Studies on the thermal-hydraulic performance of zigzag channel with supercritical pressure CO2

Accepted Manuscript Title: Studies on the thermal-hydraulic performance of zigzag channel with supercritical pressure CO2 Authors: Haiyan Zhang, Jiangfeng Guo, Xiulan Huai, Keyong Cheng, Xinying Cui PII: DOI: Reference:

S0896-8446(19)30011-7 https://doi.org/10.1016/j.supflu.2019.03.003 SUPFLU 4491

To appear in:

J. of Supercritical Fluids

Received date: Revised date: Accepted date:

7 January 2019 1 March 2019 2 March 2019

Please cite this article as: Zhang H, Guo J, Huai X, Cheng K, Cui X, Studies on the thermal-hydraulic performance of zigzag channel with supercritical pressure CO2 , The Journal of Supercritical Fluids (2019), https://doi.org/10.1016/j.supflu.2019.03.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Studies on the thermal-hydraulic performance of zigzag channel with supercritical pressure CO2

Haiyan Zhang1,2, Jiangfeng Guo1,2, , Xiulan Huai1,2, Keyong Cheng1,2, Xinying Cui1,2 1Institute

of engineering science, University of Chinese Academy of Sciences, Beijing 100049, China

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2School

of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100190, China

Corresponding author. Tel: +86 10 82543035; fax: +86 10 82543033. E-mail address: [email protected] (J. Guo)

Graphical Abstract

0.75

Fc

1.5

1.0

80

100

120

140

160

180

0.00

0.002

A4

A3

A2

A1

0.000

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Highlights

A5

0.001

ED

Bend (degree)

Zigzag channels with bend angles between 100˚ and 130˚ show optimum performance.

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

0.003

0.25

M

0.5 60

0.50

0.004

Reverse flow

N

Bend angle

Bend=115° m=0.1 g/s Cold side

Ns1

(j/j0)/(f/f0)1/3

2.0

Hot — — 0.1 g/s - - - - 0.2 g/s Cold ——— 0.1 g/s - - - - - 0.2 g/s

A

Optimum values

U

1.00

2.5

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*

Field synergy principle could explain flow and heat transfer behavior of SCO2 very well.



Reverse flows near bend corner boost field synergy and reduce entropy generation.

A





Secondary flows improve field synergy but enlarge irreversibility near the wall.

Abstract The numerical investigations of zigzag channel showed that the reduction of the zigzag bend angle 1

improves heat transfer performance but worsens hydraulic performance. The heat transfer enhancement has much to do with the decrease of the entropy generation and the increase of the synergy between the velocity and the temperature gradient. The larger heat effectiveness corresponds to better convective heat transfer effectiveness, and a very small heat capacity ratio

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could lead to slight reduction of the heat effectiveness. The reverse flow boosts the field synergy and diminishes the heat transfer entropy generation, and the secondary flow would improve the

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synergy between the velocity and temperature gradient in zigzag channels as well as enlarges the

entropy generation near the wall. The zigzag channels with the bend angles between 110° to 130°

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show the best comprehensive performance in terms of both the first and second laws of

N

thermodynamics.

A

Keywords: Printed circuit heat exchanger (PCHE); Entropy generation; Field synergy principle;

A

CC E

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ED

M

Thermal-hydraulic performance; Supercritical pressure CO2 (SCO2); Heat transfer.

2

Nomenclature

Q

q Rc Re S g'''

T U U

A

CC E

V

V

W

3

N

U

SC R

β ε λ μ μt ρ Φ

dimensionless volume heat capacity (kW/K) Geek symbols synergy angle between velocity and temperature gradient (°) synergy angle between velocity and velocity gradient (°) heat effectiveness thermal conductivity [W/(m K)] dynamic viscosity (Pa s) turbulence viscosity (Pa s) density (kg/m3) energy dissipation due to viscosity (W/m2) dimensionless inlet section length Subscripts bulk cold hot tensor indices inlet mean the nth sub heat exchanger outlet solid x, y, z directions

