# Interstage Pressures of a Multistage Compressor with Intercooling

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## Abstract

**:**

## 1. Introduction

_{2}refrigeration cycle, the estimated interstage pressure from the suction and discharge pressures geometric mean underestimates the actual interstage pressure of the cycle. Jekel and Reindl [10] explore single- versus two-stage compression arrangements from an operating efficiency perspective. They find that the optimum operating efficiency for each system is obtained when the real interstage pressure is smaller than that obtained from the geometric mean. Özgür [11] and Romeo et al. [19] use directly and indirectly the geometric mean as the basis for their initial designs used in their performance studies of refrigeration cycles with two and three compression stages with intercooling, respectively. Srinivasan [20] shows that the criterion of equal discharge temperatures of each stage is a good criterion for the choice of interstage pressure for CO

_{2}compressors used in low (−30 °C) and medium temperature (−5 °C) refrigeration. Lugo-Leyte et al. [21] study the performance of complex gas turbine cycles with multistage compression. They determined that the optimum pressure ratios are in an acceptable range, between 8.1 and 23.1 for the maximum power and between 17.4 and 32.2 for the maximum thermal efficiency. Lewins [22] models and optimizes a two-stage compressor with an intercooler considering the ideal gas model. He uses the Lagrange optimization method to find the operating conditions to achieve the maximum work in the gas turbine. Furthermore, he shows the optimum condition can be calculated based on the isentropic efficiencies of the compressors and the efficiency of the intercoolers. Azizifar and Banooni [23] model and optimize the power consumption of a two-stage compressed air system considering the ideal gas model. The system includes two centrifugal compressors, a casing, and a tube intercooler. The power consumption is expressed in terms of the isentropic efficiencies and thermal effectiveness of the intercooler. The isentropic efficiencies of the compressors are considered as functions of the inlet temperature, and the thermal effectiveness of the intercooler is considered as a function of the inlet air temperature, inlet water temperature of the intercooler, and inlet volumetric air flow rate of the system.

## 2. System Description and Assumptions

- A constant mass flow rate of a working fluid behaving as an ideal gas with constant heat capacities is compressed.
- The gas undergoes a pressure drop in each j-intercooler—see Figure 2. The pressure drop coefficient across the j-intercooler is defined as$${\epsilon}_{j}=\frac{{P}_{2j}-{P}_{2j+1}}{{P}_{2j}},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}j=1,\dots ,{N}_{c}-1$$$${P}_{2j-1}=\left(1-{\epsilon}_{j-1}\right){P}_{2\left(j-1\right)},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}j=1,\dots ,{N}_{c}$$The above equation is valid if we define ${\epsilon}_{0}=0$ and therefore ${P}_{0}={P}_{1}$.
- The gas temperature at the inlet of each compressor is not assumed to be the same. However, the compressed gas outlet temperature of each intercooler is close to ${T}_{1}$.$${T}_{2j+1}\ne {T}_{1},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}j=1,\dots ,{N}_{c}-1$$
- The isentropic efficiencies of the individual compressors are assumed to be different, and the compression from $2j-1$ to $2j$ is considered to occur at constant isentropic efficiency,$${\eta}_{SIC,j}=\frac{{w}_{j,s}}{{w}_{j}},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}j=1,...,{N}_{c}-1$$

