Multi-objective optimization of geometric parameters for the helically coiled tube using Markowitz optimization theory

Journal Pre-proof Multi-objective optimization of geometric parameters for the helically coiled tube using Markowitz optimization theory

Yong HAN, Xue-sheng WANG, Zhao ZHANG, Hao-nan ZHANG PII:

S0360-5442(19)32262-5

DOI:

https://doi.org/10.1016/j.energy.2019.116567

Reference:

EGY 116567

To appear in:

Energy

Received Date:

13 June 2019

Accepted Date:

16 November 2019

Please cite this article as: Yong HAN, Xue-sheng WANG, Zhao ZHANG, Hao-nan ZHANG, Multiobjective optimization of geometric parameters for the helically coiled tube using Markowitz optimization theory, Energy (2019), https://doi.org/10.1016/j.energy.2019.116567

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Journal Pre-proof Multi-objective optimization of geometric parameters for the helically coiled tube using Markowitz optimization theory Yong HAN, Xue-sheng WANG*, Zhao ZHANG, Hao-nan ZHANG Key Laboratory of Pressure System and Safety, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China.

Abstract: In this research, a novel multi-objective optimization of helically coiled tube (HCT) using Markowitz effective boundary theory was studied. Firstly, the expressions of modified entropy generation number (EGN, Ns,c, Ns,p, Ns) for the HCT was derived. Secondly, the mathematical relation between the heat transfer EGN (Ns,c) and NTU was validated. Then, a novel method to acquire the effective boundary was proposed. Finally, the results of Markowitz optimization were compared with MOGA. The results show that, in comparison between the 2 optimization methods, relative error of heat transfer coefficient using both optimization methods are below ±2%; as for the flow resistance, the relative error of Markowitz effective boundary is still below ±2%, the relative error of MOGA is more than ±2%; there are 3 optimal points in each optimization; when heat transfer coefficient (h) and pressure drop (|∆p|) are selected as the objective functions, the PEC of Markowitz optimization increases by 1.63%, 3.36% and 0.4%, respectively; when heat transfer coefficient (h), pressure drop (|∆p|) and total EGN (Ns) are selected as the objective functions, the PEC of Markowitz optimization increases by 3.25%, 25.47% and 21.97%, respectively. Therefore, the Markowitz optimization is more effective than the MOGA of ANSYS. Keywords: The helically coiled tube (HCT); Multi-objective optimization; Markowitz optimization; Effective boundary; Entropy generation number

1. Introduction Due to advantages of compactness in structure, ease of manufacture and high heat transfer efficiency, helically coiled tube heat exchangers (HCTHXs) are utilized extensively in many industrial applications and processes, such as food, nuclear, aerospace, refrigeration, power generation, heat recovery industries, space heating and air-conditioning processes. Helically coiled tubes (HCTs) are the most important component of the HCTHXs. Laminar flow heat transfer of the HCT was experimentally studied by Janssen and Hoogendoorn [1], who proposed the Nu correlations of the laminar flow with a certain application range. The estimated accuracy of the local heat transfer coefficients was 10-15%. The accuracy of the overall heat-transfer coefficients was estimated to be 10-20%. Ito [2] firstly defined the critical Nusselt number (Recr) to distinguish the flow state in the HCT, and he also proposed the correlation to predict the flow resistance performances when the Reynolds number (Re) was less than Recr. Schmidt [3] experimentally studied the heat transfer and flow resistance performances of the HCT. The flow state was divided into 3 regimes: the laminar flow regime (100 ≤ Re ≤ Recr), the turbulent flow regime (Recr < Re ≤ 22000) and the highly turbulent flow regime (20000 ≤ Re ≤ 150000). Austen and Soliman [4] studied flow resistance and heat transfer of laminar flow of the HCT with substantially different pitch ratio using water as the working fluid, who also proposed the correlation to predict the flow resistance of the HCT with a good agreement(±20%). Pawar et al.[5, 6] experimentally studied the heat transfer performances of the HCT under isothermal steady and non-isothermal unsteady states with Newtonian and non-Newtonian fluids in laminar and turbulent regimes, where the heat transfer correlations have reliability margins within ±10%. The HCT with spherical corrugation was proposed by Zhang et al. [7], who numerically investigated heat transfer and flow resistance. the augmentation on heat transfer performance is about 1.05–1.7 times as compared to the smooth HCT, while friction factor sharply increases 1.01–1.24 times. Beigzadeh and Rahimi [8] brought in the Adaptive NeuroFuzzy Inference System and genetic algorithm methods to modify correlations to predict the heat transfer and flow resistance in the HCT with geometrical parameters (coil diameter and pitch). The mean relative errors (MRE) of the developed ANFIS models for estimation of Nu and f are 6.24% and 3.54%, respectively. for empirical correlations, MRE of 8.06% was found for prediction Nu while MRE of 5.03% was obtained for f. Moawed [9] experimentally studied the forced convection from HCT with different parameters (Re, Dco/di and P/di), which have important effects on the heat transfer coefficient. Moawed also proposed the correlation to predict the Nusselt number. The maximum deviation between the experimental data and the correlation is ±10%. With the rapid development of the heat transfer and thermodynamic theories, the mechanism of the HCTHXs illustrated by the thermodynamic laws causes great interests of the scientists. The application of the entropy theory *

Corresponding author: Xuesheng WANG.

Tel.: +86 21 64253105.

E-mail address: [email protected].

Journal Pre-proof in heat transfer and thermal design was firstly described in detail by Bejan [10, 11], who also proposed the EGM method (entropy generation minimization method), and utilized it to a straight tube with smooth surface [12]. Ko and Ting [13-15] reported the optimized Reynolds number for the rectangular curved tubes based on the EGM method. They also employed the CFD method to investigate the entropy generation of the curved tubes and those with ribs [16, 17]. Satapathy [18] proposed a theoretical method to optimize the heat load and structure of the HCTHXs for both laminar and turbulent flow regimes using the EGM method. Shokouhmand et al. [19, 20] utilized the EGM method on fully developed laminar forced convection in HCTHX with uniform wall temperature. It was revealed that the optimum Reynolds numbers decreased as the curvature ratio increased except in the low ranges of curvature ratios. And the optimum Reynolds number was acquired under certain conditions. Ahadi and Abbassi [21] investigated effects of length and heat flux of the helical coils on entropy generation rates and optimal state of operation analytically. They found that the entropy generation rates increase with augmentation in combined length, heat flux and inlet temperature. Ali Abdous et al. [22] studied the entropy generation process in helically coiled concentric tube-in-tube heat exchanger with HFC-134a under flow boiling condition. It was obtained that only the tube and coil diameters can reach the optimum under certain conditions. Also, the effects of the velocity, the inlet vapor quality, saturation temperature, and the heat flux on the entropy generation were obtained; however, there was no optimum. Shi[23] combined the dimensionless entropy generation of the rotating HCTHXs with the multi-objective optimization simultaneously, where the results showed that the combined entropy generation therefore can go through a decreasing-increasing procedure yielding a local minimum with all dimensionless parameters with the certain B0, Q and Pr. Based on the above researches, the HCTHXs with large heat transfer coefficient and small irreversible loss would have large flow resistance. The optimization design of the HCTHX is highly conflicting multi-objective. Thus, the optimization theory provides a more effective methodology for the heat exchangers. The multi-objective genetic algorithm (MOGA) has been widely utilized in the multi-objective optimization, which enables researchers to find the extreme values of objective function accurately through iterations. The optimization for the geometric parameters of the shell side of the HCTHX using multi-objective genetic algorithm was investigated by Wang et al.[24, 25], who also utilized the entransy-dissipation-based thermal resistance as the objective function for the HCTHX optimization. Compared with original structure, the comprehensive performance evaluation factor (Nu/f1/3) of traditional optimal results was improved by an average of 41.02%, while that of optimal structures obtained from entransy theory was strengthened by an average of 76.64%. Wen et al. [26] carried out the optimization study on shell-side performances of a shell and tube heat exchanger with staggered baffles. The performance was influenced by structural parameters, especially helix angle and overlapped degree. Using the MOGA optimization, Wang [27] studied multi-objective of shell side of the HCTHX with maximum performance evaluation criterion (PEC) and maximum field synergy number (Fc) as the objective functions. Compared with the initial structure under the same working condition, the Fc based on the field synergy principle (FSP) indicated that the comprehensive performance was improved by 36.84%, and PEC showed 33.42% improvement in the comprehensive performance. Liu et al.[28] utilized the MOGA optimization to optimize the design of plate-fin heat exchanger, where the Colburn factor (j) and the friction factor (f) were selected as objective functions. The optimization results indicated that the Colburn factor (j) increased by 12.83% and the friction factor (f) decreased by 26.91%. The application of MOGA optimization shows the intelligence and efficiency of the optimization theory. The optimal solution of the MOGA optimization is obtained by solving Pareto front through iterations. Pareto front is the distribution of optimal solutions of multi-objective optimization [29]. There are two convergence conditions in MOGA optimization [30]: (1) The Allowable Pareto Percentage represents a specified ratio of Pareto points per number of samples per iteration; when the Maximum Allowable Pareto Percentage criterion is reached, the optimization is converged. (2) The Convergence Stability Percentage criterion looks for population stability; when a population is stable with regards to the previous one, the optimization is converged. However, the population of the MOGA is generated by multi-level iterations, where the crossover and mutation could cause the difference between the predicted and calculated results. This difference would affect the accuracy of MOGA optimization. Khan et al.[31] proposed a technique, named Adaptive Simulated Annealing algorithms, to improve the robustness of population based algorithms. Can other optimization methods be better than the MOGA? For this purpose, Markowitz effective boundary theory is brought in. The original Markowitz effective boundary theory describes the relationship between 2

