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The optimum design and arrangement of a steam generator in an integral pressurized water reactor
International Journal of Heat and Mass Transfer 94 (2016) 211–221
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
The optimum design and arrangement of a steam generator in an integral pressurized water reactor Xinyu Wei a,⇑, Chunhui Dai b, Pengfei Wang a, Fuyu Zhao a a b
Department of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China Lab. on Steam Power System, Wuhan Second Ship Des. & Rec. Institute, Wuhan, Hubei 430064, PR China
a r t i c l e
i n f o
Article history: Received 4 June 2015 Received in revised form 10 November 2015 Accepted 19 November 2015
Keywords: Double-tube once-through steam generator (DOTSG) Pitch Arrangement Two-level optimization method
a b s t r a c t This paper presents a double-tube once-through steam generator (DOTSG), whose tube unit includes an outer straight tube and an inner helical tube, in an integral pressurized water reactor (IPWR). To obtain the optimum structure of the inner helical tube and the arrangement of DOTSGs in the reactor pressure vessel, a two-level optimization method is used, aiming at the lower pumping power needed, and a smaller reactor pressure vessel volume is used. The pitch of inner helical tubes and the central distance of outer tubes are considered design parameters when minimizing the pumping power with the genetic algorithm in the bottom level, while the number of tube units in a single DOTSG and the number of DOTSGs in the reactor pressure vessel (RPV) are optimized to obtain the minimum volume of the IPWR, which is conducted in the top level. The optimum pitch of the helical tube varies in the subcooled region, boiling region and superheated region. The results show that the smaller pitch brings a shorter tube length and a higher pressure drop, and the effects are strong in the sub-cooled region and the superheated region but weak in the boiling region. In this way, the optimal structure of the inner helical tube is a small pitch in the single-phase region and a large pitch in the boiling region. According to the bottom level results, the optimum arrangement of DOTSGs in the pressure vessel is determined. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The integral pressure water reactor (iPWR) has advantages in the aspects of volume, weight, cost, and safety, which make it has been widely used in nuclear engineering, such as nuclear power propelling, heat supply, seawater desalination and integrated nuclear power station of next generation [1–4]. In iPWR, the main equipment of the primary loop system, including reactor core, steam generators, main coolant pump, and pressurizer, is housed in the reactor pressure vessel (RPV). This compact structure eliminates the pipe connection and essentially excludes the occurrence of large break loss of coolant accidents (LBLOCA). The compactness and thermal hydraulic characteristics of the oncethrough steam generator (OTSG), that transfers heat from the primary side to secondary side and supplies superheated steam to the turbine, contribute a lot to the advantages of iPWR, such as simple mechanical structure, smaller size, and higher heat transfer efficiency. So, the OTSG is one of the most widely used stream generator in iPWR [5–8]. ⇑ Corresponding author. Tel.: +86 13772096784 (cell), +86 029 82668648 (o). E-mail addresses: [email protected] (X. Wei), [email protected] (C. Dai), [email protected] (P. Wang), [email protected] (F. Zhao). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.11.070 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
This paper presents a double-tube once-through steam generator (DOTSG) whose tube unit includes an outer straight tube and an inner helical tube, which is showed in Fig. 1. The DOTSG adopts a vertical, counterflow, shell-and-tube heat exchanger, which is an existing double tube type heat transfer structure [9,10]. In the primary side, water flows through the inner helical tube and the shell side, while in the secondary side water/steam flows through the lunate channel between the outer straight tube and the inner helical tube. In the both sides of the inner helical tube, the flow is spirally, therefore, the heat transfer is enhanced, and this is the first advantage of the inner helical tube. The osculating structure of the helical tube and the outer straight tube reduces the tube vibration, and this is the secondary advantage. But the helical tube results in the additional flow resistance. In RPV, a certain number of units constitute a DOTSG and a certain number of DOTSGs are located around and above the reactor core. The structure, size, and arrangement of the tube units and DOTSGs in the RPV are key factors to the size of the whole iPWR. In order to design an iPWR with compact structure, the choice of an appropriate design parameter is considered as an optimization problem and solved by the two-level optimization method in this study.
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2. Problem formulation
A
Outer straight tube
Inner helical tube
A Section A-A
(a)
(b)
Fig. 1. The structure of tube unit of DOTSG. (a) Perspective view, (b) axial cross section.
Many scholars have studied on the OTSG equipped in iPWR. However, most of their works studied on the thermal hydraulic characteristics [11–13], model and transient performance [14– 16]. There are few researches aiming at the structure design and arrangement in the reactor pressure vessel of the OTSG. Chen et al. [17] designed and optimized a double-tube heat exchanger. They used the smallest difference of outlet temperatures of two primary sides to appraise the best flow distribution. In order to reach the optimized flow distribution, the gap width changes with the tube dimensions. Yu and Jia [18] developed an immovable enthalpy boundary model for heat transfer calculation of the DOTSG, and their results indicated that the heat transfer was enhanced by the inner helical tube. But the pressure drop was not considered in their study. As to the optimization of the heat exchanger, some authors considered the cost of heat transfer surface area or capital investment as an objective function to be minimized [19,20], while others considered the sum of entropy generation of streams as an objective function was also reported in [21,22]. We have optimized the single tube unit of DOTSG use multi-objective optimization method and Maximum principle respectively [23,24]. However, the previous work only focus on the single tube unit. When it comes to the design of whole iPWR, the situation is different: (1) the whole heat transfer area must be calculated, that means the number of the tube units is needed to be calculated; (2) as the arrangement of DOTSGs determines the diameter of the RPV, how many DOTSGs equipped in the vessel, and how many tube units in every DOTSG should be optimized. Therefore, we propel the studies to overcome these problems. In this paper, a two-level optimization method is proposed. At the top level, the objective is to minimize the volume of the RPV, while at the bottom level, the objective is to minimize the total pressure drop of the DOTSG. This paper is structured as follows. Sections 2 illustrate the DOTSG in iPWR, and introduces the heat transfer process and the pressure drop calculation for DOTSG. Section 3 establishes optimization problem that is optimized later by the two-level optimization method. The results are discussed in Section 4. Finally, conclusion is given in Section 5.
We design a concept iPWR, which adopts the DOTSG as steam generator. The systematic diagram of a typical iPWR is shown in Fig. 2. The core is located at the lower part of the RPV, and the DOTSGs are symmetrically arranged along the annular region at the upper part of the RPV. The pressurizer is located in the top of the RPV. Main coolant pumps (MCPs) are equipped above the DOTSG. Once the heat in the core is removed, the primary coolant flows upward through the upper region, and enters at the top of the DOTSGs. It then travels downward through the DOTSGs, where the heat will be transferred to the secondary coolant. Finally, the primary coolant exits at the bottom of the DOTSGs and back into the reactor core. In the secondary side, the feed water enters the bottom of the DOTSGs, then flows upward inside the thimble tube to remove the heat from the primary coolant and exits the DOTSGs as superheated steam which flows to the turbine. Fig. 3 is the structure diagram of the DOTSGs. In the RPV, the DOTSGs are symmetrically arranged along the annular region between the core support barrel and the RPV wall. Fig. 3(a) shows the relative location of the reactor core and DOTSGs from the cross section. There is a gap between the core and DOTSGs, which is accommodated the fixed devices and accessory equipment. Fig. 3 (b) is a single DOTSG, the tube units are arranged staggered, which can make the DOTSGs and the whole iPWR more compact. The details of the staggered arrangement of tube units are showed in Fig. 3(c). It is feasible and convenient to assume that every steam generator tube has identical flow. The dashed line in Fig. 3(c) is the assuming shell side boundary of one single tube unit. It can be seen that there are three flow channel in the tube unit from Fig. 3 (c) and (d), Channel A is the primary side in the inner helical tube. Channel B is the secondary side, Channel C is the primary side in shell side. In a single tube unit of the DOTSG, the primary water flows in Channel A and C, while the secondary fluid flows in Channel B, which is constituted by the outer straight tube and the inner helical tube. The secondary fluid obtains heat from the primary water in both of the inner helical tube and shell side. Fig. 3(d) is the vertical section of the tube unit. The pitch of the inner helical tube is the axial length of one coil of the helical tube, which is denoted by S in the figure. 2.1. Reactor volume As shown in Figs. 2 and 3(a), the volume of iPWR can be calculated by formula (1).
VR ¼
p 4
Dc þ 2Dsg þ 2Dd
2
Lc þ Lsg þ Ld þ Lt þ Lb
ð1Þ
where Dc , Dsg , Dd are the diameters of reactor core and DOTSGs, and the radial gap distance between the DOTSG and reactor core. Lt is the top part height of RPV above the DOTSG, Lsg is the height of DOTSG, Ld is the axial distance of reactor core and DOTSG, Lc is the height of reactor core, Lb is the bottom part height of RPV under the reactor core. Lc , Lb are considered constant in this study. Dsg is calculated by formula (2)
Dsg ¼ pt Nd
ð2Þ
where pt is the central axes distance of outer tubes, N d is the number of the tube units along the diagonal in the single DOTSG. 2.2. Heat transfer model For the single tube unit of DOTSG, it is divided into three regions according to the secondary water/steam status: sub-cooled region, boiling region, and superheated region. In the sub-cooled and superheated regions, the secondary fluids are sub-cooled water
X. Wei et al. / International Journal of Heat and Mass Transfer 94 (2016) 211–221
213
Pressurizer
Lt Main coolant pump
DOTSG
L sg
DOTSG
to Turbine
from Feed water pump
Ld Reactor core
Lc
Lb
Secondary coolant
Primary coolant
Fig. 2. A typical iPWR system with DOTSG. Lt is the top part height of RPV above the DOTSG, Lsg is the height of DOTSG, Ld is the axial distance of reactor core and DOTSG, Lc is the height of reactor core, Lb is the bottom part height of RPV under the reactor core.
and superheated steam respectively, both of which are single phase. While in the boiling region, the secondary fluid is the mixture of steam and water, which is two-phase. In a single tube unit, the primary water flows in Channel A and C, while the secondary fluid flows in Channel B in Fig. 3. The division of the heat transfer region and axially micro-unit are showed in Fig. 4. The heat transfer rate per unit length dz are
G1
dH1 ¼ k12 ðt 1 t 2 Þ þ k13 ðt 1 t3 Þ dz
ð3Þ
G2
dH2 ¼ k12 ðt 1 t 2 Þ þ k32 ðt 3 t2 Þ dz
ð4Þ
G3
dH3 ¼ k32 ðt 3 t 2 Þ k13 ðt 1 t3 Þ dz
ð5Þ
where H1 , H2 , and H3 are specific enthalpy of fluid in Channel A, B, C; t1 , t 2 , and t 3 are temperature of fluid in Channel A, B, C; G1 , G2 , and G3 are mass flow rate of fluid in Channel A, B, C, which can be calculated by formula (6)–(8); k12 is the overall heat transfer coefficient of Channel A to B, k32 is the overall heat transfer coefficient of Channel C to B, and k13 is the overall heat transfer coefficient of Channel C to A.
