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

: A one-dimensional dual-pressure steam turbine (ST) model for the marine Rankine cycle is built in this paper. Based on constructal theory, the optimal design of the dual-pressure ST is performed with a fixed total volume of the high-and low-pressure STs. The total power output (PO) of the dual-pressure ST is maximized. Seventeen parameters, including the dimensionless average diameters (DADs) of the stages, steam inlet angles (SIAs) of the stages, average reaction degrees (ARDs) of the stages, and volume ratio of the high-pressure ST are taken as optimization variables. The optimal structure parameters of the stages are gained. It reveals that the total PO of the dual-pressure ST is increased by 2.59% by optimizing the average diameter of the Curtis stage, and the change in the total PO is not obvious by optimizing the average diameter of the third stage of the low-pressure ST. Both the total PO and the corresponding efficiency of the dual-pressure ST are increased by 10.8% after simultaneously optimizing 17 variables with the help of the Matlab optimization toolbox. The novelty of this paper is introducing constructal theory into turbine performance optimization by varying seventeen structure, thermal and flow parameters, and the result shows that the constructal optimization effect is remarkable. Optimal designs of practical STs can be guided by the optimization results gained in this paper.


Introduction
The steam turbine (ST) [1,2] is an important component for the energy conversion process of a steam Rankine cycle, which converts the thermal energy of the steam into kinetic energy and mechanical energy in turn.There are many parameters for a complex multistage ST, and parameter optimization is one effective way to elevate the performance of the ST.
Many scholars have conducted various performance optimizations for STs.Chen et al. [3] performed multi-optimization of a marine ST stage and analyzed the effect of stage number on the ST efficiency.Qin et al. [4] maximized the stage efficiency of an axial flow ST by varying the geometric and steam flow parameters of the flow passage section and elevated the stage efficiency by up to 1.8%.Ni et al. [5] built a segemented lumped parameter model for a dual-pressure ST (DPST) and compared the simulation and measured results to validate its correctness.Abadi et al. [6] built a turbine blade cascade model with two-phase flows and augmented its efficiency by up to 2.1% after optimization.Anđelić et al. [7] analyzed the performance of a marine ST under different loads and found that the losses and efficiency of the ST augmented with the augment of the turbine load.Zhao et al. [8] considered the guide ring in a low-pressure 300MW ST model and pointed out that the eddy current was abated, and the efficiency of the low-pressure cylinder was augmented by introducing the guide ring.Vedran et al. [9] calculated the energy efficiency of a three-cylinder ST and pointed out that the energy efficiencies of the three cylinders were 95.08%, 95.02% and 94.92% sorted by the reduction of the cylinder pressure.Moreover, the performance of turbines with organic working fluids has also been investigated by some scholars [10][11][12][13].
The essence of constructal theory is constructal law, which can be described as follows: "For a finite-size flow system to persist in time (to live), its configuration must change in time such that it provides easier and easier access to its currents" [14,15].Structure optimization based on constructal law is called constructal optimization.Different from land turbines, both PO and finite volume should be considered in the design of marine turbines.For this paper, according to the constructal law, in the condition of the fixed total volume (finite size) of the DPST, structure optimization of the ST will be conducted with multi-variable optimization.The PO of the DPST will be maximized.The optimal design variables will be obtained.From this point of view, this paper belongs to constructal design work.In addition, Bejan and colleagues performed similar work [42].This work is inspired by [42].This is the biggest difference between this paper and the existing literature.The marine turbine is always composed of two STs with different pressures, and it is important to elevate turbine performance by optimizing its structure under finite size.Based on Refs.[46,47], a one-dimensional DPST model for the marine steam Rankine cycle will be built in this paper.Based on constructal theory, with a fixed total volume of the high-and lowpressure STs, the optimal design of the DPST will be conducted by varying the structure parameters of the stages to search for the maximum PO.The optimization results of the DPST gained by single-and multiple-variable optimizations will be compared.The first novelty of this paper is the introduction of constructal theory into turbine performance optimization.Another novelty of this paper is optimizing the multistage DPST by simultaneously varying seventeen structure, thermal and flow parameters.The performance of the multistage DPST is hoped to be improved by applying these methods.

Marine DPST Model
Figure 1 shows a schematic diagram of a marine one-dimensional DPST.The ST is composed of high-and low-pressure axial flow STs in series.The superheated steam generated by a boiler successively enters the high-and low-pressure STs to do work, and finally, the torque generated by the STs is transmitted to the propeller through the reduction gear and shaft.The high-pressure ST is an impulse internal bypass composed of a Curtis stage and nine single-stage STs [5].The design point condition of the STs is considered in this paper, and only the Curtis stage and the last three single stages are working at this condition.The low-pressure ST is an impulse double-path one, which is composed of five impulse stages in each path [5].The low-pressure ST is symmetrical, and exhaust steam is discharged into the condenser from its middle.It is assumed that the stable flow in the ST is insulated from the environment and that the parameters of the steam are only changed along the axis.Thus, the steam in the ST can be viewed as a one-dimensional steady adiabatic flow.

