# 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 CO$_2$-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 a$_{el}$ which is needed to generate the combined product electrical energy W (work) and a$_{th}$ for the thermal energy H (useful heat) respectively, can be calculated accordingly to the first and second laws of thermodynamics as follows:
a$_{el}$= (1 · η$_{el}$) / (η$_{el}$ + η$_c$ · η$_{th}$)
a$_{th}$= (η$_c$ · η$_{th}$) / (η$_{el}$ + η$_c$ · η$_{th}$)
Note: a$_{el}$ + a$_{th}$ = 1
with a$_{el}$: allocation factor for electrical energy, i.e. the share of the fuel input which is allocated to electricity production a$_{th}$: allocation factor for thermal energy, i.e. the share of the fuel input which is allocated to heat production
η$_{el}$ = W/Q$_F$
η$_{th}$ = H/Q$_F$
W: electrical work
H: useful heat
Q$_F$: Total heat, fuel or primary energy input
and
η$_c$: Carnot factor 1-T$_i$/T$_s$ (Carnot factor for electrical energy is 1) T$_i$: lower temperature, inferior (ambient)
T$_s$: 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. T$_s$ = (T$_{FF}$+T$_{RF}$) / 2
or - if more thermodynamic precision is needed - the logarithmic mean temperature is used
T$_s$ = (T$_{FF}$-T$_{RF}$) / ln(T$_{FF}$/T$_{RF}$)
If process steam is delivered which condenses and evaporates at the same temperature, T$_s$ is the temperature of the saturated steam of a given pressure.

## Fuel factor

The fuel intensity or the fuel factor for electrical energy f$_{F,el}$ resp. thermal energy f$_{F,th}$ is the relation of specific input to output.
f$_{F,el}$= a$_{el}$ / η$_{el}$ = 1 / (η$_{el}$ + η$_c$ · η$_{th}$)
f$_{F,th}$= a$_{th}$ / η$_{th}$ = η$_c$ / (η$_{el}$ + η$_c$ · η$_{th}$)

## Primary energy factor

To obtain the primary energy factors of cogenerated heat and electricity, the energy prechain needs to be considered.
f$_{PE,el}$ = f$_{F,el}$ · f$_{PE,F}$
f$_{PE,th}$ = f$_{F,th}$ · f$_{PE,F}$
with
f$_{PE,F}$: 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.
η$_{el,eff}$ = η$_{el}$ / a$_{el}$ = 1 / f$_{F,el}$
η$_{th,eff}$ = η$_{th}$ / a$_{th}$ = 1 / f$_{F,th}$
with
η$_{el,eff}$: effective efficiency of electricity generation within the CHP process η$_{th,eff}$: 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 η$_x$ 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:
η$_{x,total}$ = η$_{el}$ + η$_c$ · η$_{th}$
The allocation with the Carnot method always results in:
η$_{x,total}$ = η$_{x,el}$ = η$_{x,th}$
with η$_{x,total}$ = exergetic efficiency of the combined process η$_{x,el}$ = exergetic efficiency of the virtual electricity-only process η$_{x,th}$ = 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 O$_1$ and a second output O$_2$. 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
f$_i$ · I = f$_1$ · O$_1$ + f$_2$ · O$_2$
The factor for the input f$_i$ and the quantities of I, O$_1$, and O$_2$ are known. An equation with two unknowns f$_1$ and f$_2$ has to be solved, which is possible with a lot of adequate tuples. As second equation, the physical transformation of product O$_1$ in O$_2$ and vice versa is used.
O$_1$ = η$_{21}$ · O$_2$
η$_{21}$ is the transformation factor from O$_2$ into O$_1$, the inverse 1/η$_{21}$=η$_{12}$ describes the backward transformation. A reversible transformation is assumed, in order not to favour any of the two directions. Because of the exchangeability of O$_1$ and O$_2$, the assessment of the two sides of the equation above with the two factors f$_1$ and f$_2$ should therefore result in an equivalent outcome. Output O$_2$ evaluated with f$_2$ shall be the same as the amount of O$_1$ generated from O$_2$ and evaluated with f$_1$.
f$_1$ · (η$_{21}$ · O$_2$) = f$_2$ · O$_2$
If we put this into the first equation, we see the following steps:
f$_i$ · I = f$_1$ · O$_1$ + f$_1$ · (η$_{21}$ × O$_2$)
f$_i$ · I = f$_1$ · (O$_1$ + η$_{21}$ · O$_2$)
f$_i$ = f$_1$ · (O$_1$/I + η$_{21}$ · O$_2$/I)
f$_i$ = f$_1$ · (η$_1$ + η$_{21}$ · η$_2$)
f$_1$ = f$_i$ / (η$_1$ + η$_{21}$ · η$_2$) or respectively f$_2$ = η$_{21}$ · f$_i$ / (η$_1$ + η$_{21}$ · η$_2$)
with η$_1$ = O$_1$/I and η$_2$ = O$_2$/I