On the Determination of Efficiency of a Gas Compressor

Abstract

For a gas undergoing a compression process, it is more appropriate to think of either isentropic or polytropic efficiency as process-defining parameters indicating that a given end state of compression has been achieved, rather than a measure of effectiveness of conversion of one form of energy into another. The polytropic efficiency, as defined in ASME PTC-10 standard for compressor field trials and acceptance tests, actually involves the comparison of two distinct compression processes, neither of which are actually connected to the performance of the compressors producing them. Consequently, it is not rational to compare the ASME PTC-10 polytropic efficiency of a compressor designed to compress a gas predominantly adiabatically with that for a compressor designed to compress a gas predominantly isothermally. A framework correcting this situation is set out and is illustrated with several numerical examples. Suggestions for maintaining backward compatibility with ASME PTC-10 are also put forward.

1. Introduction

The work reported here arose through a need to make comparisons of the relative effectiveness of novel gas compressors systems with incumbent compressor systems for which published compressor efficiency values were available or were determinable through the application of ASME PTC-10 [1]. This technical note discusses methods of computation of the efficiency of a gas compressor, and, in particular, aims at identifying the most appropriate method by which the efficiency of a conventional mechanical compressor may be compared with that of a hydraulic air compressor (HAC).

1.1. Scope of Compressor Types Considered in This Work

Kayode Coker [2] has usefully brought together detailed review materials of reciprocating and centrifugal compressors and more abbreviated descriptions of rotary positive displacement compressors (screws, Roots, sliding vane) that focus on mechanical details and compressor components. It is a comprehensive reference source and signposting document that contains valuable data on typical performance characteristics (adiabatic and polytropic efficiencies), gas parameters and coefficients that enter compressor calculations for power, pressure and flow. Fans are treated as low pressure ratio axial or centrifugal compressors.

1.2. Review of Frameworks for Compressor Efficiency Determination

There are relatively few sources in the existing literature that aim to explain a general synthesis and framework for the description of the efficiency of such equipment. In an effort to address this problem, Pandeya and Soedel [3] open with statements concerning a “multiplicity of definitions [for efficiency]” creating “a lot of confusion in analyzing the performance of a compressor, especially if one were to compare the performance of two different designs of compressors”. Importantly, when there is nil leakage or circulation of gas or vapour being compressed in a compressor, the ‘efficiency of performance’ proposed therein actually requires independent quantification of the heat transferred to, or removed from, the gas being compressed during the compression process in characterisation of compressor performance. Pandeya and Soedel [3] close by stating that the none of the definitions being used at that time, 1978, gave a clear picture of the various losses occurring in the compressor that affect its performance.

Rasmussen and Jakobsen [4] is a paper frequently cited by others discussing compressor efficiency. They refer to the selection of parameters (isentropic, adiabatic, isothermal, exergetic and thermal efficiencies) to quantify the energetic performance (the efficiency) of a compressor as a choice of reference process, against which actual performance is compared. Such definitions can be frustrating in design tasks where the expectation may be that the efficiency is described as a ratio of actual pneumatic power delivered by a compressor to the actual electrical power input to that compressor. Only with descriptions of compressor efficiency of the latter nature can selection of a compressor type and duty be rationally made on how economically the input energy is used in achieving the design objective; comparison of pneumatic power delivered against an idealized reference process is not very helpful. Tellingly, Rassmussen and Jakobsen conclude that “the thermal performance of a compressor is usually not very well described in compressor models, if described at all”. It is only with integrated characterization of their ‘thermal performance’ of a compressor with their ‘energetic performance’ of a compressor that the rationally most appropriate ideal reference process can be identified, if one is forced to recourse to that approach in design and selection tasks.

The motivation of the contribution of Ueno et al. [5] is to make a recommendation for the most appropriate definition of compressor efficiency to a compressor manufacturer competing in an industry that faces with a multiplicity of ‘standard’ efficiency definitions, frequently referred to interchangeably and formulated on varying assumptions which lead to incorrect comparisons between compressors offered by others in the marketplace. Ueno et al. [5] state that this precludes those procuring compressors “from obtaining a fair and accurate assessment of their compressor needs”. Importantly, in their development, Ueno et al. [5] identify that the power delivered by a compressor depends on the heat added or removed by the compressor system installation as the gas or vapour undergoes its compression process. The many and various evolutions of ASME PTC-10 [1] through the decades have been recently (2023) summarized by Panchal and Hohlweg [6]. Given the earlier widely cited contributions of Pandeya and Soedel [3] and Rassmussen and Jakobsen [4], exemplified by the challenges addressed by Ueno et al. [5], it remains surprising that systematic characterization of heat transfers across compressor system boundaries (or equivalently irreversibilities arising within the actual compressors) are not necessarily required in compressor performance determination codes.

In 2007 Casey [7] put forward a general, rational framework for compressor efficiency description in the context of the development of a new computer code. Descriptions of loss and efficiency therein apply to general turbomachines as well as to both liquids and gases and vapours, and enter Casey’s formulation through Gibb’s equation. There is an emphasis on loss accounting and the characterization of loss from various sources using entropy terms and entropy-based polytropic and isothermal efficiencies are offered. But these should not simply be alternatives for adoption amongst the multiplicity of definitions already available. Entropy change, irreversibility and heat transfer are shown to be directly linked through thermodynamic laws in this work and directly affect compressor efficiency determinations.

The literature invoking compressor efficiency definitions apply linguistic constructs such as “…efficiency is usually defined…”, or “…efficiency can be defined…”, that ignore or even contradict the actual conditions subsequently applying for the compressor or compression process under study. As will be seen, subtleties in understanding of the conditions applying to the compression process are required to arrive at a compressor performance parameter that does not ultimately become ‘disconnected’ from performance of the actual compressor under test.

1.3. Uncertainty of Compressor Efficiency Determination

This paper is about correctness or appropriateness of formulations for compressor efficiency expressions. Introducing analysis of uncertainty may obfuscate this central objective of the manuscript. Thus, uncertainty associated with determinations of compressor efficiency arising from selection of an ideal, perfect or real gas model for the fluid being compressed, or arising from measurement errors of temperature or pressure variables, is not a concern of this work. For the former issue, it is assumed that a real gas model is adopted. For the second issue, it is presumed that pressure and temperature values used in the examples set out are completely free from measurement error. The reader interested in such uncertainty issues may be referred to Lou et al. [8].

To deal with measurement uncertainty, the laws of propagation of variance may be applied to the performance measure arithmetic, but the difficult aspect of such an approach is proper characterization of the input probability distributions for the measured variables. Once this is done, with the power of personal computing available in the mid-2020s when this paper was authored, a Monte Carlo Simulation approach may lead to a much more rapid determination that accounts for the model complexities without need for simplifications (e.g., dropping of second order terms).

This work aims to highlight how the absence of a general framework for compressor efficiency determination leads not to uncertainty, but to error or inappropriate comparisons, and puts forward rationale to correct this.

1.4. Framework for Analysis

The schematic diagram of Figure 1 shows the variables used to describe a general flow process, contextualized to a compressor flow process. Inlet fluid condition is characterised by two thermodynamic state variables, the fluid absolute pressure, $P_1$ (Pa), and absolute temperature, $T_1$ (K), with the outlet conditions denoted with the subscript 3. Specification of just two thermodynamic state variables for the fluid permits the determination of the remaining thermodynamic state variables of the fluid including: (i) the density, $\rho$ (kg/m³), the specific enthalpy, $h$ (J/kg), the specific entropy $s$ (J kg⁻¹ K⁻¹) and the specific internal energy $e$ (J/kg).

The outlet fluid state maps from the inlet fluid state via work and heat transfers, $w_{13}$ and $q_{13}$ respectively, and internal irreversibility, $F_{13}$, which includes that arising from friction and minor losses. For the fluid system, with the assumption of 1D analysis, the steady flow energy equation (e.g., McPherson, [9]) applies:

$$\frac{u_1^2-u_3^2}{2}+g\left(z_1-z_3\right)+w_{13}={\displaystyle {\int }_1^{3}VdP}+F_{13}=h_3-h_1-q_{13}$$

(1)

The units of each term in (1) are J/kg, and $w_{13}$ and $q_{13}$ are taken as positive when added to the fluid in the system. The integral term in (1), ${\displaystyle \int VdP}$, is the useful flow work imparted to the fluid, or an ideal representation of the so-called indicated work. The term $w_{13}$ is the work input to the system in order to provide the useful indicated work.

