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Preliminary design optimization of a steam generator
Energy Conversion and Management 43 (2002) 1651–1661 www.elsevier.com/locate/enconman
Preliminary design optimization of a steam generator Lingen Chen a,*, Shengbing Zhou a, Fengrui Sun a, Chih Wu b b
a Faculty 306, Naval University of Engineering, Wuhan 430033, China Department of Mechanical Engineering, US Naval Academy, Annapolis, MD 21402, USA
Received 21 February 2001; accepted 15 June 2001
Abstract In the present work, a procedure for preliminary design optimization of a steam generator has been developed with the objective of minimizing the weight of the generator. Some real engineering constraints are considered in the problem formulation. A method of evaluating the objective function and constraints of the problem is presented. The problem has been solved numerically by using the exterior SUMT in which the Powell unconstrained minimization technique improved by Sargent with the parabolic interpolation method of one-dimensional minimization, is employed. The results of the optimum design and a sensitivity analysis conducted about the optimum point have been reported. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Steam generator; Design optimization; Weight minimization
1. Introduction The steam generator is one of the key equipments in the pressurized water reactor nuclear propulsion plant. It is necessary that optimization techniques be applied to the preliminary design of the steam generator. In the present work, an attempt has been made to optimize the weight of a steam generator by considering thermodynamic and geometric constraints. 2. Formulation of the optimum design problem Any optimization problem involves the identification of design variables, objective function and constraints of the problem. *
Corresponding author. E-mail address: [email protected] (L. Chen).
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For the preliminary design of a steam generator, the following parameters are taken as the design variables: x1 ¼ inlet and outlet temperature difference of the primary cycle water through the generator ðd1 Þ, x2 ¼ flow velocity of the primary cycle water in the generator (W1 ), x3 ¼ inner diameter of the U-type tube in the generator ðdi Þ, ðx4 Þ ¼ wall thickness of the U-type tube ðdd Þ, x5 ¼ ratio of tube spacing to outer diameter of the U-type tube in the generator ðtj =do Þ. Therefore, the vector of design variables X becomes: *
X ¼ ½X1 ; X2 ; X3 ; X4 ; X5 T ¼ ½d1 ; W1 ; di ; dd ; tj =do T
ð1Þ
The weight is a very important design index of a marine propulsion plant. For the present case, the total weight ðWT Þ of the steam generator has been taken as the objective function to be minimized. In a steam generator design, generally, the following requirements are to be met from considerations of thermodynamic, strength and suitability requirements of the model: 1. The inlet and outlet temperature difference ðx1 Þ of the primary cycle water should be within the specified bounds. 2. The flow velocity ðx2 Þ of the primary cycle water should be within specified bounds. 3. The inner diameter ðx3 Þ of the U-type tube should be within specified bounds. 4. The wall thickness ðx4 Þ of the U-type tube should be within specified bounds. 5. The ratio ðx5 Þ of tube spacing to outer diameter of the U-type should be within specified bounds. 6. The pitch temperature difference ðdt0 Þ between the primary cycle and secondary loop should be greater than a specified value. 7. The tube plate thickness (t) should be within specified value. 8. The tube bundle diameter ðd1 Þ should be within specified bounds. 9. The steam generator height ðH0 Þ should be within specified bounds. 10. The steam generator diameter ðd2 Þ should be within specified bounds. 11. The total stress ðru Þ developed at the root of the U-type tube should be less than the permissible value. 12. The total stress ðrs Þ developed at the tube plate should be less than the permissible value.
3. Evaluation of objective function Thermodynamic and geometric parameters of the steam generator are calculated by the following steps: 1. 2. 3. 4. 5. 6.
Evaluates Evaluates Evaluates Evaluates Evaluates Evaluates
the the the the the the
heat transfer coefficients. steam generation quantity and heat transfer surface area. sizes of the steam–water separator. number of U-type tubes. sizes of the tube bundle. tube plate thickness.
