Exergy analysis¶

Performing thermodynamic cycle analyses making use of the second law of thermodynamics provides further process information and uncovers potentials for improvement in power plant engineering. Therefore, TESPy’s analyses module provides you with an inbuilt and fully automatic exergy analysis.

We have published a paper with the features described in this section. The publication is licensed under an open-access license, download the pdf at https://doi.org/10.3390/en15114087, also see [10].

Fundamentals of exergy analysis¶

Energy is a concept of the first law of thermodynamics. It cannot be destroyed. But regarding the design and analysis of thermal systems, the idea that something can be destroyed is useful. According to the second law of thermodynamics, the conversion of heat and internal energy into work is limited. This constraint and the idea of destruction are applied to introduce a new concept: “Exergy”.

Exergy can be destroyed due to irreversibility and is able to describe the quality of different energy forms. The difference in quality of different forms of energy shall be illustrated by the following example. 1 kJ of electrical energy is clearly more valuable than 1 kJ of energy in a glass of water at ambient temperature [13].

In literature, exergy is defined as follows:

“An opportunity for doing useful work exists whenever two systems at different states are placed in communication, for in principle work can be developed as the two are allowed to come into equilibrium. When one of the two systems is a suitably idealized system called an environment and the other is some system of interest, exergy is the maximum theoretical useful work (shaft work or electrical work) obtainable as the systems interact to equilibrium, heat transfer occuring with the environment only.” [13]

Terminology¶

The definitions and nomenclature of the exergy analysis in TESPy are based on [36]. The exergy destruction ratios are described in more detail in [13]. Since the current version of the exergy analysis in TESPy only focuses on physical exergy and does not include reaction processes yet, chemical exergy is not considered. Changes in kinetic and potential exergy are neglected and therefore not considered as well.

Terminology¶
variable	name	symbol	description
`ex_physical`, `Ex_physical`	(specific) physical exergy	\(e^\mathrm{PH}\), \(E^\mathrm{PH}\)	due to the deviation of the temperature and pressure of the system from those of the environment
`ex_therm`, `Ex_therm`	(specific) thermal exergy	\(e^\mathrm{T}\), \(E^\mathrm{T}\)	associated with the system temperature
`ex_mech`, `Ex_mech`	(specific) mechanical exergy	\(e^\mathrm{M}\), \(E^\mathrm{M}\)	associated with the system pressure
`ex_chemical`, `Ex_chemical`	(specific) chemical exergy	\(e^\mathrm{CH}\), \(E^\mathrm{CH}\)	based on standard chemical exergy in ambient model, the tespy.data module provides three different datasets for standard exergy based on various sources, i.e. Ahrendts [30, 31, 32], Szargut1988 [33] and Szargut2007 [34, 35].
`E_P`	product exergy	\(\dot{E}_\mathrm{P}\)	represents the desired result(expressed in terms of exergy) generated by the system being considered represents the resources (expressed in terms of exergy)
`E_F`	fuel exergy	\(\dot{E}_\mathrm{F}\)	represents the resources (expressed in terms of exergy) expended to provide the product exergy
`E_D`	exergy destruction	\(\dot{E}_\mathrm{D}\)	thermodynamic inefficiencies associated with the irreversibility (entropy generation) within the system boundaries
`E_L`	exergy loss	\(\dot{E}_\mathrm{L}\)	thermodynamic inefficiencies associated with the transfer of exergy through material and energy streams to the surroundings
`epsilon`	exergetic efficiency	\(\varepsilon\)	ratio between product exergy and fuel exergy
`y_D,k`	exergy destruction ratio	\(y_\mathrm{D}\)	rate of exergy destruction in a component compared to the exergy rate of the fuel provided to the overall system
`y*_D,k`	exergy destruction ratio	\(y^*_\mathrm{D}\)	rate of exergy destruction in a component compared to the total exergy destruction rate within the system

Note

The generic exergy analysis balance equations have not yet been fully implemented and tested for the components FuelCell, WaterElectrolzer and CombustionEngine.