G1 ¼
ewW p Nsg N tu
ð6Þ
G2 ¼
wW s Nsg N tu
ð7Þ
G3 ¼
wð1 eÞW p Nsg Ntu
ð8Þ
where Wp and Ws are the total primary and secondary mass flow rate, respectively; Nsg is the number of DOTSGs arranged in VPR;
Ntu is the total number tube units in one DOTSG, which can be calculated by formula (7); w is a coefficient which indicate the safety allowance of the design, about 10% number of additional tubes are usually arranged in the steam generators to guarantee the normal operation when some tubes are out of order, so here w is set equals to 0.9; e is the ratio of mass flow rate of the primary side in the inner helical tube to the total mass flow rate of the primary side. Because the primary fluid is from the reactor core, both of the primary side passages are parallel pipeline, the value of e is adjusted to equalize the total pressure drops between Channel A and C.
Ntu ¼ k
i 3 h 2 Nd 1 þ 1 4
ð9Þ
where N d the number of the tube units along the diagonal in a single DOTSG. The cross section of DOTSG is usually designed circularly while the tubes are arranged hexagonally, hence some additional tube units could be placed between the circle and hexagon if N tu is large enough, k is introduced as a correction factor
k¼
1
ðNd 6 7Þ
1:11 1:14 ðNd > 7Þ
ð10Þ
The DOTSG is divided into three regions according to the secondary water/steam status: sub-cooled region, boiling region, and superheated region, as shown in Fig. 4. S in these regions are denoted as Sc , Sb and Sh respectively. In order to reduce the pumping power and save the space of the reactor power plant, the optimum structure of the heat exchanger and the acceptable arrangement of the DOTSGs should be obtained, in other words, the Sc , Sb , and Sh should be optimized under the constraint conditions. 2.3. Pumping power The total pumping power in iPWR mainly contains the power provided by main coolant pump and feed water pump. The function
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DOTSG DOTSG
Reactor Core
¦Á
Ds
DOTSG
tube units
DOTSG
g
Dd Dc
DOTSG
Nd
(b)
(a) Channel C
Channel A Channel C Channel A
rS
d1 d2
S
d3 d4 Primary sides
Assuming shell side boundary
Channel B
Secondary side
pt
Channel B
(c)
(d)
Fig. 3. Structure diagram of the DOTSG. (a) Arrangement of the DOTSGs in RPV (cross section), Dsg is the diameter of the DOTSG, Dc is the diameter of the reactor core, Dd is the radial gap distance between the DOTSG and reactor core. (b) Arrangement of the tube units in the single DOTSG, N d is the number of the tube units along the diagonal in a single DOTSG. (c) Details of the staggered arrangement of tube units, the dashed line is the assuming shell side boundary of one tube unit, d1 is the inside diameter of the inner tube, d2 is the outside diameter of the inner tube, d3 is the inside diameter of the outer tube, d4 is the outside diameter of the outer tube, pt is the central axes distance of outer tubes. (d) Vertical section of the tube unit. S is the pitch of the inner helical tube. Channel A is the primary side in the inner helical tube. Channel B is the secondary side, Channel C is the primary side in shell side.
of the two type of pumps is to work to overcome the pressure drop of the fluid in Channel A, B, and C. The pressure drop contains the frictional pressure drop, acceleration pressure drop, and gravitational pressure drop. Since the design parameters mentioned in this paper mainly affects the frictional pressure drop, but not the acceleration pressure drop and gravitational pressure drop, only the frictional pressure drop is considered in this work. Thus, we will only focus on the frictional pressure drop, and the ‘‘pressure drop” means the ‘‘frictional pressure drop” hereinafter. The total pumping power can be obtained as formula (11).
Ppump ¼
Z 3 X Gi i¼1
Lsg 0
f i x2i dz 2dei
ð11Þ
where Lsg is the total length of DOTSG. In order to obtain the value of the unknown Lsg , f i , and xi , some calculation should be carried
out on the basis of the heat transfer process which expressed as formula (3)–(5). However, the phase of secondary fluid will change when it achieves the saturation temperature of water or steam, this takes errors in the calculation because the phase change may takes place in a micro-unit dz while the heat transfer coefficients are set constant in the whole dz. As the inlet temperature is known in this research, and the outlet steam temperature and pressure must meet the requirement of the turbine, H2 is set fixed and divided as the compute node. Substitute formula (4) into formula (3) and (5) respectively, we have
dH1 G2 k12 ðt 1 t 2 Þ þ k13 ðt 1 t 3 Þ ¼ dH2 G1 k12 ðt 1 t 2 Þ þ k32 ðt 3 t 2 Þ
ð12Þ
dH3 G2 k32 ðt 3 t 2 Þ k13 ðt 1 t 3 Þ ¼ dH2 G3 k12 ðt 1 t 2 Þ þ k32 ðt 3 t 2 Þ
ð13Þ
X. Wei et al. / International Journal of Heat and Mass Transfer 94 (2016) 211–221
STEP (3) ti ðk þ 1Þ are transformed from Hi ðk þ 1Þ according to IAPWS IF-97 and t 2 ðk þ 1Þ is chosen for the judgment of the second cycle. If t 2 ðk þ 1Þ tsj 6 0:1 satisfy, reset the helical pitch S according to the value of j, and start the next region calculation, otherwise go to the following calculation point. STEP (4) The standard temperatures tsj are different in three regions of DOTSG. When j = 1, it means the fluid is superheated and ts1 is the saturated steam temperature of the pressure p2 ðk þ 1Þ; ts2 denotes the saturated water temperature; ts3 , the criterion of finishing the third calculation cycle, is the inlet temperature of the secondary fluid, that means if t2 ðk þ 1Þ ts3 6 0:1 satisfies, this cycle is completed, we save the total step k as k0,j. STEP (5) The pressure of the primary side in the inner tube is compared by that in the shell side respectively, if jp1 p3j 6 0:01 is satisfied, the two pressure drops are thought equal and the fourth cycle is stopped, otherwise e is changed by another value e0 and repeat the above calculation process. Dpfi ðkÞ is computed based on 2Dz ¼ zðk þ 1Þ zðk 1Þ, so the Ppump is finally calculated through formula (16).
z
Superheated Region
Boiling Region dz
Sub-cooled Region Ppump
r
Rewrite formula (4) as
ð14Þ
According to IAPWS IF-97 [25], the temperature can be mutually transformed with the specific enthalpy and the pressure. Thus, P pump could be obtained via numerical calculation, the procedure is shown in Fig. 5. STEP (1) Initiation. Giving the initial e = 0.5, the calculation step k = 1, the j is used to indicate the three regions of the inner helical tubes,
8 > < Sh ; ðj ¼ 1Þ Sj ¼ Sb ; ðj ¼ 2Þ > : Sc ; ðj ¼ 3Þ
STEP (2) The physical parameters such as Hi ð1Þ, t i ð1Þ, pi ð1Þ at the initial calculation point are set so that Hi ð2Þ and zð2Þ can be computed through Euler method
8 H1 ð2Þ ¼ H1 ð1Þ þ DH2 F 1 ½t 1 ð1Þ; t 2 ð1Þ; t 3 ð1Þ > > > < H2 ð2Þ ¼ H2 ð1Þ þ DH2 > zð2Þ ¼ zð1Þ þ DH2 F 2 ½t1 ð1Þ; t 2 ð1Þ; t3 ð1Þ > > : H3 ð2Þ ¼ H3 ð1Þ þ DH2 F 3 ½t 1 ð1Þ; t 2 ð1Þ; t 3 ð1Þ
" # k0;j 1 3 X 3 X X Ppumpi ðkÞ ¼ Ppumpi ð1Þ þ Ppumpi ðkÞ þ 2 i¼1 j¼1 k¼2
ð16Þ
2.4. Coefficients calculation
Fig. 4. The division of the heat transfer region and axially micro-unit.
dz G2 ¼ dH2 k12 ðt1 t2 Þ þ k32 ðt 3 t 2 Þ
215
ð15Þ
where F i ði ¼ 1; 2; 3Þ represent formula (12)–(14) respectively. For increasing the calculation accuracy, the following calculation points are computed through the method shown in Fig. 5.
2.4.1. Heat transfer coefficients The heat transfer coefficients of the primary fluid a1 and a3 are estimated as a modified Dittus–Boelter equation as formula (17) [18] and the Dittus–Boelter equation respectively.
a1 ¼
k1 0:3 1:3 0:023Re0:8 1 Pr 1 de1
ð17Þ
where k1 and is the thermal conductivity, Re1 is the Reynolds number, Pr1 is the Prandtl number. According to the experimental results [9], the heat transfer convection heat transfer coefficient of the secondary fluid based on the pitch is obtained from the experiment at the condition: non-adiabatic flow, P ¼ 3:0 6:5 MPa, qv ¼ 50 450 kg=m2 s, q ¼ 0:04 0:25 MW=m2 . And the results showed that the convection heat transfer coefficient of the secondary fluid varied in different heat transfer regions. In the single-phase regions (sub-cooled region and superheated region)
a2 ¼
k2 Nu2 de2
ð18Þ
while in the two-phase region
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a a2 ¼ aL 1 þ 0:6 0
aL
ð19Þ
where k2 is the thermal conductivity of the secondary fluid, Nu2 is the Nusselt number of the secondary fluid, aL is the convective heat transfer coefficient when the secondary fluid is saturated, and a0 is the convective heat transfer coefficient when the inner tube is straight. Nu2 and a0 are estimated as follow [9].