## 3. Theoretical Model

#### 3.1. Optimal Interstage Pressures for Minimum Compression Specific Work

#### 3.2. Minimum Compression Specific Work

## 4. Applications

#### 4.1. Estimation of the Number of Compression Stages

#### 4.2. Interstage Pressure Estimation of a Natural Gas Compression System

- ${P}_{2}=\sqrt{{P}_{1}{P}_{4}}$, same suction temperatures (${T}_{3}\approx {T}_{1}$) and no pressure drops in the intercooler (${\epsilon}_{1}\approx 0$);
- ${P}_{2}=\sqrt{\frac{{P}_{1}{P}_{4}}{1-{\epsilon}_{1}}}$, same suction temperatures (${T}_{3}\approx {T}_{1}$) and pressure drops in the intercooler;
- ${P}_{2}=\sqrt{{\left(\frac{{T}_{3}}{{T}_{1}}\right)}^{\frac{1}{x}}{P}_{1}{P}_{4}}$, different suction temperatures (${T}_{3}\ne {T}_{1}$) and no pressure drops in the intercooler (${\epsilon}_{1}\approx 0$);
- ${P}_{2}=\sqrt{{\left(\frac{{T}_{3}}{{T}_{1}}\right)}^{\frac{1}{x}}\frac{{P}_{1}{P}_{4}}{1-{\epsilon}_{1}}}$, different suction temperatures (${T}_{3}\ne {T}_{1}$) and pressure drops in the intercooler;
- ${P}_{2}=\sqrt{{\left(\frac{{T}_{3}}{{T}_{1}}\xb7\frac{{\eta}_{SIC,1}}{{\eta}_{SIC,2}}\right)}^{\frac{1}{x}}\frac{{P}_{1}{P}_{4}}{1-{\epsilon}_{1}}}$, different suction temperatures (${T}_{3}\ne {T}_{1}$) and pressure drops in the intercooler.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${c}_{P}$ | specific heat at constant pressure, kJ kg${}^{-1}$ K${}^{-1}$ |

${c}_{v}$ | specific heat at constant volume, kJ kg${}^{-1}$ K${}^{-1}$ |

K | real constant defined by $K={\left(\frac{{T}_{2j-1}}{{\eta}_{SIC,j}}\right)}^{\frac{1}{x}}\frac{{P}_{2j}}{\left(1-{\epsilon}_{j-1}\right){P}_{2\left(j-1\right)}}$ for $j=1,\dots ,{N}_{c}$ |

${N}_{c}$ | compression stages |

P | pressures, bar |

R | specific gas constant, kJ kg${}^{-1}$ K${}^{-1}$ |

s | specific entropy, kJ kg${}^{-1}$ K${}^{-1}$ |

T | temperature, K or ${}^{\circ}$C |

w | specific work, kJ kg${}^{-1}$ |

x | $x=R/{c}_{P}=\left(\gamma -1\right)/\gamma $ |

Greek symbols | |

${\alpha}_{j}$ | ${\alpha}_{j}^{x}=\frac{{T}_{2j-1}}{{\eta}_{SIC,j}}$, K${}^{\frac{1}{x}}$ |

$\mathsf{\Delta}$ | drop or increment |

${\epsilon}_{j}$ | pressure drop coefficient across the j-intercooler, ${\epsilon}_{j}=\frac{{P}_{2j}-{P}_{2j+1}}{{P}_{2j}}$ |

${\u03f5}_{g}$ | geometric mean for the set $\left\{1-{\epsilon}_{j-1},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}j=1,\dots ,{N}_{c}\right\}$ |

$\gamma $ | adiabatic index or specific heat ratio |

$\eta $ | efficiency |

$\pi $ | pressure ratio |

${\tau}_{a}$ | arithmetic mean for the set $\left\{{T}_{2j-1}/{\eta}_{SIC,j},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}j=1,\dots ,{N}_{c}\right\}$, K |

${\tau}_{g}$ | geometric mean for the set $\left\{{T}_{2j-1}/{\eta}_{SIC,j},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}j=1,\dots ,{N}_{c}\right\}$, K |

Subscripts | |

c | compressor |

d | discharge state |

$i,\phantom{\rule{3.33333pt}{0ex}}j,\phantom{\rule{3.33333pt}{0ex}}k,\phantom{\rule{3.33333pt}{0ex}}m$ | thermodynamic states |