Journal Pre-proof expected return and investment risk, that is, gaining maximum expected return with minimum investment risk [32]. In essence, Markowitz effective boundary theory is an optimization problem of finding the optimal value of two-objective functions under constraints. Nowadays, Markowitz optimization has been extended. A new analytical approach combining Markowitz model with Euclidean vector spaces was proposed by Rambaud et al. [33], which gave us the enlightenment to acquire the effective boundary by calculating the Euclidean distance. Stempien et al. [34] utilized the Markowitz’s model to analyze energy trilemma (energy security, energy sustainability and energy equity), which proves the Markowitz’s model could solve the optimization issues with more than two objective functions. Georgalos et al. [35] utilized Markowitz model to explain risky choice of heterogeneity of preferences in experimental research. Algarvio et al. [36] investigated the risk management and the optimization of the portfolios in liberalized electricity markets using Markowitz’s model. However, the Markowitz optimization has not been utilized in optimization design of heat exchanger yet. In this study, the Markowitz effective boundary theory is brought in for the optimization design of the HCTHXs. As for the heat exchanger, it is aimed to find an optimized structure which has the maximum heat transfer coefficient, minimum flow resistance and minimum irreversible loss. Thus, the heat transfer coefficient can be seen as the expected return, while the flow resistance and irreversible loss can be seen as the investment risk. To acquire the effective boundary, a new method that can transfer the multi-objective optimization into the calculation of minimum Euclidean distance was proposed. The Markowitz optimization is solved by calculating the minimum Euclidean distance. The main characteristics are that: (1) the calculation is to obtain the effective boundary, not the Pareto front; (2) the calculation is to do the reversible mathematical transformation, not the crossover and mutation calculation; (3) there is no iteration. Thus, this calculation is not complicated, and it is also easy to implement. Consequently, to investigate whether the multi-objective Markowitz optimization is suitable for optimization design of HCTHXs, the multi-objective Markowitz optimization for the HCT is studied. The main contents are as follows: (1) the modified entropy generation numbers (EGN, Ns,c, Ns,p, Ns) of the HCT was derived; (2) the mathematical relation between the heat transfer EGN (Ns,c) and NTU was validated; (3) the multi-objective Markowitz optimization for the HCT was investigated; (4) the optimization results were compared with MOGA optimization.

2. Calculation model of the HCT 2.1. Geometric model

(b)

(a) Fig.1.

Geometric model of the HCT

Table. 1. Geometric parameters of the HCT Parameters Symbol Value Inner diameter di 5 ~ 7.5 (mm) Coil diameter Dco 58~110 (mm) Coil pitch P 17~110 (mm) Longitudinal length E 150~500 (mm)

The geometric structure of the HCTs is shown in Fig.1. The main geometric parameters are as follows: tube diameter (di), coil diameter (Dco), coil pitch (P) and number of coils (N). Through an in-depth study, the helically winding angle (ε), the longitudinal length (E) and the total length of the HCT (Lco) can be expressed as: 3

Journal Pre-proof 12

  atan  P  Dco  , Lco  N   Dco   P 2    2

 E sin    , N  E P

(1)

2.2. Physical model Some assumptions are employed as follows: (1) the physical properties of working fluid is determined by ANSYS FLUENT; (2) the gravity of the fluid is neglected; (3) thermal radiation and natural convection are overlooked; (4) water serves as working fluid and is considered incompressible, steady state, homogeneous and neglect of viscous heating; (5) according to the previous researches (Ref.[2, 3]), the heat transfer process is at turbulent regime ( Re  20000 ~ 30000 ). To make the numerical simulation close to the actual engineering application, the convection boundary (or the third boundary condition) is adopted, which is set as the constant main stream temperature and the constant heat transfer coefficient. The HCTs are applied in the highly efficient condenser; water serves as the refrigerant in the tube-side, while the condensation is ongoing outside the wall. The main boundary conditions are as follows: (1) The inlet is set as the velocity inlet ( uin  Re      di  , Re  20000 ~ 30000 ). (2) The outlet is set as the pressure outlet (the gauge pressure of the outlet is set as: pout = 0 Pa). (3) The HCTs are applied in the highly efficient condenser; the equivalent mean temperature [37] of the mainstream outside the wall is set as Ta,o  330 K ( Ta,o  Tout,o  Tin,o  ln Tout,o Tin,o  ); also, the recommendation of condensation in Ref. [38] is 2000 ~ 6000 W∙m-2∙K-1; thus, the convection heat transfer coefficient of the mainstream outside wall is set as ho  5000 W  m 2  K -1 (4) The thickness of the wall is neglected ( Rwall  0 ).

2.3. CFD model validation The governing equations of the single-phase homogeneous flow are: Continuous equation,

  u j  x j

Momentum equation,

   ui u j  x j

Energy equation,



p   x j x j

   u jT  x j



 x j

0

(2)

  u u j   i    x j xi  k  T     c  x j

    

(3)

    

(4)

SIMPLE algorithm is applied for the pressure-velocity coupling. Also, second-order upwind scheme is used to solve the momentum and energy equations. Furthermore, PRESTO scheme is adopted for the pressure correction equation. The convergence criteria are set to 10−8 for energy equation and 10−5 for other equations. Having tried to present a regular plan to step forward, the validation of the turbulent model is carried out upon the comparison of two important factors, Nusselt number (Nu) and friction factor (f), which are: Nu 

h  di



, f 

2di  p

(5)

 u 2 Lco

200

0.034

180

0.032 0.030

160

f

Nu

0.028

140

0.026 0.024

120

0.022

100 0

5

10

15

20

Mesh quantity ×10-5

(a) Nu

25

30

0.020

0

5

10

15

20

Mesh quantity ×10-5

25

30

(b) f

4

Journal Pre-proof Fig 2. Mesh independent validation Table. 2 The empirical correlation compared in this research Expression Applied range

Nu  0.359 Re0.781 0.933 0.172 Pr 0.016

Beigzadeh correlation

150 140

4000  Re  48000

f  2.32 Re 0.311 0.467 0.074 0.045

Beigzadeh correlation k-ε standard k-ε realizable Reynolds stress

0.040

Beigzadeh correlation k-ε standard k-ε realizable Reynolds stress

f

Nu

130 120 110

0.035

0.030

100

0.025

20000 22000 24000 26000 28000 30000

(a) Variation trend of Nu

135

k-ε standard k-ε realizable Reynolds stress

(b) Variation trend of f 5.0

+10%

-10%

120

105

90 90

MRE = 5.71%, RMSE = 8.30 MRE = 6.42%, RMSE = 8.76 MRE = 6.85%, RMSE = 9.04

105

120

135

150

f-Numerical value (×10-2)