ST Model with Axial Flow
The stage is the smallest unit for the axial flow ST to work. Figure 2 shows the thermodynamic process diagram of a single stage.
where n h   is the ideal stagnation enthalpy drop of the steam in the nozzle.
Figure 3 further shows the thermodynamic process diagram of a Curtis stage with a certain reaction degree.Each thermodynamic process of the Curtis stage is similar to that of the single stage, which is not repeated here.
when the steam flows through the nozzle, its actual outlet speed ( 1 c ) is less than the ideal speed ( 1t c ) due to the friction and vortex in the nozzle, i.e., where  is the nozzle velocity coefficient, whose variation range is from 0.92 to 0.98.

The energy loss (
, n l h  ) of the nozzle with the actual flow is equal to the kinetic energy at the nozzle outlet under the ideal condition minus that under the actual condition, i.e., where n is the rotational speed of the ST.
Figure 4 shows the velocity triangle of the single stage.The relative velocity ( 1 w ) and relative inlet angle ( 1  ) of the steam flow at the inlet of the rotating blade can be calculated according to the velocity triangle.
where 1  is the absolute angle of the steam flow at the inlet of the rotating blade.According to the energy balance equations of the inlet and outlet of the rotating blade, the ideal relative velocity ( 2t w ) of the steam flow at the outlet of the rotating blade can be calculated as where ) is the ideal stagnation enthalpy drop of the rotating blade.
The absolute velocity ( 2 c ) and absolute angle ( 2  ) of the steam flow at the outlet of the rotating blade can be given as where 2 w and 2  are the ideal relative velocity and relative angle of the steam flow at the outlet of the rotating blade, respectively.Similarly, energy loss exists when steam goes through the rotating cascade.The actual relative velocity ( 2 w ) at the rotating blade outlet is less than the ideal relative velocity, i.e., where  is the speed coefficient of the rotating blade, whose variation range is from 0.85 to 0.95.The energy loss ( , b l h  ) of the steam going through the rotating blade, named the ro- tating blade loss, can be expressed as The residual speed loss ( 2 c h  ) of the steam flow caused by the kinetic energy of the exhaust steam is where 2 c is the absolute speed at the outlet of the rotating blade.
In the multistage ST, the kinetic energy carried by the residual speed can be used by the next stage, and its degree of utilization can be expressed by the utilization coefficient After considering the energy losses of the nozzle, rotating blade and residual speed, the effective specific enthalpy ( u h  ) of a turbine stage is expressed as where 0  is the utilization coefficient of the residual speed.
Figure 5 further shows the velocity triangle of the Curtis stage, and its velocity relationship is similar to Figure 4, which is not repeated here.

Loss Model in the Stage
In addition to nozzle loss, rotating blade loss and residual speed loss, there exist other losses in the stage, such as spanwise loss, sector loss, impeller friction loss, partial inlet steam loss, steam leakage loss and wet steam loss.
(1) Spanwise loss ( The spanwise loss is the additional loss along the height of the blade, which is often calculated by the semi-empirical formula [51], i.e., where the empirical coefficient a is usually 1.2 for the single stage and 2.0 for the Curtis stage; the cascade height l is the nozzle height for the single stage and the average height of the cascades for the Curtis stage; u h  is the effective specific enthalpy drop of the stage. (2) Sector loss ( h   ) Sector loss is the additional loss caused by the deviation from the design condition, which is often calculated by the semi-empirical formula [51], i.e., where b l is the height of the rotating blade, and 0 E is the ideal energy of the stage.
(3) Impeller friction loss ( The impeller friction loss is the additional loss caused by the friction movement between the impeller surface and steam, which is often calculated by the semi-empirical formula [51], i.e., where f p  is the friction power consumption of the impeller, st m  is the steam mass flow rate of the stage, 1 k is the empirical coefficient, m D is the average diameter of the stage, and v is the average specific volume of the steam.
The admission loss is composed of the blast loss ( w h  ) and steam rejection loss ( s h  ).
w h  can be calculated as [51]: where c e is the ratio of the arc length of the protective cover to the whole circumference length; e is the admission degree; and the stage type coefficient ( e B ) of the ST is set as 0.15 for the single stage and 0.55 for the Curtis stage.
s h  can be calculated as [51]: where n Z is the group number of the nozzles; n D is the average diameter of the station- ary cascade; and the coefficient ( s c ) related to the stage type of the ST is set as 0.012 for the single stage and 0.016 for the Curtis stage.Therefore, the total admission loss in this stage is: given as [51]: where p  is the discharge coefficient of the steam seal, p A is the clearance area of the steam seal, 1t v is the ideal specific volume of the steam at the outlet of the steam seal teeth, and p z is the teeth number of the steam seal.
p h  can be calculated as [51]: The leakage mass flow rate ( t m  ) of the blade top is given as [51]: where t  is the discharge coefficient of the blade top clearance, n  is the discharge co- efficient of the nozzle, t  is the reaction degree of the blade top, and t  is the equiva- lent clearance of the blade top for steam leakage.t h  can be calculated as [51]: Therefore, the total steam leakage loss ( h   ) in the stage is: When the steam flows into the last few stages of the turbine, a wet steam area is generated.The wet steam loss is usually calculated using the following empirical formula [51]: where m x is the average steam dryness.

