# Optimal Design of a Dual-Pressure Steam Turbine for Rankine Cycle Based on Constructal Theory

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## Abstract

**:**

## 1. Introduction

## 2. Marine DPST Model

#### 2.1. ST Model with Axial Flow

#### 2.1.1. Expansion Process of the Steam in the Nozzle

#### 2.1.2. Flow and Energy Conversion Processes of the Steam in a Rotating Cascade

#### 2.1.3. Loss Model in the Stage

#### 2.1.4. Internal Power of the Stage

#### 2.1.5. Volume of the DPST

#### 2.2. Performance of the DPST

## 3. Optimal Design of the DPST Based on Constructal Theory

^{−20}. The maximum iteration number is set as 200. Figure 14 shows the relationship between ${\tilde{P}}_{\mathrm{t}}$ and iteration number with different initial values in the function of “fmincon”. This indicates that the optimization results are slightly influenced by the initial values. For this reason and the local optimization solver of the “fmincon” function, different initial variable values are tried to ensure the stability of the optimization results.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$a$ | Empirical coefficient |

${B}_{e}$ | Stage type coefficient |

$c$ | Absolute velocity (m/s) |

${D}_{\mathrm{a}}$ | Average diameter of the turbine casing (m) |

${D}_{b}$ | Average diameter of the rotating cascade (m) |

${D}_{m}$ | Average diameter of the stage (m) |

${D}_{n}$ | Average diameter of the stationary cascade (m) |

${e}_{c}$ | Length ratio |

${k}_{1}$ | Empirical coefficient |

${L}_{\mathrm{a}}$ | Maximum length of the ST (m) |

$l$ | Cascade height (m) |

${l}_{b}$ | Height of the rotating blade (m) |

${\dot{m}}_{st}$ | Steam mass flow rate of the stage (kg/s) |

$n$ | Rotational speed (Revolutions/s) |

${P}_{i}$ | Internal power (kW) |

${P}_{\mathrm{T},\mathrm{H}}$ | Power output of the high-pressure ST (kW) |

${P}_{\mathrm{T},\mathrm{L}}$ | Power output of the low-pressure ST (kW) |

${P}_{\mathrm{t}}$ | Total power output of the multistage ST (kW) |

${P}_{\mathrm{the}}$ | Theoretical power output of the ST (kW) |

${p}_{0}$ | Pressure (Pa) |

$u$ | Circumferential velocity (m/s) |

${V}_{\mathrm{HT}}$ | Volume of the high-pressure ST (m^{3}) |

${V}_{\mathrm{LT}}$ | Volume of the low-pressure ST (m^{3}) |

${V}_{\mathrm{T}}$ | Total volume of the dual-pressure ST (m^{3}) |

$v$ | Average specific volume of the steam (m^{−3}) |

$w$ | Relative velocity (m/s) |

${x}_{m}$ | Average dryness of the steam |

${x}_{\mathrm{v}}$ | Volume ratio of the high-pressure ST |

${Z}_{n}$ | Group number of the nozzles |

Greek symbols | |

$\alpha $ | Absolute angle (degree) |

$\beta $ | Relative angle (degree) |

${\eta}_{\mathrm{t}}$ | Efficiency of the dual-pressure ST |

$\mu $ | Utilization coefficient |

${\mu}_{p}$ | Discharge coefficient |

$\phi $ | Nozzle velocity coefficient |

$\psi $ | Speed coefficient |

$\Delta {h}_{n}$ | Enthalpy drop (kJ) |