8 h i 0:25 0:21 d2 0:4 0:6½1þ0:0737ðS=d3 Þ0:73 Pr2 > ðd3 Þ Re2 Pr 0:43 ; > 2 > 0:002 1 þ 68:2ðS=d3 Þ Prw > > > h i 1:68 > 0:4 0:6½1þ0:0165ðS=d3 Þ > 0:43 Pr2 0:25 > < 0:002 1 þ 1:25 106 ðS=d3 Þ6:37 dd2 Re2 Pr 2 ; Prw 3 Nu2 ¼ h i 0:35 0:25 > 0:9 0:43 Pr 2 d2 > > Re0:8 ; > 0:015 1 þ 5:39ðS=d3 Þ 2 Pr 2 d3 Prw > > > h i > 0:35 0:25 > 3:04 0:43 Pr2 : d2 0:015 1 þ 182ðS=d3 Þ Re0:8 ; 2 Pr 2 d3 Prw
2 103 6 Re2 6 Re2;cr ; 3:08 6 S=d3 6 4:92 2 103 6 Re2 6 Re2;cr ; S=d3 P 4:92 Re2;cr 6 Re2 6 4 104 ; 3:08 6 S=d3 6 4:92 Re2;cr 6 Re2 6 4 104 ; S=d3 P 4:92
ð20Þ
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Fig. 5. Flow chart of the calculation for Ppump.
a0 ¼ 4:6q0:7 p0:2
where Re2 is the Reynolds number of the secondary fluid, Pr 2 is the Prandtl number of secondary fluid, Prw is the Prandtl number of secondary fluid at the temperature of the wall, q is the heat flux, p is the pressure Re2;cr is the critical Reynolds number that describes the transition of the laminar flow to the turbulent flow,
8
1:3 0:54 > S > > 1 þ dr2s Re0:55 < 3:5 1 þ 30 rs 1
f1 ¼ 0:46 3 > 3 > > 1 þ dr2s Re0:25 : 0:316 1 þ 1:2 10 rSs 1
0
ð21Þ Re2;cr
10:3
d3 d2 B C ¼ 18; 500@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 S =4 þ ðd3 d2 Þ
ð22Þ
2.4.2. The friction factors The friction factors of the primary fluid in the helical tube are obtained by formula (23) [10]
2 103 6 Re1 6 Re1;cr ð23Þ Re1;cr 6 Re1 6 8 104
X. Wei et al. / International Journal of Heat and Mass Transfer 94 (2016) 211–221
where rs is helical radius of the inner helical tube, as show in Fig. 3 (d), and Re1,cr is the critical Reynolds number of the helical tube which is computed from:
Re1;cr ¼ 22; 000
0:3 S rs
ð24Þ
The friction factor f3 which indicates the friction pressure drop of the fluid in the shell side is computed by Blausius and modified by the Sieder–Tate formula respectively. When the secondary fluid is single-phase, the friction factor of the secondary fluid can be calculated as follow [10].
8 h i 0:64 0:25 1þ0:92ðS=d Þ0:3 ½ 3 1:6 d2 > > 0:5 1 þ 28ðS=d3 Þ Re2 ; > d3 > > > h i 0:64 > 1:2 > > 0:25½1þ4:5ðS=d3 Þ < 0:5 1 þ 706ðS=d3 Þ3:6 d2 Re2 ; d3 f2 ¼ h i 0:46 > > 2 d2 > Re0:25 ; > 2 > 0:316 1 þ 10ðS=d3 Þ d3 > > > > 0:46 : 0:316½1 þ 12:6ðS=d Þ2:1 ðd2 Þ Re0:25 ; 3
d3
2
qG
Re2 ¼ 2 103 Re2;cr ; S=d3 P 4:92
4 X An xn
ð25Þ
Re2 ¼ Re2;cr 4 104 ; S=d3 ¼ 3:08 4:92 Re2 ¼ Re2;cr 4 104 ; S=d3 P 4:92
ð26Þ
where Dp0 is the single-phase frictional pressure drop of the saturated water at the temperature of boiling region, and can be calculated according to formula (25), qL, qG are the density of the saturated water and the saturated steam at the temperature of boiling region respectively, x is the quality of the secondary fluid, which is mixture of water and steam, w is the non-uniform coefficient, which calculates as formula (27)
w ¼ A0 þ
in the two-level optimization method was a decomposition of the design task into subtasks performed independently, see Fig. 6, and a system-level giving rise to a two-level optimization. In general, decomposition was motivated by the obvious need to distribute work over many people and computers to compress the task calendar time. An equally important benefit from the decomposition is granting an autonomy to the groups of engineers responsible for each particular subtask in choosing their methods and tools for the subtask execution. It has been studied by some researchers, Bartheley [26] used the problem matrix method to describe the relationship between the
Re2 ¼ 2 103 Re2;cr ; S=d3 ¼ 3:08 4:92
While in the boiling region, the frictional pressure drop is obtained from the following formula:
q Dp2 ¼ w 1 þ x L 1 Dp0
217
ð27Þ
n¼1
where Ai (i = 0, 1, 2, 3, 4) are constants, if P = 4 MPa, A0 = 1, A1 = 2.28, A2 = 5.91, A3 = 7.94, A4 = 3.94; if P = 6 MPa, A0 = 1.32, A1 = 5.3, A2 = 24.1, A3 = 35.1, A4 = 16.8. 3. Two-level optimization In order to simplify and accelerate the calculation process, a two-level method is used to solve the problem. The key concept
objective functions and variables. Haftka [27] investigated two important problems in multilevel optimization: decomposition and co-ordination. In [28], the multilevel genetic algorithm (MLGA) was proposed and an actively controlled tower building subjected to earthquake excitations was considered to investigate the effectiveness of MLGA. The objective of this study contains the optimum structure of DOTSG for the single tube unit, the optimum arrangement of tube units in DOTSG and the optimum arrangement of DOTSGs in RPV. For the purpose of achieving these three goals, two sub-objectives are to minimize the volume of RPV and to minimize the total pumping power needed, as shown in formula (28). Design parameters or decision variables for the optimization process are the pitch of the inner helical tube, the central distance of the neighboring tube units, the number of DOTSGs in the RPV, and the number of tube units along the diagonal in a single DOTSG.
Min V R
ðSj ; pt Þ
Min Ppump
ðNsg ; N d Þ
ð28Þ
In order to simplify and accelerate the calculation process, a two-level method is used to solve the problem. As shown in Fig. 7, the problem which is solved by the two-level optimization method in this study is decomposed into one bottom level and one top level. The bottom level is the structure optimization of
Fig. 6. Scheme of the two-level optimization method.
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T ¼ ½V 11 V 21 V N1 V 12 V 22 V N2 V 13 V 23 V N3 V 14 V 24 V N4 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Sc
Sb
Sh
ð30Þ
pt
Following the initialization of chromosome, a decoding program converts each vector to the input file of a code such as pt in order to calculate the fitness function (Ppump). This procedure is based on the constraint conditions which provide the upper and lower boundaries of the decision variables. As the restriction of manufacturing technology and the range of experiment parameters, the pitch should in the range of 40–120 mm. The calculation region of helical pitches are set as formula (31).
8 > < 40 < Sc < 120 40 < Sb < 120 > : 40 < Sh < 120
ð31Þ
The pt should be larger than the outside diameter of the outer tube, the upper boundary denoted as ptmax is depended on the relationship between the reactor core and steam generator. As shown in Fig. 3(a) the DOTSGs are arranged around the reactor core, the relationship of a and N sg can be described as formula (32).
2a 6
2p Nsg
where
a ¼ arcsin Fig. 7. Structure and arrangement optimization of DOTSG by two-level optimization method.
DOTSG, the objective is minimize the pumping power, and the decision variables are the pitch distribution of inner helical tube (Sj ) and the central distance of the neighboring tube units (pt ). As the optimization objective is to find an optimum combination of Sj and pt of the tube unit, the genetic algorithm method (GA) is used in the bottom level. While in the top level, the objective is the arrangement optimization of DOTSG in RPV according the optimization results of bottom level, the objective is minimize the volume of RPV, and the decision variables are the number of tube units in a single DOTSG and the number of DOTSGs in the RPV. A normal constrained non-linear optimization method is used in top level. The key advantage of this two-level optimization is the interactive between the two levels, which can simplify and accelerate the calculation process. In the bottom level, the calculation needs the parameters of N sg and N d from the top level. While in the top level, the calculation needs the parameters of Lsg and pt from the bottom level. In other words, the interactive is an iterated process between the top level and the bottom level.
ð32Þ
Dsg Dsg þ Dc þ 2Dd
ð33Þ
substitute formula (33) into formula (32), the maximum reasonable pt, which is noted as ptmax, can be obtained. In this way, the constraint of the objective in bottom level, which shown as formula (34).
8 40 < Sc < 120 > > > < 40 < S < 120 b s:t: > 40 < Sh < 120 > > : d4 < pt 6 pt;max
ð34Þ
Considering the constraints, the decode calculation can be shown as formula (35).
