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Entransy-Based Depletion Index and Its Application for Assessing Efficiency and Sustainability

  • Research Article-Mechanical Engineering
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Abstract

Despite the importance of variation in depletion index, nothing is known on the changes in the Optimized Exergy Depletion Index (XDI) with cost (USD) and Entransy Depletion Index (NDI) with cost (USD). The aim of this paper is to stablish a new depletion index based on entransy and to applicate this concept to the evaluation of efficiency and sustainability. The efficiency and sustainability of a thermal system can be evaluated through the Depletion Index. It was developed a new mathematical expression for assessing the depletion index based on entransy and it was compared with the conventional index based on exergy. Deductive inductive methods were used to obtain the new depletion index based on entransy theory. A multi-objective optimization is proposed considering as criteria the cost and efficiency based on exergy and entransy. Depletion index values based on entransy dissipation are similar than those ones based on exergy destruction and then the value of efficiency obtained by the two concepts is similar. Until recently, the generation of entropy was used for the Shell and tube heat exchangers optimization, however, the entransy dissipation or some function involving it can also be used for this purpose, being a new approach for assessing the depletion index. The optimization results obtained based on each of the efficiency indices were very similar mainly in relation to the total heat exchange rate. The Pareto fronts obtained in the multi-objective optimization allow to find and match optimum designs adjustable to costs and to the space available to install the equipment and the auxiliary services.

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Fig. 2

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Abbreviations

T:

Temperature (K)

\({\dot{\text{E}}\text{x}}_{{\text{D}}}\) :

Exergy destruction rate (W)

\(\dot{S}_{{{\text{gen}}}}\) :

Entropy generation rate in a process or equipment (W/K)

\(T_{0}\) :

Reference ambient temperature (often taken as 298 K or local temperature)

\({\dot{\text{E}}\text{x}}_{{{\text{in}}}}\) :

Total rate of energy consumption (W)

\(Q_{{{\text{vh}}}}\) :

Thermal energy of heat stored in an object with constant volume, which can be referred to as thermal load (W)

\(V_{{{\text{lin}}}}\) :

Fluid lineal velocity (m/s)

\(U_{{\text{h}}}\) or \(T\) :

Thermal potential (temperature) (K)

\(M\) :

Body mass (kg)

\(c_{{\text{v}}}\) :

Specific heat at constant volume (J/kg K)

\(G_{\Delta T}\) :

Entransy dissipation (kW °C)

\(\dot{m}\) :

Mass flow rate (kg/s)

\(c_{{\text{p}}}\) :

Specific heat at constant pressure (J/kg K)

H:

Hot fluid

C:

Cold fluid

\(T_{{{\text{hi}}}}\) :

Hot fluid inlet temperature (K)

\(T_{{{\text{ho}}}}\) :

Hot fluid outlet temperature (K)

\(T_{{{\text{ci}}}}\) :

Cold fluid inlet temperature (K)

\(T_{{{\text{co}}}}\) :

Cold fluid outlet temperature (K)

\(\dot{Q}\) :

Heat flow in the heat exchanger (kW).

\(U_{{{\text{o}}\;{\text{cal}}}} \) :

Total heat transfer coefficient calculated (W/m2°C)

\(\varepsilon\) :

Heat exchanger effectiveness

\(C_{\min }\) :

Minimal calorific capacity (J/K)

\(A_{{\text{T}}}\) :

Total area of heat transfer (m2)

LMTD:

Logarithmic mean temperature difference (°C)

\(\lambda\) :

Latent heat of vapor phase change (hg − hf) (kJ/kg)

\(T_{v}\) :

Steam saturation temperature (K)

\(\dot{m}_{v}\) :

Mass flow of saturated steam (kg/h)

\( P_{v}\) :

Saturated steam pressure (kPa)

\(\overline{v}\) :

Specific volume of saturated steam (m3/kg)

\(\dot{m}\) :

Mass flow (kg/s)

L :

Tube length (m)

N :

Number of tubes

Dt:

Tube diameter (mm)

B :

Space between baffles (mm)

Ds:

Shell diameter (mm)

NTU:

Number of transfer units

\(U_{{\text{o}}}\) :

Total heat transfer coefficient (W/m2 °C)

\(\Delta P_{{\text{t}}}\) :

Presure drop in tube side (kPa)

\(\Delta P_{{\text{s}}}\) :

Pressure drop in shell side (kPa)

C i :

Installation cost ($)

C o :

Operation cost ($)

C :

Total cost ($)

\(\psi_{{{\text{ex}}}}\) :

Efficciency according exergy (%)

\(\psi_{{{\text{ent}}}}\) :

Efficciency according entransy (%)

ΔT :

Temperature difference (K)

\({\text{c}}\) :

Cold fluid

\({\text{h}}\) :

Hot fluid

\({\text{i}}\) :

Inlet

\({\text{o}}\) :

Outlet

\({\text{s}}\) :

Shell-side

\({\text{t}}\) :

Tube side

\({\text{w}}\) :

Wall

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Acknowledgements

The research work has been supported by the Ministry of Education of Brazil under the program for improvement of higher education personnel scholarship. The authors would like to thank the CNPq Productivity of Research Funds Processes 301811/2019-9 and 301478/2018-0 and to PNPD/CAPES N° 88887.314336/2019-00 PEI-UFBA.

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Authors and Affiliations

  1. Graduate Program in Industrial Engineering, Polytechnic Institute, Federal University of Bahia, Salvador, Brazil

    Maida Bárbara Reyes Rodríguez, Jorge Laureano Moya Rodríguez & Cristiano de Hora Fontes

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  1. Maida Bárbara Reyes Rodríguez
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Correspondence to Jorge Laureano Moya Rodríguez.

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Reyes Rodríguez, M.B., Moya Rodríguez, J.L. & de Hora Fontes, C. Entransy-Based Depletion Index and Its Application for Assessing Efficiency and Sustainability. Arab J Sci Eng 48, 3339–3349 (2023). https://doi.org/10.1007/s13369-022-07111-x

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