$\Delta {m}_{p}$ | Leakage mass flow rate of the steam (m/s) |

$\Delta {p}_{f}$ | Friction power consumption of the impeller (kW) |

${\Omega}_{m}$ | Average reaction degree |

Superscript | |

$~$ | Dimensionless |

* | Stagnation state |

. | Rate |

Subscripts | |

b | Rotating blade |

DSH | Curtis stage of the high-pressure |

H | High-pressure |

i | Internal |

L | Low-pressure |

m | Middle |

max | Maximum |

n | nozzle |

T | Total |

the | Theoretical |

0 | State point at the inlet of the nozzle |

1, 2 | State points at the inlet and outlet of the rotating blade |

Abbreviations | |

ARD | Average reaction degree |

DAD | Dimensionless average diameter |

DPST | Dual-pressure steam turbine |

PO | Power output |

SIA | Steam inlet angle |

ST | Steam turbine |

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**Figure 8.**Relationship between ${\tilde{P}}_{\mathrm{t}}$ and ${\tilde{D}}_{\mathrm{m},\mathrm{DSH}}$ with different ${\alpha}_{1,\mathrm{DSH}}$.

**Figure 9.**Relationship between ${\tilde{P}}_{\mathrm{t}}$ and ${\tilde{D}}_{\mathrm{m},\mathrm{H}1}$ with different ${\alpha}_{\mathrm{H}1}$.

**Figure 10.**Relationship between ${\tilde{P}}_{\mathrm{t}}$ and ${\tilde{D}}_{\mathrm{m},\mathrm{H}3}$ with different ${\alpha}_{\mathrm{H}3}$.

**Figure 11.**Relationship between ${\tilde{P}}_{\mathrm{t}}$ and ${\tilde{D}}_{\mathrm{m},\mathrm{L}1}$ with different ${\alpha}_{\mathrm{L}1}$.

**Figure 12.**Relationship between ${\tilde{P}}_{\mathrm{t}}$ and ${\tilde{D}}_{\mathrm{m},\mathrm{L}4}$ with different ${\alpha}_{\mathrm{L}4}$.

**Figure 13.**Relationship between ${\tilde{P}}_{\mathrm{t}}$ and ${\tilde{D}}_{\mathrm{m},\mathrm{L}5}$ with different ${\alpha}_{\mathrm{L}5}$.

**Figure 14.**Relationship between ${\tilde{P}}_{\mathrm{t}}$ and iteration number with different initial values.

Items | Expressions | Equation Numbers |
---|---|---|

Nozzle energy loss | $\Delta {h}_{n,l}=\frac{1}{2}\left({c}_{1t}^{2}-{c}_{1}^{2}\right)=\frac{1}{2}{c}_{1t}^{2}\left(1-{\phi}^{2}\right)=\left(1-{\phi}^{2}\right)\Delta {h}_{n}^{*}$ | Equation (4) |

Rotating blade loss | $\Delta {h}_{b,l}=\frac{1}{2}\left({w}_{2t}^{2}-{w}_{2}^{2}\right)=\left(1-{\psi}^{2}\right)\Delta {h}_{b}^{\ast}$ | Equation (13) |

Residual speed loss | $\Delta {h}_{c2}=\frac{1}{2}{c}_{2}^{2}$ | Equation (14) |

Spanwise loss | $\Delta {h}_{l}=\frac{a}{l}\Delta {h}_{u}$ | Equation (16) |

Sector loss | $\Delta {h}_{\theta}={E}_{0}{\zeta}_{\theta}$ | Equation (17) |

Impeller friction loss | $\Delta {h}_{f}=\frac{3600\Delta {p}_{f}}{{\dot{m}}_{st}}$ | Equation (19) |

Admission loss | $\Delta {h}_{e}={B}_{e}\frac{1}{e}\left(1-e-0.5{e}_{c}\right){E}_{0}{x}_{a}^{3}+{c}_{s}\frac{1}{e}\frac{{Z}_{n}}{{D}_{n}}{E}_{0}{x}_{a}$ | Equation (23) |

Steam leakage loss | $\Delta {h}_{\delta}=\frac{\Delta {m}_{p}}{{m}_{st}}\Delta {h}_{u}+\frac{\Delta {m}_{t}}{{m}_{t}}\Delta {h}_{u}$ | Equation (25) |