From the second law, the combined heat added to the system arising from heat transfer, internal irreversibility and friction is:

$$q_c={\displaystyle {\int }_1^{3}Tds}=F_{13}+q_{13}$$

(2)

The inlet and outlet states of the fluid may be shown on thermodynamic process diagrams as in Figure 2. On the PV diagram, the area to the left of the emboldened process line may be identified with the flow work, ${\displaystyle {\int }_1^{3}VdP}$, and on the Ts diagram, the area beneath the emboldened process line may be identified with the combined heat, ${\displaystyle {\int }_1^{3}Tds}$.

Geographic elevations of inlet and outlet, $z_1,z_3$ are presumed equal. Design choices of the compressor inlet and outlet pipe diameters, delivering and discharging the fluid, are presumed to be such that the inlet and outlet fluid velocities, $u_1$ and $u_3$ respectively, have similar magnitudes.

1.5. Determination of Work, Heat Transfer and Process Irreversibility

With these simplifications, Equations (1) and (2) together define the following system of equations, applying simultaneously:

$$\begin{array}{l}{w}_{13}+q_{13}=\left(h_3-h_1\right)\\ w_{13}-F_{13}={\displaystyle {\int }_1^{3}VdP}\\ q_{13}+F_{13}={\displaystyle {\int }_1^{3}Tds}\end{array}$$

which can be cast:

$$\left[\begin{array}{ccc}1& 1& 0\\ 1& 0& -1\\ 0& 1& 1\end{array}\right]\cdot \left[\begin{array}{c}{w}_{13}\\ q_{13}\\ F_{13}\end{array}\right]=\left[\begin{array}{c}\left(h_3-h_1\right)\\ {\displaystyle {\int }_1^{3}VdP}\\ {\displaystyle {\int }_1^{3}Tds}\end{array}\right]=\left[\begin{array}{c}\left(h_3-h_1\right)\\ {\displaystyle {\int }_1^{3}VdP}\\ \left(h_3-h_1\right)-{\displaystyle {\int }_1^{3}VdP}\end{array}\right]$$

(3)

or, more compactly:

$$A^{\prime }x=b^{\prime }.$$

(4)

As the determinant of $A^{\prime }$ is zero, the solution for the work, heat transfer and irreversibility cannot be obtained through computation of $A^{\prime -1}$. Physically, the indeterminate state of this set of equations is indicative of the fact that some additional physical condition must be taken to hold, so that the system becomes mathematically determined, and solvable. Examples of such conditions are:

(a)

The physical system is adiabatic (no heat transfer): $q_{13}=0$

(b)

The physical system introduces a known amount of heat transfer: $q_{13}=J$

(c)

The physical system introduces a known amount of irreversibility: $F_{13}=X$

(d)

A known amount of input specific work is applied to the system: $w_{13}=G$

(e)

The physical system is ideal (no irreversibilities): $F_{13}=0\hspace{1em}\because q_{13}={\displaystyle {\int }_1^{2}Tds}$

When one or more of the conditions (a) to (e) hold, the system of Equation (3) may be augmented and the unknown, work, heat transfers and irreversibilities solved for systematically using a least squares formulation. As an example, condition (a) may be taken to imply an adiabatic process holds so that:

$$\left[\begin{array}{ccc}1& 1& 0\\ 1& 0& -1\\ 0& 1& 1\\ 0& 1& 0\end{array}\right]\cdot \left[\begin{array}{c}{w}_{13}\\ q_{13}\\ F_{13}\end{array}\right]=\left[\begin{array}{c}\left(h_3-h_1\right)\\ {\displaystyle {\int }_1^{3}VdP}\\ \left(h_3-h_1\right)-{\displaystyle {\int }_1^{3}VdP}\\ 0\end{array}\right]$$

(5)

or, more compactly:

$$Ax=b.$$

(6)

so that the unknown vector, $x$, may be solved for in a least squares formulation:

$$x={\left(A^{T}A\right)}^{-1}{A}^{T}b.$$

(7)

through the assembly and computation of a so-called pseudo-inverse: $p{A}^{-1}={\left(A^{T}A\right)}^{-1}{A}^T$. If condition (c), $F_{13}=X$, held instead, meaning a process with friction, Equation (5) would be replaced with:

$$\left[\begin{array}{ccc}1& 1& 0\\ 1& 0& -1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{c}{w}_{13}\\ q_{13}\\ F_{13}\end{array}\right]=\left[\begin{array}{c}\left(h_3-h_1\right)\\ {\displaystyle {\int }_1^{3}VdP}\\ \left(h_3-h_1\right)-{\displaystyle {\int }_1^{3}VdP}\\ X\end{array}\right]$$

(8)

but Equation (7) would still be used to solve the system thereafter.

Assumptions about the character of the fluid to which Equation (3) plus constraints (a) to (e) may be applied can lead to variations in the numerical values obtained for work, heat transfer and irreversibility, as well as the thermodynamic state variables obtained as part of such determinations.

1.6. Flow Work Determination for Perfect, Ideal and Real Gas

A perfect gas assumption implies that the ideal gas equation:

$$PV=RT$$

(9)

may be taken to hold, with the further condition that the specific heats, $C_v$ and $C_p$, of the fluid do not vary with temperature.

An ideal gas assumption further permits the specific heats to vary with temperature.

A real gas assumption involves the adoption of a much more complex equation of state for the gas. In this work, the REFPROP 9.1 library by Lemmon, Huber and McLinden, ref. [10] for fluid properties is relied on heavily, and its adoption implies an assumption of a real gas following the GERG-2008 equation of state described in Kunz et al. [11]. Consequently, in determination of the quantities that are used in the assembly of the coefficients and constant terms of Equation (3), a need for consistency with the choice of equation of state adopted arises.

If an ideal gas assumption is taken to hold, density $\rho =1/V$ should be evaluated with Equation (9), and use of:

$${\displaystyle {\int }_1^{3}VdP}=\frac{n}{n-1}\left(P_3{V}_3-P_1{V}_1\right)$$

(10)

$${\displaystyle {\int }_1^{3}VdP}=\frac{n}{n-1}R\left(T_3-T_1\right)=\frac{\ln \left(P_3/P_1\right)}{\ln \left(T_3/T_1\right)}R\left(T_3-T_1\right)$$

(11)

will lead to consistent results, with a process governed by a law $P{V}^n=C$ for virtually all process conditions. The exception to this is when the process is isothermal ($n=1$); $n/\left(n-1\right)$ and $\ln \left(T_3/T_1\right)$ are undefined and the integral cannot be evaluated. Under such a situation, and with an ideal gas assumption still holding:

$${\displaystyle {\int }_1^{3}VdP}=P_1{V}_1\ln \left(\frac{P_2}{P_1}\right)$$

(12)

In use of (10), (11) or (12), if known values of $P$ and $T$ are used with REFPROP to compute the specific volume of the fluid, $V$, instead of Equation (9), then because an equation of state that is not the ideal gas law is used to do this, an inconsistency (a mixture of different fluid character assumptions) arises. This will not matter much for gas states relatively close to standard conditions, but the inconsistency will become more significant as the fluid approaches critical conditions.

If the fluid must be considered a real gas, and an equation of state for a real gas is adopted, use of an analytical form for the integral may not faithfully represent the infinite number of intermediate states of the fluid that exist between inlet and outlet conditions. Real gas equations of state are typically sufficiently complex that the only effective recourse is: (i) to recognise that for $P{V}^n=C$:

$${\displaystyle {\int }_1^{3}VdP}={\displaystyle {\int }_1^3{\left(\frac{C}{P}\right)}^{1/n}dP},$$

(13)

holds, where: (ii) $n$ can be estimated from the end states, using the equation of state via $V$ or $\rho$ as follows:

$$P_1{V}_1^n=P_2{V}_2^n\Rightarrow n=\frac{\ln \left(P_2/P_1\right)}{\ln \left(V_1/V_2\right)}=\frac{\ln \left(P_2/P_1\right)}{\ln \left({\rho }_2/{\rho }_1\right)}$$

(14)

and to integrate numerically. If $P{V}^n\ne C$, that is the value of $n$ varies in the process, then Equation (14) must be considered an approximation. $n$ is known as the polytropic index and can be seen, in Equation (13), to act as a process-defining parameter.

