## A2 The Concepts of First and Second Law Efficiency

The basic difference between both concepts is that while first law efficiency solely relates to the specific technology applied in order to satisfy certain tasks, for example, illumination and space heating, the concept of second law efficiency takes into account both the actual technology employed and the technology that would theoretically be the best for the same task. Our following example on whether to heat a house with either electric furnaces or heat pumps illustrates the difference between these two concepts.

49 This appendix is prepared by Manuel Frondel and Dirk Rubbelke.

50 Nakicenovic 1996.

51 Gilli 1995.

Consider a house that shall be heated up to the temperature of Tin = 21°C, while the outside temperature remains Tout = 1°C. Among a great variety of technologies available for this task, heating with furnaces is one of the methods mostly used, whereas heat pumps are rarely employed. When, for instance, an electric furnace is used, electric energy is transformed into heat. In practice, a part of the electric energy provided by a power plant cannot be transformed into the desired heat. That means, depending upon the furnace, more or less of the electric energy is wasted.

The concept of first law efficiency measures the ratio of the minimum energy per time, such as for instance, the minimum power that is theoretically required to the power that is actually necessary to maintain a constant temperature with this electric furnace:

Pmin (electric furnace) -Pcmal (electric furnace)

The power that is actually necessary, Pactual (electric furnace), may be split into two parts: the minimum power Pmin (electric furnace) that is necessarily required and an amount of power that is wasted. Only if no power is wasted the first law efficiency will amount to 100 per cent. An ideally working furnace is an example. In practice, however, E is smaller than 100 per cent. Typically, first law efficiencies of electric furnaces are in the range of 70 to 80 per cent, with best performing electric furnaces operating with efficiencies greater than 90 per cent.

For the sake of simplicity, we assume that our electric furnace is perfect in terms of first law efficiencies: £1 = 100 per cent. In this case, the minimum electric power is required to maintain the temperature level Tin (Figure A4.1), because the electric power Pel delivered is transformed completely into heat power, Pheat. Moreover, an entropy flux S is produced by the furnace, providing the heat service ultimately desired. Due to imperfect insulation, the entropy flux produced is exported gradually, requiring a permanent heating. This heat or entropy production is similar to the

Figure A4.1 Comparison of Energy and Entropy Flows of (a) Electric Furnace and (b) Heat Pump

(a) Electric furnace (b) Heat pump

Pimp = Pel

Source: Reynolds and Lunnas 1978.

Entropy flux

Energy flux

Pimp = Pel

Source: Reynolds and Lunnas 1978.

entropy-producing rubbing of hands in order to warm them. A person who rubs his/her hands on a cold winter day experiences what entropy production is.

Unlike an ideal electric furnace, a heat pump ideally does not produce entropy. Rather, the heat pump merely imports entropy from outside the house. Yet, without a heat pump, entropy, especially heat, would not flow from the lower temperature Tout to the higher temperature Tin by itself. By itself entropy, especially heat, flows from higher to lower temperatures. This is the reason why a heat pump is indispensable for delivering the heat power desired.

As do electric furnaces, heat pumps run on electricity. Recall that, in our example, the heat pump is assumed to work ideally, that is, its first law efficiency equals 100 per cent. In other words, the electric power which is necessary for heating the house with a perfect heat pump equals precisely the minimum electric energy. Hence, both the ideal electric furnace and the ideal heat pump work equally well in terms of first law efficiency. Nevertheless, the furnace technology is inferior to that of the heat pump. For Tin = 21°C = 294 K and Tout = 1°C = 274 K, we obtain from the laws of physics.53

Pel (heat pump) = (T„ - T0M) • S = _20K_ = 0 073 Pei (furnace) T • S 294 K . (2)

The minimum electric power required by the heat pump is lower than 8 per cent of the electric power required by the furnace for the same amount of heat power desired. Theoretically, a heat pump is the best technology. It needs only a minimum amount of electric power for the task of space heating because the heat pump imports almost all the heat from outdoors. On the other hand, an electric furnace is the worst technology. It requires the maximum amount of electric power for the same task, as no heat is imported at all (see Figure A4.1). Rather, the entire heat is produced inside.

Inspired by equation (4.2) Nakicenovic54 as well as Frondel and Rubbelke55 define second law efficiency as the ratio of minimum power required by the theoretically best technology to the minimum power necessary for the technology actually employed in order to achieve the same task:56

/^¡„(theoretically best technology)

By definition, second law efficiency refers to both the technology actually employed and to the technology that would, at least theoretically, be the best. By contrast, first law efficiency relates only to the technology actually employed.

53 See Frondel and Rubbelke 2002, 290 and Figure 1.

54 Nakicenovic 1996.

### 55 Frondel and Rubbelke 2002.

56 Nakicenovic (1996, 81) define second law efficiency as the ratio of theoretical minimum energy consumption for a particular task to the actual energy consumption for the same task.

The miserable second law efficiency performance of electric furnaces reflects the poor match of the technology employed and the purpose for which energy is used: Heating with the help of electricity is a sin in terms of entropy because electricity has to be generated with some effort and is accompanied by a production of entropy in the form of waste heat. The high-quality energy in the form of electricity is misused for heating purposes, producing entropy in the form of heat once again, albeit desired now. With respect to entropy considerations, it would be better to use gas and to produce entropy in the form of desired heat only once.

As far as first law efficiencies are concerned; such questions are not an issue. No alternatives, exclusively the technology actually employed, are taken into account. From this limited point of view and given the high first law efficiencies with which electric energy is transformed into the desired heat, heating with electricity seems to be a big deal. However, the fact that the second law efficiency is lower than 8 per cent in this example stresses the mismatch of the application of high-quality energy for low-quality energy services.

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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