# The Physical Optimum as an Ideal Reference Value for Balancing Thermodynamic Processes Integrating the Exergetic Evaluation by the Example of Heat Supply

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## Abstract

**:**

## 1. Introduction

- BDP: best demonstrated practice;
- OEO: operational energy optimum;
- PEO: plant energy optimum;
- TEO: theoretical energy optimum;
- Best practice observed;
- BAT: best available technology;

_{PhO}as the corresponding indicators and integrates the already common methods of efficiency and exergy analysis, using the methods in combination. The particularity of the application of the PhO is the reference to an ultimate value. This allows the analysis of losses, aiming to distinguish whether they are avoidable or not because an optimization can only reduce avoidable losses. Other studies such as [4,5] outlined the differences between the PhO and other evaluation methods, while this study aims to integrate, in particular, the state value exergy into the PhO.

## 2. Materials and Methods

#### 2.1. Efficiency

#### 2.2. Exergy Analysis

_{m}of the heat transfer is used to calculate the amount of exergy transferred with the heat. For ideal gases, as well as incompressible fluids with an approximately constant specific isobaric heat capacity, the average thermodynamic temperature can be calculated as follows [6] (pp. 119–120).

#### 2.3. Physical Optimum

_{PhO}, resulting in a value higher than one at all times (according to the initial approach in [1] to focus on the demand of processes, the symbol “B” is used, as in [3]).

_{PhO}is defined and calculated based on an ideal reference process.

_{B}:

_{ideal}→ 1), the efficiency outlines only the avoidable loss ∆Ω

_{B}because the unavoidable loss equals zero. In any other case (for η

_{ideal}< 1), the efficiency does not enable conclusions regarding the degree to which the process can be optimized because unavoidable losses are included in the overall loss. Below, a practical example of a wood pellet-fired boiler illustrates this problem.

## 3. Results

#### 3.1. Application of PhO to Boiler Process

_{B}and combustion air m

_{l}are fed to the system. In this model, a complete combustion is assumed.

_{B}, the combustion heat Q

_{B}is fed to the system. Apart from the useful heat Q

_{N}, the following losses leave the system:

- Q
_{fg}exhaust loss; - Q
_{ash}ash loss; - Q
_{CO}losses by incompletely burned material; - Q
_{S}surface loss.

_{s}—the combustion efficiency η

_{f}

_{CO}⟶ 0, q

_{ash}⟶ 0). The combustion efficiency is simplified to

_{gf}is defined as

_{mp,i}is approximated for 100 °C and the condensation enthalpy of water for 25 °C of r

_{H}

_{2O, 25 °C}= 2411 kJ/kg.

#### 3.1.1. Example

_{2,fg}= 7%; ϑ

_{fg}= 130 °C. The real λ results in being

_{PhO}would be 1 (complete combustion):

_{1}= ϑ

_{fg}= 60 °C, terminal temperature difference TTD = 0. Figure 3 shows the temperature of the two fluids over the surface of the heat exchanger in the real case and the PhO.

_{a,Hs,PhO}= 9.25% for this case. Thus, in the PhO, if 100 units of fuel are fed to the system, Q

_{N,PhO}= 90.75 units of useful heat can be provided. For the real case, the standard operating point (SOP) is shown in Figure 4. The exhaust loss sums up to q

_{a,Hs}= 13.64%, which means that it is q

_{a,Hs,}

_{Δ}

_{Ω}= 4.39% higher than the ideal case. This equals the potential for optimization. Figure 5 and Figure 6 illustrate the losses. If the input is assumed to be constant, a higher amount of useful heat can be gained for the PhO. Furthermore, the distinction between unavoidable and avoidable losses is shown.

_{f,Hs,SOP}= 1 − 0.8636 → Potential for optimization ≠ 13.64%

_{s}/H

_{i}= 1.084. The combustion efficiency amounts to

#### 3.1.2. PhO Factor Based on Exergy Analysis

_{amb}= 293 K. In this study, the outlet temperature of hot water was set to ϑ

_{2}= 80 °C, and the return flow temperature was set to ϑ

_{1}= 60 °C. Thus, the resulting thermodynamic middle temperature is

#### 3.2. Comparison of Boiler and Heat Pump with PhO

#### 3.2.1. PhO Factor and Efficiency

- For B
_{PhO}: Assuming a TTD of 0 K, the ideal HP operates between 20 and 80 °C, with a COP of 5.88. - For $\Delta {\Omega}_{1}$: For the real HP, the liquefaction temperature is set to 85 °C, and the evaporation temperature is set to 15 °C (TTD = 5 K). With a Carnot grade of approximately 0.6, the COP of the real HP would be 3.07.
- For $\Delta {\mathsf{\Omega}}_{2}$: Furthermore, the efficiency of the power plant (PP) delivering the electric energy is assumed to be 40%.

_{ind}is shown in the middle representing the absolute minimum exergy input for the boiler as well as the HP. The anergy of the useful heat is added in white. In the case of the HP, this anergy is taken from the environment. For the boiler, it is exergy turned into anergy in the process. On the left and right in gray, the consumption in the real case is given for each option. In between, the losses are given as a cascade adding to the PhO

_{ind}until the real expense is reached. Thus, the impact of a specific loss on the total losses can be quantified with this form of presentation.

_{ind}is explained below.