Nu-Numerical value

150

20000 22000 24000 26000 28000 30000

Re

Re

4.5

k-ε standard k-ε realizable Reynolds stress

+20%

4.0 3.5

-20%

3.0 MRE = 11.98%, RMSE = 0.0042 MRE = 14.04%, RMSE = 0.0050 MRE = 14.70%, RMSE = 0.0053

2.5 2.0 2.0

2.5

3.0

3.5

4.0

4.5

Nu-Beigzadeh correlation f-Beigzadeh correlation (×10-2) (c) Error of Nu (d) Error of f Fig.3. Turbulent model validation

5.0

Nonuniform mesh strategy with mesh refinement near the wall is used, which is shown in Fig.1(b). According Ref. [39], there is relation as y+ ≥ 15 for the standard wall function. Through the calculation, the first layer of the grid is smaller than 0.15 mm. Thus, to satisfy the standard wall function, the first layer of boundary grid is set as: 0.1 mm, 0.07 mm, 0.05 mm, 0.02 mm and 0.01 mm. The growth ratio is set as 1.2 for the near-wall mesh, which grows slowly equal to the dimension of the grids in main flow region (4 mm, 3 mm and 2 mm). Consequently, the total mesh quantity can be acquired as: 400000, 700000, 1000000, 1609000 and 3009000. Mesh independent validation is shown in Fig.2, where the values of Nu and f at a good stability for the mesh quantity of 1609000 and 3009000 with the maximum difference (-1.04%). Hence, the case with 1609000 grids is selected for calculations. MRE 

t y 1  i t i , RMSE  n i

1 2   ti  yi  n

(6)

According the recommendation in Ref.[40], 3 CFD models (k-ε standard, k-ε realizable and Reynolds stress) are studied, which are shown in Fig.3. Since the applied range and the geometric parameters of the HCT are close to Beigzadeh’s research, the Beigzadeh correlation (shown in Table.2) is employed for the validation [8]. In addition, Mean Relative Errors (MRE) and Root Mean Square Error (RMSE) (shown as Eq.(6)) are employed to evaluate the accuracy of different turbulent models, which is shown in Fig.3. The k-ε standard model should be adopted, which has the minimum MRE (Nu, 5.71%; f, 11.98%) and RMSE (Nu,8.30; f, 0.0042). There are further two additional equations as:

t  k       kui        Gk  Gb  YM  Sk  xi x j   k  x j        ui      t xi x j  

    2  S   C1  Gk  G3 Gb   C2  k k  x j 

(7) (8)

The turbulent viscosity, t , is computed by combining k and ε as follows: 5

Journal Pre-proof t   C   k 2  

(9)

Where C1 , C2 , C ,  k and   are constants which have the following values [39]:

C1  1.44 , C2  1.92 , C  0.09 ,  k  1.0 ,    1.3

(10)

3. Criterions for the HCT 3.1. Heat transfer coefficient and pressure drop It is well known that the traditional criterions for heat exchanger are the overall heat transfer coefficient ( K ) and the pressure drop ( p ) [41]. The overall heat transfer coefficient ( K ) can be seen as the criterion for heat transfer performance, which is, K  Q ATln

(11)

Q is the heat transfer rate in the HCT; Tln is the logarithmic mean temperature difference. The expressions are as:

Q  mc Tout  Tin  , Tln 

T  T   T ln T  T  T a,o

a,o

out

out

a,o a,o

 Tin 

 Tin  

(12)

Tout is the outlet temperature of the HCT; Tin is the inlet temperature of the HCT; Ta,o is the main stream

temperature outside the HCT (also, the equivalent mean temperature outside the HCT). The convection wall boundary is utilized, but the wall thickness is neglected. Then, the overall heat transfer coefficient ( K ) is: (13) 1 K  1 h  1 ho

h is the heat transfer coefficient in the HCT; ho is the heat transfer coefficient outside the HCT. Since ho is set as constant in the physical model, the heat transfer coefficient in the HCT ( h ) can represent the overall heat transfer coefficient ( K ), which can be expressed as: h  1 K  1 ho 

1

(14)

The pressure drop ( p ) can be seen as the criterion for flow resistance, which can be expressed as:

p  pout  pin

(15)

pin and pout are inlet and outlet pressure of the HCT, respectively.

3.2. Modified entropy generation number 3.2.1. Expression of entropy generation rate for the internal flow The internal flow of the HCT is shown in Fig.4. The entropy generation rate in the HCT can be expressed as [12]:

 Sg  mdsC   Q T  T   mdsp

(16)

 Sg   Sg,c   Sg,p

(17)

Fig.4. Internal flow of the HCT

The internal flow with a heat source in the HCT can be seen as an open system with a heat source but no 6

Journal Pre-proof output power. The thermodynamic mathematical relations of this system is [42]: di  TdsC  vdp , v  1 

(18)

di is the enthalpy change; dsC is the entropy change rate of heat transfer; dp is the pressure difference; The general expression of the enthalpy change ( di ) is [43]:

  v   di  cdT   v  T    dp   T  p   The expression of the enthalpy change is [43]: di  cdT  vdp

(19)

(20)

The entropy change rate of heat transfer ( dsC ) is [43]: cdT (21) T Assuming the fluid flowing through an arbitrary cross section, the flow is caused solely by the pressure difference between the two points along the fluid path. If the enthalpy change contribution to heat transfer can be neglected (steady and adiabatic flow), there is relation as [44]: (22) 0  di  Tdsp  vdp dsC 

dsp is entropy change rate of flow resistance. Here the entropy change rate of flow resistance can be seen as

the irreversible loss of the flow resistance, which can be expressed as [44]: vdp dp dsp    T T

(23)

Therefore, entropy generation rate can be obtained as:

 Sg  m  Sg,c  m

cdT Q mdp   T T  T  T

cdT Q mdp  ,  Sg,p   T T   T T  

(24) (25)

 Sg is the total entropy generation rate;  Sg,c is the entropy generation rate of heat transfer;  Sg,p is the entropy generation rate of flow resistance. The energy conservation of the heat transfer process can be expressed as [43]:  Q  KAT  dEU  pdv  mcdT

(26)

EU is the internal energy; K is the overall heat transfer coefficient; A is the heat transfer area. Substituting

Eq.(26) into Eq.(25), yields,

 Sg,c  m

cdT T T T  T 

(27)

The term T T can be negligible as compared to unity, it can be simplified as [12]:

 Sg,c  m

cdT T T T  T 

(28)

With Eq.(26), the temperature difference between the wall and the stream temperature ( T ) can be derived as T  mcdT  K  dA  . Then, 2

1  mcdT     K  dA Also, the entropy generation rate can be expressed as:

 Sg,c    T

Sg,c   Sc   2

1 m p , Sg,p  KA Ta

(29)

(30)

The heat transfer entropy generation rate is not only corelated with entropy change rate of heat transfer, but 7

Journal Pre-proof also is correlated with the overall heat transfer coefficient and the heat transfer area. Ta is the equivalent mean temperature of working fluid, which is [37]:

Ta 

Tout  Tin ln Tout Tin 

There are mathematical expressions as: mc Tout  Tin  T Q , Sc  mc ln out , p  pin  pout KA   Tin Tln Tln

(31)

(32)

With Eq.(32), it can be obtained as: 2

 T  Tln m Sg  mc  ln out    p T T  T  Ta  in  out in

(33)

2

 T  m Tln , Sg,p  p Sg,c  mc  ln out   Ta  Tin  Tout  Tin

(34)

3.2.2. Expression of modified entropy generation number Bejan obtained the entropy generation number by dividing minimum heat capacity rate mcmin, which is Sg Ns  (35) mcmin However, the entropy generation number in Eq.(35) is not suitable for assessing the heat transfer performance. This is because the Ns proposed by Bejan would lead to the phenomenon of ‘entropy generation paradox’ [45]. Hesselgreaves proposed the modified entropy generation number as [46]:

N s,c 

Tin Sg Q



Tin Sg cm Tout  Tin

(36)

However, there is still not sufficient to consider the ambient temperature ( T0 ) in Eq.(36). Another expression of modified entropy generation numbers (EGNs) are proposed as: Sg Sg,c Sg,p Ns  , N s,c  , N s,p  Sc Sc Sc N s  N s,c  N s,p

(37) (38)

Where Ns is the total EGN of the heat transfer process; Ns,c is the heat transfer EGN caused by heat transfer; Ns,p is the flow resistance EGN caused by flow resistance. Energy is divided into two parts, one can be utilized, defined as exergy; the other cannot be utilized, defined as anergy [47].The exergy that transformed into anergy is defined as irreversible loss [47]. If T0 is the ambient temperature, the mathematical expressions of anergy (An), exergy (Ex) and irreversible loss (or exergy loss, Ig) can be shown as follows:

An  T0 S , Ex  E  T0 S , I g  T0 Sg

(39)

With Eq.(37) ~(39), it can be obtained as:

Ns 

Sg Sc



Ig An

(40)

The total EGN (Ns) can express the proportion of the irreversible loss in the anergy. When Ns becomes larger, the proportion of irreversible loss becomes larger. The entropy generation numbers (EGNs) can also be selected as the performance criteria.

4. Markowitz optimization method 4.1. General expression As is depicted above, Markowitz effective boundary theory is an optimization problem of finding the optimal value of two-objective functions under constraint design space. In this research, the application of Markowitz 8

Journal Pre-proof effective boundary is extended into m objective functions. It is assumed that the independent variable is a ndimensional vector x =  x1 , x2 , , xn  , the definition interval is as x j   L, j ,  H, j  ( j  1, 2, , n ). The quantity T

of objective functions is m, which can be expressed as yi ( yi  0 , i  1, 2, , m ). Thus, with the regression, the mathematical expression can be obtained as follows:

 

yi  yi xT , i  1, 2, , m

(41)

Firstly, the maximum and minimum of the objective function should be calculated as:

   ,

yi ,max  max yi xT

  

yi ,min  min yi xT

(42)

Then, the boundary of objective function can be acquired as:

y max   y1,max , y2,max , , ym,max  , y min   y1,min , y2,min , , ym,min 

(43)

If the optimization is to get the maximum, a normalized function as i    yi  ( i  1, 2, , m ) is brought to obtain the normalization of yi , which is, i    yi  

yi  yi ,max yi ,max  yi ,min

, i  1, 2, , m

(44)

The normalized interval can be derived as i   1, 0 . When yi  yi ,max , there is relation as i  0 ; when

yi  yi ,min ,there is relation as i =  1 . If the optimization is to get the minimum, an intermediate function as yˆi   yi ( yˆi  0 , i  1, 2, , m ) is brought to do the transformation:

 

yˆi   yi   yi xT

(45)

The boundary of objective function can be acquired as:

yˆ max   yˆ1,max , yˆ 2,max , , yˆ m,max    y1,min ,  y2,min , ,  ym,min  yˆ min   yˆ1,min , yˆ 2,min , , yˆ m,min    y1,max ,  y2,max , ,  ym,max 

(46)

Also, a normalized function as i ( i  1, 2, , m ) is brought to obtain the normalization of yˆi , which is: i  i  yˆi    i   i   yi  

yˆi  yˆi ,max yˆi ,max  yˆi ,min



yi ,min  yi yi ,max  yi ,min

, i  1, 2, , m

(47)

The normalized interval also can be obtained as i   1, 0 . When yˆi  yˆi,max , there is relation as yi  yi ,min ,

 i  0 ; when yˆi  yˆi,min ,there is relation as yi  yi ,max , i  1 . Consequently, the distribution of objective functions can be transformed into a m-dimensional normalized distribution as   1 , 2 , , m  1  i  0, i  1, 2, , m  . The optimized objective function can be seen as an objective point, obj   0, 0, , 0  . Thus, the Markowitz effective boundary is the nearest boundary of the normalized distribution to the objective point. Therefore, the multi-objective optimization can be stated that, to find an optimized point ( opt ) which has the minimum Euclidian distance to the objective point ( obj ).The multiobjective optimization function (Euclidian distance) can be expressed as: EDmin

1  m  2  2   min   2   min   i   , i  1, 2, , m  i 1    

(48)

m

Furthermore, the weighted factors ( ai , 0  ai  1 ,  ai  1 , i  1, 2, , m ) is defined to distinguish the i 1

importance of different objective functions. Then, the multi-objective optimization function can be obtained as:

EDw,min

1  m  m 2 2   2 2  min   ai  i    or EDw,min  min  ai  i   , i  1, 2, , m    i 1   i 1  

(49)

9

Journal Pre-proof In general, the general Markowitz optimization can be expressed as:  m 2 2 2 min EDw : EDw  min   ai  i   , i  1    m  ai   0,1 ,  ai  1, i  1, 2, , m i 1  Subject to:  x =  x1 , x2 , , xn T , x j   L, j ,  H, j  , j  1, 2, , n   yi  yi  xT  , yi   yi ,min , yi ,max  , i  1, 2, , m  yi  yi ,max  , i   1, 0 i    yi   yi ,max  yi ,min 

(50)

4.2. Application in the optimization of the HCT 4.2.1. Description of the optimization The optimization design for heat exchanger is gaining maximum heat transfer coefficient with minimum pressure drop. In addition, the minimum total EGN always corresponds to the best energy utilization [37]. Thus, the multi-objective optimization for geometric parameters of the HCT under constant Reynolds number can be formally written as follows: Maximize : h  h  di , Dco , P, E   Minimize : p  p  p  di , Dco , P, E   Minimize : N s  N s  di , Dco , P, E   Subject to: d   d , d  , D  D , ,D  i  i,min i,max  co  co,max co,min   P   Pmin , Pmax  , E   Emin , Emax  

(51)

4.2.2. Genetic Aggregation regression To investigate the relationship between the objective functions and the input parameters, the Genetic Aggregation regression [41] would be implemented before optimization calculation. Firstly, the Optimal SpaceFilling Distributive Entropy of Euclidian Distance methodology [48-50] is used to acquire the design points of the input parameters. 100 design points in total were generated and calculated using ANSYS FLUENT. Re is set as: Re = 25000, The definition intervals of the geometric parameters are as: di  5(mm), 7.5(mm)  , Dco  58(mm),110(mm)  , E  150(mm),500(mm)  , P  17(mm),110(mm)  (52) Then, the Genetic Aggregation regression is employed to obtain the objective regression functions. The Genetic Aggregation regression includes both the interpolation and the regression methodologies. Through the mutation and cross-over interpolation, the few design points would grow into hundreds. Finally, the least square method is used to obtain the objective functions. The input parameters vector of the HCT is x =  di , Dco , P, E  . T

Therefore, heat transfer coefficient (h), flow resistance (|∆p|) and the total EGN (Ns) can be obtained as:

 

 

 

h  h xT , p  p  p xT , N s  N s xT

(53)

The Genetic Aggregation is employed to obtain the objective regression functions in the form of Eq.(53). Fig.5 shows the validity between regression-predicted and simulated data. Turning to Fig.5 (a), it reveals a comparison between the calculated and regression-predicted values of h. With a similar approach, the validation of flow resistance (|∆p|) is illustrated in Fig.5 (b); while the validation of the total ENG (Ns) is illustrated in Fig.5(c). It indicates that the predicted values for all design points are close to the calculated values, where acceptable difference in error (less than ±5%) for all the regression functions proves the validity.