8 N X > > > V i1 ð2i1 Þ 12040 > Sc ¼ 40 þ > 210 1 > > i¼1 > > > > N X > > > > Sb ¼ 40 þ V i2 ð2i1 Þ 12040 > 210 1 < i¼1
N > X > > > S ¼ 40 þ V i3 ð2i1 Þ 12040 > h > 210 1 > > i¼1 > > > > N X > > p d4 > V i4 ð2i1 Þ t;max > pt ¼ d4 þ : 210 1
ð35Þ
i¼1
3.1. The bottom level optimization
3.2. The top level optimization
Aiming at obtaining the lowest pumping power Ppump (considered as the objective function in this study), the helical pitches (Sc, Sb, Sh) of the inner tube and the central distance of outer tubes (pt) are regarded as the decision variables in the lower level optimization. The problem can be described as formula (29).
The DOTSGs are used to take the energy out of the reactor core, however the total the pressure vessel volume of an iPWR should be paid special attention so that it could not only save the space but also arrange enough heat transfer tubes to guarantee the safe operation. In this study, N sg and N d are considered as the decision variables in the top level optimization, and VR which could be calculated by formula (1) is optimized as the objective. Therefore, the objective and constrains in top level can be described as formula (36) and (37). Then the problem can be solved by using the fmincon routine based on sequential quadratic programming (SQP) algorithm in the optimization toolbox of Matlab.
Min V RPV ðSc ; Sb ; Sh ; Pt Þ
ð29Þ
The optimization problem is optimized by GA in this level. The decision variables are encoded into a chromosome which consists of sub regions that imply the decision variables respectively, as shown in formula (30).
X. Wei et al. / International Journal of Heat and Mass Transfer 94 (2016) 211–221 Table 1 Major parameters of the reactor core. Reactor power (MW) Primary side Operating pressure (MPa) Cooled/heated temperature (°C) Coolant mass flow rate (t/h) Core height (m) Core diameter (m)
38 15.41 245.0/331.27 289.98 0.568 0.694
Secondary side Operating pressure (MPa) Inlet/outlet temperature (°C)
4 106/280
219
of Lsg, so that the VR is reduced. The minimum VR is 5:1273 m3 when Nsg and Nd are 13 and 8, with the corresponding lowest Ppump equals to 915.87 W. To consider the RPV Volume and pumping power at the same time, formula (38) is introduced to obtain the final optimization results.
F ¼ c1
VR P pump þ c2 V0 P0
ð38Þ
where c1 and c2 are two weighting coefficients, as VR is regarded more important, c1 should be a larger value 0.75, while c2 equals to 0.25. V0 and P0 are normative quantities to make VR and Ppump dimensionless. In this study, the values of VR and Ppump when Nsg equals to 15 and Nd is 5 are chosen as the V0 and P0
V 0 ¼ 5:4776 P0 ¼ 6204:7
ð39Þ
The final calculation results of F with different Nsg and Nd are listed in Table 2. As shown in Table 2, F gets a minimum value 0.70364, so the combination which Nsg is 12 and Nd equals to 9 is regarded as the final optimization. The optimum VR and Ppump are 5:1415 m3 and 648.51 W respectively. 4. Discussion
Fig. 8. The relationship of VR, Nsg, and Nd.
Min V R ðNsg ; Nd Þ
ð36Þ
Nsg 2 INT s:t: Nd 2 INT
ð37Þ
When a combination of N sg and N d is given, the Gi (i = 1, 2, 3), Dsg, and ptmax could be obtained according to a certain reactor core using formula (6)–(8), formula (2) and formula (32). Then the bottom level optimization based on the GA is performed to get the optimum S and pt, at the same time, Lsg which is a variable in computing the top level object VR through formula (1) is obtained. 3.3. Optimization results Take a tight-lattice integral reactor as the computing core [29], the design parameters are listed in Table 1. In the bottom level, the optimization structure of the tube unit is can be obtained by GA with different interactive parameters from top level. Nsg and Nd are mainly related to the flow rate of every tube unit. When Nsg and Nd are 12 and 9, the optimum S and pt are Sc = 59.6 mm, Sb = 114.5 mm, Sh = 64.4 mm, and pt = 18.1 mm. As a result, the interactive parameters from bottom to top level are Lsg = 1.06 m, and pt = 18.1 mm. Actually, the top level optimization is a geometry relationship of RPV among VR, Nsg and Nd. The VR with different Nsg and Nd are shown in Fig. 8. If Nsg is constant, VR will firstly decrease and then increase with the increasing of Nd, in other words, there is an optimum Nd for every Nsg. On the other hand, when a single steam generator is determined (Nd is changeless), the VR is reduced when more steam generators are arranged around the reactor core, because the addition of steam generator will lead to the decrease
The temperature distributions of fluid in the inner helical tube, secondary side, and the shell side are shown in Fig. 9, on which the arrows and dashed lines denote the flow direction of each fluid and the boundaries of three regions respectively. As little heat is transferred between the two primary fluids and the secondary superheated steam in the superheated region, T1 and T2 are nearly the same until the end of boiling region (namely the start of superheated region) after 0.0789 m. Lasting about 0.6945 m, T1 is higher and higher than T3 in the boiling region because the heat transfer area between secondary and outer tube fluid is larger than that in the inner tube. In this region, T2 equals to the saturation temperature which decreases a little from the downward to the upward of the tubes as a result of the pressure decreasing. The both primary fluids go on to flow to the bottom, the difference of T1 and T3 is reduced gradually due to the heat exchange between themselves. The three optimal pitches vary in different heat transfer region. This is mainly because the effects of the helical tube to the heat transfer capacity and pressure drop are different in the three heat transfer region. As stated before, the inner helical tube can increase the heat transfer surface area and add the turbulence of the fluids. These effects should be strong when the pitch is small, and vice versa. From Eq. (18) we can obtain that increase the fi, wi, and Lsg can all result in the Ppump increasing. When the pitch is small, fi is big and the tube length is short, on the other hand if the pitch is large the friction factor is small but the tube length will be long. Thus there should be some optimum values of S, making the Ppump minimum. As the boiling heat transfer is stronger than that between single-phase fluids in sub-cooled and superheated region and the effects of Sb to the tube length is weak, besides, f2 of the boiling fluid may be bigger, the optimum Sb is larger than Sc and Sh. In addition, the flow rate in superheated region is higher than that in sub-cooled region, so Sc could be a little less than Sh without causing larger P pump . As stated before, the inner helical tube can increase the heat transfer surface area and add the turbulence of the fluids. But in boiling, the turbulence of the mixture of water and steam is already strong. In the nature of things, the strong turbulence brings the big pressure drop which affects the. So the pressure drop is bigger than that in sub-cooled region and superheated region. As the
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Table 2 The value of F with different Nsg and Nd. Nd
5
6
7
8
9
10
11
Nsg 3 4 5 6 7 8 9 10 11 12 13 14 15
– – – – – – – – – – – 1.0799 1
– – – – – – 1.2181 1.1195 1.0117 0.94223 0.88607 0.8363 0.79409
– – – 1.4188 1.2225 1.0944 1.0025 0.92693 0.86366 0.82282 0.77841 0.74625 –
– 1.7675 1.3911 1.2017 1.0609 0.97209 0.90508 0.84868 0.80854 0.7663 0.71135 – –
1.7806 1.5017 1.232 1.0861 0.99338 0.91599 0.85907 0.81746 0.78308 0.70364 – – –
1.5891 1.319 1.1574 1.0306 0.95183 0.88661 0.85411 0.81014 0.73819 – – – –
1.4625 1.3274 1.1041 1.0139 0.94103 0.89863 0.85852 0.81809 – – – – –
12
13
14
15
16
17
18
1.6438 1.2314 1.0948 1.0056 0.94747 0.90226 0.86929
1.3866 1.2216 1.133 1.0395 0.96903 0.92612 0.8582
1.5931 1.2198 1.1249 1.0519 0.99279 0.96453 –
1.3992 1.2298 1.1346 1.07 0.99115 0.97668 –
1.4075 1.2565 1.1667 1.0793 1.0194 – –
1.4298 1.2935 1.2072 1.0866 1.066 – –
1.4773 1.316 1.2037 1.1596 1.1131 – –
3 4 5 6 7 8 9
from the differences of heat transfer capacity and flow state in different region. Smaller pitch takes better heat transfer, but at the same time, the pressure drop will be increased. The boiling heat transfer is stronger than that between single-phase fluids, this means the effects of Sb to heat tube length is weak, and so the optimum Sb is the largest. In addition, Sc could be a little less than Sh because of the flow rate which affects the Ppump is higher in superheated region. As to the arrangement of the arrangement of DOTSGs in the RPV, it is a geometry problem to obtain a smaller volume of RPV. It is clear that a small Nd gives a small diameter of single DOTSG, and result a big Nsg for a given number of tube units. The result shows that the minimum VR is 5:1273 m3 when Nsg and Nd are 13 and 8. However, the corresponding minimum Ppump is large (915.87 W), so an evaluation formula with two weighting coefficients is introduced to adjust the conflict between VR and Ppump. As VR is considered more important than Ppump, 12 and 9 are chosen as the final optimum Nsg and Nd, making the VR and Ppump equals to 5:1415 m3 and 648.51 W respectively, with the corresponding bottom level optimization shows that the parameters are Sc = 59.6 mm, Sb = 114.5 mm, Sh = 64.4 mm, pt = 18.1 mm, and Lsg = 1.06 m. Fig. 9. The axial temperature distribution of DOTSG.
narrow distribution of the tube length and the wide distribution of the pressure drop, the bigger pitch is reasonable in boiling region.
5. Conclusion A two-level optimization method is applied to optimize DOTSG with outer straight tube and inner helical tube. In the bottom level optimization, after studying the flow performances of DOTSG, the numerical calculation model is established. Ppump is then minimized by GA algorithm, considering Sh, Sb, Sc, and pt as the decision variables in the bottom level with the interactive parameters from the top level. Based on the Lsg and pt obtained in the bottom level optimization, different combinations of Nsg and Nd which determine the arrangement of DOTSG are regarded as the variables to compute the VR in the top level optimization. While a normal constrained non-linear optimization method is used in top level. The optimum pitch varies in three heat transfer regions mainly result
Acknowledgment This research is supported by National Natural Science Foundation of China (11405125), China Postdoctoral Science Foundation Fund Project (2014M562420), and the Fundamental Research Funds for the Central Universities.
