluation of the input = evaluation of the output

fi · I =  f1 · O1 + f2 · O2

The factor for the input fi and the quantities of I, O1, and O2 are known. An equation with two unknowns f1 and f2 has to be solved, which is possible with a lot of adequate tuples. As second equation, the physical transformation of product O1 in O2 and vice versa is used.

O1 = η21 · O2

η21 is the transformation factor from O2 into O1, the inverse 1/η21=η12 describes the backward transformation. A reversible transformation is assumed, in order not to favour any of the two directions. Because of the exchangeability of O1 and O2, the assessment of the two sides of the equation above with the two factors f1 and f2 should therefore result in an equivalent outcome. Output O2 evaluated with f2 shall be the same as the amount of O1 generated from O2 and evaluated with f1.

f1 · (η21 · O2) = f2 · O2

If we put this into the first equation, we see the following steps:

fi · I = f1 · O1 + f1 · (η21 × O2)

fi · I = f1 · (O1 + η21 · O2)

fi = f1 · (O1/I + η21 · O2/I)

fi = f1 · (η1 + η21 · η2)

f1 = fi / (η1 + η21 · η2) or respectively f2 = η21 · fi / (η1 + η21 · η2)

with η1 = O1/I and η2 = O2/I