10

18

0.11

360

+5%

15

-5% 12

9 9

12

15

18

h-predicted (kW·m-2·K-1) (a) h

+5%

300

-5%

240

-5%

0.09

180 120

0.08

60 0

+5%

0.10

Ns-simulated

|Δp|-simulated (kPa)

h-simulated (kW·m-2·K-1)

Journal Pre-proof

0.07 0

60

120

180

240

300

360

|Δp|-predicted (kPa) (b) |∆p| Fig.5. The validity of the regression functions

0.07

0.08

0.09

Ns-predicted (c) Ns

0.10

0.11

4.2.3. Optimization method The boundary of each objective function under definition design space should be calculated as:

h   hmin , hmax  , p   pmin , pmax  , N s   N s,min , N s,max 

(54)

The normalized function as h  h  h  is brought to obtain the normalization of heat transfer coefficient, which is

h  h  h  

h  hmax hmax  hmin

(55)

There are additional 2 normalized functions are brought to obtain the normalizations for pressure drop (|Δp|) and the total EGN (Ns), which are N s,min  N s p p ,  Ns   Ns  N s   (56)  p   p  p   min N s,max  N s,min pmax  pmin





It can be obtained as h , p , Ns   1, 0 .When h  hmax , there is relation as h  0 ; when h  hmin ,there is relation as h  1 . When p  pmin , there is relation as p  0 ; when p  pmax , there is relation as p  1 . When N s  N s,max , there is relation as Ns  1 ; when N s  N s,min , there is relation as Ns  0 .

Fig.6. The projection of distribution for normalized objective functions

Fig.6 shows the projection of distribution of normalized objective functions. Region Φ is the normalized A distribution region; A (0,0) is the objective point; the red boundary (Curve CAD ) is the Markowitz effective A boundary; the optimized point Gopt ( h ,opt , p ,opt ) is on the Markowitz effective boundary (Curve CAD ). The final

multi-objective optimization of h, |∆p| and Ns for the HCT becomes that solving the minimum Euclidian distance between the optimized point G ( h ,opt , p ,opt ) and the objective point A (0,0). Consequently, the multi-objective optimization can be stated that, to find an optimized point Gopt ( h ,opt ,

 p ,opt , Ns ,opt ), which has the minimum Euclidian distance to the objective point A ( h,max , p ,max , Ns ,max ) ( h ,max   p ,max   Ns ,max  0 ). The Euclidian distance can be calculated as:



ED  h 2   p 2   N s 2



1 2

or ED 2  h 2   p 2   N s 2

(57) 11

Journal Pre-proof The Markowitz optimization of geometric parameters for the HCT can be expressed as: Minimize : ED 2   h 2   p 2   N s 2  Subject to:  N s,min  p  h  h  hmax , p  pmin  p , N  s  hmax  hmin pmax  pmin N s,max  N s,min  T T T h  h  x  , p  p  x  , Ns  Ns  x   T x =  di , Dco , P, E   di   di,min , di,max  , Dco   Dco,max , Dco,min  ,   P   Pmin , Pmax  , E   Emin , Emax  

(58)

START Generate design point population x = (di, P, Dco, E)T Calculate values of objective functions

Using Genetic Aggregation regression to acquire the mathematical expression of the objective functions h = h(xT), |∆p| = pt(xT), |Ns| = Ns(xT) Get the maximum and minimum values of the objective functions (hmin, hmax), (pt,min, pt,max), (ps,min, ps,max) The objective functions are transferred into the standard objective functions (φh,min, φh,max), (ψp,min, ψp,max), (ψNs,min, ψNs,max) Finding the minimum Euclidian distance between optimized point Gopt (φh,opt, ψp,opt, ψNs,opt) and objective point A (φh,max, ψp,min, ψNs,min) = (0,0,0) EDmin = (φh2+ψp2+ψNs2)½ or ED2min = φh2+ψp2+ψNs2

Acquire the distribution corresponding to the minimum Euclidian distance

Plot the distribution of the Markowitz effective boundary

Finding the minimum Euclidian distance between objective point (0,0,0) and optimized points on the Markowitz effective boundary

Acquire the final optimization results

END

Fig.7. Workflow of the Markowitz optimization

Based on the above discussion, the Markowitz optimization of heat transfer coefficient (h), flow resistance (|∆p|) and the entropy generation number (Ns) for the HCT is preliminary established. The workflow of the Markowitz optimization is shown in Fig.7. This optimization is implemented by Python language. The steps involved are described as follows: a) Generate the design points (100 design points); calculate the objective functions (h, |∆p| and Ns) corresponding to design space, x =  di , Dco , P, E  . T

12

Journal Pre-proof b) c) d)

Utilize Genetic Aggregation regression to acquire the mathematical expression of the objective functions; get the maximum and minimum of the objective functions; the objective functions are transferred into the normalized objective functions. Solve the normalized objective functions; plot the objective point A (0,0,0); calculating the minimum Euclidian distance between the point Gopt (φopt, ψp,opt, ψNs,opt) and objective point A(0,0,0) (Eq.(57)); the point set (Gopt) constructs the Markowitz effective boundary. Comparing the Euclidian distance between objective point A (0,0,0) and the points on the Markowitz effective boundary; then, the final result can be acquired.

5. Results and discussion 5.1. Variations of output parameters versus input parameters 5.1.1. Variations of the heat transfer EGN (Ns,c), h and NTU The variations of h versus Dco, P and E are shown in Fig.8. The main cause of the heat transfer enhancement in the HCT is the centrifugal force. When Dco or P increases, the curvature of the HCT decreases. Then, the centrifugal force decreases, which causes that the flow state in the HCT is close to the straight tube. Thus, as is shown in Fig.8 (a) and (b), h decreases as Dco or P increases. The total length of the HCT and heat transfer coefficient are independent of each other. When other geometric parameters (di, Dco and P) are set as constants, the total length of the HCT (ΔLco) has no effect on the heat transfer coefficient, which means there is no relation between E and h. In Fig.8 (c), when P =63.5 mm, Dco = 84 mm, h maintains at a constant. In general, P and Dco have negative effects on heat transfer enhancement, but E has no effect on heat transfer enhancement. The variations of Ns,c versus Dco, P and E are shown in Fig.9. The HCT with small coil pitch (P), large coil diameter (Dco) and large longitude length (E) has a large heat transfer area, and the large outlet temperature. As is shown in Eq.(34), the heat transfer EGN (Ns,c) is negatively corelated with the outlet temperature. Thus, In Fig.9 (a), Ns,c decreases as Dco increases. Fig.9 (c) shows that Ns,c decreases as E increases. Inversely, Fig.9 (b) shows that Ns,c increases as P increases. From the variation of the heat transfer coefficient (h) and the heat transfer EGN (Ns,c), the heat transfer EGN (Ns,c) does not always decrease as the heat transfer coefficient (h) increases. Therefore, the heat transfer EGN cannot fully represent the heat transfer enhancement of the HCT.

16 14 12 10

14

60

70

80

90

Dco (mm) (a) E = 325 mm

11

0.08

Ns,c

0.07 P = 17 mm P = 40.25 mm P = 63.5 mm P = 86.75 mm P = 110 mm

0.06 0.05 0.04 60

70

80

90

Dco (mm) (a) E = 325 mm

12 10 8 6 4 2

0

20

40

60

80

100

120

Dco =58 mm Dco =84 mm Dco =110 mm

Dco =71 mm Dco =97 mm

100 150 200 250 300 350 400 450 500 550

P (mm) E(mm) (b) Dco = 84 mm (c) P = 63.5 mm Fig.8. When Re =25000, di = 6.5 mm, the variation of h versus Dco, E and P

0.09

Ns,c

12

9

100 110 120

0.10

0.03 50

13

14

10

8 6 50

16 E = 150 mm E = 237.5 mm E = 325 mm E = 412.5 mm E = 500 mm

0.10

0.100

0.09

0.095 0.090

0.08 E = 150 mm E = 237.5 mm E = 325 mm E = 412.5 mm E = 500 mm

0.07 0.06 100 110 120

0.05

Ns,c

h (kW·m-2·K-1)

18

15

h (kW·m-2·K-1)

P = 17 mm P = 40.25 mm P = 63.5 mm P = 86.75 mm P = 110 mm

20

h (kW·m-2·K-1)