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7. Evaluates the total sizes of the generator (length, width and height). 8. Evaluates the total volume (VT ) and the total weight (WT ) of the generator. 4. Solution procedure Once the system variables, the objective function and the constraints are defined, a suitable method has to be adopted to determine the values of the design variables that minimize the objective function while satisfying the given constraints. The present optimization model is a nonlinear programming problem with constrained functions and several variables. For such problems, the exterior SUMT, in which the Powell unconstrained minimization technique improved by Sargent with the parabolic interpolation method of one-dimensional minimization, is employed, has been found to be quite efficient [1–4]. Accordingly, this algorithm has also been employed in the present work.
5. Numerical example The numerical example considered here is a design plan of steam generator for a nuclear propulsion demonstration. The limitations of the constraints used in the example are given in Table 1. The mass flow rate of the second loop water, i.e. the quantity of steam generation is D ¼ 8260 g/h. For the weight minimization design objective, it is important whether the total volume of the steam generator is fixed or not. Thus, two different cases are considered herein. 5.1. Case A. Weight minimization with the fixed volume constraint The optimum results for different inlet water temperature of the primary cycle ðTin Þ with the same data pertinent to the design of the generator and the fixed total volume (VT ¼ VTF ¼ 1:53 m3 ) are shown in Table 2 and Fig. 1, in which the ordinate is non-dimensional based on the original design values with Tin ¼ 295°C. In Table 2, A is the heat transfer area, and K is the total heat transfer coefficient. The progress of the optimization path showing the cumulative number of objective function evaluation iteration steps versus the penalty function is shown in Fig. 2. In this figure, each point corresponds to one step of one-dimensional minimization. There is an additional equivalence constraint other than h1 ¼ VT VTF ¼ 0 in addition to the constraints shown in Table 1 in this case. The results of optimization show that there is a 19.45% reduction in the Table 1 The limitations of constraints g1 g2 g3 g4 g5 g6 g7
¼ dt 2:3 P 0 ¼ 80:0 dt P 0 ¼ W1 0:9 P 0 ¼ 5:5 W1 P 0 ¼ di 0:0089 P 0 ¼ 0:025 di P 0 ¼ dd 0:001 P 0
g8 ¼ 0:003 dd P 0 g9 ¼ tj =do 1:25 P 0 g10 ¼ 2:0 tj =do P 0 g11 ¼ dt0 10:0 P 0 g12 ¼ t 0:09 P 0 g13 ¼ 0:15 t P 0 g14 ¼ d1 0:4 P 0
g15 g16 g17 g18 g19 g20 g21
¼ 0:7 d1 P 0 ¼ H0 2:5 P 0 ¼ 6:0 H0 P 0 ¼ d2 0:5 P 0 ¼ 1:0 d2 P 0 ¼ 11:5 ru P 0 ¼ 12:8 rs P 0
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Table 2 Optimum results for case A Parameter Tin (°C) d1 (m) d2 (m) x5 A (m2 ) K (W/m2 °C) x1 (°C) VT (T) G1 (T/h) WT (T)
Plans Original
Optimum
295 0.5871 0.6837 1.4670 58.9440 2370.8500 62.0000 1.5300 67.7260 2.6789
295 0.5365 0.6269 1.8671 30.671 3154.7600 52.8710 1.5300 75.6580 2.1578
310 0.5387 0.6273 1.8740 26.1730 3172.4600 54.3920 1.5300 73.1650 2.1436
320 0.5391 0.6278 1.8752 23.5420 3198.9500 61.2726 1.5300 65.729 2.1251
340 0.5611 0.6534 1.9540 8.6570 3251.9400 69.890 1.5300 58.0680 2.0163
Fig. 1. Optimum results for case A.
objective function with the same inlet water temperature of the primary cycle Tin ¼ 295°C. The optimum point corresponds to an increase in the mass flow rate of the primary cycle water ðG1 Þ by 14.9%. This means that the reduction of weight is at the price of the increase of the mass flow rate of the primary cycle water. On the other hand, the total weight of the steam generator, the mass flow rate of the primary cycle water and the heat transfer surface area decrease with the increase of the inlet temperature ðTin Þ of the primary cycle water.
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Fig. 2. Progress of optimization path for case A.