Tutorial¶

In this short tutorial, an exergy analysis is carried out for the so-called “Solar Energy Generating System” (SEGS). The full python script is available on GitHub in an individual repository: https://github.com/fwitte/SEGS_exergy.

Tip

Two other full code examples are to be found at:

Supercritical CO₂ power cycle: https://github.com/fwitte/sCO2_exergy
Refrigeration machine: https://github.com/fwitte/refrigeration_cycle_exergy

SEGS consists of three main systems, the solar field, the steam cycle and the cooling water system. In the solar field Therminol VP1 (TVP1) is used as heat transfer fluid. In the steam generator and reheater the TVP1 is cooled down to evaporate and overheat/reheat the water of the steam cycle. The turbine is divided in a high pressure turbine and a low pressure turbine, which are further subdivided in 2 parts (high pressure turbine) and 5 parts. In between the stages steam is extracted for preheating. Finally, the main condenser of the steam cycle is connected to an air cooling tower. The figure below shows the topology of the model.

Topology of the Solar Energy Generating System (SEGS)

The input data are based on literature [37], which provides measured data. Some parameters are however taken from a follow-up publication, as the original data show some inconsistencies, e.g. higher enthalpy at the low pressure turbine’s last stage outlet than at its inlet [38]. As mentioned, you can find all data in the respective GitHub repository.

TESPy model¶

The TESPy model consists of 53 components. The feed water tank serves as mixing preheater, thus can be modeled using a merge. All other components are modeled highlighted in the flowsheet. The preheaters and the main condenser are modeled as Condenser instances, while all other heat exchangers are modeled using HeatExchanger instances. For the solar field a parabolic trough is implemented, calculating the surface area required for the provision of the heat input at optimal conditions.

All components are flagged with the fkt_group parameter, which will automatically create functional groups (component groups) for the exergy analysis Grassmann diagram. The specification of this parameter is not required for the exergy analysis itself, but helps to simplify the automatically generated diagram. Components not assigned to any functional group will form their respective group.

Regarding parameter specification, the following parameters are specified:

isentropic efficiency values
electrical conversion efficiencies of motors and generators
terminal temperature difference values at preheaters
pressure values of steam extraction
pressure values in the preheating route
pressure losses in the heat exchangers
solar fluid temperature
steam cycle live steam and reheat temperatures
some temperature values in the cooling water system

The ambient state is defined as follows:

pamb = 1.013
Tamb = 25

Pressure and temperature of the ambient air in the cooling tower are equal to these values in the script provided.

For the exact values of the component parameters please see in the referenced python script.

Due to the complexity of the plant, the solver sometimes struggles when given bad starting values. Therefore, the TESPy model is built in two steps. After solving the initial setup without both of the high pressure preheater subcoolers, the missing connections and components are added in a second step and the model is again solved.

Analysis setup¶

After the simulation of the plant, the exergy analysis can be carried out. To perform it, all exergy streams leaving or entering the network’s system boundaries have to be defined by the user. These are:

fuel exergy E_F
product exergy E_P
exergy loss streams E_L
internal exergy streams not bound to connections internal_busses

In case of the solar thermal power plant, the fuel exergy is the heat input at the parabolic trough. The product is the electricity produced by the plant, i.e. the electricity generated by the turbine generators minus the electricity consumed by the pumps and the fan. Lastly, exergy loss streams are the hot air leaving the cooling tower and the cold air entering the cooling tower fan from the ambient. Similar to the electricity consumption of the fan and pumps the cold air will be taken into account as negative value for the total exergy loss.