Internal Power of the Stage
According to the internal losses of the stage, the effective enthalpy drop ( i h  ) of the stage can be given as The calculation process of the internal power of the stage is shown in Figure 6.Finally, the internal power ( i P ) of a stage is given as To illustrate the ST model more intuitively, the loss models and internal power of the stage are further listed in Table 1.Table 1.Lists of the loss models and the internal power of the stage.

Items Expressions Equation Numbers
Nozzle energy loss Rotating blade loss Residual speed loss 16) Sector loss Impeller friction loss Admission loss Steam leakage loss Wet steam loss Effective enthalpy drop of the stage a, j j sp, j where sp, j L is the flow passage length of the ST, and j  and j  are the coefficients of the average diameter and length, respectively.The volumes ( HT V and LT V ) of the high-and low-pressure STs can be approximately calculated as The total volume ( T V ) of the DPST is

Performance of the DPST
The new steam generated by the boiler enters the high-and low-pressure STs to do the work.Under the design point working condition, the steam only goes through the Curtis stage and last three single stages in the high-pressure ST.Therefore, the PO ( T, H P ) of the high-pressure ST is the sum of the POs of these four stages.
The calculation process of the PO of the DPST is shown in Figure 7.The total PO ( t

P
) of the multistage ST is the sum of the POs of the high-and low-pressure STs.

Optimal Design of the DPST Based on Constructal Theory
In this section, with a fixed total volume of the DPST, the constructal design of the double-pressure ST will be conducted subjected to the geometric constraints of increasing dimensionless average diameters (DADs) of the stages along the direction of steam flow.The dimensionless average diameter is defined as the ratio of the current value of the average diameter to its initial value.The effects of the DADs and steam inlet angle (SIA) of the stages on the total PO of the DPST will be analyzed first, and the results can guide the following multivariable optimization.To illustrate the multi-variable optimization problem more intuitively, the constant parameters, design variables, optimization objective, and constraints of the model are listed in Table 2.Then, the total PO will be optimized by simultaneously varying the design variables in Table 2. Due to the parameter settings and geometric constraints of the DPST, the available variation range of one parameter will be different in the following figures.Moreover, the difference between the theoretical and actual total POs of the DPST under the initial parameters is 0.3%, which validates the correctness of the theoretical model. , but the increment gradually becomes small.x ) of the high-pressure ST.Among these, the form of the DAD for each stage is dimensionless, which is divided by its initial value.The selected optimization variables and their variation ranges are listed in Table 3.The Matlab software function of "fmincon" is applied to search for the maximum total PO, and the "interior-point" algorithm is adopted in this function.Both the total tolerances of the variables and the optimization objective are set as 10 −20 .The maximum iteration number is set as 200. Figure 14 shows the relationship between t P  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.Volume ratio of the high-pressure ST 0.05~0.4 After simultaneously optimizing the above seventeen variables, the dimensionless maximum PO and the corresponding results of the DPST are obtained, which are listed in Table 4.In Table 4, after constructal optimization, both the total PO and the corresponding efficiency of the DPST are augmented by 10.8%.This reveals that the structure parameter optimization with multiple variables significantly improves the performance of the DPST compared to that with a single variable.