22

0

20

40

60

80

100

0.085 0.080 0.075

120

Dco =58 mm Dco =71 mm Dco =84 mm Dco =97 mm Dco =110 mm

0.070 100 150 200 250 300 350 400 450 500 550

P (mm) E (mm) (b) Dco = 84 mm (c) P = 63.5 mm Fig.9. When Re =25000, di = 6.5 mm, the variation of Ns,p versus Dco, E and P

13

Journal Pre-proof Dco =58 mm Dco =84 mm Dco =110 mm

1.0

0.8

NTU

NTU

E = 150 mm E = 237.5 mm E = 325 mm E = 412.5 mm E = 500 mm

1.0

0.8 0.6

0.4

0.2

0.2

60

70

80

90

Dco (mm)

(a) E = 325 mm

100

110

0.4 0.3

0.6

0.4

0.0 50

0.5

1.2

Dco =71 mm Dco =97 mm

NTU

1.2

0.0

Dco =58 mm Dco =71 mm Dco =84 mm Dco =97 mm Dco =110 mm

0.2 0.1

0

20

40

60

P (mm)

80

100

120

0.0 100 150 200 250 300 350 400 450 500 550

E (mm)

(b) Dco = 84 mm (c) P = 63.5 mm Fig.10. When Re =25000, di = 6.5 mm, the variation of Ns,p versus Dco, E and P

Furthermore, the convection boundary condition is utilized in this study. The heat transfer capacity of the heat transfer process (Number of Heat Transfer Unit, NTU) of a single HCT can be obtained. The mathematical expression of the NTU is as follows: (59) NTU  KA mc It can be observed that NTU is the dimensionless quantity of (KA), which can represent the heat transfer capacity of the heat transfer process. The NTU contains not only the heat transfer coefficient, but also the heat transfer area. The variations of NTU versus Dco, P and E are shown in Fig.10. Compared with the heat transfer ENG (Fig.9), the NTU has just the opposite tendency. NTU increases as Dco increases, which is also affected by P. NTU also increases as E increases. Inversely, NTU decreases as P increases. It can be found that the heat transfer EGN (Ns,c) is perfectly corresponding with the NTU. In other words, the heat transfer EGN (Ns,c) decreases as the NTU increases. Therefore, the heat transfer EGN may represent the heat transfer capacity of the HCT. Also, there must be some relation between the heat transfer EGN and the NTU, which is derived in the following part. 5.1.2. Mathematical relation between the heat transfer EGN (Ns,c) and NTU Based on the above discussion, the heat transfer rate in the HCT can be expressed as follows: Q  KATln  cm Tout  Tin 

(60)

Also, there is relation as: Tln   T1  T2  ln  T1 T2  , T1  Ta,o  Tin , T2  Ta,o  Tout , T1  T2  Tout  Tin 

(61)

Substituting Eq.(61) into Eq.(60), yields, KA ln  T1 T2   cm

(62)

And it can be obtained that,

T2  T1 exp  KA cm or Tout  Ta,o  Ta,o  Tin  exp  KA cm

From Eq.(25), Eq.(30) and Eq.(37), the heat transfer EGN can be expressed as:



ln Tout Tin  ln Ta,o  T1 exp  NTU  Tin 1   KA NTU NTU Some dimensionless intermediate quantities are brought in, which are, N s,c   Sc  

 a,o  Ta,o Tin ,  out  Tout Tin , T T1  Ta,o  T1  T1   a ,o  1



(63)

(64)

(65)

Therefore, the mathematical relation between the heat transfer EGN (Ns,c) and NTU can be expressed as: N s,c 

ln  a,o   a,o  1 exp  NTU 

(66) NTU It can be acquired the variation of the heat transfer EGN (Ns,c) along with the NTU, which is shown in Fig.11. Fig.11(a) shows the theoretical distribution at different τa,o. The possibly maximum heat transfer ENG is up to τa,o With the increase of τa,o, it would result in the increase of the possibly maximum heat transfer ENG. When Ta,o=330 K, Tin=300 K, ho=5000 W∙m-2∙K-1, the theoretical distribution and the calculated results from the numerical simulation are shown in Fig.11(b). There are 100 calculated results from the numerical simulation. And all the calculated results are distributed along the theoretical line, which means there is a regular correspondence between the heat transfer ENG and the NTU. In general, the heat transfer ENG cannot fully represent the enhancement of 14

Journal Pre-proof the convection heat transfer in the HCT, but it can represent the heat transfer capacity of the heat transfer process in the HCT. 0.10

0.40

τa,o =1.1 τa,o =1.2 τa,o =1.3 τa,o =1.4

0.35 0.30

0.06

Ns,c

Ns,c

0.25

0.08

0.20

0.04

0.15 0.10

Theoretical distribution Calculated results

0.02

0.05 0.00

0

2

4

6

8

0.00 0.0

10

0.2

0.4

0.6

0.8

1.0

NTU NTU (b) (a) Fig. 11. The heat transfer EGN (Ns,c) along with NTU with the convection heat transfer boundary condition

1.2

5.1.3. Variations of the flow resistance EGN (Ns,p) and |Δp| Fig.12 shows the variations of |Δp| versus Dco, P and E. The HCT with small coil pitch (P), large coil diameter (Dco) and large longitude length (E) has a large total length (Lco), which would result in the large flow resistance (|Δp|). It is known that the main cause of the flow resistance in the HCT is Lco. Lco increases as Dco increases, Lco decreases as P increases. Also, the direct way to extend Lco is to extend E. In Fig.12 (a), when E =325 mm, P = 17 mm, |Δp| increases from 161.57 (kPa) to 247.82 (kPa); but it is affected by the coil pitch (P). In Fig.12 (b), when E = 325 mm, Dco = 84 mm, |Δp| decreases from 210.64 (kPa) to 35.927 (kPa) as P increases from 17 (mm) to 110 (mm). Inversely, in Fig.12 (c), when P =63.5 mm, Dco = 58 mm, |Δp| increases from around 27.056 (kPa) to 85.252 (kPa) as E increases from 150 mm to 500 mm. In general, E and Dco have positive effects on flow resistance, but P has negative effect on flow resistance (|Δp|).

300 250 200 150

150

70

80

90

100

0.0035

20

40

60

80

100

0.0030

0.0032

0.0028 0.0026

0.0028

0.0025 70

80

90

100

110

0.0020

0.0024 0.0022

0.0024 60

120

0.0030

E = 150 mm E = 237.5 mm E = 325 mm E = 412.5 mm E = 500 mm

0.0036

Ns,p

0.0040

0

0.0040

P = 17 mm P = 40.25 mm P = 63.5 mm P = 86.75 mm P = 110 mm

0.0045

0

110

110 Dco =58 mm 100 Dco =71 mm 90 Dco =84 mm 80 Dco =97 mm 70 Dco =110 mm 60 50 40 30 20 10 100 150 200 250 300 350 400 450 500 550

Dco (mm) P (mm) E (mm) (a) E = 325 mm (b) Dco = 84 mm (c) P = 63.5 mm Fig.12. When Re =25000, di = 6.5 mm, the variation of |∆p| versus Dco, E and P

Ns,p

60

0.0050

Ns,p

200

50

50

0.0020 50

250

100

100 0 50

E = 150 mm E = 237.5 mm E = 325 mm E = 412.5 mm E = 500 mm

300

|Δp| (kPa)

350

|Δp| (kPa)

350

P = 17 mm P = 40.25 mm P = 63.5 mm P = 86.75 mm P = 110 mm

400

|Δp| (kPa)

450

0.0020

0

20

40

60

80

100

120

Dco =58 mm Dco =71 mm Dco =84 mm Dco =97 mm Dco =110 mm

0.0018 100 150 200 250 300 350 400 450 500 550

Dco (mm) E (mm) P (mm) (a) E = 325 mm (b) Dco = 84 mm (c) P = 63.5 mm Fig.13. When Re =25000, di = 6.5 mm, the variation of Ns,p versus Dco, E and P

Fig.13 shows the variations of Ns,p versus Dco, P and E. In Fig.13 (a), when E =325 mm, P = 63.5 mm, Ns,p decreases from more than 0.0027 to 0.002527 as Dco increases from 58 to 110 mm; but it is affected by P. In Fig.13 (b), firstly, Ns,p decreases dramatically as P increases, when P > 60 mm, the tendency becomes smooth. Inversely, In Fig.13 (c), when P =63.5 mm, Dco = 84 mm, Ns,p increases from 0.00246 to 0.0027 as E increases from 150 mm to 500 mm. In general, E have positive effects on flow resistance, but P and Dco have negative effect 15

Journal Pre-proof on Δp|. From the variation of|Δp| and Ns,p, it can be found that Ns,p does not always correspond to |Δp|. Thus, Ns,p cannot fully represent the flow resistance performance of the HCT. In other words, the flow resistance can be divided into 2 parts, the necessary consumed power maintaining the fluid flow and the irreversible loss, respectively.