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α

χ

A

P Pr

M

Nu Ns1

ED

m

heat transfer area (mm2) specific heat at constant pressure [kJ/(kg K)] hydraulic diameter (mm) Euler number Fanning friction factor field synergy number heat transfer coefficient [kW/(m2 K)] specific enthalpy (kJ/kg) Colburn j factor the inlet length of the flow (mm) flow length (mm) mass flow rate (g/s) Nusselt number dimensionless entropy generation number pressure (MPa) Prandtl number heat transfer rate (kW) heat flux (kW/m2) heat capacity ratio Reynolds number volumetric entropy generation rate [W/(m3 K)] temperature (K) velocity (m/s) dimensionless velocity volume (mm3)

PT

A cp D Eu f Fc h H j l L

b c h i, j, k in m n out s X, Y, Z

1. Introduction With the intensification of global warming and environmental deterioration, considerable attention has been paid to find clean and efficient energy substitution in recent years. As CO2 is non-toxic, non-flammable, chemically stable and

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environmentally friendly, it has been widely used in many advanced heat transport and energy conversion systems such as high temperature solar power stations, high

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efficiency supercritical CO2 Brayton cycles and innovative trans-critical CO2

refrigeration and heat pump systems [1-3]. In a supercritical pressure CO2 (SCO2)

U

power cycle, the recuperator is one of the most essential components which has

N

important influences on the whole energy transformation efficiency and should be

A

with high effectiveness and compactness. Printed circuit heat exchanger (PCHE) is a

M

novel compact heat exchanger commercially provided by HEATRIC company [4] and

ED

shows superior heat transfer performance to traditional shell and tube heat exchangers. A PCHE is made of diffusion-bonded stacks of plates in which channels are fabricated

PT

by the chemical etching, resulting that the whole exchanger offers almost the same

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strength and properties as the parent metal. During the last two decades, lots of experimental and numerical studies on

thermal-hydraulic performance of PCHEs using CO2 as working fluid have been

A

proposed. Ishizuka et al. [5] and Nikitin et al. [6] experimentally investigated flow and heat transfer characters of zigzag PCHEs, they obtained the data in the CO2-CO2 loops and developed new correlations. Ngo et al. [7] numerically promoted a new PCHE with S-shaped fins for hot water supplier, and then experimentally confirmed 4

that this new shape PCHE provides much lower pressure drop than zigzag channel PCHEs [8]. Tsuzuki et al. [9] evaluated performances of PCHEs with some new configuration

channels

based on simulations.

Afterwards, considering the

performance improvement in terms of the rib shape in PCHEs, Kim et al. [10]

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proposed a new shape PCHE based on NACA airfoil fins which results in great reductions of the pressure drop. Cui et al. [11] further proposed two novel fins based

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on the configuration of NACA 0020 airfoil fins, and their analysis showed that one of

the novel fins has better flow and heat transfer performance than the NACA fins. As

U

for optimization method, Pieve [12] proposed design methods for some compact heat

N

exchangers including the PCHE. Lee and Kim [13] conducted a multi-object

A

optimization with multi-objective genetic algorithm and five optimal designs were

M

selected. They [14] also optimized PCHE zigzag channels based on the exergy

ED

analysis, and the optimal channel showed greater thermal-hydraulic performance with respect to the reference one. Kwon et al. [15] proposed a cost-based object function

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and gave some suggestions for the configuration of airfoil fins. Since channel

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configurations, arrangements and boundary conditions have great influence on the flow and heat transfer behaviors of PCHEs, Lee and Kim [16] compared performances of four channel cross-section shapes and four channel configurations of

A

zigzag channels, the results showed that the rectangular channel provides the best thermal performance but the worst hydraulic performance, and one of the four configuration shows the best convective heat transfer performance, the another one has the highest heat effectiveness. They also found that the effectiveness reaches the 5

maximum when the hot and cold sides have the similar angle in zigzag PCHEs [17]. Kim et al. [18] carried out studies on the arrangement optimization of airfoil fins and the optimal arrangement was obtained when the staggered number equals to 1. Meshram et al. [19] analyzed the effect of the channel structure on the overall heat

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transfer and pressure drop of SCO2, the results showed that the zigzag channel with a larger bend angle and smaller linear pitch has greater comprehensive performance.