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
The optimum design and arrangement of a steam generator in an integral pressurized water reactor Xinyu Wei a,⇑, Chunhui Dai b, Pengfei Wang a, Fuyu Zhao a a b
Department of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China Lab. on Steam Power System, Wuhan Second Ship Des. & Rec. Institute, Wuhan, Hubei 430064, PR China
a r t i c l e
i n f o
Article history: Received 4 June 2015 Received in revised form 10 November 2015 Accepted 19 November 2015
Keywords: Double-tube once-through steam generator (DOTSG) Pitch Arrangement Two-level optimization method
a b s t r a c t This paper presents a double-tube once-through steam generator (DOTSG), whose tube unit includes an outer straight tube and an inner helical tube, in an integral pressurized water reactor (IPWR). To obtain the optimum structure of the inner helical tube and the arrangement of DOTSGs in the reactor pressure vessel, a two-level optimization method is used, aiming at the lower pumping power needed, and a smaller reactor pressure vessel volume is used. The pitch of inner helical tubes and the central distance of outer tubes are considered design parameters when minimizing the pumping power with the genetic algorithm in the bottom level, while the number of tube units in a single DOTSG and the number of DOTSGs in the reactor pressure vessel (RPV) are optimized to obtain the minimum volume of the IPWR, which is conducted in the top level. The optimum pitch of the helical tube varies in the subcooled region, boiling region and superheated region. The results show that the smaller pitch brings a shorter tube length and a higher pressure drop, and the effects are strong in the sub-cooled region and the superheated region but weak in the boiling region. In this way, the optimal structure of the inner helical tube is a small pitch in the single-phase region and a large pitch in the boiling region. According to the bottom level results, the optimum arrangement of DOTSGs in the pressure vessel is determined. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The integral pressure water reactor (iPWR) has advantages in the aspects of volume, weight, cost, and safety, which make it has been widely used in nuclear engineering, such as nuclear power propelling, heat supply, seawater desalination and integrated nuclear power station of next generation [1–4]. In iPWR, the main equipment of the primary loop system, including reactor core, steam generators, main coolant pump, and pressurizer, is housed in the reactor pressure vessel (RPV). This compact structure eliminates the pipe connection and essentially excludes the occurrence of large break loss of coolant accidents (LBLOCA). The compactness and thermal hydraulic characteristics of the oncethrough steam generator (OTSG), that transfers heat from the primary side to secondary side and supplies superheated steam to the turbine, contribute a lot to the advantages of iPWR, such as simple mechanical structure, smaller size, and higher heat transfer efficiency. So, the OTSG is one of the most widely used stream generator in iPWR [5–8]. ⇑ Corresponding author. Tel.: +86 13772096784 (cell), +86 029 82668648 (o). E-mail addresses: [email protected] (X. Wei), [email protected] (C. Dai), [email protected] (P. Wang), [email protected] (F. Zhao). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.11.070 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
This paper presents a double-tube once-through steam generator (DOTSG) whose tube unit includes an outer straight tube and an inner helical tube, which is showed in Fig. 1. The DOTSG adopts a vertical, counterflow, shell-and-tube heat exchanger, which is an existing double tube type heat transfer structure [9,10]. In the primary side, water flows through the inner helical tube and the shell side, while in the secondary side water/steam flows through the lunate channel between the outer straight tube and the inner helical tube. In the both sides of the inner helical tube, the flow is spirally, therefore, the heat transfer is enhanced, and this is the first advantage of the inner helical tube. The osculating structure of the helical tube and the outer straight tube reduces the tube vibration, and this is the secondary advantage. But the helical tube results in the additional flow resistance. In RPV, a certain number of units constitute a DOTSG and a certain number of DOTSGs are located around and above the reactor core. The structure, size, and arrangement of the tube units and DOTSGs in the RPV are key factors to the size of the whole iPWR. In order to design an iPWR with compact structure, the choice of an appropriate design parameter is considered as an optimization problem and solved by the two-level optimization method in this study.
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2. Problem formulation
A
Outer straight tube
Inner helical tube
A Section A-A
(a)
(b)
Fig. 1. The structure of tube unit of DOTSG. (a) Perspective view, (b) axial cross section.
Many scholars have studied on the OTSG equipped in iPWR. However, most of their works studied on the thermal hydraulic characteristics [11–13], model and transient performance [14– 16]. There are few researches aiming at the structure design and arrangement in the reactor pressure vessel of the OTSG. Chen et al. [17] designed and optimized a double-tube heat exchanger. They used the smallest difference of outlet temperatures of two primary sides to appraise the best flow distribution. In order to reach the optimized flow distribution, the gap width changes with the tube dimensions. Yu and Jia [18] developed an immovable enthalpy boundary model for heat transfer calculation of the DOTSG, and their results indicated that the heat transfer was enhanced by the inner helical tube. But the pressure drop was not considered in their study. As to the optimization of the heat exchanger, some authors considered the cost of heat transfer surface area or capital investment as an objective function to be minimized [19,20], while others considered the sum of entropy generation of streams as an objective function was also reported in [21,22]. We have optimized the single tube unit of DOTSG use multi-objective optimization method and Maximum principle respectively [23,24]. However, the previous work only focus on the single tube unit. When it comes to the design of whole iPWR, the situation is different: (1) the whole heat transfer area must be calculated, that means the number of the tube units is needed to be calculated; (2) as the arrangement of DOTSGs determines the diameter of the RPV, how many DOTSGs equipped in the vessel, and how many tube units in every DOTSG should be optimized. Therefore, we propel the studies to overcome these problems. In this paper, a two-level optimization method is proposed. At the top level, the objective is to minimize the volume of the RPV, while at the bottom level, the objective is to minimize the total pressure drop of the DOTSG. This paper is structured as follows. Sections 2 illustrate the DOTSG in iPWR, and introduces the heat transfer process and the pressure drop calculation for DOTSG. Section 3 establishes optimization problem that is optimized later by the two-level optimization method. The results are discussed in Section 4. Finally, conclusion is given in Section 5.
We design a concept iPWR, which adopts the DOTSG as steam generator. The systematic diagram of a typical iPWR is shown in Fig. 2. The core is located at the lower part of the RPV, and the DOTSGs are symmetrically arranged along the annular region at the upper part of the RPV. The pressurizer is located in the top of the RPV. Main coolant pumps (MCPs) are equipped above the DOTSG. Once the heat in the core is removed, the primary coolant flows upward through the upper region, and enters at the top of the DOTSGs. It then travels downward through the DOTSGs, where the heat will be transferred to the secondary coolant. Finally, the primary coolant exits at the bottom of the DOTSGs and back into the reactor core. In the secondary side, the feed water enters the bottom of the DOTSGs, then flows upward inside the thimble tube to remove the heat from the primary coolant and exits the DOTSGs as superheated steam which flows to the turbine. Fig. 3 is the structure diagram of the DOTSGs. In the RPV, the DOTSGs are symmetrically arranged along the annular region between the core support barrel and the RPV wall. Fig. 3(a) shows the relative location of the reactor core and DOTSGs from the cross section. There is a gap between the core and DOTSGs, which is accommodated the fixed devices and accessory equipment. Fig. 3 (b) is a single DOTSG, the tube units are arranged staggered, which can make the DOTSGs and the whole iPWR more compact. The details of the staggered arrangement of tube units are showed in Fig. 3(c). It is feasible and convenient to assume that every steam generator tube has identical flow. The dashed line in Fig. 3(c) is the assuming shell side boundary of one single tube unit. It can be seen that there are three flow channel in the tube unit from Fig. 3 (c) and (d), Channel A is the primary side in the inner helical tube. Channel B is the secondary side, Channel C is the primary side in shell side. In a single tube unit of the DOTSG, the primary water flows in Channel A and C, while the secondary fluid flows in Channel B, which is constituted by the outer straight tube and the inner helical tube. The secondary fluid obtains heat from the primary water in both of the inner helical tube and shell side. Fig. 3(d) is the vertical section of the tube unit. The pitch of the inner helical tube is the axial length of one coil of the helical tube, which is denoted by S in the figure. 2.1. Reactor volume As shown in Figs. 2 and 3(a), the volume of iPWR can be calculated by formula (1).
VR ¼
p 4
Dc þ 2Dsg þ 2Dd
2
Lc þ Lsg þ Ld þ Lt þ Lb
ð1Þ
where Dc , Dsg , Dd are the diameters of reactor core and DOTSGs, and the radial gap distance between the DOTSG and reactor core. Lt is the top part height of RPV above the DOTSG, Lsg is the height of DOTSG, Ld is the axial distance of reactor core and DOTSG, Lc is the height of reactor core, Lb is the bottom part height of RPV under the reactor core. Lc , Lb are considered constant in this study. Dsg is calculated by formula (2)
Dsg ¼ pt Nd
ð2Þ
where pt is the central axes distance of outer tubes, N d is the number of the tube units along the diagonal in the single DOTSG. 2.2. Heat transfer model For the single tube unit of DOTSG, it is divided into three regions according to the secondary water/steam status: sub-cooled region, boiling region, and superheated region. In the sub-cooled and superheated regions, the secondary fluids are sub-cooled water
X. Wei et al. / International Journal of Heat and Mass Transfer 94 (2016) 211–221
213
Pressurizer
Lt Main coolant pump
DOTSG
L sg
DOTSG
to Turbine
from Feed water pump
Ld Reactor core
Lc
Lb
Secondary coolant
Primary coolant
Fig. 2. A typical iPWR system with DOTSG. Lt is the top part height of RPV above the DOTSG, Lsg is the height of DOTSG, Ld is the axial distance of reactor core and DOTSG, Lc is the height of reactor core, Lb is the bottom part height of RPV under the reactor core.
and superheated steam respectively, both of which are single phase. While in the boiling region, the secondary fluid is the mixture of steam and water, which is two-phase. In a single tube unit, the primary water flows in Channel A and C, while the secondary fluid flows in Channel B in Fig. 3. The division of the heat transfer region and axially micro-unit are showed in Fig. 4. The heat transfer rate per unit length dz are
G1
dH1 ¼ k12 ðt 1 t 2 Þ þ k13 ðt 1 t3 Þ dz
ð3Þ
G2
dH2 ¼ k12 ðt 1 t 2 Þ þ k32 ðt 3 t2 Þ dz
ð4Þ
G3
dH3 ¼ k32 ðt 3 t 2 Þ k13 ðt 1 t3 Þ dz
ð5Þ
where H1 , H2 , and H3 are specific enthalpy of fluid in Channel A, B, C; t1 , t 2 , and t 3 are temperature of fluid in Channel A, B, C; G1 , G2 , and G3 are mass flow rate of fluid in Channel A, B, C, which can be calculated by formula (6)–(8); k12 is the overall heat transfer coefficient of Channel A to B, k32 is the overall heat transfer coefficient of Channel C to B, and k13 is the overall heat transfer coefficient of Channel C to A.