In this work, the Simpson’s Rule routine qsimp from Press et al. [12] is used with the REFPROP equation of state and Equation (13) to determine the flow work with a defined precision (EPS = 0.000001) and with real gas assumptions for all process conditions, including isothermal conditions. The flow work term is then used to populate Equation (3) and ${\displaystyle \int Tds}$ is established with further calls to the equation of state for the end point enthalpies.

2. ASME PTC-10, the Standard Method of Determination of Compressor Efficiency

2.1. Polytropic Efficiency

The pioneering work of Schulz [13] in 1962 on polytropic compression of compressors laid the foundations for ASME Performance Test Code (PTC) 10 [1] which sets out the industry standard procedure for the evaluation of compressor performance. Schultz’ work explained a definition of a so-called polytropic process as one where the quantity:

$$\frac{1}{\rho }\frac{dP}{dH}=\mathrm{e}$$

(15)

remains constant throughout the process (Note that it is frequently the case that a polytropic process is denoted, $P{V}^n=C$, and this works well for perfect and ideal gases, but for real gases, the polytropic index $n$ can vary between process end states. Schultz’ definition of a polytropic process uses Equation (15) as a conditioning requirement, irrespective of the nature of the gas). In the context of Figure 2, the constant $\mathrm{e}$ then effectively defines a process trajectory across a thermodynamic diagram, that begins from the location of the inlet conditions on the diagram, ($P_1$, $T_1$).

Schulz [14] also shows the quantity $\mathrm{e}$ to be equal to the polytropic efficiency, as further defined in Equation (16) and numerically illustrated, albeit crudely, in Cases 4 to 9 of

Table 1. Air delivery temperatures and work, heat and irreversibility terms computed using P₁ = 101.325 kPa, t₁ = 20 °C, r_p = 3 and η_isen = 77.00% or η_poly,PTC10 = 80.19% (cases 1 and 2, respectively). Case 3 is identical to Case 2, with the exception that it is reversible and not adiabatic. Cases 4 to 9 decompose Case 2 into six incremental stages, to illustrate calculation of e by Equation (15).

Case			1	2	3	4	5	6	7	8	9	Sum of
Efficiency type			Isentropic	Polytropic	Polytropic	Polytropic	Polytropic	Polytropic	Polytropic	Polytropic	Polytropic	Cases
Defined value			77.00%	80.19%	80.19%	80.19%	80.19%	80.19%	80.19%	80.19%	80.19%	4 to 9
Initial air state
	P1	kPa	101.325	101.325	101.325	101.325	121.691	146.151	175.528	210.809	253.181
	t1	°C	20.00	20.00	20.00	20.00	39.77	60.84	83.30	107.23	132.68
	T1	K	293.15	293.15	293.15	293.15	312.92	333.99	356.45	380.38	405.83
Pressure ratio
	rp		3	3	3	1.201	1.201	1.201	1.201	1.201	1.201
Final air state
	P3	kPa	303.975	303.975	303.975	121.691	146.151	175.528	210.809	253.181	304.071
	t3	°C	159.67	159.67	159.67	39.77	60.84	83.30	107.23	132.68	159.75
	T3	K	432.82	432.82	432.82	312.92	333.99	356.45	380.38	405.83	432.90
Additional condition
			qAB	qAB	FAB	qAB	qAB	qAB	qAB	qAB	qAB
			0	0	0	0	0	0	0	0	0
Work, Heat and Irreversibility (Equation (5))
	dH	kJ/kg	141.061	141.060	141.060	19.860	21.201	22.632	24.157	25.781	27.511	141.142
	IntVdP	kJ/kg	113.101	113.101	113.101	15.925	17.001	18.148	19.371	20.674	22.061	113.180
	IntTdS	kJ/kg	27.960	27.958	27.958	3.934	4.200	4.483	4.786	5.107	5.450	27.960
	w13	kJ/kg	141.061	141.060	113.101	19.860	21.201	22.632	24.157	25.781	27.511	141.142
	q13	kJ/kg	0.000	0.000	27.958	0.000	0.000	0.000	0.000	0.000	0.000	0.000
	F13	kJ/kg	27.960	27.958	0.000	3.934	4.200	4.483	4.786	5.107	5.450	27.960
Backcalculated polytropic efficiency
	ASME PTC 10		80.19%	80.19%	80.19%	80.19%	80.19%	80.19%	80.19%	80.19%	80.19%
	IntVdP/w13		80.18%	80.18%	100.00%	80.19%	80.19%	80.19%	80.19%	80.19%	80.19%
Incremental polytropic efficiency
Mean density		kg/m3				1.280	1.439	1.620	1.822	2.051	2.308
dP/dH		m3/kg				1.025	1.154	1.298	1.460	1.644	1.850
$\mathrm{e}$(Equation (15))						80.15%	80.15%	80.15%	80.15%	80.15%	80.15%

. Although ASME PTC-10 has recently found to be deficient in computation of the polytropic efficiency of compressors compressing some blends of hydrocarbon gases (Sandberg and Colby, [13]), the standard has been reaffirmed on several occasions since publication in 1997, and still stands as the accepted method of factory acceptance and field performance testing. Sandberg and Colby [13] point out that despite this, the isentropic efficiency can frequently be adopted as the basis used for air compression specifically, which clearly can lead to confusion, if the basis—polytropic or isentropic—of a stated level of compressor performance is not declared.

In Schulz’ original 1962 paper, the polytropic efficiency is defined:

$${\eta }_{poly,PTC10}=\frac{\mathrm{reversible}\hspace{0.33em}\mathrm{mechanical}\hspace{0.33em}\mathrm{energy}\hspace{0.33em}\mathrm{input}}{\mathrm{work}\hspace{0.33em}\mathrm{done}\hspace{0.33em}\mathrm{by}\hspace{0.33em}\mathrm{adiabatic}\hspace{0.33em}\mathrm{compressor}}.$$

(16)

With simplifying assumptions for elevation difference and velocity at compressor inlet and outlet, and accounting for the reversible nature of the process ($F_{13}=0$), from Equation (1), the numerator is:

$$w_{13,rev}={\displaystyle {\int }_1^{3}VdP}+0=h_3-h_1-{\left.q_{13}\right|}_{F_{13}=0}$$

(17)

and the denominator is:

$$w_{13,adia}={\displaystyle {\int }_1^{3}VdP}+{\left.F_{13}\right|}_{q_{13}=0}=h_3-h_1-0$$

(18)

so that:

$${\eta }_{poly,PTC10}=\frac{w_{13,rev}}{w_{13,adia}}=\frac{{\displaystyle {\int }_1^{3}VdP}}{h_3-h_1}\hspace{1em}\left[=\frac{{\displaystyle {\int }_1^{3}VdP}}{{\displaystyle {\int }_1^{3}VdP}+{\displaystyle {\int }_1^{3}Tds}}=\frac{{\displaystyle {\int }_1^{3}VdP}}{{\displaystyle {\int }_1^{3}VdP}+{\left.F_{13}\right|}_{q_{13}=0}}\right]$$

(19)

Thus, in the context of compressors and compression processes, ${\eta }_{poly,PTC10}$ holds special significance. It is a constant that reflects the capacity of a given compressor to deliver a particular end state given the inlet conditions; that is, it is a process-defining parameter.

As terms ${\displaystyle {\int }_1^{3}VdP}$ and $h_3-h_1$ can be quantified with knowledge of the process end states only, ($P_1$, $T_1$) and ($P_3$, $T_3$), and because there are an infinite number of ways to arrive at the outlet state from the inlet state in practice, it may then become appreciated that ${\eta }_{poly,PTC10}$ does not actually depend on the characteristics of the compressor, providing the compression process.