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Baik et al. [20] intentionally used the round corner in zigzag channels and resulted in

much better hydraulic performance than the traditional sharp corner. Baik et al. [21]

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investigated effects of the channel size and waviness factors on wavy-channeled

N

PCHEs, the results showed that the channel size has less impacts on the thermal

A

performance than waviness factors. Chen et al. [22] compared overall performance of

M

NACA 00XX series airfoil fins in PCHEs and proved that NACA 0010 airfoil fins

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have the best comprehensive performance. As an accurate correlation for predicting heat transfer and pressure drop in the PCHE is necessary in practical applications,

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Kim et al. [23] and Saeed et al. [24] explored validity of existing correlations in a

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wider range of Re and gave some recommendations. Furthermore, there were a few studies on cross flow [25, 26] and counter flow [27] in straight PCHE channels, and

A

PCHEs with non-CO2 working fluid like helium [28, 29], water [30] and LNG [31]. The above literature review demonstrates that most of the existing papers focused

on the configuration optimization and thermal-hydraulic performance improvement of the PCHE channels based on the first law of thermodynamics. To further understand the mechanism of heat transfer, Guo et al. [32] proposed the field synergy principle, in 6

which the convective heat transfer process is analogized as a conductive problem with inner heat source, and they concluded that the convection heat transfer depends not only on the velocity and the temperature gradient but also on their synergy. This principle has been proved in applications to some traditional heat exchangers using

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the constant properties fluid as the working medium [33, 34]. The energy is conserved in the process of heat transfer, while the heat transfer

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process is inherently irreversible, which is directly related to the second law of

thermodynamics. Bejan [35, 36] pointed out that the second law analysis is extremely

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important in the heat transfer analysis and proposed an entropy generation number Ns

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for the performance evaluation and optimization of heat exchanger. Due to the

A

‘entropy generation paradox’ presented by Bejan’s Ns, Hesselgreaves [37] proposed a

M

new entropy generation number which provides more accurate optimization of heat

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exchangers. Guo et al. [38] numerically investigated the thermal-hydraulic performance of a curved rectangle channel based on the entropy generation using

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water as the working fluid whose properties are almost constant.

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As one of the most widely used constructions in PCHEs, zigzag channels could obtain heat transfer advantages of more than 1.2 times over straight channels although they cost pressure drop of around 1.5 times [39]. This leads to much better

A

comprehensive performance of zigzag channels than straight channels [29]. Yoon et al. [40] also pointed that the zigzag PCHE is the best option in the laminar region because of its lowest pressure drop and relatively high heat transfer area. Thus, the zigzag channels with SCO2 is numerically investigated in terms of the field synergy 7

principle and entropy generation in the present work. The influences of some important parameters on the overall and local thermal-hydraulic performance of zigzag channels are discussed and analyzed. The present work may provide a guideline significance on the design and optimization of SCO2 PCHEs with zigzag

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channels.

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2. Numerical analysis 2.1. Physical model

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A physical model coupled two semicircular channels is built in the present work as

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shown in Fig. 1. To facilitate periodic boundary conditions, the same bend angle is

A

used in two fluid domains. The whole model consists of zigzag sections with 13.5

M

pitches and two extended straight sections with a fixed length of 20 mm. The straight

ED

sections are added at the ends of the zigzag channel for flow fully development. Stainless steel 316L with the thermal conductivity of 16. 2 [W/(m K)] is employed as

PT

the material of solid domain. In the previous studies [6, 20, 23], the zigzag channel

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with the bend of 115° and the pitch of 9 mm has been widely used in PCHEs. Based on that, different channel structures could be obtained by changing bend angle solely. The total lengths and single pitch lengths of different structures vary slightly for

A

maintaining the same flow length. The diameter of the semicircular channel for both sides is fixed as 1.8 mm and the overall cross-section size is fixed at 2.5 mm (LX) ×3.3 mm (LZ) as shown in Fig. 1. Thus, hydraulic diameter Dh, plate thickness tp and wall thickness tw are calculated to be 1.1 mm, 1.65 mm and 0.7 mm, respectively. For fluid 8

domains, inlet temperatures and outlet pressures of hot and cold side are set as 390.15 K, 295.15 K and 21 MPa and 8.5 MPa, which ensures that the SCO2 of both sides operating near the pseudocritical point. The inlet mass flow rate keeps the same in both sides. For solid domain, the adiabatic boundary condition is set in left-right and

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front-back walls and the periodic boundary condition is set in top-bottom walls. The effect of the gravity is ignored in the current work. Detailed geometry parameters and

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inlet mass flow rates are presented in Table 1. The Reynolds number ranges from

1000-9000 in the present work, which means most of the calculations are under the

N

2.2. Governing equation and mesh independence

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turbulence condition.