power = Bus('total output power')
power.add_comps(
    {'comp': hpt1, 'char': 0.97, 'base': 'component'},
    {'comp': hpt2, 'char': 0.97, 'base': 'component'},
    {'comp': lpt1, 'char': 0.97, 'base': 'component'},
    {'comp': lpt2, 'char': 0.97, 'base': 'component'},
    {'comp': lpt3, 'char': 0.97, 'base': 'component'},
    {'comp': lpt4, 'char': 0.97, 'base': 'component'},
    {'comp': lpt5, 'char': 0.97, 'base': 'component'},
    {'comp': fwp, 'char': 0.95, 'base': 'bus'},
    {'comp': condpump, 'char': 0.95, 'base': 'bus'},
    {'comp': ptpump, 'char': 0.95, 'base': 'bus'},
    {'comp': cwp, 'char': 0.95, 'base': 'bus'},
    {'comp': fan, 'char': 0.95, 'base': 'bus'}
)

heat_input_bus = Bus('heat input')
heat_input_bus.add_comps({'comp': pt, 'base': 'bus'})

exergy_loss_bus = Bus('exergy loss')
exergy_loss_bus.add_comps({'comp': air_in, 'base': 'bus'}, {'comp': air_out})

SEGSvi.add_busses(power, heat_input_bus, exergy_loss_bus)

In order to define these values a list of busses representing the individual exergy streams is passed when creating the ExergyAnalysis instance.

ean = ExergyAnalysis(SEGSvi, E_P=[power], E_F=[heat_input_bus], E_L=[exergy_loss_bus])

In this case, the Bus power represents the product exergy, the Bus heat_input_bus the fuel exergy of the solar field and the Bus exergy_loss_bus the exergy lost with the hot air leaving the cooling tower. An example application using the internal_busses can be found in the API documentation of class tespy.tools.analyses.ExergyAnalysis.

After the setup of the exergy analysis, the tespy.tools.analyses.ExergyAnalysis.analyse() method expects the definition of the ambient state, thus ambient temperature and ambient pressure. With this information, the analysis is carried out automatically. The value of the ambient conditions is passed in the network’s (nw) corresponding units.

ean.analyse(pamb=pamb, Tamb=Tamb)

Using the same tespy.tools.analyses.ExergyAnalysis instance, it is possible to run the analysis again with a different ambient state. The data generated by the analysis will automatically update, e.g. changing the ambient state temperature value to 15 °C.

ean.analyse(pamb=pamb, Tamb=15)

Note

If the network’s topology changed a new instance of the ExergyAnalysis class needs to be defined.

Checking consistency¶

An automatic check of consistency is performed by the analysis. The sum of all exergy destruction values of the network’s components and the exergy destruction on the respective busses is calculated. On top of that, fuel and product exergy values as well as exergy loss are determined. The total exergy destruction must therefore be equal to the fuel exergy minus product exergy and minus exergy loss. The deviation of that equation is then calculated and checked versus a threshold value of \(10^{-3}\) (to compensate for rounding errors).

\[ \begin{align}\begin{aligned}\dot{E}_\mathrm{D} = \dot{E}_\mathrm{F} - \dot{E}_\mathrm{P} - \dot{E}_\mathrm{L}\\\Delta \dot{E} = \dot{E}_\mathrm{F} - \dot{E}_\mathrm{P} - \dot{E}_\mathrm{L} - \dot{E}_\mathrm{D}\\\Delta \dot{E} \leq 10^{-3}\end{aligned}\end{align} \]

Note

If the exergy analysis is carried out on a converged simulation and the analysis is set up correctly, this equation must be True. Otherwise, an error will be printed to the console, which means:

The simulation of your plant did not converge or
the exergy analysis has not been set up correctly. You should check, if the definition of the exergy streams E_F, E_P, E_L and internal_busses is correct.

If you suspect a bug in the calculation, you are welcome to submit an issue on our GitHub page.