2.2. Isentropic Efficiency

In Schulz’ original 1962 paper [14], the isentropic efficiency is defined:

$${\eta }_{isen}=\frac{\mathrm{isentropic}\hspace{0.33em}\mathrm{energy}\hspace{0.33em}\mathrm{input}}{\mathrm{work}\hspace{0.33em}\mathrm{done}\hspace{0.33em}\mathrm{by}\hspace{0.33em}\mathrm{adiabatic}\hspace{0.33em}\mathrm{compressor}}$$

(20)

With simplifying assumptions for elevation difference and velocity at compressor inlet and outlet, and accounting for the reversible, adiabatic nature of the process, from Equation (1), the numerator is:

$$w_{14,isen}={\displaystyle {\int }_1^{4}VdP}=h_4-h_1$$

(21)

so that:

$${\eta }_{isen}=\frac{w_{14,isen}}{w_{13,adia}}=\frac{{\displaystyle {\int }_1^{4}VdP}}{h_3-h_1}$$

(22)

The isentropic efficiency also takes on a special meaning in that it is a process-defining parameter. This is because it defines a trajectory on a process diagram, where that trajectory is expressed as an ‘extent of deviation’ of the adiabatic process from the trajectory of the isentropic process on the same diagram.

3. A Practical Application of Efficiency Values

In the previous section, it was explained that both the isentropic efficiency and the polytropic efficiency are quantities that are process-defining parameters. Why, then, are they also referred to as ‘efficiencies’? It is a rhetorical question. The minds of those who first conceived and set out the definitions and terminologies for those terms cannot be known with certainty. However, one very important thing that design engineers use efficiencies for is to size required equipment. For example, an isentropic efficiency value for an air compressor, looked up from a database of recorded, historical compressor performance, together with design input gas conditions and pressure ratio, may be used to estimate the required motor shaft work input, and hence the size of the motor that would be needed to drive the compressor.

This lends the isentropic efficiency an additional special utility, over and above simply being a process-defining parameter. This additional character is shared with the polytropic efficiency; database values can be adopted, together with inlet conditions and a pressure ratio, to permit sizing of the motor to drive the compressor. These processes are explained in the following sub-sections. In each case, the intermediate step is to compute a delivery temperature of the air, after the assumed compression process.

3.1. Real Process Defined with an Isentropic Efficiency

Suppose that $P_1$, $T_1$, $r_p=P_3/P_1=P_4/P_1$ and ${\eta }_{isen}$ are known. Firstly, $P_1$ and $T_1$ being known, together imply that $s_1$ and $h_1$ are known, through the equation of state. For the isentropic process, $s_4=s_1$ so that $P_4$ and $s_4$ known together mean that $h_4$ and $T_4$ become known too, again, through the equation of state. As the states of the gas at the process endpoints are now established for the isentropic process ($P_1$, $T_1$) and ($P_4$,$T_4$), the numerator of Equation (22), ${\displaystyle {\int }_1^{4}VdP}$, can be quantified. This can be done through determination of $n$ using Equation (14), followed by numerical integration for Equation (13). As process 1 to 4 is a reversible ($F_{14}=0$) adiabatic ($q_{14}=0$), the isentropic flow work can also be obtained directly through ${\displaystyle {\int }_1^{4}VdP=h_4-h_1}$ and is the better method to adopt if the gas is assumed real, because it is not necessary to assume that $P{V}^n=C$ holds through the compression process as required by the application of Equation (13). Either way, to establish the delivery temperature of the real compression process, the actual delivery enthalpy is found from Equation (22):

$$h_3=\frac{{\displaystyle {\int }_1^{4}VdP}}{{\eta }_{isen}}+h_1=\frac{h_4-h_1}{{\eta }_{isen}}+h_1$$

(23)

With ($P_3$,$h_3$) defined, the actual delivery temperature $T_3$ is obtained from the equation of state.

3.2. Real Process Defined with a Polytropic Efficiency

Suppose that $P_1$, $T_1$, $r_p=P_3/P_1=P_4/P_1$ and ${\eta }_{poly,PTC10}$ are known. In this case, the reference process is polytropic. An approximate, initial value of $T_3$ may be produced, for numerical processes only, using:

$$T_3=T_1{\left(r_p\right)}^{(k-1)/k}$$

(24)

where:

$$k=C_p/C_v.$$

(25)

Thereafter the approximate value of $T_3$ may be used in computation of an estimate of the polytropic efficiency, as follows:

$P_1$, $T_1$ and $P_3$,$T_3$ permits computation of $n$ using Equation (14). $P_1$, $T_1$ and $r_p$ permits $T_4$ to be computed, as in the previous section. For an isentropic process, $n=k$, and thus $k$, can be computed with $P_1$, $T_1$ and $P_4$,$T_4$ also using Equation (14). Then, the ASME PTC-10-recommended equation may be adopted to compute the polytopic efficiency:

$${\eta }_{poly,PTC10}^{\ast }=\frac{n/\left(n-1\right)}{k/\left(k-1\right)}\cdot \frac{h_4-h_1}{h_3-h_1}\cdot \frac{\left(P_3/{\rho }_3-P_1/{\rho }_1\right)}{\left(P_4/{\rho }_4-P_1/{\rho }_1\right)}$$

(26)

The error in the polytropic efficiency so determined from the initial value of $T_3$ is:

$$\epsilon =\left({\eta }_{poly,PTC10}-{\eta }_{poly,PTC10}^{\ast }\right).$$

(27)

and $\epsilon$ may be reduced by making corrections to $T_3$ using numerical root finding techniques such as the Newton–Raphson method, so that the value of $T_3$ consistent with the process-defining parameter is found by minimising Equation (27). In Equation (26), to obtain the value of $T_3$ that strictly adheres to ASME PTC-10, $q_{13}=0$ (because the denominator of Equation (20) is adiabatic). Equation (26) arises from adoption of the correction:

$$f=\frac{h_4-h_1}{k/\left(k-1\right)\left(P_4/{\rho }_4-P_1/{\rho }_1\right)}$$

(28)

that enters into the ASME PTC-10 scheme, due to Schulz [14], as explained by Wei [15] and evaluates to very close to unity. In Equation (26), ${\displaystyle {\int }_1^{3}VdP}$ may also be obtained numerically starting with the approximate value of $T_3$. Effectively, Equation (26) defines the relationship between alternative process-defining parameters, ${\eta }_{poly,PTC10}$ and $n$.

3.3. A Numerical Example

The first two numerical example cases in Table 1 consider the same real air compression process (1 to 3, in Figure 2), differing only in that for Case 1, the reference process is an isentropic process (1 to 4, in Figure 2) and the process-defining parameter is ${\eta }_{isen}$, whereas for Case 2, the reference process is polytropic (1 to 3, in Figure 2) and the process-defining parameter is ${\eta }_{poly,PTC10}$. $T_3$ is determined according to the process set out in Section 3.1 for Case 1 and according to the process set out in Section 3.2 for all remaining cases. The back-calculated polytropic efficiency value (ASME PTC-10) is determined directly from the process end states and the condition that $q_{13}=0$, to simulate the process end states being measured in a test.

There are no differences in the computed values between Case 1 and Case 2; adopting either ${\eta }_{isen}$ or ${\eta }_{poly,PTC10}$ as the process-defining parameter leads to the same value of the input work, $w_{13}$, and consequently, the same motor selection would be made to match the compressor in either case. It may also be concluded that the process-defining parameter ${\eta }_{isen}=77.00\%$ is equivalent to the process-defining parameter ${\eta }_{poly,PTC10}=80.19\%$. This relationship between the two process-defining parameters is usefully articulated in the approximate expression presented by Manzoor [16]:

$${\eta }_{poly}=\frac{\ln \left(r_p{\left(k-1\right)}^{/k}\right)}{\ln \left(\frac{r_p{\left(k-1\right)}^{/k}-1}{{\eta }_{isen}}+1\right)}$$

(29)

and illustrates clearly that both are independent of the compressor delivering the process.

Case 3 of Table 1 considers a reversible polytropic process between the same two end states of the process to explicitly permit identification of the reversible mechanical energy input $w_{13,rev}$ (17) as is required to be used in the numerator of Equation (16). For the conditions between Cases 2 and 3 shown, this is numerically identical to the adiabatic mechanical energy input; ${\displaystyle {\int }_1^{3}VdP}$ depends only on the end states of the process. Case 2 explicitly permits identification of the adiabatic input work, $w_{13,adia}$ (18), also adopting a process-defining parameter, ${\eta }_{poly,PTC10}$. The ratio:

$$\frac{w_{13,rev}}{w_{13,adia}}=\frac{{\left[{\displaystyle {\int }_1^{3}VdP}\right]}_{Case3}}{{\left[h_3-h_1\right]}_{Case2}}=\frac{113.101}{141.060}=80.18\%$$

(30)

correctly leads to a value for ${\eta }_{poly,PTC10}$ consistent with that defined to characterise the process, as required by Equation (16), but Equation (30) highlights that Equation (16) actually considers two distinct processes with different conditions. Note that in Case 3, the heat transfer has been exchanged for reversibility, and the opposite is true for Case 2. While Equation (30), and the need to consider two separate processes to obtain ${\eta }_{poly,PTC10}$ may seem overly elaborated, as will be seen, the need for this degree of precision of conformance with the ASME PTC-10 definition (Equation (16)) will become apparent when more general compression processes need to be compared to those of ASME PTC-10.