A

Software ANSYS CFX 15.0 is employed in the present calculation. For steady

M

simulations, the governing equations for the fluid and solid domains are described as

ED

follows [41]:

( u j ) x j

0



x j

(  u j c p T)



 x j

 T    x j

    

(2)

(3)

A

x j

(1)

Pi ui u j 2 uk       ) (   t )( xi x j  x j xi 3 xk 

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(  ui u j )

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For fluid domain, continuity, momentum and energy equations:

For solid domain, energy equation:  x j

 T   s   0  x j  

(4)

where μ is dynamic viscosity and μt is turbulence viscosity, Φ represents the energy 9

dissipation caused by viscosity, λs is the thermal conductivity of the solid. The Shear Stress Transport k-ω (SST k-ω) turbulence model [42] is adopted in the calculation since it combines the merits of k-ε and k-ω models. A real gas property (RGP) file is generated by using thermal properties of SCO2 from the NIST database [43] for the

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current simulation in CFX. Structured meshes for numerical models are generated by ANSYS ICEM 15.0. ‘O’

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type grids are employed in the radial direction of two fluid domains, and the overall

grid quality could be larger than 0.5. Meshes are also refined in the corner and near

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wall regions as illustrated in Fig. 1 because great changes of flow and heat transfer

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take place in these locations. Since the SST k-ω is a ‘y+ sensitive’ model, the mesh

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refinement near the wall also ensures that the dimensionless wall distance y+ is always

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less than 1. Besides, all the computations get converged with the residual of 10-5.

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A grid independence test is carried out with four meshes for the channel of bend=115°. Mesh details are shown in Table 2, in which y0 represents the first layer

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distance from the wall of two fluid domains. Temperature and pressure distributions

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of the cold side calculated by four grids are compared in Fig. 2. Since Grid 3 and Grid 4 provide almost the same results, and the maximum relative error between the two grids is about 0.94%, Grid 3 is selected for current simulations for accuracy and time

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saving.

2.3. Validation of numerical analysis To validate the reliability of present simulations for zigzag channels, a simplified model with one cold channel and two hot channels based on the configuration given 10

by Ishizuka et al. [5] is established. The overall size of this validation model is 4.89 mm (LZ) × 6.55 mm (LX) × 846 mm (LY). The inlet temperatures, inlet mass flow rates and outlet pressures of both hot and cold sides are set the same as the experiment, and top-bottom and left-right walls are set as the periodic boundary condition. The

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comparison results between experimental and numerical results are listed in Table 3. The relative error of the pressure difference in the hot side is the largest and is about

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8.16%. As the thermal properties of SCO2 changes fiercely near the critical point, experimental results of local heat transfer coefficient in a straight semicircular tube

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proposed by Li et al. [44] are also employed to further validate the present simulation.

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The comparisons between the experimental and numerical results are presented in Fig.

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3, and the maximum relative error under the heating and cooling conditions are about

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12.8% and 8.5%, respectively. The above validations indicate that the numerical

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model employed in the present work is reliable and acceptable because it could predict the flow and heat transfer in zigzag channels as well as the effect of sharply

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changed properties on the local heat transfer very well.

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3. Theoretical analysis The changes of the thermal-hydraulic performance along the flow direction due to

the variable thermal properties of SCO2 cannot be ignored in zigzag channels. The

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whole model is divided into N sub heat exchangers based on one pitch as shown in Fig. 4 to obtain local performance. The local heat flux can be calculated as follows:

 H hn,out )  mc ( H cn,out  H cn,in )  1 1  m (H qn  (qhn  qcn )   h hn,in  2 2 An 

(5)

where qn represents the heat flux in the nth sub heat exchanger, m and H represent 11

mass flow rate and specific enthalpy, respectively, An is the local heat transfer area, subscripts h, c, in and out stand for hot side, cold side, inlet and outlet, respectively. The local heat transfer coefficient for the hot and cold side could be obtained: qn qn  Tn Tb,n  Tw,n

(6)

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hn 

where Tb and Tw represent the bulk temperature and wall temperature, respectively.