G1 ¼
ewW p Nsg N tu
ð6Þ
G2 ¼
wW s Nsg N tu
ð7Þ
G3 ¼
wð1 eÞW p Nsg Ntu
ð8Þ
where Wp and Ws are the total primary and secondary mass flow rate, respectively; Nsg is the number of DOTSGs arranged in VPR;
Ntu is the total number tube units in one DOTSG, which can be calculated by formula (7); w is a coefficient which indicate the safety allowance of the design, about 10% number of additional tubes are usually arranged in the steam generators to guarantee the normal operation when some tubes are out of order, so here w is set equals to 0.9; e is the ratio of mass flow rate of the primary side in the inner helical tube to the total mass flow rate of the primary side. Because the primary fluid is from the reactor core, both of the primary side passages are parallel pipeline, the value of e is adjusted to equalize the total pressure drops between Channel A and C.
Ntu ¼ k
i 3 h 2 Nd 1 þ 1 4
ð9Þ
where N d the number of the tube units along the diagonal in a single DOTSG. The cross section of DOTSG is usually designed circularly while the tubes are arranged hexagonally, hence some additional tube units could be placed between the circle and hexagon if N tu is large enough, k is introduced as a correction factor
k¼
1
ðNd 6 7Þ
1:11 1:14 ðNd > 7Þ
ð10Þ
The DOTSG is divided into three regions according to the secondary water/steam status: sub-cooled region, boiling region, and superheated region, as shown in Fig. 4. S in these regions are denoted as Sc , Sb and Sh respectively. In order to reduce the pumping power and save the space of the reactor power plant, the optimum structure of the heat exchanger and the acceptable arrangement of the DOTSGs should be obtained, in other words, the Sc , Sb , and Sh should be optimized under the constraint conditions. 2.3. Pumping power The total pumping power in iPWR mainly contains the power provided by main coolant pump and feed water pump. The function
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DOTSG DOTSG
Reactor Core
¦Á
Ds
DOTSG
tube units
DOTSG
g
Dd Dc
DOTSG
Nd
(b)
(a) Channel C
Channel A Channel C Channel A
rS
d1 d2
S
d3 d4 Primary sides
Assuming shell side boundary
Channel B
Secondary side
pt
Channel B
(c)
(d)
Fig. 3. Structure diagram of the DOTSG. (a) Arrangement of the DOTSGs in RPV (cross section), Dsg is the diameter of the DOTSG, Dc is the diameter of the reactor core, Dd is the radial gap distance between the DOTSG and reactor core. (b) Arrangement of the tube units in the single DOTSG, N d is the number of the tube units along the diagonal in a single DOTSG. (c) Details of the staggered arrangement of tube units, the dashed line is the assuming shell side boundary of one tube unit, d1 is the inside diameter of the inner tube, d2 is the outside diameter of the inner tube, d3 is the inside diameter of the outer tube, d4 is the outside diameter of the outer tube, pt is the central axes distance of outer tubes. (d) Vertical section of the tube unit. S is the pitch of the inner helical tube. Channel A is the primary side in the inner helical tube. Channel B is the secondary side, Channel C is the primary side in shell side.
of the two type of pumps is to work to overcome the pressure drop of the fluid in Channel A, B, and C. The pressure drop contains the frictional pressure drop, acceleration pressure drop, and gravitational pressure drop. Since the design parameters mentioned in this paper mainly affects the frictional pressure drop, but not the acceleration pressure drop and gravitational pressure drop, only the frictional pressure drop is considered in this work. Thus, we will only focus on the frictional pressure drop, and the ‘‘pressure drop” means the ‘‘frictional pressure drop” hereinafter. The total pumping power can be obtained as formula (11).
Ppump ¼
Z 3 X Gi i¼1
Lsg 0
f i x2i dz 2dei
ð11Þ
where Lsg is the total length of DOTSG. In order to obtain the value of the unknown Lsg , f i , and xi , some calculation should be carried
out on the basis of the heat transfer process which expressed as formula (3)–(5). However, the phase of secondary fluid will change when it achieves the saturation temperature of water or steam, this takes errors in the calculation because the phase change may takes place in a micro-unit dz while the heat transfer coefficients are set constant in the whole dz. As the inlet temperature is known in this research, and the outlet steam temperature and pressure must meet the requirement of the turbine, H2 is set fixed and divided as the compute node. Substitute formula (4) into formula (3) and (5) respectively, we have
dH1 G2 k12 ðt 1 t 2 Þ þ k13 ðt 1 t 3 Þ ¼ dH2 G1 k12 ðt 1 t 2 Þ þ k32 ðt 3 t 2 Þ
ð12Þ
dH3 G2 k32 ðt 3 t 2 Þ k13 ðt 1 t 3 Þ ¼ dH2 G3 k12 ðt 1 t 2 Þ þ k32 ðt 3 t 2 Þ
ð13Þ
X. Wei et al. / International Journal of Heat and Mass Transfer 94 (2016) 211–221
STEP (3) ti ðk þ 1Þ are transformed from Hi ðk þ 1Þ according to IAPWS IF-97 and t 2 ðk þ 1Þ is chosen for the judgment of the second cycle. If t 2 ðk þ 1Þ tsj 6 0:1 satisfy, reset the helical pitch S according to the value of j, and start the next region calculation, otherwise go to the following calculation point. STEP (4) The standard temperatures tsj are different in three regions of DOTSG. When j = 1, it means the fluid is superheated and ts1 is the saturated steam temperature of the pressure p2 ðk þ 1Þ; ts2 denotes the saturated water temperature; ts3 , the criterion of finishing the third calculation cycle, is the inlet temperature of the secondary fluid, that means if t2 ðk þ 1Þ ts3 6 0:1 satisfies, this cycle is completed, we save the total step k as k0,j. STEP (5) The pressure of the primary side in the inner tube is compared by that in the shell side respectively, if jp1 p3j 6 0:01 is satisfied, the two pressure drops are thought equal and the fourth cycle is stopped, otherwise e is changed by another value e0 and repeat the above calculation process. Dpfi ðkÞ is computed based on 2Dz ¼ zðk þ 1Þ zðk 1Þ, so the Ppump is finally calculated through formula (16).
z
Superheated Region
Boiling Region dz
Sub-cooled Region Ppump
r
Rewrite formula (4) as
ð14Þ
According to IAPWS IF-97 [25], the temperature can be mutually transformed with the specific enthalpy and the pressure. Thus, P pump could be obtained via numerical calculation, the procedure is shown in Fig. 5. STEP (1) Initiation. Giving the initial e = 0.5, the calculation step k = 1, the j is used to indicate the three regions of the inner helical tubes,
8 > < Sh ; ðj ¼ 1Þ Sj ¼ Sb ; ðj ¼ 2Þ > : Sc ; ðj ¼ 3Þ
STEP (2) The physical parameters such as Hi ð1Þ, t i ð1Þ, pi ð1Þ at the initial calculation point are set so that Hi ð2Þ and zð2Þ can be computed through Euler method
8 H1 ð2Þ ¼ H1 ð1Þ þ DH2 F 1 ½t 1 ð1Þ; t 2 ð1Þ; t 3 ð1Þ > > > < H2 ð2Þ ¼ H2 ð1Þ þ DH2 > zð2Þ ¼ zð1Þ þ DH2 F 2 ½t1 ð1Þ; t 2 ð1Þ; t3 ð1Þ > > : H3 ð2Þ ¼ H3 ð1Þ þ DH2 F 3 ½t 1 ð1Þ; t 2 ð1Þ; t 3 ð1Þ
" # k0;j 1 3 X 3 X X Ppumpi ðkÞ ¼ Ppumpi ð1Þ þ Ppumpi ðkÞ þ 2 i¼1 j¼1 k¼2
ð16Þ
2.4. Coefficients calculation
Fig. 4. The division of the heat transfer region and axially micro-unit.
dz G2 ¼ dH2 k12 ðt1 t2 Þ þ k32 ðt 3 t 2 Þ
215
ð15Þ
where F i ði ¼ 1; 2; 3Þ represent formula (12)–(14) respectively. For increasing the calculation accuracy, the following calculation points are computed through the method shown in Fig. 5.
2.4.1. Heat transfer coefficients The heat transfer coefficients of the primary fluid a1 and a3 are estimated as a modified Dittus–Boelter equation as formula (17) [18] and the Dittus–Boelter equation respectively.
a1 ¼
k1 0:3 1:3 0:023Re0:8 1 Pr 1 de1
ð17Þ
where k1 and is the thermal conductivity, Re1 is the Reynolds number, Pr1 is the Prandtl number. According to the experimental results [9], the heat transfer convection heat transfer coefficient of the secondary fluid based on the pitch is obtained from the experiment at the condition: non-adiabatic flow, P ¼ 3:0 6:5 MPa, qv ¼ 50 450 kg=m2 s, q ¼ 0:04 0:25 MW=m2 . And the results showed that the convection heat transfer coefficient of the secondary fluid varied in different heat transfer regions. In the single-phase regions (sub-cooled region and superheated region)
a2 ¼
k2 Nu2 de2
ð18Þ
while in the two-phase region
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a a2 ¼ aL 1 þ 0:6 0
aL
ð19Þ
where k2 is the thermal conductivity of the secondary fluid, Nu2 is the Nusselt number of the secondary fluid, aL is the convective heat transfer coefficient when the secondary fluid is saturated, and a0 is the convective heat transfer coefficient when the inner tube is straight. Nu2 and a0 are estimated as follow [9].