4. The Problem with ASME PTC-10

It needs to be far better recognised that Schulz’ 1962 formulation [14] actually uses two process-defining parameters, rather than one. The first is ${\eta }_{poly,PTC10}$; the second is easy to miss, but is in fact that the heat transferred in the denominator of Equation (16) is zero; $q_{13}=0$. In effect, Equation (16) assesses the useful, indicated, work defined by the process end states, and normalizes this, not by the actual compressor work $w_{13}=h_3-h_1-q_{13}$, but by the work done by a hypothetical compressor that follows a polytropic process yet permits no heat transfer from system to surroundings, $w_{13,adia}=h_3-h_1$. In adopting the ASME PTC-10 standard, in effect, this leads to the performance of the actual compressor being removed from the defined metric of compressor performance, and this is the problem with ASME PTC-10.

In mechanical compressors, where the velocity of the gas is typically high relative to the flow path through the machine that the gas takes, the assumption that the actual process that the gas experiences is adiabatic (the denominators of either Equation (19) or Equation (22)) is generally quite a good assumption. Close to adiabatic conditions generally mean that the temperature at the outlet of the compressor will be elevated appreciably from the temperature at inlet, and that further, this rise in temperature will generally mean that a temperature gradient is set up in the compression process to drive heat from the fluid across the compressor casing or block to the surroundings. For a pipe with gases flowing on outer and inner surfaces, the overall heat transfer co-efficient, ${\psi }_0$, across the pipe wall may be estimated as follows:

$$\frac{1}{{\psi }_0}=\frac{r_2}{r_1{\psi }_1}+\frac{r_2}{\kappa }\ln \frac{r_2}{r_1}+\frac{1}{{\psi }_2}$$

(31)

where $\psi$ is a convective heat transfer coefficient, $\kappa$ is the thermal conductivity of the metal from which the pipe, radius $r$, has been fabricated, and subscripts 1 and 2 denote inner surface of and outer surface of the pipe, respectively. If the pipe is installed in slowly moving or still air of ambient temperature, ${\psi }_0$ will be dominated by the term containing ${\psi }_2$ which will be very low. Even when the ambient air is forced over the outer pipe surface at great speed so that ${\psi }_2$ becomes larger, if the pipe has appreciable thickness (for example, 12.5 mm), the logarithmic term will be low and will dominate ${\psi }_0$ instead. The insulating effects of thick compressor casing and still ambient air are sufficient to render the assumption of adiabatic compression excellent, even with large LMTD. However some mechanical compressors are equipped with water jackets, and some are designed to compress while simultaneously transferring heat, and in these cases the assumption of adiabatic compression is very poor.

A second problem with the definition of polytropic efficiency in Schultz’ 1962 formulation, and the ASME PTC-10 derived from it, is that for compressors capable of near isothermal compression, which are far from adiabatic, the denominators of either Equation (19) or Equation (22) can be negative because $h_3$ can be lower than $h_1$ for the real, nearly isothermal process, and negative efficiency values result.

It is also problematic to evaluate the performance of multi-stage compressors with intercooling and/or aftercooling with a ${\eta }_{poly,PTC10}$ for similar reasons. In the latter situation, where the objective is to approach an isothermal process more closely, recourse is made to the definition of the isothermal efficiency, where the reference process is isothermal, and the actual performance of the compressor system is quantified with $w_{13}=h_3-h_1-q_{13}$. This is consistent with definitions of isothermal efficiencies cited by other authors, e.g., Heo et al. [17]:

$${\eta }_{iso}=\frac{w_{15}}{w_{13}}=\frac{h_5-h_1-q_{15}}{h_3-h_1-q_{13}}$$

(32)

The numerator of Equation (32) is the ideal isothermal process (see Figure 1), and the denominator is the overall actual enthalpy difference between inlet and outlet of the multi-stage compressor, less $q_{13}$, the total heat transferred in the inter/aftercoolers. As both the reference process is different from Schulz’ polytrope and the real process includes the heat transferred, direct comparison of ${\eta }_{poly,PTC10}$ and ${\eta }_{iso}$ is not rational. However, it is desirable to have a method of direct comparison of compression machines designed and operated nearly adiabatically with compression machines designed and operated nearly isothermally. At the same time, it is undesirable and impractical to re-evaluate performance measures of compression processes that have already been evaluated using the ASME PTC-10 standard. Since 1962, thousands of compressors have been tested against Schultz’ work and manufactures have assembled valuable databases of performance.

5. A Suggested Improvement for ASME PTC-10

5.1. Revised Formulation

ASME PTC-10 effectively has two process-defining parameters: ${\eta }_{poly,PTC10}$ and $q_{13}=0$. The central suggested improvement to AMSE PTC-10 is to modify the second of these, so that:

$$q_{13}=J$$

(33)

where $J$ is the heat transferred from surroundings to system during the process, whether this be determined via calculation, measurement via calorimetry or other means. Secondly, a modified polytropic efficiency is defined, differing only from Equation (19) or Equation (26) in that it adopts the real rather than the adiabatic process in the denominator:

$${\eta }_{poly,real}=\frac{w_{13,rev}}{w_{13,real}}=\frac{{\displaystyle {\int }_1^{3}VdP}}{h_3-h_1-q_{13}}\hspace{1em}\left[=\frac{h_3-h_1-{\left.q_{13}\right|}_{F_{13}=0}}{h_3-h_1-q_{13}}\right]$$

(34)

The quotient in square brackets in Equation (34) is presented to show that in retaining $q_{13}$ in the denominator of Equation (34), the result does not reduce to unity as: $q_{13}\ne {\left.q_{13}\right|}_{F_{13}=0}$. It should be reiterated, Equation (20), and now Equation (34), consider two distinct processes. However, the numerator, ${\displaystyle {\int }_1^{3}VdP}$, only depends on the end states of the process, whether the process is real or reversible, and means that, effectively, numerator and denominator are for the same process. It is believed that this represents a further conceptual advantage over PTC-10. The polytropic efficiency of Equation (34) is precisely equivalent to:

$${\eta }_{poly,real}=\frac{w_{13,rev}}{w_{13,real}}=\frac{{\displaystyle {\int }_1^{3}VdP}}{{\displaystyle {\int }_1^{3}VdP}+F_{13}}$$

(35)

from which it may be appreciated that the role of compressor irreversibility is more prominent than in ASME PTC-10. If $q_{13}$ can be estimated numerically, analytically or from calorimetry, then Equations (3), (7) and (33) may be used to obtain $F_{13}$, and this provides a more objective basis for the comparison of performance, properly reflecting the actual compressor irreversibility, rather than permitting comparison of compression processes.

Permitting $q_{13}\ne 0$ renders some auxiliary effects that must be allowed for. Firstly, the delivery temperature, $T_3$, should reflect the heat transfer now permitted. This is simply done by modifying the terms in Equation (29) that reflected the adiabatic condition so that they instead reflect the real process:

$${\eta }_{poly,real}^{\ast }=\frac{n/\left(n-1\right)}{k/\left(k-1\right)}\cdot \frac{h_4-h_1}{h_3-h_1-q_{13}}\cdot \frac{\left(P_3/{\rho }_3-P_1/{\rho }_1\right)}{\left(P_4/{\rho }_4-P_1/{\rho }_1\right)}$$

(36)

and otherwise following the numerical procedure of Section 3.2 exactly.