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Then, Nusselt number Nu, Colburn factor j and Fanning friction factor f are calculated by: hn  Dh

Nun Ren Prn1/3

fn 

( Pn,in  Pn,out ) Dh  2U mn 2 n Ln

A

jn 

N

U

n

(7)

(8)

(9)

M

Nun 

ED

where Dh is hydraulic diameter and λ is the thermal conductivity of SCO2, Re is Reynolds number and Pr is Prandtl number, Um represents the mean velocity in the

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sub heat exchanger model, Ln represents the local flow length.

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The specific heat cp of SCO2 changes rapidly along the flow direction as shown in Fig. 5(a), which has significant influences on the local heat capacity ratio and heat

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effectiveness: Rc ,n 

n 

Wn,min

Wn,max



(mh c pn,h , mc c pn,c )min

(10)

(mh c pn,h , mc c pn,c )max

mc pn Tn,in  Tn,out Qn  Qn,max (mh c pn,h , mc c pn ,c )min (Thn ,in  Tcn,in )

(11)

where W is heat capacity and Q is heat transfer rate. The overall parameters are 12

calculated by averaging local parameters. According to the field synergy principle [32], Nusselt number can be expressed as: Nu  RePr  (U T )dV  RePr  U  T cos dV V

(12)

V

with U T dV , T  , dV  Um (Tb  Tw ) / ( Dh / 2) V

(13)

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U

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where V is the volume of the model. From Eq. (12), the smaller the synergy angle α is,

the greater the convective heat transfer will be under the same other conditions, which indicates that the heat transfer performance depends not only on the values of the

N

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velocity and the temperature gradient but also on their synergy. As Prandtl number

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and Reynolds number change rapidly along the flow direction as shown in Fig. 5(b)

Nu  U  T cos  dV RePr  V

(14)

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Fc 

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and (c). Eq. (12) can be further written as [45]:

Based on the field synergy for the heat transfer, Liu et al. [46] proposed synergy

P  UYm 2

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Eu  P 

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principle for flow resistance. The dimensionless pressure drop can be expressed as: 0.646  3(1   )    (U UY )dV Re Re L1 / ( D / 2) V

(15)

with

A



UY l , UY  L UYm / ( Dh / 2)

(16)

and the dot product can be written as: (17)

U UY  U  UY cos 

where l is inlet length of the flow, L is whole flow length, UY is velocity along the 13

main flow direction which is along the Y-axis in this work. It can be seen from Eq. (17) that the hydraulic performance depends not only on the values of the flow velocity and the main flow velocity gradient but also on their synergy angle β. The local entropy generation caused by heat transfer and fluid friction can be

S g,P 

2 2 2    T   T   T  

       T 2   x   y   z  

(18)

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S g,T 

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expressed as follows [35]:

  ui

u j  ui    T  x j xi  x j

U

and the total volumetric entropy generation rate is:

(20)

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S g  S g,T  S g,P

(19)

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Thus, the dimensionless entropy generation number proposed by Hesselgreaves [37] is:

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 S dV g

V

Q / Tin

(21)

ED

N s1 

4. Results and discussion

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4.1 Overall flow and heat transfer characteristics

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The overall heat transfer coefficient h, friction factor f and heat effectiveness ε for different channel structures at different mass flow rates has been shown in Fig. 6. They decrease as the bend angle increases in both sides obviously. The heat transfer

A

coefficient is larger at m =0.2 g/s than that at m =0.1 g/s, while the friction factor is almost the same in both mass flow rates conditions for the same bend angle, as shown in Fig. 6(a) and (b). The heat effectiveness in Fig. (c) is slightly smaller at smaller mass flow rate. Since the reduction of the bend angle and the increase of the mass 14

flow rate could strengthen the turbulence intensity, the enhanced convective heat transfer is obtained. The reduction of the bend angle significantly increases the flow resistance in channels, while the mass flow rate has unobvious influences on the friction factor. A suitable bend angle is essential for the flow and heat transfer in