5.2. Case B. Weight minimization without the fixed volume constraint The optimum results for different Tin without the fixed volume constraint are shown in Table 3 and Fig. 3. The progress of the optimization path showing the cumulative number of objective function evaluation iteration steps versus the penalty function is shown in Fig. 4. The results of optimization show that there are a 30.9% reduction in the total volume, a 36.8% reduction in the objective function (total weight), an 18.6% reduction in the heat transfer surface area and a 5.4% increase in Table 3 Optimum results for case B Parameter Tin (°C) d1 (m) d2 (m) A (m2 ) K (W/m2 °C) x1 (°C) x2 VT (T) G1 (T/h) WT (T)
Plans Original
Optimum
295 0.5871 0.6837 58.9440 2370.8530 62 1.7050 1.5300 62.7260 2.6789
295 0.4452 0.5184 44.6797 2842.93 58.3672 2.9059 1.0710 69.2810 1.6916
310 0.4452 0.5184 29.3829 2956.1200 57.9453 2.90373 0.7053 69.2810 1.3353
320 0.4060 0.4727 23.0564 3141.8500 57.641 4.3526 0.6449 69.2960 1.1883
340 0.4060 0.4727 17.1606 3202.9400 57.5000 4.3266 0.5136 68.8440 1.0289
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Fig. 3. Optimum results for case B.
Fig. 4. Progress of optimization path for case B.
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Fig. 5. Influence of W1 lim on the optimum results.
Fig. 6. Influence of H0 on the optimum results.
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Fig. 7. Influence of variable x1 .
Fig. 8. Influence of variable x2 .
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Fig. 9. Influence of variable x3 .
Fig. 10. Influence of variable x4 .
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the mass flow rate of the primary cycle water. Because the fixed volume constraint is relaxed, the number of objective function evaluation iteration steps in this case is less than that in case A. 6. Sensitivity analysis In practice, a designer would be interesting in knowing how the response quantities vary with a change in the design variables and design constraints. This type of sensitivity analysis will help the designer in manipulating the design variables to suit some specific requirements. Further, in some cases, the results obtained from the optimization procedure may have to be rounded to the nearest practical values of the design variables. Hence, sensitivity analysis of the response quantities with respect to the various design variables and design constraint limitations is conducted. The case A is taken as an example. 6.1. Influence of the flow velocity limitation of the primary cycle water in the generator on the optimum results In the design of case A, the constraint of flow velocity of the primary cycle water is always an effective constraint. The flow velocity limitation, W1 lim is varied on the negative and positive sides of the reference (case A) values (8%), and the optimum results for different Tin with the two velocity limitations are shown in Fig. 5.
Fig. 11. Influence of variable x5 .
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6.2. Influence of the generator height limitation on the optimum results In the design of case A, the constraint of steam generator height limitation, H, is varied on the negative and positive sides of the reference (case A) values (10%), and the optimum results for different Tin with the two height limitations are shown in Fig. 6. 6.3. Influence of the variations of the design variables on the performance of the generator In this analysis, the reference design is taken as the optimum design of case A with Tin ¼ 295°C. The design variables are varied on the negative and positive sides of the reference (optimum, case A) values (15%), and the magnitudes of the response quantities are plotted against the percentage changes of the design variables in Figs. 7–11. In these figures, N is the number of U-type tubes.
7. Conclusion A preliminary design optimization procedure for the steam generator of a nuclear propulsion plant has been presented in this paper. The numerical examples show the procedure is effective. The results of sensitivity analysis would provide more full guidance for the real designer. The procedure can be used for the multi-objective optimization of the steam generator.
References [1] Chen L, Wu C, Blank D, Sun F. Preliminary design optimization of marine dual tandem gear. Int J Power Energy Syst 1997;17(3):218–22. [2] Chen L, Wu C, Ni N, Cao Y, Sun F. Optimum design of centrifugal compressor stages. Int J Power Energy Syst 1998;18(1):12–5. [3] Chen L, Wu C, Blank D, Sun F. Multi-objective optimum design method for a radial-axial flow turbine with optimum criteria of blade twist at outlet of blades. Int J Power Energy Syst 1998;18(1):16–20. [4] Chen L, Zhang J, Wu C, Blank D, Sun F. Analysis of multi-objective decision-making for marine steam turbine stage. Int J Power Energy Syst 1998;18(2):96–101.