$T_3$ then is used in the numerical integration of Equation (13) using Equation (14), as before, to determine ${\displaystyle {\int }_1^{3}VdP}$, which will have a different value in comparison to the case where the adiabatic polytrope was considered as the reference process. This is very important, because when one owns and operates a compressor, what one pays for is the pressure of a gas at $P_1$ to be raised to $P_3$. In achieving this engineering objective, if ${\displaystyle {\int }_1^{3}VdP}$ is lower because, by design, heat has been permitted to leave the compression system, so that the delivery temperature is lower, then one will pay less to compress.

With Equation (33) augmenting Equation (3), using Equation (7) then permits $w_{13,real}$ and $F_{13,real}$ to be established.

5.2. Backward Compatibility with ASME PTC-10

The purpose of the proposed modification to ASME PTC-10 is to permit direct comparison of compressors designed to allow heat transfer with those, principally the incumbent fleet (irrespective of manufacturer), that do not. A key question is, for the incumbents, what database value of ${\eta }_{poly,real}$ should be used as the target value to converge upon in the numerical determination of $T_3$ as per Section 3.2, when no prior database of values of ${\eta }_{poly,real}$ exist?

The answer is, conveniently, to use the existing database value of ${\eta }_{poly,PTC10}$, the value of ${\eta }_{poly,real}$ obtained if the compressor under test was perfectly insulated. Practically, ${\eta }_{poly,PTC10}$ needs to be identified, only, as a process-defining parameter. This ‘works’ because the effect of permitting $q_{13}\ne 0$ on the delivery temperature calculation (and hence the ${\displaystyle {\int }_1^{3}VdP}$ integration), as well as the evaluation of the real work input to the compressor is counterbalancing; ${\eta }_{poly,real}$ is invariant under $q_{13}=J$ for any $J$, including $J=0$.

Thus, the most appropriate discriminating parameter of compressor performance is not ${\eta }_{poly,real}$, but is $w_{13,real}$ (J/kg). This makes sense because it is the work input required in the real process to raise the pressure of 1 kg of gas from $P_1$ to $P_3$. It is a parameter, ‘normalized’ by the mass flow of gas being compressed, so that it readily facilitates the comparison of compressors of different types and different scales.

5.3. Summary of the Benefits

Where does this leave us? The suggested modification to ASME PTC-10 can be summarised by recognising that there are two process-defining parameters for the compressor performance: ${\eta }_{poly,PTC10}={\eta }_{poly,real}$ and $q_{13}=J$. Furthermore:

• The overall effect is to extend the ASME PTC-10 scheme, so that actual compressor performance can be assessed, rather than the performance of a compression process.
• Compressor irreversibility becomes prominent as a comparison parameter, whereas beforehand, the standard did not consider it. Establishing $q_{13}=J$ effectively also quantifies $F_{13}$ of the compressor.
• The modification permits the consistent, and rational analysis of compressors capable of near isothermal performance, so that their performance can be sensibly compared with compressors designed to perform near adiabatically, whereas this was not possible with the ASME PTC-10 metric with an imputed $q_{13}=0$.

5.4. A Further Numerical Example

To illustrate these points, it is worth inspecting the additional numerical example provided in

Table 2. Calculations of final air state and work, heat and irreversibility for a compressor with P₁ = 101.325 kPa, t₁ = 20 °C, r_p = 3, η_isen = 77.00% or η_poly,PTC10 = 80.19% while varying q₁₃= J.

Case			1	2	3	4	5	6	7	7	8
Efficiency type			Polytropic	Polytropic	Polytropic	Polytropic	Polytropic	Polytropic	Polytropic	Polytropic	Polytropic
Defined value			80.19%	80.19%	80.19%	80.19%	80.19%	80.19%	80.19%	80.19%	80.19%
Initial air state
	P1	kPa	101.325	101.325	101.325	101.325	101.325	101.325	101.325	101.325	101.325
	t1	°C	20.00	20.00	20.00	20.00	20.00	20.00	20.00	20.00	20.00
	T1	K	293.15	293.15	293.15	293.15	293.15	293.15	293.15	293.15	293.15
Pressure ratio
	rp		3	3	3	3	3	3	3	3	3
Final air state
	P3	kPa	303.975	303.975	303.975	303.975	303.975	303.975	303.975	303.975	303.975
	t3	°C	159.67	159.67	147.83	127.84	100.04	69.81	39.32	27.05	20.03
	T3	K	432.82	432.82	420.98	400.99	373.19	342.96	312.47	300.20	293.18
Additional condition
			qAB	qAB	qAB	qAB	qAB	qAB	qAB	qAB	qAB
			0	−10	−10	−26.8	−50	−75	−100	−110	−115.7
Work, Heat and Irreversibility (Equation (5))
	dH	kJ/kg	141.060	141.060	128.991	108.656	80.442	49.849	19.033	6.638	−0.447
	IntVdP	kJ/kg	113.101	113.101	111.443	108.609	104.588	100.103	95.441	93.520	92.409
	IntTdS	kJ/kg	27.958	27.958	17.548	0.048	−24.146	−50.255	−76.407	−86.882	−92.857
	w13	kJ/kg	141.060	141.060	138.991	135.456	130.442	124.849	119.033	116.638	115.253
	q13	kJ/kg	0.000	0.000	−10.000	−26.800	−50.000	−75.000	−100.000	−110.000	−115.700
	F13	kJ/kg	27.958	27.958	27.548	26.848	25.854	24.745	23.593	23.118	22.843
Backcalculated polytropic efficiency
	ASME PTC 10		80.19%	80.19%	86.41%	99.97%	130.03%	200.8%	501.5%	1409%	−20,663%
	IntVdP/w13		80.18%	80.18%	80.18%	80.18%	80.18%	80.18%	80.18%	80.18%	80.18%

. Cases 1 and 2 show calculations strictly according to ASME PTC-10, which demonstrate undesirable insensitivity to $q_{13}=J$. Case 3 should be compared to Case 2, when heat of 10 kJ/kg is known to leave the system. Cases 3 to 8 illustrate progressively greater amounts of heat leaving the compressor during compression, with Case 8 corresponding to a nearly isothermal compressor exhibiting ΔT₁₃ = 30 mK. Back-calculated values of ${\eta }_{poly,PTC10}$ strictly should not be computed for Cases 2 to 8 because for these cases $q_{13}\ne 0$, which violates the ASME PTC-10 conditions. These are thus in error because they have been misapplied, but nevertheless, the evaluated quantities are provided to illustrate the problems ${\eta }_{poly,PTC10}$ exhibits when compressors are known to transfer heat to their surroundings. For this reason, the erroneously applied values are presented in italics in Table 2.

As a discriminator of compressor performance, ${\eta }_{poly,real}$ is diminished in value due to its insensitivity to heat transferred. However, it can still be applied to compute the required motor input work to match to the compressor under test, providing it is applied to the appropriate value of indicated work, that is:

$$w_{13,real}=\frac{{\left.{\displaystyle {\int }_1^{3}VdP}\right|}_{q_{13}=J}}{{\eta }_{poly,real}}.$$

(37)

Importantly, it is useful to highlight the backward compatibility:

$$w_{13,real}=\frac{{\left.{\displaystyle {\int }_1^{3}VdP}\right|}_{q_{13}=J}}{{\left.{\eta }_{poly,PTC10}\right|}_{q_{13}=0}}$$

(38)

${\eta }_{poly,real}$ in Equation (37) or ${\eta }_{poly,PTC10}$ in Equation (38) thus retain their usual special meaning in a practical, engineering, sense as an ‘efficiency’. However, the more material values to examine to discriminate compressor performance are the actual work input, $w_{13,real}$ or compressor irreversibility,$F_{13}$, with larger values meaning worse, for both measures. Under this modified scheme, it cannot be said that near isothermal compressors are more ‘efficient’, but it can be said that they consume less power per unit mass flow to take the same mass flow of inlet air to the same delivery pressure. Their superiority in this respect is clear.

6. Two Illustrative Case Studies

6.1. Multi-Stage Polytropic Air Compression

In this section, data from a ~9 MW rated, 3-stage, intercooled and aftercooled centrifugal compressor forming part of an air liquefaction plant is considered initially, and then, for comparison purposes, it is supposed that the same compressor is retrofitted with water jackets.