Wet steam loss | $\Delta {h}_{x}=\left(1-{x}_{m}\right)\Delta {h}_{u}$ | Equation (29) |

Effective enthalpy drop of the stage | $\Delta {h}_{i}=\Delta {h}_{u}-(\Delta {h}_{l}+\Delta {h}_{\theta}+\Delta {h}_{f}+\Delta {h}_{e}+\Delta {h}_{\delta}+\Delta {h}_{x})$ | Equation (30) |

Internal power (${P}_{i}$) of the stage | ${P}_{i}={\dot{m}}_{st}\Delta {h}_{i}$ | Equation (31) |

**Table 2.**Constant parameters, design variables, optimization objective, and constraints of the model.

Items | Contents |
---|---|

Constant parameters | Steam mass flow rate (${\dot{m}}_{\mathrm{st}}$), pressure (${p}_{0}$), temperature (${T}_{0}$), total volume (${V}_{\mathrm{T}}$) and rotational speed ($n$) |

Design variables | DADs (${\tilde{D}}_{\mathrm{m},\mathrm{DSH}}$, ${\tilde{D}}_{\mathrm{m},\mathrm{H}1}$, ${\tilde{D}}_{\mathrm{m},\mathrm{H}3}$, ${\tilde{D}}_{\mathrm{m},\mathrm{L}1}$, ${\tilde{D}}_{\mathrm{m},\mathrm{L}3}$, ${\tilde{D}}_{\mathrm{m},\mathrm{L}4}$ and ${\tilde{D}}_{\mathrm{m},\mathrm{L}5}$), SIAs (${\alpha}_{1,\mathrm{DSH}}$, ${\alpha}_{3,\mathrm{DSH}}$, ${\alpha}_{\mathrm{H}1}$ and ${\alpha}_{\mathrm{L}1}$), ARDs (${\Omega}_{\mathrm{b}1}$, ${\Omega}_{\mathrm{gb}}$, ${\Omega}_{b2}$, ${\Omega}_{\mathrm{H}}$ and ${\Omega}_{\mathrm{L}}$) and volume ratio (${x}_{\mathrm{v}}$) |

Optimization objective | Total PO (${P}_{\mathrm{t}}$) of the multistage DPST |

Constraints | Total volume (${V}_{\mathrm{T}}$) of the DPST and increasing average DADs of the stages along the flow direction |

Number | Variables | Names of the Variables | Variation Ranges |
---|---|---|---|

1 | ${\tilde{D}}_{\mathrm{m},\mathrm{DSH}}$ | DAD of the Curtis stage | 0.69~1.32 |

2 | ${\tilde{D}}_{\mathrm{m},\mathrm{H}1}$ | DAD of the third stage from last of the high-pressure ST | 0.69~1.32 |

3 | ${\tilde{D}}_{\mathrm{m},\mathrm{H}3}$ | DAD of the last stage of the high-pressure ST | 0.67~1.27 |

4 | ${\tilde{D}}_{\mathrm{m},\mathrm{L}1}$ | DAD of the first stage of the low-pressure ST | 0.86~1.36 |

5 | ${\tilde{D}}_{\mathrm{m},\mathrm{L}3}$ | DAD of the third stage of the low-pressure ST | 0.80~1.26 |

6 | ${\tilde{D}}_{\mathrm{m},\mathrm{L}4}$ | DAD of the fourth stage of the low-pressure ST | 0.76~1.20 |

7 | ${\tilde{D}}_{\mathrm{m},\mathrm{L}5}$ | DAD of the fifth stage of the low-pressure ST | 0.70~1.0 |

8 | ${\alpha}_{1,\mathrm{DSH}}$ | SIA of the first row of the rotating blade for the Curtis stage | 8~25° |

9 | ${\alpha}_{3,\mathrm{DSH}}$ | SIA of the second row of the rotating blade for the Curtis stage | 8~25° |