s | suction state |

$SIC$ | isentropic compression |

## Appendix A. Optimal Interstage Pressures in Terms of Their Predecessor and Successor Pressures

## Appendix B. Optimal Interstage Pressures Obtained by Successive Substitutions

## Appendix C. Induction Proof of Equation (A4)

- Base case: When $k=1$, Equation (A4) corresponds to Equation (7), proving Equation (A4) is true for $k=1$.
- Induction hypothesis: In this step, we assume Equation (A4) is valid for k.
- Inductive step: When $j=j+k$, Equation (7) raised to the power of $k+1$ becomes$${P}_{2\left(j+k\right)}^{2\left(k+1\right)}={\left(\frac{{\alpha}_{j+k+1}^{1/x}}{{\alpha}_{j+k}^{1/x}}\right)}^{k+1}{\left(\frac{1-{\epsilon}_{j+k-1}}{1-{\epsilon}_{j+k}}\right)}^{k+1}{P}_{2\left(j+k-1\right)}^{k+1}{P}_{2\left(j+k+1\right)}^{k+1}$$The substitution of Equation (A4), corresponding to the induction hypothesis, into the left-hand side of Equation (A10) leads to$${P}_{2\left(j+k\right)}^{2\left(k+1\right)}={\left(\frac{{\alpha}_{j+k+1}^{1/x}}{{\alpha}_{j+k}^{1/x}}\right)}^{k+1}{\left(\frac{1-{\epsilon}_{j+k-1}}{1-{\epsilon}_{j+k}}\right)}^{k+1}{\left(\frac{{\alpha}_{j+k}^{1/x}}{1-{\epsilon}_{j+k-1}}\right)}^{k}\prod _{i=1}^{i=k}\frac{1-{\epsilon}_{j+\left(i-2\right)}}{{\alpha}_{j+\left(i-1\right)}^{1/x}}{P}_{2\left(j-1\right)}{P}_{2\left(j+k\right)}^{k}{P}_{2\left(j+k+1\right)}^{k+1}$$After performing some algebraic steps in the above equation, we obtain the following expression:$${P}_{2\left(j+k\right)}^{k+2}={\left(\frac{{\alpha}_{j+k+1}^{1/x}}{1-{\epsilon}_{j+k}}\right)}^{k+1}\prod _{i=1}^{i=k+1}\frac{1-{\epsilon}_{j+\left(i-2\right)}}{{\alpha}_{j+\left(i-1\right)}^{1/x}}{P}_{2\left(j-1\right)}{P}_{2\left(j+k+1\right)}^{k+1}$$Thus, Equation (A4) holds for $k+1$, and the proof of induction step is complete.