Base case: Free air delivered: 120,700 Nm³/h; power draw: 9068.8 kW. Discharge Air (gauge) Pressure: 1st Stage—102.5 kPa, 2nd stage—221.3 kPa, 3rd stage—424.8 kPa; Discharge Air Temperature: 1st Stage—51.8 °C, 2nd stage 71.2 °C, 3rd stage 74.0 °C. From the 3rd stage, the compressed air was passed to a direct contact aftercooler which produced a discharge temperature of 11.8 °C. On the day of operations, the intake atmospheric air temperature and pressure was (minus) −22.9 °C and 102.6 kPa. Intercooler water was circulated to a forced convection cooling tower that discharged heat to atmosphere. The return water temperatures to the intercoolers were unknown.

At ‘normal’ conditions, with negligible humidity, inlet air density is ~1.293 kg/m³ leading to the mass flow rate of air being 43.351 kg/s. Assuming air inlet temperature of 8.0 °C to Stage 2 and Stage 3, due to effective operation of the cooling system on a very cold day, the performance of the compressor installation is reported in

Table 3. Performance of case study multi-stage polytropic compressor (LHS) and revised performance when assumed that each stage is equipped with a water jacket (RHS). Grey shaded cells show the input required for each set of calculations.

Observed Operating Conditions				Equipped with Water Jacket
	Stage 1	Stage 2	Stage 3		Stage 1	Stage 2	Stage 3
Inlet condition				Inlet condition
P1 (kPa)	101.051	205.1	323.9	P1 (kPa)	101.051	205.1	323.9
t1 (°C)	−22.9	8.0	8.0	t1 (°C)	−22.9	8.0	8.0
T1 (K)	250.3	281.2	281.2	T1 (K)	250.3	281.2	281.2
Delivery condition				Delivery condition
P3 (kPa)	205.1	323.9	527.4	P3 (kPa)	205.1	323.9	527.4
t3_actual (°C)	51.8	71.2	74.0
T3_actual (K)	325.0	344.4	347.2
Process-defining parameters				Process-defining parameters
	Adiabatic	Adiabatic	Adiabatic		Real	Real	Real
qAB (kJ/kg)	0	0	0	qAB (kJ/kg)	−38.000	−57.000	−60.000
eta_poly,adia	77.47%	64.31%	65.97%	eta_poly,adia	77.47%	64.31%	65.97%
Stage performance				Stage performance
T3_poly (K)	325.0	344.3	347.1	T3_poly (K)	282.0	281.8	281.2
dH (kJ/kg)	74.982	63.560	66.371	dH (kJ/kg)	31.667	0.356	−0.453
IntVdP (kJ/kg)	58.090	40.873	43.783	IntVdP (kJ/kg)	53.972	36.883	39.282
IntTdS (kJ/kg)	16.892	22.687	22.587	IntTdS (kJ/kg)	−22.306	−36.527	−39.735
w13 (kJ/kg)	74.982	63.560	66.371	w13 (kJ/kg)	69.667	57.356	59.547
q13 (kJ/kg)	0.000	0.000	0.000	q13 (kJ/kg)	−38.000	−57.000	−60.000
F13 (kJ/kg)	16.892	22.687	22.587	F13 (kJ/kg)	15.694	20.473	20.265
Motor sizing				Motor sizing
Motor Eff	98.0%	98.0%	98.0%	Motor Eff	98.0%	98.0%	98.0%
W13e (kJ/kg)	76.512	64.857	67.726	W13e (kJ/kg)	71.089	58.527	60.762
Motor Pwr (kW)	3316	2811	2935	Motor Pwr (kW)	3081	2537	2634
Intercooler/Aftercooler processes				Intercooler/Aftercooler processes
P_i (kPa)	205.1	323.9	527.4	P_i (kPa)	205.1	323.9	527.4
t_i (oC)	8	8	11.8	t_i (oC)	8	8	11.8
T_i (K)	281.15	281.15	284.95	T_i (K)	281.15	281.15	284.95
h_i (kJ/kg)	325.241	344.631	347.134	h_i (kJ/kg)	281.926	281.427	280.311
h_i_del (kJ/kg)	281.071	280.764	284.092	h_i_del (kJ/kg)	281.071	280.764	284.092
q _i (kJ/kg)	−44.170	−63.867	−63.043	q _i (kJ/kg)	−0.854	−0.663	3.781
Summary				Summary
Total indicated work		142.746	(kJ/kg)	Total indicated work		130.137	(kJ/kg)
Total input work		204.913	(kJ/kg)	Total work		186.570	(kJ/kg)
Heat transferred in stages		0.000	(kJ/kg)	Heat transferred in stages		−155.000	(kJ/kg)
Heat transferred in coolers		−171.080	(kJ/kg)	Heat transferred in coolers		2.264	(kJ/kg)
Indicated work/Total work		0.697		Indicated work/Total work		0.698
Total electrical work		209.095	(kJ/kg)	Total electrical work		190.378	(kJ/kg)
Total electrical power		9063	(kW)	Total electrical power		8251	(kW)

, on the left-hand side (LHS).

The in-service ${\eta }_{poly,PTC10}$ values, computed with observed air delivery temperatures, indicate good compression efficiency, the highest being for the first stage where the inlet air is very cold due to wintery conditions. These are treated as stage ‘database’ values for this compressor. Across the three adiabatic compression stages, the total indicated work is 143 kJ/kg, and across the intercoolers and aftercooler, the total heat transferred is −171 kJ/kg (heat leaves the gas system). The total power predicted to be consumed by the compressor motor is 9063 kW with motor efficiency of 98% which compares well with the measured value of 9069 kW, and serves to verify the assumed inlet temperatures of 8 °C to stages 2 and 3. The ratio of total indicated work to total input work is 69.7% indicating good overall efficiency. If Compressed Air and Gas Institute (CAGI) standard conditions of 100 kPa and 20 °C were adopted for reporting, for the delivery pressure of 61.7 psi(g), the performance would be reported 11.74 kW/100 scfm (CAGI).

On the right-hand side (RHS) of Table 3, the performance of the same compressor is considered, with the same air inlet conditions and pressure ratios, the difference being that it is supposed the compression stages are fitted with water jackets so that chilled water from the cooling tower may cool the gas as it is being compressed, as well as offering cooling in the intercoolers and aftercoolers. Further, it is supposed that the temperatures of the water entering and leaving the water jackets, as well as the cooling water mass flow rates, are monitored so that the amount of heat transferred during the compression stages can be quantified, and provided as input to the calculations. The magnitudes of heat transferred shown in of Table 3 correspond to the delivery air temperatures being just under 9 °C at the end of each compression stage, so that the intercoolers do not have much intercooling to do and the aftercooler actually heats the air up to 11.8 °C. Database values for ${\eta }_{poly,PTC10}$ are used as process-defining parameters for the compression stages. The state of the air at the outlet of the aftercooler is the same across the two situations: P₃ = 527 kPa, T₃ = 285 K.

Across the three non-adiabatic compression stages, the total indicated work done is 130 kJ/kg and the heat transferred to the water jackets is −155 kJ/kg during compression, and 2 kJ/kg are transferred in the intercoolers and aftercooler. The ratio of indicated power to input power is 69.8%, practically identical to that of the LHS compressor with standard intercooling and aftercooling. However, the total power predicted to be consumed by the water-jacketed multi-stage compressor motor is 8251 kW with motor efficiency of 98%. This is 812 kW lower than the multi-stage, intercooled and aftercooled, compressor with adiabatic stages. The polytropic efficiency effectively only informs that the same process is being considered in each case. It is only by further inspection of the work input required, that the superiority of one of the two cases over the other can be rationally discerned. With a load factor of 98% (allowing for maintenance downtime) and electricity cost of USD 80/MWh, the value of the electrical energy savings between the two cases is USD 558,000 per annum (around 9%). Using CAGI standard conditions, the performance of the compressor system would be reported as 10.68 kW/100 scfm (CAGI) for a delivery pressure of 61.7 psi(g).

The principal value in this case study is not to illustrate how the suggested modifications to ASME PTC-10 lead to greater efficiency and cost savings, because they do not do so. Instead, the case study illustrates rational comparison between a compressor design that does not permit the transfer of compression heat as the compression process takes place with a design that does. It can be seen that there are substantial energy efficiency savings to be made, even against a multi-stage compression process, where the design idea in multi-staging is to bring about a process that is closer to the isothermal ideal. As the intercooler/aftercooler sub-systems do not achieve much in the non-adiabatic multi-stage case, they could be dispensed with, saving capital cost.