Preliminary design optimization of a steam generator Lingen Chen a,*, Shengbing Zhou a, Fengrui Sun a, Chih Wu b b
a Faculty 306, Naval University of Engineering, Wuhan 430033, China Department of Mechanical Engineering, US Naval Academy, Annapolis, MD 21402, USA
Received 21 February 2001; accepted 15 June 2001
Abstract In the present work, a procedure for preliminary design optimization of a steam generator has been developed with the objective of minimizing the weight of the generator. Some real engineering constraints are considered in the problem formulation. A method of evaluating the objective function and constraints of the problem is presented. The problem has been solved numerically by using the exterior SUMT in which the Powell unconstrained minimization technique improved by Sargent with the parabolic interpolation method of one-dimensional minimization, is employed. The results of the optimum design and a sensitivity analysis conducted about the optimum point have been reported. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Steam generator; Design optimization; Weight minimization
1. Introduction The steam generator is one of the key equipments in the pressurized water reactor nuclear propulsion plant. It is necessary that optimization techniques be applied to the preliminary design of the steam generator. In the present work, an attempt has been made to optimize the weight of a steam generator by considering thermodynamic and geometric constraints. 2. Formulation of the optimum design problem Any optimization problem involves the identification of design variables, objective function and constraints of the problem. *
Corresponding author. E-mail address: [email protected] (L. Chen).
0196-8904/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 1 ) 0 0 1 1 7 - 0
1652
L. Chen et al. / Energy Conversion and Management 43 (2002) 1651–1661
For the preliminary design of a steam generator, the following parameters are taken as the design variables: x1 ¼ inlet and outlet temperature difference of the primary cycle water through the generator ðd1 Þ, x2 ¼ flow velocity of the primary cycle water in the generator (W1 ), x3 ¼ inner diameter of the U-type tube in the generator ðdi Þ, ðx4 Þ ¼ wall thickness of the U-type tube ðdd Þ, x5 ¼ ratio of tube spacing to outer diameter of the U-type tube in the generator ðtj =do Þ. Therefore, the vector of design variables X becomes: *
X ¼ ½X1 ; X2 ; X3 ; X4 ; X5 T ¼ ½d1 ; W1 ; di ; dd ; tj =do T
ð1Þ
The weight is a very important design index of a marine propulsion plant. For the present case, the total weight ðWT Þ of the steam generator has been taken as the objective function to be minimized. In a steam generator design, generally, the following requirements are to be met from considerations of thermodynamic, strength and suitability requirements of the model: 1. The inlet and outlet temperature difference ðx1 Þ of the primary cycle water should be within the specified bounds. 2. The flow velocity ðx2 Þ of the primary cycle water should be within specified bounds. 3. The inner diameter ðx3 Þ of the U-type tube should be within specified bounds. 4. The wall thickness ðx4 Þ of the U-type tube should be within specified bounds. 5. The ratio ðx5 Þ of tube spacing to outer diameter of the U-type should be within specified bounds. 6. The pitch temperature difference ðdt0 Þ between the primary cycle and secondary loop should be greater than a specified value. 7. The tube plate thickness (t) should be within specified value. 8. The tube bundle diameter ðd1 Þ should be within specified bounds. 9. The steam generator height ðH0 Þ should be within specified bounds. 10. The steam generator diameter ðd2 Þ should be within specified bounds. 11. The total stress ðru Þ developed at the root of the U-type tube should be less than the permissible value. 12. The total stress ðrs Þ developed at the tube plate should be less than the permissible value.
3. Evaluation of objective function Thermodynamic and geometric parameters of the steam generator are calculated by the following steps: 1. 2. 3. 4. 5. 6.
Evaluates Evaluates Evaluates Evaluates Evaluates Evaluates
the the the the the the
heat transfer coefficients. steam generation quantity and heat transfer surface area. sizes of the steam–water separator. number of U-type tubes. sizes of the tube bundle. tube plate thickness.