5.2. Optimization results with h and |∆p| as objective functions In this section, h and |∆p| are selected as the objective functions. A weighted factor (a) is brought to distinguish the importance of objective functions. Furthermore, Re is set as a constant (Re = 25000). This optimization problem can be expressed as: Minimize : ED 2  a h 2  1  a  p 2  Subject to:  h  hmax pmin  p  h  , p  hmax  hmin pmax  pmin   T T h  h  x  , p  p  x   T x =  di , Dco , P, E    di  5(mm), 7.5(mm)  , Dco  58(mm),110(mm)  ,  P  17(mm),110(mm) , E  150(mm),500(mm)      

(67)

To validate the applicability of Markowitz effective boundary, the optimization results are compared with the MOGA optimization with the same input parameters. The settings of MOGA are as follows: Pareto percentage is 70%; Convergence Stability Percentage is 0.02; the number of design points in population of each iteration is 100. 180 Markowitz effective boundary Pareto front (MOGA)

160

|Δp| (kPa)

140 120 100 80 60 40 20 0 12

13

14

15

16

17

18

19

h (kW·m-2·K-1) Fig. 14. Distribution of Markowitz effective boundary with h and |∆p| as objective functions when Re = 25000

Fig.14 shows the effective boundary with h and |∆p|. Also, the Pareto front calculated by MOGA optimization is shown in Fig.14. The effective boundary and Pareto front have the same distribution and same variation trend. The difference is that the best result of Pareto front is at around h=17.5 (kW∙m-2∙K-1), |∆p|=60 (kPa), but the best result of effective boundary is at around h=18 kW∙m-2∙K-1, |∆p|=60 kPa. The predicted optimization results of MOGA and Markowitz effective boundary are shown in Table.4. MOGA optimization gives 3 candidate points (A1, A2 and A3). According to the different weighted factors, there are 3 candidate points (A4, A5 and A6) given by Markowitz optimization. The predicted Markowitz optimization results are almost the same as that of MOGA optimization. The distribution of Markowitz effective boundary is consistent with Pareto front. Table.4. Geometrical optimization results of MOGA and Markowitz effective boundary h (kW∙m-2∙K-1) MOGA optimization Optimal point A1 Optimal point A2 Optimal point A3 Markowitz optimization Optimal point A4 (a = 0.95) Optimal point A5 (a = 0.7) Optimal point A6 (a = 0.5) Optimal point A7 (a = 0.3)

|∆p| (kPa)

di (mm)

Dco (mm)

P (mm)

E (mm)

17.594 16.516 15.533

58.946 43.825 28.498

7.29 7.15 7.39

58.31 58.64 58.20

17.14 27.01 39.59

152.16 151.32 152

17.998 16.609 15.576 14.607

60.433 38.261 27.055 19.459

7.40 7.40 7.40 7.40

58.00 58.00 58.00 58.00

17.00 27.11 39.39 57.22

150 150 150 150

16

Journal Pre-proof The predicted optimization results with 2 objective functions are validated by numerical simulation, which is shown in Table.5. The comprehensive performance evaluation factor (PEC, Nu/f1/3) is brought to evaluate the comprehensive performances. The predicted heat transfer coefficient is almost the same as that of the numerical simulation. The relative error of MOGA is -0.02%~-1.84%; while the relative error of Markowitz effective boundary is -0.01%~0.01%. The relative errors of both optimization methods are below ±2%. As for the flow resistance, the relative error of Markowitz optimization is still below ±2% (0.04%~0.89%), but the relative error of MOGA is more than ±2% (3.54%~-7.31%). As for the PEC, the relative error of Markowitz optimization is rather low (-0.02%~-0.30%), but the relative error of MOGA is larger (-1.19%~-2.38%). Thus, the predicted optimization results of Markowitz effective boundary are more accurate than MOGA of ANSYS. Table. 5. Validity of predicted optimization results o MOGA and Markowitz effective boundary with 2 objective functions h (kW∙m-2∙K-1)

Nu/f1/3

Simulated

Relative error

Predicted

Simulated

Relative error

Predicted

Simulated

Relative error

17.594 16.516 15.533

17.917 16.520 15.523

-1.84% -0.02% 0.06%

58.946 43.825 28.498

63.255 42.272 27.135

-7.31% 3.54% 4.78%

627.10 555.56 539.01

612.54 562.27 547.88

2.38% -1.19% -1.62%

17.998 16.609 15.576

17.997 16.610 15.575

0.01% -0.01% 0.01%

60.433 38.261 27.055

59.410 38.093 26.813

0.04% 0.44% 0.89%

622.45 580.35 548.38

622.55 581.18 550.05

-0.02% -0.14% -0.30%

Predicted MOGA optimization Optimal point A1 Optimal point A2 Optimal point A3 Markowitz optimization Optimal point A4 (a = 0.95) Optimal point A5 (a = 0.7) Optimal point A6 (a = 0.5)

|∆p| (kPa)

In the comparison of optimization results (A1 and A4, A2 and A5, A3 and A6), the optimization results of Markowitz effective boundary are superior than those of MOGA. The PEC increases by 1.63%, 3.36% and 0.4%, respectively; the flow resistance decreases by 6.08%, 9.89% and 1.19%, respectively.

5.3. Optimization results with h, |∆p| and Ns as objective functions In this section, h, |∆p| and Ns are selected as the objective functions. Three weighted factors (b1, b2 and b3, b1  b 2  b3  1 ) are brought to distinguish the importance of objective functions. Also, when Re is set as a constant (Re = 25000), this optimization problem can be expressed as: Minimize : ED 2  b1   h 2  b 2  p 2  b3  N s 2  b1  b 2  b3  1 Subject to:  N s,min  p  h  hmax pmin  p , p  , Ns   h  hmax  hmin pmax  pmin N s,max  N s,min   h  h  xT  , p  p  xT  , N s  N s  xT   x =  d , D , P, E T i co  di  5(mm), 7.5(mm)  , Dco  58(mm),110(mm)  ,   P  17(mm),110(mm)  , E  150(mm),500(mm) 

(68)

The optimization results are still compared with the MOGA optimization with the same input parameters by ANSYS workbench. The settings of MOGA are still the same as Section 5.2. Fig.15 shows the Markowitz effective boundary with h, |∆p| and Ns. Also, the Pareto front calculated by MOGA optimization is shown in Fig.15. Most points on Pareto front surface coincide with the effective boundary. The difference is that the distribution range of the effective boundary is larger than the Pareto front of MOGA in ANSYS. The predicted optimization results of MOGA and Markowitz optimization are shown in Table.6. The comprehensive performance evaluation criteria factor (PEC, Nu/f1/3) is also brought to evaluate comprehensive performances. MOGA optimization gives 3 candidate points (B1, B2 and B3). According to the different weighted factors, there are 3 candidate points (B4, B5 and B6) given by Markowitz optimization. When it comes to 3 objective functions, the EGNs (0.07084~0.09022) predicted by Markowitz optimization are almost the same as those of MOGA (0.07157~0.09252). However, the heat transfer coefficient and the flow resistance of MOGA are very different from those of Markowitz optimization. The heat transfer coefficients predicted by Markowitz optimization (17.724 ~ 17.806 kW∙m-2∙K-1) are larger than those of MOGA (15.105 ~ 17.867 kW∙m-2∙K-1); the flow resistances predicted by Markowitz effective boundary (56.722~201.18 kPa) are smaller than those of MOGA (101.75~227.97 kPa). 17

Journal Pre-proof The predicted optimization results with 3 objective functions are validated by numerical simulation, which is shown in Table.7. Similar to Section 5.2, the predicted heat transfer coefficient is almost the same as that of the numerical simulation. The relative error of MOGA is -0.25%~-1.88%; while the relative error of Markowitz optimization is -0.01%~0.01%. The relative error of heat transfer coefficient using each optimization method is below ±2%. As for the flow resistance, the relative error of MOGA is more than ±2% (-2.39%~-7.41%); while the relative error of Markowitz optimization is still below ±2% (-0.05%~0.57%), Thus, the predicted optimization results of Markowitz optimization are more accurate than MOGA of ANSYS.

