The Carnot method is an allocation procedure for dividing up fuel input (
primary energy
Primary energy (PE) is an energy form found in nature that has not been subjected to any human engineered conversion process. It is energy contained in raw fuels, and other forms of energy, including waste, received as input to a system. Prim ...
, end energy) in joint production processes that generate two or more energy products in one process (e.g.
cogeneration
Cogeneration or combined heat and power (CHP) is the use of a heat engine or power station to generate electricity and useful heat at the same time.
Cogeneration is a more efficient use of fuel or heat, because otherwise- wasted heat from elec ...
or trigeneration). It is also suited to allocate other streams such as
CO2-emissions or variable costs. The potential to provide physical work (
exergy
In thermodynamics, the exergy of a system is the maximum useful work possible during a process that brings the system into equilibrium with a heat reservoir, reaching maximum entropy. When the surroundings are the reservoir, exergy is the pot ...
) 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
The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various paramet ...
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
[
{{Citation , last=Tereshchenko , first=Tymofii , last2=Nord , first2=Natasa
, title=Uncertainty of the allocation factors of heat and electricity production of combined cycle power plant , doi=10.1016/j.applthermaleng.2014.11.019
, date=2015-02-05
, journal=Applied Thermal Engineering , volume=76 , publisher=Elsevier , location=Amsterdam , page=410–422 , hdl=11250/2581526 , hdl-access=free ]
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
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
,
exergy
In thermodynamics, the exergy of a system is the maximum useful work possible during a process that brings the system into equilibrium with a heat reservoir, reaching maximum entropy. When the surroundings are the reservoir, exergy is the pot ...
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
Waste heat is heat that is produced by a machine, or other process that uses energy, as a byproduct of doing work. All such processes give off some waste heat as a fundamental result of the laws of thermodynamics. Waste heat has lower utility ...
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''
See also
*
Cogeneration
Cogeneration or combined heat and power (CHP) is the use of a heat engine or power station to generate electricity and useful heat at the same time.
Cogeneration is a more efficient use of fuel or heat, because otherwise- wasted heat from elec ...
*
Variable cost
Variable costs are costs that change as the quantity of the good or service that a business produces changes.Garrison, Noreen, Brewer. Ch 2 - Managerial Accounting and Costs Concepts, pp 48 Variable costs are the sum of marginal costs over all un ...
*
Power loss factor
*
Joint product pricing
In microeconomics, joint product pricing is the firm's problem of choosing prices for joint products, which are two or more products produced from the same process or operation, each considered to be of value. Pricing for joint products is more co ...
*
Nicolas Léonard Sadi Carnot
*
Second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...
References
Further reading
* Marc Rosen
Allocating carbon dioxide emissions from cogeneration systems: descriptions of selected output-based methods Journal of Cleaner Production, Volume 16, Issue 2, January 2008, p. 171–177.
* Andrej Jentsch: The Carnot-Method for Allocation of Fuel and Emissions
EuroHeat&Power Vol 12 II, 2015, p. 26-28.
* Andrej Jentsch
A novel exergy-based concept of thermodynamic quality and its application to energy system evaluation and process analysis dissertation, TU Berlin, 2010.
* Verein Deutscher Ingenieure
VDI-Guideline 4608 Part 2 Energy systems - Combined heat and power - Allocation and evaluation, Juli 2008.
* EN 15316-4-5:2017 Energy performance of buildings - Method for calculation of system energy requirements and system efficiencies - Part 4-5: District heating and cooling
Directive (EU) 2018/2001 on the promotion of the use of energy from renewable sources 2018-12-11. Annex V, C. Methodology, b) and Annex VI, B. Methodology, d)
Cogeneration
Energy conversion
Pricing