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7. Evaluates the total sizes of the generator (length, width and height). 8. Evaluates the total volume (VT ) and the total weight (WT ) of the generator. 4. Solution procedure Once the system variables, the objective function and the constraints are defined, a suitable method has to be adopted to determine the values of the design variables that minimize the objective function while satisfying the given constraints. The present optimization model is a nonlinear programming problem with constrained functions and several variables. For such problems, the exterior SUMT, in which the Powell unconstrained minimization technique improved by Sargent with the parabolic interpolation method of one-dimensional minimization, is employed, has been found to be quite efficient [1–4]. Accordingly, this algorithm has also been employed in the present work.
5. Numerical example The numerical example considered here is a design plan of steam generator for a nuclear propulsion demonstration. The limitations of the constraints used in the example are given in Table 1. The mass flow rate of the second loop water, i.e. the quantity of steam generation is D ¼ 8260 g/h. For the weight minimization design objective, it is important whether the total volume of the steam generator is fixed or not. Thus, two different cases are considered herein. 5.1. Case A. Weight minimization with the fixed volume constraint The optimum results for different inlet water temperature of the primary cycle ðTin Þ with the same data pertinent to the design of the generator and the fixed total volume (VT ¼ VTF ¼ 1:53 m3 ) are shown in Table 2 and Fig. 1, in which the ordinate is non-dimensional based on the original design values with Tin ¼ 295°C. In Table 2, A is the heat transfer area, and K is the total heat transfer coefficient. The progress of the optimization path showing the cumulative number of objective function evaluation iteration steps versus the penalty function is shown in Fig. 2. In this figure, each point corresponds to one step of one-dimensional minimization. There is an additional equivalence constraint other than h1 ¼ VT VTF ¼ 0 in addition to the constraints shown in Table 1 in this case. The results of optimization show that there is a 19.45% reduction in the Table 1 The limitations of constraints g1 g2 g3 g4 g5 g6 g7
¼ dt 2:3 P 0 ¼ 80:0 dt P 0 ¼ W1 0:9 P 0 ¼ 5:5 W1 P 0 ¼ di 0:0089 P 0 ¼ 0:025 di P 0 ¼ dd 0:001 P 0
g8 ¼ 0:003 dd P 0 g9 ¼ tj =do 1:25 P 0 g10 ¼ 2:0 tj =do P 0 g11 ¼ dt0 10:0 P 0 g12 ¼ t 0:09 P 0 g13 ¼ 0:15 t P 0 g14 ¼ d1 0:4 P 0
g15 g16 g17 g18 g19 g20 g21
¼ 0:7 d1 P 0 ¼ H0 2:5 P 0 ¼ 6:0 H0 P 0 ¼ d2 0:5 P 0 ¼ 1:0 d2 P 0 ¼ 11:5 ru P 0 ¼ 12:8 rs P 0
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Table 2 Optimum results for case A Parameter Tin (°C) d1 (m) d2 (m) x5 A (m2 ) K (W/m2 °C) x1 (°C) VT (T) G1 (T/h) WT (T)
Plans Original
Optimum
295 0.5871 0.6837 1.4670 58.9440 2370.8500 62.0000 1.5300 67.7260 2.6789
295 0.5365 0.6269 1.8671 30.671 3154.7600 52.8710 1.5300 75.6580 2.1578
310 0.5387 0.6273 1.8740 26.1730 3172.4600 54.3920 1.5300 73.1650 2.1436
320 0.5391 0.6278 1.8752 23.5420 3198.9500 61.2726 1.5300 65.729 2.1251
340 0.5611 0.6534 1.9540 8.6570 3251.9400 69.890 1.5300 58.0680 2.0163
Fig. 1. Optimum results for case A.
objective function with the same inlet water temperature of the primary cycle Tin ¼ 295°C. The optimum point corresponds to an increase in the mass flow rate of the primary cycle water ðG1 Þ by 14.9%. This means that the reduction of weight is at the price of the increase of the mass flow rate of the primary cycle water. On the other hand, the total weight of the steam generator, the mass flow rate of the primary cycle water and the heat transfer surface area decrease with the increase of the inlet temperature ðTin Þ of the primary cycle water.
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Fig. 2. Progress of optimization path for case A.