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zigzag channels under a wide range of mass flow rate. When the bend angle is smaller than about 110°, the growth of the heat transfer coefficient and heat effectiveness both

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get slow down at m =0.1 g/s, while the h and ε keep almost liner growth at m =0.2 g/s. According to the flow and heat transfer performance in Fig. 6, the larger heat

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transfer coefficient has much to do with the larger heat effectiveness with the decrease

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of the bend angle at the same mass flow rate.

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Fig. 7 illustrates the relations of field synergy number Fc, synergy angle α, Euler

M

number Eu and synergy angle β with the bend angle in both sides. From Fig. 7(a), one

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can see that the larger the bend angle is, the smaller the field synergy number and the larger the field synergy angle would be. The field synergy number is larger at m =0.2

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g/s than at m =0.1 g/s under the same bend angle. Thus, the larger the field synergy

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number is, the greater the convective heat transfer performance becomes. The convective heat transfer enhancement could be attributed to the improvement of the synergy between the velocity and the temperature gradient with the bend angle

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decreases. Fig. 7(b) shows that the changing trends of the Eu and β are completely opposite and the Euler number has no significant difference between m =0.1 g/s and m =0.2 g/s in both sides. It is notable that the field synergy principle could explain the

flow and heat transfer characteristics in zigzag channels using fluids with variable 15

properties very well. To compare the combined performance of flow and heat transfer among different bend angle structures, a combined performance criterion (CPC) is employed, which is defined as: j / j0 ( f / f 0 )1/3

(1)

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CPC 

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where j0 and f0 is the reference values calculated by the straight channel model. As shown in Fig. 8(a), the CPC is larger in the cold side than that in the hot side, and it is

larger at m =0.1 g/s than at m =0.2 g/s under the same bend angle, which may

N

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indicate that the zigzag channel shows relatively good thermal-hydraulic behavior at

A

small mass flow rate. As the changes of the thermal properties in the cold side are

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more intense than that in the hot side as shown in Fig. 5, the dramatic properties variations may be the main reason for the better combined performance of zigzag

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channel structures in the cold side. The CPC increases at first and then decreases with

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the increase in bend angle, and the peaks of CPC occur when the bend angle is larger than 110° and smaller than 150°. It can be explained that when the bend angle is

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larger than 150°, the heat transfer performance is not very good although the flow resistance in channels may be quite small. When the bend angle is smaller than 110°,

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the decrease of bend angle leads to a small improvement of heat transfer behavior but a large increase in flow resistance. In addition to the first law analysis of thermodynamics, Fig. 8(b) shows the relationships of the overall entropy generation number to the bend angle in hot and cold sides at m =0.1 g/s and 0.2 g/s. With the increase of the bend angle, the entropy 16

generation number increases in both sides, and the increasing range gets smaller in the hot side while gets larger in the cold side. The increase of mass flow rate has relatively slight influence on the entropy generation number. Comparing to the results in Fig. 6, the decrease of the total entropy generation corresponds to the enhancement

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of the heat transfer performance at small bend angles, especially in the cold side. So, channels with smaller bend angle could have larger heat transfer coefficient and

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smaller irreversibility loss at the same time. When the bend angles are smaller than 130°, the channels have relatively small irreversible losses.

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Therefore, zigzag channels with bend angles between 110° and 130° have relatively

N

good comprehensive performance in ranges of parameters selected in the present work

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considering both the first and second laws of thermodynamics.