## Appendix D. Induction proof of Equation (A7)

- Base case: When $j=1$ and $k={N}_{c}-1$, Equation (A4) corresponds to Equation (A7), proving Equation (A7) holds for $m=1$.
- Induction hypothesis: In this step, we assume Equation (A7) is valid for m.
- Inductive step: For $m=m+1$, Equation (A6) becomes$$\begin{array}{cc}\hfill {P}_{2\left({N}_{c}-m-1\right)}^{{N}_{c}-m}\phantom{\rule{1.em}{0ex}}& ={\left(\frac{{\alpha}_{{N}_{c}-m}^{1/x}}{1-{\epsilon}_{{N}_{c}-m-1}}\right)}^{{N}_{c}-m-1}\prod _{i=1}^{i=m+1}\frac{{\alpha}_{{N}_{c}-i+1}^{1/x}}{1-{\epsilon}_{{N}_{c}-i}}\prod _{i=1}^{i={N}_{c}}\frac{1-{\epsilon}_{i-1}}{{\alpha}_{i}^{1/x}}{P}_{0}{P}_{2\left({N}_{c}-m\right)}^{{N}_{c}-m-1}\hfill \end{array}$$Raising Equation (A7), corresponding to the induction hypothesis, to the power of $\left({N}_{c}-m-1\right)/{N}_{c}$, leads to$$\begin{array}{c}\hfill {P}_{2\left({N}_{c}-m\right)}^{{N}_{c}-m-1}={\left(\prod _{i=1}^{i=m}\frac{{\alpha}_{{N}_{c}-i+1}^{1/x}}{1-{\epsilon}_{{N}_{c}-i}}\right)}^{{N}_{c}-m-1}{\left(\prod _{i=1}^{i={N}_{c}}\frac{1-{\epsilon}_{i-1}}{{\alpha}_{i}^{1/x}}\right)}^{\frac{m\left({N}_{c}-m-1\right)}{{N}_{c}}}{P}_{0}^{\frac{m\left({N}_{c}-m-1\right)}{{N}_{c}}}{P}_{2{N}_{c}}^{\left({N}_{c}-m\right)\frac{{N}_{c}-m-1}{{N}_{c}}}\end{array}$$Substituting Equation (A7) into the left-hand side of Equation (A15)$$\begin{array}{c}{P}_{2\left({N}_{c}-m-1\right)}^{{N}_{c}-m}=\prod _{i=1}^{i=m+1}\frac{{\alpha}_{{N}_{c}-i+1}^{1/x}}{1-{\epsilon}_{{N}_{c}-i}}{\left(\frac{{\alpha}_{{N}_{c}-m}^{1/x}}{1-{\epsilon}_{{N}_{c}-m-1}}\prod _{i=1}^{i=m}\frac{{\alpha}_{{N}_{c}-i+1}^{1/x}}{1-{\epsilon}_{{N}_{c}-i}}\right)}^{{N}_{c}-m-1}\hfill \\ \hfill {\left(\prod _{i=1}^{i={N}_{c}}\frac{1-{\epsilon}_{i-1}}{{\alpha}_{i}^{1/x}}\right)}^{\frac{m\left({N}_{c}-m-1\right)}{{N}_{c}}+1}{P}_{0}^{\frac{m\left({N}_{c}-m-1\right)}{{N}_{c}}+1}{P}_{2{N}_{c}}^{\frac{\left({N}_{c}-m\right)\left({N}_{c}-m-1\right)}{{N}_{c}}}\end{array}$$After carrying out some algebra with the above equation, we derive the following expression:$${P}_{2\left({N}_{c}-m-1\right)}^{{N}_{c}-m}={\left(\prod _{i=1}^{i=m+1}\frac{{\alpha}_{{N}_{c}-i+1}^{1/x}}{1-{\epsilon}_{{N}_{c}-i}}\right)}^{{N}_{c}-m}{\left[{\left(\prod _{i=1}^{i={N}_{c}}\frac{1-{\epsilon}_{i-1}}{{\alpha}_{i}^{1/x}}\right)}^{m+1}{P}_{0}^{m+1}{P}_{2{N}_{c}}^{{N}_{c}-\left(m+1\right)}\right]}^{\frac{{N}_{c}-m}{{N}_{c}}}$$Raising Equation (A16) to the power of ${N}_{c}/\left({N}_{c}-m\right)$, we may write$${P}_{2\left[{N}_{c}-\left(m+1\right)\right]}^{{N}_{c}}={\left(\prod _{i=1}^{i=m+1}\frac{{\alpha}_{{N}_{c}-i+1}^{1/x}}{1-{\epsilon}_{{N}_{c}-i}}\right)}^{{N}_{c}}{\left(\prod _{i=1}^{i={N}_{c}}\frac{1-{\epsilon}_{i-1}}{{\alpha}_{i}^{1/x}}\right)}^{m+1}{P}_{0}^{m+1}{P}_{2{N}_{c}}^{{N}_{c}-\left(m+1\right)}$$Thus, Equation (A7) holds for $m+1$, and the proof of induction step is complete.

## Appendix E. Two-Stage Compression System With Intercooling

**Table A1.**Thermodynamic states of the natural gas two-stage compression system obtained from ASPEN-HYSIS simulations for different shaft speeds.

6134 rpm | 6114 rpm | 6074 rpm | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\dot{\mathbf{m}}$ $\left(\frac{\mathbf{kg}}{\mathbf{h}}\right)$ | T (°C) | P (bar) | ρ $\left(\frac{\mathbf{kg}}{{\mathbf{m}}^{\mathbf{3}}}\right)$ | c_{p}$\left(\frac{\mathbf{kJ}}{\mathbf{kg}\mathbf{K}}\right)$ | MW $\left(\frac{\mathbf{kg}}{\mathbf{kmol}}\right)$ | Z (-) | T (°C) | P (bar) | ρ $\left(\frac{\mathbf{kg}}{{\mathbf{m}}^{\mathbf{3}}}\right)$ | c_{p}$\left(\frac{\mathbf{kJ}}{\mathbf{kg}\mathbf{K}}\right)$ | MW $\left(\frac{\mathbf{kg}}{\mathbf{kmol}}\right)$ | Z (-) | T (°C) | P (bar) | ρ $\left(\frac{\mathbf{kg}}{{\mathbf{m}}^{\mathbf{3}}}\right)$ | c_{p}$\left(\frac{\mathbf{kJ}}{\mathbf{kg}\mathbf{K}}\right)$ | MW $\left(\frac{\mathbf{kg}}{\mathbf{kmol}}\right)$ | Z (-) | |