8 h i 0:25 0:21 d2 0:4 0:6½1þ0:0737ðS=d3 Þ0:73 Pr2 > ðd3 Þ Re2 Pr 0:43 ; > 2 > 0:002 1 þ 68:2ðS=d3 Þ Prw > > > h i 1:68 > 0:4 0:6½1þ0:0165ðS=d3 Þ > 0:43 Pr2 0:25 > < 0:002 1 þ 1:25 106 ðS=d3 Þ6:37 dd2 Re2 Pr 2 ; Prw 3 Nu2 ¼ h i 0:35 0:25 > 0:9 0:43 Pr 2 d2 > > Re0:8 ; > 0:015 1 þ 5:39ðS=d3 Þ 2 Pr 2 d3 Prw > > > h i > 0:35 0:25 > 3:04 0:43 Pr2 : d2 0:015 1 þ 182ðS=d3 Þ Re0:8 ; 2 Pr 2 d3 Prw
2 103 6 Re2 6 Re2;cr ; 3:08 6 S=d3 6 4:92 2 103 6 Re2 6 Re2;cr ; S=d3 P 4:92 Re2;cr 6 Re2 6 4 104 ; 3:08 6 S=d3 6 4:92 Re2;cr 6 Re2 6 4 104 ; S=d3 P 4:92
ð20Þ
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Fig. 5. Flow chart of the calculation for Ppump.
a0 ¼ 4:6q0:7 p0:2
where Re2 is the Reynolds number of the secondary fluid, Pr 2 is the Prandtl number of secondary fluid, Prw is the Prandtl number of secondary fluid at the temperature of the wall, q is the heat flux, p is the pressure Re2;cr is the critical Reynolds number that describes the transition of the laminar flow to the turbulent flow,
8
1:3 0:54 > S > > 1 þ dr2s Re0:55 < 3:5 1 þ 30 rs 1
f1 ¼ 0:46 3 > 3 > > 1 þ dr2s Re0:25 : 0:316 1 þ 1:2 10 rSs 1
0
ð21Þ Re2;cr
10:3
d3 d2 B C ¼ 18; 500@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 S =4 þ ðd3 d2 Þ
ð22Þ
2.4.2. The friction factors The friction factors of the primary fluid in the helical tube are obtained by formula (23) [10]
2 103 6 Re1 6 Re1;cr ð23Þ Re1;cr 6 Re1 6 8 104
X. Wei et al. / International Journal of Heat and Mass Transfer 94 (2016) 211–221
where rs is helical radius of the inner helical tube, as show in Fig. 3 (d), and Re1,cr is the critical Reynolds number of the helical tube which is computed from:
Re1;cr ¼ 22; 000
0:3 S rs
ð24Þ
The friction factor f3 which indicates the friction pressure drop of the fluid in the shell side is computed by Blausius and modified by the Sieder–Tate formula respectively. When the secondary fluid is single-phase, the friction factor of the secondary fluid can be calculated as follow [10].
8 h i 0:64 0:25 1þ0:92ðS=d Þ0:3 ½ 3 1:6 d2 > > 0:5 1 þ 28ðS=d3 Þ Re2 ; > d3 > > > h i 0:64 > 1:2 > > 0:25½1þ4:5ðS=d3 Þ < 0:5 1 þ 706ðS=d3 Þ3:6 d2 Re2 ; d3 f2 ¼ h i 0:46 > > 2 d2 > Re0:25 ; > 2 > 0:316 1 þ 10ðS=d3 Þ d3 > > > > 0:46 : 0:316½1 þ 12:6ðS=d Þ2:1 ðd2 Þ Re0:25 ; 3
d3
2
qG
Re2 ¼ 2 103 Re2;cr ; S=d3 P 4:92
4 X An xn
ð25Þ
Re2 ¼ Re2;cr 4 104 ; S=d3 ¼ 3:08 4:92 Re2 ¼ Re2;cr 4 104 ; S=d3 P 4:92
ð26Þ
where Dp0 is the single-phase frictional pressure drop of the saturated water at the temperature of boiling region, and can be calculated according to formula (25), qL, qG are the density of the saturated water and the saturated steam at the temperature of boiling region respectively, x is the quality of the secondary fluid, which is mixture of water and steam, w is the non-uniform coefficient, which calculates as formula (27)
w ¼ A0 þ
in the two-level optimization method was a decomposition of the design task into subtasks performed independently, see Fig. 6, and a system-level giving rise to a two-level optimization. In general, decomposition was motivated by the obvious need to distribute work over many people and computers to compress the task calendar time. An equally important benefit from the decomposition is granting an autonomy to the groups of engineers responsible for each particular subtask in choosing their methods and tools for the subtask execution. It has been studied by some researchers, Bartheley [26] used the problem matrix method to describe the relationship between the
Re2 ¼ 2 103 Re2;cr ; S=d3 ¼ 3:08 4:92
While in the boiling region, the frictional pressure drop is obtained from the following formula:
q Dp2 ¼ w 1 þ x L 1 Dp0
217
ð27Þ
n¼1
where Ai (i = 0, 1, 2, 3, 4) are constants, if P = 4 MPa, A0 = 1, A1 = 2.28, A2 = 5.91, A3 = 7.94, A4 = 3.94; if P = 6 MPa, A0 = 1.32, A1 = 5.3, A2 = 24.1, A3 = 35.1, A4 = 16.8. 3. Two-level optimization In order to simplify and accelerate the calculation process, a two-level method is used to solve the problem. The key concept
objective functions and variables. Haftka [27] investigated two important problems in multilevel optimization: decomposition and co-ordination. In [28], the multilevel genetic algorithm (MLGA) was proposed and an actively controlled tower building subjected to earthquake excitations was considered to investigate the effectiveness of MLGA. The objective of this study contains the optimum structure of DOTSG for the single tube unit, the optimum arrangement of tube units in DOTSG and the optimum arrangement of DOTSGs in RPV. For the purpose of achieving these three goals, two sub-objectives are to minimize the volume of RPV and to minimize the total pumping power needed, as shown in formula (28). Design parameters or decision variables for the optimization process are the pitch of the inner helical tube, the central distance of the neighboring tube units, the number of DOTSGs in the RPV, and the number of tube units along the diagonal in a single DOTSG.
Min V R
ðSj ; pt Þ
Min Ppump
ðNsg ; N d Þ
ð28Þ
In order to simplify and accelerate the calculation process, a two-level method is used to solve the problem. As shown in Fig. 7, the problem which is solved by the two-level optimization method in this study is decomposed into one bottom level and one top level. The bottom level is the structure optimization of
Fig. 6. Scheme of the two-level optimization method.
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T ¼ ½V 11 V 21 V N1 V 12 V 22 V N2 V 13 V 23 V N3 V 14 V 24 V N4 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Sc
Sb
Sh
ð30Þ
pt
Following the initialization of chromosome, a decoding program converts each vector to the input file of a code such as pt in order to calculate the fitness function (Ppump). This procedure is based on the constraint conditions which provide the upper and lower boundaries of the decision variables. As the restriction of manufacturing technology and the range of experiment parameters, the pitch should in the range of 40–120 mm. The calculation region of helical pitches are set as formula (31).
8 > < 40 < Sc < 120 40 < Sb < 120 > : 40 < Sh < 120
ð31Þ
The pt should be larger than the outside diameter of the outer tube, the upper boundary denoted as ptmax is depended on the relationship between the reactor core and steam generator. As shown in Fig. 3(a) the DOTSGs are arranged around the reactor core, the relationship of a and N sg can be described as formula (32).
2a 6
2p Nsg
where
a ¼ arcsin Fig. 7. Structure and arrangement optimization of DOTSG by two-level optimization method.
DOTSG, the objective is minimize the pumping power, and the decision variables are the pitch distribution of inner helical tube (Sj ) and the central distance of the neighboring tube units (pt ). As the optimization objective is to find an optimum combination of Sj and pt of the tube unit, the genetic algorithm method (GA) is used in the bottom level. While in the top level, the objective is the arrangement optimization of DOTSG in RPV according the optimization results of bottom level, the objective is minimize the volume of RPV, and the decision variables are the number of tube units in a single DOTSG and the number of DOTSGs in the RPV. A normal constrained non-linear optimization method is used in top level. The key advantage of this two-level optimization is the interactive between the two levels, which can simplify and accelerate the calculation process. In the bottom level, the calculation needs the parameters of N sg and N d from the top level. While in the top level, the calculation needs the parameters of Lsg and pt from the bottom level. In other words, the interactive is an iterated process between the top level and the bottom level.
ð32Þ
Dsg Dsg þ Dc þ 2Dd
ð33Þ
substitute formula (33) into formula (32), the maximum reasonable pt, which is noted as ptmax, can be obtained. In this way, the constraint of the objective in bottom level, which shown as formula (34).
8 40 < Sc < 120 > > > < 40 < S < 120 b s:t: > 40 < Sh < 120 > > : d4 < pt 6 pt;max
ð34Þ
Considering the constraints, the decode calculation can be shown as formula (35).
8 N X > > > V i1 ð2i1 Þ 12040 > Sc ¼ 40 þ > 210 1 > > i¼1 > > > > N X > > > > Sb ¼ 40 þ V i2 ð2i1 Þ 12040 > 210 1 < i¼1
N > X > > > S ¼ 40 þ V i3 ð2i1 Þ 12040 > h > 210 1 > > i¼1 > > > > N X > > p d4 > V i4 ð2i1 Þ t;max > pt ¼ d4 þ : 210 1
ð35Þ
i¼1
3.1. The bottom level optimization
3.2. The top level optimization
Aiming at obtaining the lowest pumping power Ppump (considered as the objective function in this study), the helical pitches (Sc, Sb, Sh) of the inner tube and the central distance of outer tubes (pt) are regarded as the decision variables in the lower level optimization. The problem can be described as formula (29).