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4.2 Local thermal-hydraulic performance

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As thermophysical properties change rapidly along the flow direction in channels, the local flow and heat transfer characteristics are analyzed in this section. Fig. 9

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illustrates the variations of the heat transfer coefficient with the flow length in the hot

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and cold sides for different bend angles at m =0.1 g/s. One can see that the heat transfer coefficient h increases at first and then decreases sharply in the cold side, while h increases at first and then keeps almost unchanged along the flow direction in

A

the hot side. The peaks of the h locate near L=120 mm and L=60 mm in the cold side and hot side, respectively. The peak of the heat transfer coefficient in the cold side, which is mainly ascribed to the properties variations near the pseudocritical region, becomes larger but postpones as the bend angle reduces. It is considered that better 17

heat transfer performance could be obtained when the peak value happens closer to the middle position of along the channel. Since the thermal performance has much larger changes in the cold side than in the hot side along the flow direction, the following analyses mainly focus on the cold side

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to explore the mechanism of the local heat transfer. Fig. 10 illustrates the distributions of the heat effectiveness in the cold channels and heat capacity ratio along the channel

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with different bend angles at m =0.1 g/s. The local heat effectiveness increases at first and then decreases slightly, and increases again and reduces at the end section of the

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channel. The slight fluctuation of the heat effectiveness becomes clearer in the larger

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bend angle channel. It is apparent that the relatively high heat effectiveness

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corresponds to the large heat transfer coefficient in Fig. 9. The heat capacity ratio

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decreases sharply at first and then increases, which is opposite to that of heat

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effectiveness as a whole. It is speculated that the heat capacity ratio may have great influences on the local heat effectiveness. Normally, the relatively small Rc means the

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slight maldistribution of heat capacity in both sides, which would promote the heat

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transfer from the heat side to the cold side for balance and result in comparatively great heat effectiveness. However, a very small heat capacity ratio may be bad for the field synergy in heat exchangers and the local heat effectiveness reduction may occur.

A

This may explain the increase of the heat effectiveness near the position where the valley of heat capacity ratio locates. Fig. 11 shows variations of local field synergy number and entropy generation number with different bend angles along the flow direction in cold channels at 18

m =0.1 g/s. Field synergy numbers are the largest at the entrances of channels and

then decrease swiftly along the flow direction. A valley of field synergy number appears near the exit region. The entropy generation numbers for two bend angles decrease slightly at first and increase sharply later. The valleys of the entropy

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generation number take place in the region near L=120 mm, where the convective heat transfer coefficient reaches the maximum as shown in Fig. 9. Hence, the better

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synergy between the velocity and temperature gradient corresponds to the higher local

convective heat transfer coefficient as well as the smaller entropy generation along the

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zigzag channels using SCO2 as working fluids.

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For the special structure of the zigzag channel, the thermal-hydraulic performance in

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one wave changes rapidly as well. The twelfth wave is selected, and several lines and

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sections are generated to analyze the local performance, as shown in Fig. 12. Line 1 is

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along the left wall of the upper plane and line 2 locates at the bottom of the semicircle surface. Sections from Z1 to Z4 are positions obtained along Z-axis, and A3 and A2 are

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respectively.

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two cross sections at the bend corner and in the middle position between two corners,

The distributions of local field synergy number and entropy generation number along

different lines in the cold side for the bend of 115° at m =0.1 g/s are displayed in Fig.

A

13. Along both lines, the larger field synergy number always corresponds to the smaller entropy generation number at the same location. The field synergy number along the line 1 is obviously larger than that along the line 2, and the entropy generation number along the line 1 is smaller than that along the line 2. There exist 19

several peak values of the field synergy number along line 1, most of which are near the corner region, as marked with red circles. These peaks of the field synergy number represent the superior heat transfer performance. Fig. 14 demonstrates the contours of velocity in the Y-direction for sections from

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Z1 to Z4, and the negative velocities represent reverse flows. One can see that the reverse flow area gets smaller from Z1 to Z4, which indicates that the main impact

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region of reverse flows locates near the upper plane in the semicircle channel. The

positions with strong backflows mainly locate near bend corners, where local velocity

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and temperature distributions are quite complex. This may be the reasons for the

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summits of local field synergy number found in Fig. 13. The reverse flows near the

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bend corner promote the mixture of fluids and reduce the temperature gradient locally,

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which lead to better synergy between velocity and temperature gradient as well as

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smaller heat transfer entropy generation.

Fig. 15 illustrates the profiles of velocity in Y-axis along line 1 and line 2 in the 12th

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wave with different bend angles at m =0.1 g/s. Apparent reverse flows occur in line 2,

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while they are not observed in line 1. It is obvious that the changes of the velocity for the larger bend angles are gentler than that for smaller bend angles, and the reverse flows become larger and affect larger areas with the reduction of the bend angle.