#### 3.2.2. Process-Independent PhO

_{ind}), which is also shown in Figure 8. The minimum amount of exergy needed to increase the temperature from 60 to 80 °C can be determined by the exergy in the transferred heat of 12.57 units. This equals the amount of exergy transferred to the flow of heating water in both options. In relation to the PhO

_{ind}, it is possible to determine a process-independent PhO factor F

_{PhO,ind}. In this case, the HP is the more efficient option while simultaneously consuming less energy, as shown by the factors.

_{ind}(III.).

_{cond,PhOind}therefore has to change during condensation to reach ϑ

_{2}. As this is not feasible for a phase-changing single-component fluid at constant pressure in reality, the difference between the PhO of this process and the PhO

_{ind}is an unavoidable loss. A heat pump operating between ambient temperature and the thermodynamic middle temperature ϑ

_{m}returns the result for the exergetic expense. Yet, ϑ

_{m}is below the required temperature ϑ

_{2}. A condensation at ϑ

_{m}= const. cannot supply the system. It becomes clear that in this case, the PhO

_{ind}is over-idealized, outlining the reversible heat transfer, which is not feasible in reality.

_{ind}is suitable for a process comparison and, as with the efficiency, determines the process with a lower total energy consumption. The PhO factor, in addition, outlines the efficiency of each specified process as well as the potential for optimization.

## 4. Discussion

#### 4.1. Degree of Idealization

#### 4.2. Description of the Benefit of a Process

#### 4.3. Using Exergy to Describe the PhO

#### 4.4. Research Directions: Change in Load

#### 4.5. Application of the PhO to Other Examples

_{ind}in comparison to the PhO of each process is an example for this circumstance. However, this study is limited to the example of two options for the supply of heat. In general, the principle can be transferred to other processes. The VDI 4663-1 [3] provides further examples for the application of the PhO as well as the references [1,2,4,5]. As for any other method, the application of the PhO is limited. An example is a process which does not need any input in the ideal case: Considering a room with a temperature differing from the ambient state, an expense of energy would be needed to balance the heat transfer between the environment and the room and maintain the temperature. For an ideal case, the room would be ideally isolated, resulting in an adiabatic system border. The temperature in the room would not change. Thus, the ideal input is zero. The PhO factor is not applicable, ${F}_{PhO}\to \infty $. This particular case could be a potential for improvement in the PhO method.

## 5. Conclusions

_{ind}enables a direct comparison of different processes in order to outline not only the potential for optimization but also an overall energy input. For the PhO factor, a real demand is considered in relation to an ideal input, which means that, for example, the type of energy is the same for both. The COP of an HP compares heat to pure exergy, and the combustion efficiency compares useful heat to the exergy of the fuel. On the other hand, for the PhO factor, the exergy content of the real input energy type equals that of the ideal input. Thus, the PhO

_{ind}factors can be compared directly. This also outlines the limits of optimization. For an over-idealized reference, unavoidable losses are categorized as avoidable ones.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Roman Symbols | ||

Symbol | Unit | Description |

B | e.g., J, W | Demand |

c_{p} | J/(kgK) | Specific isobaric heat capacity |

E | J | Exergy |

${F}_{PhO}^{B}$ | - | PhO factor, based on demand |

H | J | Enthalpy |

H_{i} | J/kg | Lower calorific value |

H_{s} | J/kg | Gross calorific value |

m | kg | Mass |

Q | J | Heat |

r | J/kg | Specific condensation enthalpy |

S | J | Entropy |

T | K | Kelvin temperature |

TTD | K | Terminal temperature difference |

Greek Symbols | ||

$\zeta $ | - | Exergy efficiency |

$\eta $ | - | Efficiency |

λ | - | Air ratio |

${v}_{i}$ | - | Stoichiometric number |

$\Delta {\Omega}_{B}$ | e.g., J, W | Avoidable losses |

Sub- and Superscript | ||

Symbol | Description | |

amb | Ambient | |

ash | Ashes | |

B | Fuel, demand | |

C | Carnot | |

cg | Combustion gas | |

CO | Incompletely burned | |

cond | Condensate | |

Ex | Exergy | |

fg | Flue gas (exhaust) | |

f | Combustion | |

HP | Heat pump | |

ind | Process-independent | |

irr | Irreversible | |

l | Loss; for air: dry air | |

m | Mean, average | |

N | Useful part (e.g., of heat) | |

PhO | Physical optimum | |

q | Heat | |

S | Surface | |

SOP | Standard operating point | |

tot | Total, overall | |

${\dot{\square}}_{\text{}}$ | Differentiation by time |

## References

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**Figure 10.**Temperature profile of the heat transfer in the condenser of the HP (real vs. PhO, PhO

_{ind}).

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**MDPI and ACS Style**

Volta, D.; Weber, S.A.
The Physical Optimum as an Ideal Reference Value for Balancing Thermodynamic Processes Integrating the Exergetic Evaluation by the Example of Heat Supply. *Energies* **2021**, *14*, 4426.
https://doi.org/10.3390/en14154426

**AMA Style**

Volta D, Weber SA.
The Physical Optimum as an Ideal Reference Value for Balancing Thermodynamic Processes Integrating the Exergetic Evaluation by the Example of Heat Supply. *Energies*. 2021; 14(15):4426.
https://doi.org/10.3390/en14154426

**Chicago/Turabian Style**

Volta, Dirk, and Samanta A. Weber.
2021. "The Physical Optimum as an Ideal Reference Value for Balancing Thermodynamic Processes Integrating the Exergetic Evaluation by the Example of Heat Supply" *Energies* 14, no. 15: 4426.
https://doi.org/10.3390/en14154426