5.2. Case B. Weight minimization without the fixed volume constraint The optimum results for different Tin without the fixed volume constraint are shown in Table 3 and Fig. 3. The progress of the optimization path showing the cumulative number of objective function evaluation iteration steps versus the penalty function is shown in Fig. 4. The results of optimization show that there are a 30.9% reduction in the total volume, a 36.8% reduction in the objective function (total weight), an 18.6% reduction in the heat transfer surface area and a 5.4% increase in Table 3 Optimum results for case B Parameter Tin (°C) d1 (m) d2 (m) A (m2 ) K (W/m2 °C) x1 (°C) x2 VT (T) G1 (T/h) WT (T)
Plans Original
Optimum
295 0.5871 0.6837 58.9440 2370.8530 62 1.7050 1.5300 62.7260 2.6789
295 0.4452 0.5184 44.6797 2842.93 58.3672 2.9059 1.0710 69.2810 1.6916
310 0.4452 0.5184 29.3829 2956.1200 57.9453 2.90373 0.7053 69.2810 1.3353
320 0.4060 0.4727 23.0564 3141.8500 57.641 4.3526 0.6449 69.2960 1.1883
340 0.4060 0.4727 17.1606 3202.9400 57.5000 4.3266 0.5136 68.8440 1.0289
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Fig. 3. Optimum results for case B.
Fig. 4. Progress of optimization path for case B.
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Fig. 5. Influence of W1 lim on the optimum results.
Fig. 6. Influence of H0 on the optimum results.
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Fig. 7. Influence of variable x1 .
Fig. 8. Influence of variable x2 .
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Fig. 9. Influence of variable x3 .
Fig. 10. Influence of variable x4 .
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the mass flow rate of the primary cycle water. Because the fixed volume constraint is relaxed, the number of objective function evaluation iteration steps in this case is less than that in case A. 6. Sensitivity analysis In practice, a designer would be interesting in knowing how the response quantities vary with a change in the design variables and design constraints. This type of sensitivity analysis will help the designer in manipulating the design variables to suit some specific requirements. Further, in some cases, the results obtained from the optimization procedure may have to be rounded to the nearest practical values of the design variables. Hence, sensitivity analysis of the response quantities with respect to the various design variables and design constraint limitations is conducted. The case A is taken as an example. 6.1. Influence of the flow velocity limitation of the primary cycle water in the generator on the optimum results In the design of case A, the constraint of flow velocity of the primary cycle water is always an effective constraint. The flow velocity limitation, W1 lim is varied on the negative and positive sides of the reference (case A) values (8%), and the optimum results for different Tin with the two velocity limitations are shown in Fig. 5.
Fig. 11. Influence of variable x5 .
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6.2. Influence of the generator height limitation on the optimum results In the design of case A, the constraint of steam generator height limitation, H, is varied on the negative and positive sides of the reference (case A) values (10%), and the optimum results for different Tin with the two height limitations are shown in Fig. 6. 6.3. Influence of the variations of the design variables on the performance of the generator In this analysis, the reference design is taken as the optimum design of case A with Tin ¼ 295°C. The design variables are varied on the negative and positive sides of the reference (optimum, case A) values (15%), and the magnitudes of the response quantities are plotted against the percentage changes of the design variables in Figs. 7–11. In these figures, N is the number of U-type tubes.
7. Conclusion A preliminary design optimization procedure for the steam generator of a nuclear propulsion plant has been presented in this paper. The numerical examples show the procedure is effective. The results of sensitivity analysis would provide more full guidance for the real designer. The procedure can be used for the multi-objective optimization of the steam generator.
References [1] Chen L, Wu C, Blank D, Sun F. Preliminary design optimization of marine dual tandem gear. Int J Power Energy Syst 1997;17(3):218–22. [2] Chen L, Wu C, Ni N, Cao Y, Sun F. Optimum design of centrifugal compressor stages. Int J Power Energy Syst 1998;18(1):12–5. [3] Chen L, Wu C, Blank D, Sun F. Multi-objective optimum design method for a radial-axial flow turbine with optimum criteria of blade twist at outlet of blades. Int J Power Energy Syst 1998;18(1):16–20. [4] Chen L, Zhang J, Wu C, Blank D, Sun F. Analysis of multi-objective decision-making for marine steam turbine stage. Int J Power Energy Syst 1998;18(2):96–101.