10 | ${\Omega}_{\mathrm{b}1}$ | ARD of the first row of the rotating blade for the Curtis stage | 0.05~0.2 |

11 | ${\Omega}_{\mathrm{g}\mathrm{b}}$ | ARD of the guide vane for the Curtis stage | 0.05~0.2 |

12 | ${\Omega}_{\mathrm{b}2}$ | ARD of the second row of the rotating blade for the Curtis stage | 0.05~0.2 |

13 | ${\alpha}_{\mathrm{H}1}$ | SIA of the third stage from last of the high-pressure ST | 8~25° |

14 | ${\alpha}_{\mathrm{L}1}$ | SIA of the first stage of the low-pressure ST | 8~25° |

15 | ${\Omega}_{\mathrm{H}}$ | ARD of the single row stage for the high-pressure ST | 0.05~0.2 |

16 | ${\Omega}_{\mathrm{L}}$ | ARD of the single row stage for the low-pressure ST | 0.05~0.2 |

17 | ${x}_{\mathrm{v}}$ | Volume ratio of the high-pressure ST | 0.05~0.4 |

Optimization Objectives and Variables | Before Optimization | After Optimization |
---|---|---|

${\tilde{P}}_{\mathrm{t}}$ | 1.0 | 1.108 |

${\eta}_{\mathrm{t}}$ | 0.875 | 0.97 |

${\tilde{D}}_{\mathrm{m},\mathrm{DSH}}$ | 1.0 | 0.69 |

${\tilde{D}}_{\mathrm{m},\mathrm{H}1}$ | 1.0 | 0.69 |

${\tilde{D}}_{\mathrm{m},\mathrm{H}3}$ | 1.0 | 0.77 |

${\tilde{D}}_{\mathrm{m},\mathrm{L}1}$ | 1.0 | 0.86 |

${\tilde{D}}_{\mathrm{m},\mathrm{L}3}$ | 1.0 | 1.24 |

${\tilde{D}}_{\mathrm{m},\mathrm{L}4}$ | 1.0 | 1.18 |

${\tilde{D}}_{\mathrm{m},\mathrm{L}5}$ | 1.0 | 1.1 |

${\alpha}_{1,\mathrm{DSH}}$ | 14° | 8° |

${\alpha}_{3,\mathrm{DSH}}$ | 23.4° | 8° |

${\Omega}_{\mathrm{b}1}$ | 0.12 | 0.05 |

${\Omega}_{\mathrm{gb}}$ | 0.12 | 0.05 |

${\Omega}_{\mathrm{b}2}$ | 0.12 | 0.05 |

${\alpha}_{1,\mathrm{H}}$ | 12° | 25° |

${\alpha}_{1,\mathrm{L}}$ | 23° | 25° |

${\Omega}_{\mathrm{H}}$ | 0.12 | 0.2 |

${\Omega}_{\mathrm{L}}$ | 0.12 | 0.2 |

${x}_{\mathrm{v}}$ | 0.16 | 0.08 |

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## Share and Cite

**MDPI and ACS Style**

Feng, H.; Chen, L.; Tang, W.; Ge, Y.
Optimal Design of a Dual-Pressure Steam Turbine for Rankine Cycle Based on Constructal Theory. *Energies* **2022**, *15*, 4854.
https://doi.org/10.3390/en15134854

**AMA Style**

Feng H, Chen L, Tang W, Ge Y.
Optimal Design of a Dual-Pressure Steam Turbine for Rankine Cycle Based on Constructal Theory. *Energies*. 2022; 15(13):4854.
https://doi.org/10.3390/en15134854

**Chicago/Turabian Style**

Feng, Huijun, Lingen Chen, Wei Tang, and Yanlin Ge.
2022. "Optimal Design of a Dual-Pressure Steam Turbine for Rankine Cycle Based on Constructal Theory" *Energies* 15, no. 13: 4854.
https://doi.org/10.3390/en15134854