Fundamentals of Exergy Analysis

Introduction

Exergy is bound to mass flow or to massless energy transfer such as heat transfer or power. The transport of exergy with mass can be further split into four different shares according to eq. (1), i.e.:

• physical $e^{\text{PH}}$,

• chemical $e^{\text{CH}}$,

• kinetic $e^{\text{KN}}$ and

• potential $e^{\text{PT}}$.

In thermal engineering kinetic and potential energy can usually be neglected, simplifying the respective equation:

(1)$$\dot{E}=\dot{m}(e^{\text{PH}}+e^{\text{CH}}+\cancel{e^{\text{KN}}}+\cancel{e^{\text{PT}}})$$

For heat, the share of exergy depends on the temperature level of the ambient and the temperature level of the heat:

(2)$${\dot{E}}_{\text{q,}j}=(1-\frac{T_0}{T_j})\cdot {\dot{Q}}_j$$

Work is pure exergy:

(3)$${\dot{E}}_{\text{w,}i}={\dot{W}}_{\text{t,}i}$$

Physical Exergy

Physical exergy defines the maximum work that can be generated by a mass flow when enthalpy and entropy of the flow change to the ambient state 0.

(4)$$e^{\mathrm{PH}}=(h-h(p_0,T_0))-T_0\cdot (s-s(p_0,T_0))$$

Following the definitions in [4], physical exergy can be further split into two shares:

• the mechanical exergy, eq. (6) and

• the thermal exergy, eq. (5).

The physical exergy is the sum of both shares eq. (7).

(5)$$e^{\mathrm{T}}=(h-h(p,T_0))-T_0\cdot (s-s(p,T_0))$$

(6)$$e^{\mathrm{M}}=(h(p,T_0)-h(p_0,T_0))-T_0\cdot (s(p,T_0)-s(p_0,T_0))$$

(7)$$e^{\mathrm{PH}}=e^{\mathrm{T}}+e^{\mathrm{M}}$$

Chemical Exergy

(8)$$e^{\text{CH}}={(\sum x_i{M}_i)}^{-1}\cdot (\sum x_i{\overline{e}}_i^{\text{CH}}+\overline{R}{T}_0\sum x_i\ln x_i)$$

(9)$$e^{\text{CH}}={(\sum x_i{M}_i)}^{-1}\cdot (x_{\text{gas}}(\sum x_i^{\prime }{\overline{e}}_i^{\text{CH}}+\overline{R}{T}_0\sum x_i^{\prime }\ln x_i^{\prime })+x_{{\text{H}}_2{\text{O}}_{\text{(l)}}}{\overline{e}}_{{\text{H}}_2{\text{O}}_{\text{(l)}}}^{\text{CH}})$$

Exergy Analyses

Figure 1 shows an abstract thermodynamic system and the exergy transported over the system’s boundary. Based on that, we can define an exergy balance to analyze the system, taking into account exergy transferred to or from the system. These can be heat (eq. (2)) and power (eq. (3)), incoming ${\dot{E}}_{in}$and outgoing mass flows ${\dot{E}}_{out}$ (eq. (1)) as well as the destruction of exergy ${\dot{E}}_{\text{D}}$ happening in all the system’s components.

(10)$$0=\sum _j{\dot{E}}_{\text{q,}j}+\sum _i{\dot{E}}_{\text{w,}i}+\sum _{in}{\dot{E}}_{in}-\sum _{out}{\dot{E}}_{out}-{\dot{E}}_{\text{D}}$$

Fig. 2 Overview of the exergy balance of a thermodynamic system [3].

Furthermore, it is possible to define balance equations for each of the system’s components $k$. The exergy destruction can then be calculated as the difference between fuel exergy $E_{\text{F,}k}$ and the product exergy $E_{\text{P,}k}$.

(11)$${\dot{E}}_{\text{D,}k}={\dot{E}}_{\text{F,}k}-{\dot{E}}_{\text{P,}k}$$

Der exergetische Wirkungsgrad ergibt sich aus dem bekannten Nutzen-Aufwand-Verhältnis.

(12)$${\epsilon }_k=\frac{{\dot{E}}_{\text{P,}k}}{{\dot{E}}_{\text{F,}k}}$$

Die Exergievernichtung des Gesamtprozesses ergibt sich aus der Summe der Exergievernichtungen aller Komponenten. Alle Ströme die ungenutzt über die Bilanzgrenze gehen werden als Exergieverluste betrachtet. Die Exergiebilanz für ein Gesamtprozess (total) kann damit notiert werden.

(13)$${\dot{E}}_{\text{F,tot}}={\dot{E}}_{\text{P,tot}}+{\dot{E}}_{\text{D,tot}}+{\dot{E}}_{\text{L,tot}}$$

Der exergetisiche Wirkungsgrad des Gesamtprozesses ergibt sich analog zu \cref{eq:ex-eff-k}.

(14)$${\epsilon }_{\text{tot}}=\frac{{\dot{E}}_{\text{P,tot}}}{{\dot{E}}_{\text{F,tot}}}=1-\frac{{\dot{E}}_{\text{D,tot}}+{\dot{E}}_{\text{L,tot}}}{{\dot{E}}_{\text{F,tot}}}$$

Da sich die Exergievernichtung des Gesamtprozesses aus den Beiträgen der einzelnen Komponenten zusammensetzt, kann der Einfluss der einzelnen Komponenten auf die Verringerung des Gesamtwirkungsgrad aufgezeigt werden. Der Exergievernichtungskoeffizient verbindet die Informationen aus den Komponenten und dem Gesamtprozess.

(15)$$y_{\text{D,}k}=\frac{{\dot{E}}_{\text{D,}k}}{{\dot{E}}_{\text{F,tot}}}$$