Conclusions
A one-dimensional model of a marine dual-pressure axial flow ST is researched in this paper.With fixed total volume of the high-and low-pressure STs, constructal design of the DPST is implemented by altering the DADs, SIAs and ARDs of the stages.The maximum total PO and the corresponding optimal construct of the DPST are gained.The results are summarized as follows: (1) For the single-variable optimization, within the discussion scopes of the DADs of the stages, the total PO of the DPST is augmented by 2.59% after optimizing the DAD of the Curtis stage, and the change in the total PO is not obvious after optimizing the DAD of the third stage of the low-pressure ST.Within the certain variation ranges, the total PO of the DPST can be further improved by abating the SIA of the first row of the rotating blade for the Curtis stage and SIA of the last stage of the high-pressure ST and augmenting the SIAs of the third, fourth and fifth stages of the low-pressure ST, respectively.
(2) For the multiple-variable optimization, both the total PO and the corresponding efficiency of the DPST are augmented by 10.8% after simultaneously optimizing 17 variables.This reveals that the structure parameter optimization with multiple variables significantly improves the performance of the DPST compared to that with a single variable.
The DPST model with various loss items is considered in this paper.The loss items and volume of the DPST are estimated by the empirical formulas.A more practical model of the DPST will be established in the future, and more practical guidelines will be offered for the optimal designs of axial flow STs to enhance the energy saving and consumption reduction of marine steam power plants.

Figure 1 .
Figure 1.Schematic diagram of a marine DPST model.

Figure 2 .
Figure 2. Thermodynamic process diagram of a single stage

Figure 3 .
Figure 3. Thermodynamic process diagram of the Curtis stage.

2. 1 . 1 .
Expansion Process of the Steam in the Nozzle The ideal velocity ( 1t c ) of the steam flow at the nozzle outlet is

Figure 4 .
Figure 4. Velocity triangle of a stage.

Figure 5 .
Figure 5. Velocity triangle of a Curtis stage.

( 5 )
Steam leakage loss ( h   ) The steam leakage loss is composed of the partition leakage loss ( p h  ) and blade top leakage loss ( t h  ).The leakage mass flow rate ( p m  ) of the steam from the partition is

Figure 6 .
Figure 6.Calculation process of the internal power and steam outlet parameters of the stage.

5 .D
Volume of the DPST The average stage diameters ( m, j D , j = H, L) of the high-and low-pressure STs are calculated as and z, j D are the average diameters of the first and last stages, respectively.For simplification, the average diameters of the stationary cascade and rotating cascade are approximately equal to those of the stage.The average diameters ( a, j D , j = H, L) of the turbine casing and maximum lengths ( a, j L , j = H, L) of the high-and low-pressure STs can be approximately gained based on the empirical coefficients.

Figure 7 .h
Figure 7. Calculation process of the power output and efficiency of the DPST.

DADs of the stages along the flow directionFigure 8 .Figure 9 Figure 9 ,
Figure 8 shows the relationship between the dimensionless total PO ( t P  ) of the DPST and DAD ( m, DSH D  ) of the Curtis stage of the high-pressure ST. Figure 8 shows that when the SIA ( 1, DSH 

Figure 8 .
Figure 8. Relationship between t P  and m, DSH D  with different 1, DSH 

Figure 9 .
Figure 9. Relationship between t P  and m, H1 D  with different H1 .

Figure 10 shows
Figure 10 shows the relationship between the dimensionless total PO ( t P  ) of the DPST and DAD ( m, H3 D  ) of the last stage of the high-pressure ST.As shown in Figure 10, when the SIA ( H3  ) of this stage is °12 , with the augment of m, H3

Figure 11 shows
Figure 11 shows the relationship between the dimensionless total PO ( t P  ) of the DPST and DAD ( m, L1 D  ) of the first stage of the low-pressure ST.As shown in Figure 11, when the SIA ( L1  ) of this stage is 23  , with the augment of m, L1 D  , t P  first augments and then

Figure 10 .
Figure 10.Relationship between t P  and m, H3 D  with different H3 .

Figure 11 .
Figure 11.Relationship between t P  and m, L1 D  with different L1 .

Figure 12
Figure 12 shows the relationship between the dimensionless total PO ( t P  ) of the DPST and DAD ( m, L4 D  ) of the fourth stage of the low-pressure ST.As shown in Figure 12, when the SIA ( L4  ) of this stage is 23  , with the augment of m, L4 D  , t P  continuously augments.

Figure 13 shows
Figure 13 shows the relationship between the dimensionless total PO ( t P  ) of the DPST and DAD ( m, L5 D  ) of the fifth stage of the low-pressure ST. Figure 13 shows that when the SIA ( L5  ) of this stage is 23  , with the augment of m, L5 D  , t P  continuously abates.When

Figure 12 .
Figure 12.Relationship between t P  and m, L4 D  with different L4 .

Figure 13 .
Figure 13.Relationship between t P  and m, L5 D 

Figure 14 .
Figure 14.Relationship between t P  and iteration number with different initial values.

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

Table 3 .
Optimization variables of constructal design for a DPST.
v x

Table 4 .
Multiple-variable optimization results of the DPST.