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 the distribution key. For heat, this potential can be assessed by 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 and open for discussion.

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 the distribution key. For heat, this potential can be assessed by 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 and open for discussion. a el = (1 · η el ) / (1 · η el + η c · η th ) a th = (η c · η th ) / (1 · 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) A good approximation for the upper temperature in a heating system is the average between the 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. 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 ) 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 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 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. 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

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