1 | 5.52 | 33 | 10.58 | 10.16 | 1.45 | 26.54 | 0.98 | 33 | 10.43 | 9.99 | 1.45 | 26.54 | 0.98 | 34 | 10.59 | 10.14 | 1.45 | 26.54 | 0.98 |

2 | 5.52 | 144.1 | 31.42 | 23.51 | 1.64 | 26.54 | 0.99 | 142.4 | 30.44 | 22.85 | 1.63 | 26.54 | 0.99 | 142.2 | 30.50 | 22.9 | 1.63 | 26.54 | 0.99 |

2’ | 5.52 | 35 | 30.65 | 32.38 | 1.53 | 26.54 | 0.95 | 36 | 30.41 | 31.98 | 1.53 | 26.54 | 0.95 | 37 | 30.47 | 31.92 | 1.53 | 26.54 | 0.95 |

3 | 5.50 | 35 | 30.65 | 32.28 | 1.52 | 26.58 | 0.95 | 36 | 30.41 | 31.89 | 1.52 | 26.58 | 0.95 | 37 | 30.47 | 31.83 | 1.52 | 26.58 | 0.95 |

4 | 5.50 | 135.2 | 80.49 | 63.53 | 1.71 | 26.58 | 0.98 | 133.8 | 77.76 | 61.58 | 1.70 | 26.58 | 0.98 | 132.1 | 75.76 | 60.26 | 1.70 | 26.58 | 0.98 |

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**Figure 2.**Temperature–entropy diagram of an ${N}_{c}$-multistage compression process with intercooling.

**Figure 3.**Number of compression stages and number of intercooling stages as a function: (

**a**) individual compressor pressure ratio and (

**b**) overall pressure ratio.

**Figure 4.**Natural gas two-stage centrifugal compressor: site design, and actual operating conditions.

**Figure 5.**Percentage deviations in interstage pressure models under design and actual operating conditions.

Component | CH_{4} | C_{2}H_{6} | C_{3}H_{8} | iC_{4}H_{10} | nC_{4}H_{10} | iC_{5}H_{12} | N_{2} | O_{2} | H_{2}O | CO_{2} | H_{2}S |
---|---|---|---|---|---|---|---|---|---|---|---|

${x}_{i}$ | 0.3038 | 0.0594 | 0.0328 | 0.0043 | 0.0126 | 0.0036 | 0.543 | 0.0019 | 0.007 | 0.015 | 0.0044 |

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## Share and Cite

**MDPI and ACS Style**

Lugo-Méndez, H.; Lopez-Arenas, T.; Torres-Aldaco, A.; Torres-González, E.V.; Sales-Cruz, M.; Lugo-Leyte, R.
Interstage Pressures of a Multistage Compressor with Intercooling. *Entropy* **2021**, *23*, 351.
https://doi.org/10.3390/e23030351

**AMA Style**

Lugo-Méndez H, Lopez-Arenas T, Torres-Aldaco A, Torres-González EV, Sales-Cruz M, Lugo-Leyte R.
Interstage Pressures of a Multistage Compressor with Intercooling. *Entropy*. 2021; 23(3):351.
https://doi.org/10.3390/e23030351

**Chicago/Turabian Style**

Lugo-Méndez, Helen, Teresa Lopez-Arenas, Alejandro Torres-Aldaco, Edgar Vicente Torres-González, Mauricio Sales-Cruz, and Raúl Lugo-Leyte.
2021. "Interstage Pressures of a Multistage Compressor with Intercooling" *Entropy* 23, no. 3: 351.
https://doi.org/10.3390/e23030351