Fig. 15. Distribution of Markowitz effective boundary with h, |∆p| and Ns as objective functions when Re = 25000

Table.6. Geometrical optimization results of MOGA and Markowitz effective boundary h (W∙m-2∙K-1) MOGA optimization Optimal point B1 Optimal point B2 Optimal point B3 Markowitz optimization Optimal point B4 (b1=0.7, b2=0.2, b3=0.1) Optimal point B5 (b1=0.9, b2=0.09, b3=0.01) Optimal point B6 (b1=0.7, b2=0.1, b3=0.2)

|∆p| (kPa)

Ns

di (mm)

Dco (mm)

P (mm)

E (mm)

16.626 17.867 15.105

101.75 116.4 227.97

0.08090 0.09252 0.07157

7.48 5.51 6.59

59.73 58.26 71.98

23.21 17.47 17.85

374.11 151.75 435.25

17.724

99.579

0.08233

7.30

59.25

17.0

242.80

17.806

56.722

0.09022

7.40

58.00

18.09

150

17.740

201.18

0.07084

7.20

59.25

17.0

480.33

Table.7. Optimization results of MOGA and Markowitz effective boundary with 3 objective functions h (W∙m-2∙K-1) Predicted MOGA optimization Optimal point B1 Optimal point B2 Optimal point B3 Markowitz optimization Optimal point B4 (b1=0.7, b2=0.2, b3=0.1) Optimal point B5 (b1=0.9, b2=0.09, b3=0.01) Optimal point B6 (b1=0.7, b2=0.1, b3=0.2)

Simulated

|∆p| (kPa) Relative error -0.25% -1.88% -1.02%

Nu/f1/3

Ns

Predicted

Simulated

Relative error

Predicted

Simulated

Relative error

Predicted

Simulated

Relative error

101.75 116.4 227.97

104.24 125.72 243.56

-2.39% -7.41% -6.40%

593.40 504.42 503.98

588.62 491.63 492.98

0.81% 2.60% 2.23%

0.08090 0.09252 0.07157

0.08089 0.09181 0.07024

0.01% 0.77% 1.89%

16.626 17.867 15.105

16.669 18.209 15.260

17.724

17.725

-0.01%

99.579

100.15

-0.57%

608.97

607.78

0.20%

0.08233

0.08297

-0.77%

17.806

17.805

0.01%

56.722

56.751

-0.05%

616.93

616.87

0.01%

0.09022

0.09004

0.20%

17.740

17.741

-0.01%

201.18

201.50

-0.16%

608.78

607.78

0.16%

0.07084

0.06990

1.34%

In comparison of optimization results (B1 and B4, B2 and B5, B3 and B6), the optimization results of Markowitz effective boundary are superior than those of MOGA. The PEC increases by 3.25%, 25.47% and 21.97%, respectively; the flow resistance decreases by 3.92%, 54.86% and 17.27%, respectively. Therefore, Markowitz effective boundary optimization is more effective than the MOGA of ANSYS.

6. Conclusion In this research, the multi-objective optimization of geometric parameters for the HCT using Markowitz effective boundary theory was investigated. The calculation design model was established. The mathematical relation between the heat transfer EGN and the NTU under the convection heat transfer boundary was derived. 18

Journal Pre-proof The Markowitz optimization was applied to acquire the optimal structure of the HCT under constant Reynolds number, which also was compared with the MOGA optimization. The main conclusions are as follows: (1) The distribution of Markowitz effective boundary is consistent with Pareto front. With the same regression function and the same design points, relative error of heat transfer coefficient using both optimization methods are below ±2%. As for the flow resistance, the relative error of Markowitz optimization is still below ±2%, but the relative error of MOGA is more than ±2%. Therefore, the Markowitz optimization is more accurate than the MOGA of ANSYS. (2) There are 3 optimal points in each optimization; when h and |∆p| are selected as the objective functions, the PEC of Markowitz optimization increases by 1.63%, 3.36% and 0.4%, respectively; when h, |∆p| and Ns are selected as the objective functions, the PEC of Markowitz optimization increases by 3.25%, 25.47% and 21.97%, respectively. Therefore, the Markowitz optimization is more effective than the MOGA of ANSYS. (3) The heat transfer ENG cannot fully represent the convection heat transfer enhancement, but it can represent the heat transfer capacity (NTU). The flow resistance can be divided into 2 parts, the necessary consumed power maintaining the fluid flow and the irreversible loss, respectively. (4) The Markowitz optimization for the HCT proves its applicability in the heat transfer components of the HCTHX. However, it still needs further study to validate the application in the design of the whole heat exchanger or other types.

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Journal Pre-proof

Nomenclature A An c Dco di E Ex f hi ho i Ig K ΔLco m M Ns Ns,c Ns,p Nu P Pe Pr

Area of transversal section, m2 Anergy of the working fluid in the helically coiled tube, W Specific heat of working fluid, J∙kg-1∙K-1 Coil diameter, m The inner diameter of the helically coiled tube, m Longitudinal length of the helically coiled tube, m Exergy of the working fluid in the helically coiled tube, W Friction coefficient

Δp δq

Pressure drop, Pa Heat transfer rate of the control volume, W

Re ΔS δSg

Reynolds number Entropy variation of the helically coiled tube, W∙K-1 Entropy variation of the control volume, W∙K-1

Sg

Total entropy generation of the helically coiled tube, W∙K-1

Sg,c

Convection heat transfer coefficient of the working fluid inside the helically coiled tube, W∙m-2∙K-1 Convection heat transfer coefficient of the working fluid outside the helically coiled tube, W∙m-2∙K-1 Enthalpy of working fluid, J∙kg-1 Exergy loss of the working fluid in the helically coiled tube, W Heat transfer coefficient of the helically coiled tube, W∙m-2∙K-1 Total length of the helically coiled tube, m Mass flow rate of working fluid, kg∙s-1 Quantity of coil turns Entropy Generation Coefficient (EGC) Heat transfer EGC Flow resistance EGC Nusselt number, Coil pitch, m Perimeter of the section of the helically coiled tube, m Prandtl number

Tin

Entropy generation of heat transfer of the helically coiled tube, W∙K-1∙ Entropy generation of flow resistance (pressure drop) of the helically coiled tube, W∙K-1 Inlet temperature, K

Tout

Outlet temperature, K

Ta To T

Equivalent mean temperature of working fluid, K Temperature of main stream outside the helically coiled tube, K Endothermic temperature of working fluid, K

Twall ΔTln

Wall temperature, K Logarithmic mean temperature difference, K

Sg,p

Greek symbol λ 𝜀

Thermal conductivity of working fluid, W∙m-1∙K-1 Winding angle, °

21

Journal Pre-proof

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Journal Pre-proof Highlights: 1.

The multi-objective optimization of Markowitz effective boundary theory for the helically coiled tube (the HCT) was proposed.

2. 3. 4.

The modified entropy generation numbers (EGNs, Ns, Ns,c and Ns,p) of the HCT were derived. The mathematical relation between the convection heat transfer EGN (Ns,c) and NTU was validated. The effective boundary was obtained by solving the minimum Euclidean distance.