Printing the results is possible with the tespy.tools.analyses.ExergyAnalysis.print_results() method. The results are printed in six individual tables:

connections
components
busses
aggregation (aggregation of components and the respective busses)
network
groups (functional groups)

By default, all of these tables are printed to the prompt. It is possible to deselect the tables, e.g. by passing groups=False to the method call.

ean.print_results(groups=False, connections=False)

For the component related tables, i.e. busses, components, aggregation and groups, the data are sorted in descending order for the given exergy destruction value of the individual entry. The component data contain fuel exergy, product exergy and exergy destruction values related to the component itself ignoring losses that might occur on the busses, for example, mechanical or electrical conversion losses in motors and generators. The bus data contain the respective information related to the conversion losses on the busses only. The aggregation data contain both, the component and the bus data. For instance, a turbine driving a generator will have the electrical energy delivered by the generator as product exergy value. The same component’s exergy product without considering the mechanical or electrical conversion losses is the shaft power delivered by the turbine. From the generator’s perspective, this is the fuel exergy, while the product is the electrical energy.

Note

Please note, that in contrast to the component and bus data, group data do not contain fuel and product exergy as well as exergy efficiency. Instead, all exergy streams entering the system borders of the component group and all exergy streams leaving the system borders are calculated. On this basis, a graphical representation of the exergy flows in the network can be generated in the form of a Grassmann diagram.

Accessing the data¶

The underlying data for the tabular printouts are stored in pandas DataFrames. Therefore, you can easily access and process these data. To access these use the following code snippet.

connection_data = ean.connection_data
bus_data = ean.bus_data
component_data = ean.component_data
aggregation_data = ean.aggregation_data
network_data = ean.network_data
group_data = ean.group_data

Lastly, the analysis also provides an input data generator for plotly’s sankey diagram.

Plotting¶

To use the plotly library, you’ll need to install it first. Please check the respective documentation on plotly’s documentation. Generating a sankey diagram is then easily done:

import plotly.graph_objects as go

links, nodes = ean.generate_plotly_sankey_input()

fig = go.Figure(go.Sankey(
    arrangement='snap',
    node={
        'label': nodes,
        'pad':11,
        'color': 'orange'
    },
    link=links
))
fig.show()

Sankey diagram of the Soler Energy Generating System (SEGS)

The tespy.tools.analyses.ExergyAnalysis.generate_plotly_sankey_input() method provides the links and the corresponding nodes for the diagram. Colors and node order are assigned automatically but can be changed. Additionally, a threshold value for the minimum value of an exergy stream can be specified to exclude relatively small values from display.

ean.generate_plotly_sankey_input(
    node_order=[
        'E_F', 'heat input', 'SF', 'SG', 'LPT', 'RH', 'HPT',
        'total output power', 'CW', 'LPP', 'FWP', 'HPP', 'exergy loss',
        'E_L', 'E_P', 'E_D'
    ],
    colors={'E_F': 'rgba(100, 100, 100, 0.5)'},
    display_thresold=1
)

The coloring of the links is defined by the type of the exergy stream (bound to a specific fluid, fuel exergy, product exergy, exergy loss, exergy destruction or internal exergy streams not bound to mass flows). Therefore, colors can be assigned to these types of streams.

Note

The node_order must contain all exergy streams, thus
- ALL component group labels (you can find the labels in the group data results printout),
- lables of the busses used in the definitions of the analysis and
- 'E_F', 'E_P', 'E_D' as well as 'E_L'
The colors dictionary works with the following keys:
- 'E_F', 'E_P', 'E_D', 'E_L'
- all labels of the busses used in the definition of the internal exergy streams
- all names of the network’s fluid
- 'mix' for any mixture of two or more fluids
Keys missing in the dictionary will automatically assign a color to the link.
The respective value are strings representing colors in the RGBA format, e.g. 'rgba(100, 100, 100, 0.5)'.

Note

Links with negative exergy flow, i.e. when the value of mechanical exergy is negative due to pressure lower than ambient pressure and total exergy is still negative, cannot be displayed by the sankey diagram.

The underlying exergy stream data is saved in a dictionary, if you want to handle the data by yourself.

sankey_data = ean.sankey_data

Conclusion¶

An additional example is available in the API documentation of the tespy.tools.analyses.ExergyAnalysis class. Full testing of exergy analysis at temperature levels below the ambient temperature will be implemented soon. Regarding the implementation of chemical exergy as well as exergo-economical methods, further work is required. If you are interested in contributing, please file an issue at our GitHub page.