Carnot method

The Carnot method is an allocation procedure for dividing up fuel input (primary energy) in combined production processes of energetic commodities. It is also suited to allocate other input streams such as CO₂-emissions or variable costs. The capability to provide physical work (exergy) according to the Carnot efficiency is used as the distribution key. Thereby, the Carnot method resembles to an exergy based equivalence number scheme, as same exergy content is assessed with the same value. Main application area is cogeneration, but it is useful for other combined energy products, such as a chiller generating cold and producing waste heat which could be used for low temperature heat demand. 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.

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 x η_el) / (η_el + η_c × η_th)

a_th= (η_c x 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 needed for electricity production
a_th: allocation factor for thermal energy, i.e. the share of the fuel input which is needed for heat production

η_el = W/Q_F
η_th = H/Q_F
W: electrical work
H: useful heat
Q_F: heat input by fuel
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.
T_s = (T_FF+T_RF) / 2

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 ouput.

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 efficiency of the assumed sub-process, which is responsible only for electrical or thermal energy. 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 transformation

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 is the "valuable" part of energy, therefore any energy transformation should also be assessed acording to its exergetic quality. The quality of the product "thermal energy" is fundamentally determined by the temperature level, at which this heat is delivered. Hence, the exergetic efficiency η_x describes how much of the fuel's capability to perform physical work remains in the joint energy products. With cogeneration the result is the following relation:

η_x = η_el + η_c × η_th

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

f_i × I = f₁ × O₁ + f₂ × O₂

The factor for the input f_i 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 exhangeability 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:

f_i × I = f₁ × O₁ + f₁ × (η₂₁ × O₂)

f_i × I = f₁ × (O₁ + η₂₁ × O₂)

f_i = f₁ × (O₁/I + η₂₁ × O₂/I)

f_i = f₁ × (η₁ + η₂₁ × η₂)

f₁ = f_i / (η₁ + η₂₁ × η₂) or respectively f₂ = η₂₁ × f_i / (η₁ + η₂₁ × η₂)

with η₁ = O₁/I and η₂ = O₂/I