The DOTSGs are used to take the energy out of the reactor core, however the total the pressure vessel volume of an iPWR should be paid special attention so that it could not only save the space but also arrange enough heat transfer tubes to guarantee the safe operation. In this study, N sg and N d are considered as the decision variables in the top level optimization, and VR which could be calculated by formula (1) is optimized as the objective. Therefore, the objective and constrains in top level can be described as formula (36) and (37). Then the problem can be solved by using the fmincon routine based on sequential quadratic programming (SQP) algorithm in the optimization toolbox of Matlab.
Min V RPV ðSc ; Sb ; Sh ; Pt Þ
ð29Þ
The optimization problem is optimized by GA in this level. The decision variables are encoded into a chromosome which consists of sub regions that imply the decision variables respectively, as shown in formula (30).
X. Wei et al. / International Journal of Heat and Mass Transfer 94 (2016) 211–221 Table 1 Major parameters of the reactor core. Reactor power (MW) Primary side Operating pressure (MPa) Cooled/heated temperature (°C) Coolant mass flow rate (t/h) Core height (m) Core diameter (m)
38 15.41 245.0/331.27 289.98 0.568 0.694
Secondary side Operating pressure (MPa) Inlet/outlet temperature (°C)
4 106/280
219
of Lsg, so that the VR is reduced. The minimum VR is 5:1273 m3 when Nsg and Nd are 13 and 8, with the corresponding lowest Ppump equals to 915.87 W. To consider the RPV Volume and pumping power at the same time, formula (38) is introduced to obtain the final optimization results.
F ¼ c1
VR P pump þ c2 V0 P0
ð38Þ
where c1 and c2 are two weighting coefficients, as VR is regarded more important, c1 should be a larger value 0.75, while c2 equals to 0.25. V0 and P0 are normative quantities to make VR and Ppump dimensionless. In this study, the values of VR and Ppump when Nsg equals to 15 and Nd is 5 are chosen as the V0 and P0
V 0 ¼ 5:4776 P0 ¼ 6204:7
ð39Þ
The final calculation results of F with different Nsg and Nd are listed in Table 2. As shown in Table 2, F gets a minimum value 0.70364, so the combination which Nsg is 12 and Nd equals to 9 is regarded as the final optimization. The optimum VR and Ppump are 5:1415 m3 and 648.51 W respectively. 4. Discussion
Fig. 8. The relationship of VR, Nsg, and Nd.
Min V R ðNsg ; Nd Þ
ð36Þ
Nsg 2 INT s:t: Nd 2 INT
ð37Þ
When a combination of N sg and N d is given, the Gi (i = 1, 2, 3), Dsg, and ptmax could be obtained according to a certain reactor core using formula (6)–(8), formula (2) and formula (32). Then the bottom level optimization based on the GA is performed to get the optimum S and pt, at the same time, Lsg which is a variable in computing the top level object VR through formula (1) is obtained. 3.3. Optimization results Take a tight-lattice integral reactor as the computing core [29], the design parameters are listed in Table 1. In the bottom level, the optimization structure of the tube unit is can be obtained by GA with different interactive parameters from top level. Nsg and Nd are mainly related to the flow rate of every tube unit. When Nsg and Nd are 12 and 9, the optimum S and pt are Sc = 59.6 mm, Sb = 114.5 mm, Sh = 64.4 mm, and pt = 18.1 mm. As a result, the interactive parameters from bottom to top level are Lsg = 1.06 m, and pt = 18.1 mm. Actually, the top level optimization is a geometry relationship of RPV among VR, Nsg and Nd. The VR with different Nsg and Nd are shown in Fig. 8. If Nsg is constant, VR will firstly decrease and then increase with the increasing of Nd, in other words, there is an optimum Nd for every Nsg. On the other hand, when a single steam generator is determined (Nd is changeless), the VR is reduced when more steam generators are arranged around the reactor core, because the addition of steam generator will lead to the decrease
The temperature distributions of fluid in the inner helical tube, secondary side, and the shell side are shown in Fig. 9, on which the arrows and dashed lines denote the flow direction of each fluid and the boundaries of three regions respectively. As little heat is transferred between the two primary fluids and the secondary superheated steam in the superheated region, T1 and T2 are nearly the same until the end of boiling region (namely the start of superheated region) after 0.0789 m. Lasting about 0.6945 m, T1 is higher and higher than T3 in the boiling region because the heat transfer area between secondary and outer tube fluid is larger than that in the inner tube. In this region, T2 equals to the saturation temperature which decreases a little from the downward to the upward of the tubes as a result of the pressure decreasing. The both primary fluids go on to flow to the bottom, the difference of T1 and T3 is reduced gradually due to the heat exchange between themselves. The three optimal pitches vary in different heat transfer region. This is mainly because the effects of the helical tube to the heat transfer capacity and pressure drop are different in the three heat transfer region. As stated before, the inner helical tube can increase the heat transfer surface area and add the turbulence of the fluids. These effects should be strong when the pitch is small, and vice versa. From Eq. (18) we can obtain that increase the fi, wi, and Lsg can all result in the Ppump increasing. When the pitch is small, fi is big and the tube length is short, on the other hand if the pitch is large the friction factor is small but the tube length will be long. Thus there should be some optimum values of S, making the Ppump minimum. As the boiling heat transfer is stronger than that between single-phase fluids in sub-cooled and superheated region and the effects of Sb to the tube length is weak, besides, f2 of the boiling fluid may be bigger, the optimum Sb is larger than Sc and Sh. In addition, the flow rate in superheated region is higher than that in sub-cooled region, so Sc could be a little less than Sh without causing larger P pump . As stated before, the inner helical tube can increase the heat transfer surface area and add the turbulence of the fluids. But in boiling, the turbulence of the mixture of water and steam is already strong. In the nature of things, the strong turbulence brings the big pressure drop which affects the. So the pressure drop is bigger than that in sub-cooled region and superheated region. As the
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Table 2 The value of F with different Nsg and Nd. Nd
5
6
7
8
9
10
11
Nsg 3 4 5 6 7 8 9 10 11 12 13 14 15
– – – – – – – – – – – 1.0799 1
– – – – – – 1.2181 1.1195 1.0117 0.94223 0.88607 0.8363 0.79409
– – – 1.4188 1.2225 1.0944 1.0025 0.92693 0.86366 0.82282 0.77841 0.74625 –
– 1.7675 1.3911 1.2017 1.0609 0.97209 0.90508 0.84868 0.80854 0.7663 0.71135 – –
1.7806 1.5017 1.232 1.0861 0.99338 0.91599 0.85907 0.81746 0.78308 0.70364 – – –
1.5891 1.319 1.1574 1.0306 0.95183 0.88661 0.85411 0.81014 0.73819 – – – –
1.4625 1.3274 1.1041 1.0139 0.94103 0.89863 0.85852 0.81809 – – – – –
12
13
14
15
16
17
18
1.6438 1.2314 1.0948 1.0056 0.94747 0.90226 0.86929
1.3866 1.2216 1.133 1.0395 0.96903 0.92612 0.8582
1.5931 1.2198 1.1249 1.0519 0.99279 0.96453 –
1.3992 1.2298 1.1346 1.07 0.99115 0.97668 –
1.4075 1.2565 1.1667 1.0793 1.0194 – –
1.4298 1.2935 1.2072 1.0866 1.066 – –
1.4773 1.316 1.2037 1.1596 1.1131 – –
3 4 5 6 7 8 9
from the differences of heat transfer capacity and flow state in different region. Smaller pitch takes better heat transfer, but at the same time, the pressure drop will be increased. The boiling heat transfer is stronger than that between single-phase fluids, this means the effects of Sb to heat tube length is weak, and so the optimum Sb is the largest. In addition, Sc could be a little less than Sh because of the flow rate which affects the Ppump is higher in superheated region. As to the arrangement of the arrangement of DOTSGs in the RPV, it is a geometry problem to obtain a smaller volume of RPV. It is clear that a small Nd gives a small diameter of single DOTSG, and result a big Nsg for a given number of tube units. The result shows that the minimum VR is 5:1273 m3 when Nsg and Nd are 13 and 8. However, the corresponding minimum Ppump is large (915.87 W), so an evaluation formula with two weighting coefficients is introduced to adjust the conflict between VR and Ppump. As VR is considered more important than Ppump, 12 and 9 are chosen as the final optimum Nsg and Nd, making the VR and Ppump equals to 5:1415 m3 and 648.51 W respectively, with the corresponding bottom level optimization shows that the parameters are Sc = 59.6 mm, Sb = 114.5 mm, Sh = 64.4 mm, pt = 18.1 mm, and Lsg = 1.06 m. Fig. 9. The axial temperature distribution of DOTSG.
narrow distribution of the tube length and the wide distribution of the pressure drop, the bigger pitch is reasonable in boiling region.
5. Conclusion A two-level optimization method is applied to optimize DOTSG with outer straight tube and inner helical tube. In the bottom level optimization, after studying the flow performances of DOTSG, the numerical calculation model is established. Ppump is then minimized by GA algorithm, considering Sh, Sb, Sc, and pt as the decision variables in the bottom level with the interactive parameters from the top level. Based on the Lsg and pt obtained in the bottom level optimization, different combinations of Nsg and Nd which determine the arrangement of DOTSG are regarded as the variables to compute the VR in the top level optimization. While a normal constrained non-linear optimization method is used in top level. The optimum pitch varies in three heat transfer regions mainly result
Acknowledgment This research is supported by National Natural Science Foundation of China (11405125), China Postdoctoral Science Foundation Fund Project (2014M562420), and the Fundamental Research Funds for the Central Universities.
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