6.2. Hydraulic Air Compression

A hydraulic air compressor (HAC) is an installation where the design principle exploited is exactly one where the compression process occurs as compression heat is removed. In a HAC, the gas, which takes the form of countless bubbles, is compressed with water as it descends through great elevation difference. Generally, the high surface area of the bubbly flow and the great disparity in the densities and mass flow rates of the two phases, as well as the ratio of ~4:1 in the heat capacities of water and air, specifically, are the factors that lead to the compression process being ‘nearly isothermal’ in a HAC (Pavese et al. [18]).

The same source identifies the HAC mechanical efficiency of HACs to be:

$${\eta }_{HAC}=\frac{{\dot{m}}_g{w}_{13,rev}}{{\dot{m}}_g{w}_{13,real}}$$

(39)

where ${\dot{m}}_g$ is the mass flow rate of air, and Equation (39) is clearly consistent with the suggested modification of Equation (34). In the absence of gas solubility effects, which are discussed in detail by Pavese et al. [18], Equation (39) may be written:

$${\eta }_{HAC}=\frac{{\displaystyle {\int }_1^{3}VdP}}{{\dot{m}}_{w}gH/{\dot{m}}_g}$$

(40)

where ${\dot{m}}_w$ is the mass flow rate of water flowing through the HAC at steady state and $H$ is the head that drives the water through a HAC. The denominator of Equation (40) may be appreciated to be the available hydropower (W) of the water, normalized by mass flow rate of air ${\dot{m}}_g$. The units of numerator and denominator of Equation (40) are (kJ/kg air).

A modern HAC of a scale comparable to that of the centrifugal system analysed in Section 6.1 has not yet been constructed. The largest unit that is in operation is that described in Millar and Young [19], which was designed to prove out hydrodynamic, psychrometric and solubility models formulated by Young et al. [20]. The data gathered from that facility has been published by Young and Millar [21]. More detailed understanding of the dynamics of gas solubility behaviour in HACs, further verified by Pourmahdavi et al. [22], has been embedded in these models, so that reliable forecasts of their expected performance at even larger scale can be performed.

Using the principal design input data, adopting the design concept presented in Figure 3, for intake air pressure and temperature of 101.051 kPa(a) and −22.9 °C respectively (corresponding to the same air inlet conditions as for the case study in Section 6.1, the HAC models forecast that during steady state operation, 4.059 kg/s (7235 scfm, CAGI) will be inducted and compressed to 527.138 kPa(a) (61.8 psi(g)—the same delivery pressure as in the case study of Section 6.1), when 1.969 m³/s water circulates with temperature 10 °C, driven by 3 no. Fairbanks Nijhuis Split Case Pumps of varying sizes operating in parallel while facing total dynamic head of 29.2 m. During the air–water mixing process at the point of air induction in the HAC, the inducted air will be warmed by the circulating water from −22.9 °C to 10.0 °C (more precisely to 9.985 °C as the intake air will cool the circulating water slightly). Through the compression process in the HAC downcomer, the temperature of the air will rise further, slightly, to 10.082 °C. As small as this 9.7 mK temperature rise may seem, it is entirely consistent with the ~10 mK temperature rises directly observed during comprehensive HAC performance trials reported by Millar and Young [21], and the forecasts of Pavese et al. [18] for other HAC systems. Applying Equation (40):

$${\eta }_{HAC}=\frac{126103.3}{1969.0\times 9.807\times 29.2/4.059}=0.907$$

but the real specific work on the denominator does not account for the efficiencies of motors and pumps. Pump losses (87.4% efficiency) and motor losses (96.4% efficiency) can be readily accounted for in the pump electric power consumed so that Equation (39) yields:

$${\eta }_{HAC}=\frac{4.059\times 126103.3}{\left(380.4+204.0+85.6\right)\times 1000}=0.764$$

(41)

The whole point of this paper is that this value of 0.764 can be compared directly with the values of 0.697 for the multi-stage centrifugal compressor detailed in Section 6.1 (Table 3) to identify the compressor with better performance, accounting for both the process conditions and the machine irreversibilities. If Compressed Air and Gas Institute (CAGI) standard conditions of 100 kPa and 20 °C were adopted for reporting, for the delivery pressure of 61.7 psi(g), the performance of the HAC would be reported at 9.26 kW/100 scfm (CAGI).

6.3. Comparison of Case Studies

While the established performance metrics for the compressors considered in the case studies can be rationally presented and compared as in

Table 4. Comparison of case study compressor systems. Intake air conditions: −22.9 °C and 101.051 kPa; Delivery air pressure: 527 kPa(a) (62 psi(g)).

	Air Flow Rate (kg/s, scfm (CAGI))	Delivery Air Temperature (°C)	${\mathit{\eta }}_{\mathit{p}\mathit{o}\mathit{l}\mathit{y},\mathit{r}\mathit{e}\mathit{a}\mathit{l}}$	${\mathit{w}}_{13,\mathit{r}\mathit{e}\mathit{a}\mathit{l}}$ (kJ/kg air)	CAGI Performance (kW/100 scfm (CAGI))
3-stage centrifugal compressor with intercooling and aftercooling	43.342	11.8	69.7%	209.1	11.74
3-stage centrifugal compressor with stage water jackets, intercooling and aftercooling	43.342	11.8	69.8%	190.4	10.68
Hydraulic air compressor	4.059	10.1	76.4%	165.1	9.26

, for a yet fairer comparison, work arising from compressor auxiliary systems needs to be considered too. For the 3-stage centrifugal compressors, auxiliary power due to pump work required to circulate water across the intercoolers, aftercoolers and water jackets (where applicable) should be added. For the HAC, the isothermal compression heat added to the water during compression needs to be removed from the circulating water to maintain a steady state circulating temperature of 10 °C. Thus, additional electrical power would have to be added to the denominator of Equation (41) due to (see Figure 3) (i) the auxiliary pump (P) delivering circulating water to the cooling tower/spray chamber, (ii) the fan (F) delivering cooling air to the spray chamber, and (iii) the auxiliary pump adding make up water to the reservoir.

7. Conclusions

The polytropic efficiency of a gas compressor, ${\eta }_{poly,PTC10}$, as defined in Schulz [14], and embodied in ASME PTC-10 [1] is best considered a process-defining parameter as it reflects the capacity of a given compressor to deliver a particular end state given the inlet conditions. ${\eta }_{poly,PTC10}$ does not actually depend on the characteristics of the compressor providing the compression process.

The isentropic efficiency of a gas compressor ${\eta }_{isen}$ is also considered a process-defining parameter; it defines a trajectory on a thermodynamic process diagram between the input and delivery states of gases, but as with the polytropic efficiency does not really convey anything about the characteristics of the compressor providing the compression process.

In adopting the ASME PTC-10 standard, the definition of ${\eta }_{poly,PTC10}$ assesses the useful, indicated, work defined by the process end states, and normalizes this, not by the actual compressor work, but by the work done by a hypothetical compressor that follows a polytropic process between the two end states yet permits no heat transfer from system to surroundings, $w_{13,adia}=h_3-h_1$. In effect the performance of the actual compressor is removed from the defined metric of compressor performance.

For many types of compressors, the actual work done closely approximates the adiabatic work, but this is not true for compressors designed to mediate system-surroundings heat transfers while the gas is being compressed. Quantifying ${\eta }_{poly,PTC10}$ for such compressors can produce values that are non-rational for an efficiency parameter (e.g., negative or above unity).

In the definition of ${\eta }_{poly,real}$ in Equations (34) or (35), this paper puts forward an alternative performance metric for gas compressors that aims to permit direct comparison between compressors designed to permit substantial system-surroundings heat transfers with designs that generally do not. In making this suggestion, the role of system-surroundings heat transfers and irreversibilities characteristic of the compressor become more prominent. The more material values to examine to discriminate compressor performance become the actual work input, $w_{13,real}$, or the compressor irreversibility, $F_{13}$, both (J/kg), with larger values meaning worse.

Two case studies of compressor systems designed for far from adiabatic compression, (i) a multi-stage centrifugal compressor equipped with intercooling and aftercooling and (ii) a hydraulic air compressor, have been presented to illustrate the calculation of the suggested alternative performance metrics so that the metric values can be compared rationally.