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Considering the conclusions that the reverse flow enlarges local field synergy number and reduces local entropy generation, the overall field synergy improvement in Fig. 7(a) and the entropy generation reduction in Fig. 8(b) with the decrease in the bend angle are also significantly affected by the larger reverse flows at smaller bend angle. 20

As the flow direction in zigzag channels changes greatly, obvious secondary flow is observed in cross sections. Fig. 16 presents the distributions of radial velocity, temperature gradient, synergy angle between velocity and temperature gradient as well as heat transfer entropy generation on the cross-section of L=65 mm for different

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bend angles at m =0.05 g/s in the cold side. There is almost not secondary flow in the straight channel as shown in Fig. 16 (a). With the decrease of the zigzag bend angle, the

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radial velocities increase and the vortex areas enlarge obviously. Correspondingly, the area of cross section occupied by the small temperature gradient increases as the bend

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angle decreases as shown in Fig. 16 (b). Besides, the secondary flow not only changes

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distributions of velocity and temperature gradient but also has great impacts on their

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synergy. From Fig. 16(c), the synergy angle decreases greatly as the bend angle

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decreases. It also can be seen from Fig. 16 (d) that the entropy generation caused by the

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heat transfer mainly happens in the near wall region, where synergy angle α almost equals to 90 degree in Fig. 16 (c). With the decrease of the bend angle, the apparently

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large heat transfer entropy generation rate occupies smaller area in the near wall region.

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However, the near-wall entropy generation rate is far larger with secondary flow than without secondary flow. The entropy generation near the wall becomes larger since the secondary flow reduces the boundary layer and leads to larger temperature gradient in

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the near wall region. Thus, the smaller bend angle could induce larger secondary flow, which would result in more even temperature distribution and better field synergy in the region far from the wall. While the secondary flow also significantly enlarges the entropy generation in the near wall region. 21

To further investigate the effect of the secondary flow on local entropy generation, Fig. 17 presents contours of streamlines, temperature and volumetric entropy generation rate in the twelfth wave for bend=115° at m =0.05 g/s. In Fig. 17(a), there are two obvious secondary flows with different flow directions near the A2 cross

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section in the channel, while the two flows combine and are almost in one direction when they reach the A3 cross section. The velocity in the left side are larger than that

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in the right side from A1 to A3, but more uniform velocity in the middle of two corners is obtained than that near corners. Then, more even temperature in the A2

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cross section than that in the A3 cross section in Fig. 17(b) can be explained by the

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relatively symmetry secondary flow structure. It can be seen from Fig. 17(c) that

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serious entropy generation asymmetry occurs in A3 cross section. While almost the

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same entropy generation rate is shown in two sides of the A2 cross section. As the

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secondary flow would enlarge the entropy generation rate near the wall, the larger entropy generation rate at the inner side of the corner also can be attributed to the

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obvious larger velocity in the right side of the A2 cross section. To reduce the

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irreversible losses locally, the corner of the zigzag channel should be optimized.

5. Conclusions In the present work, the overall and local thermal-hydraulic performance of zigzag

A

channels with different bend angles are investigated numerically using SCO2 as the working fluid. The entropy generation and field synergy principle are employed to analyze and discuss their effects on the performance. Reverse flows and secondary flows also have significant influence on the local performance of zigzag channels. 22

For the overall thermal-hydraulic performance, the decrease of the zigzag bend angle results in the better convective heat transfer performance but worse hydraulic performance. The better thermal performance has strong relations with the smaller entropy generation and better synergy between the velocity and the temperature

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gradient. When the synergy angle between the velocity and the main flow velocity gradient gets reduced, the worse hydraulic performance appears in the smaller bend

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angle. The larger the heat effectiveness is, the greater the convective heat transfer

coefficient also will be. Zigzag channels with bend angles between 110° and 130°

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have relatively good comprehensive performance in terms of the first and second laws

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of thermodynamics.

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For the local flow and heat transfer, there exists a peak value of the heat transfer

M

coefficient along the flow length due to drastic properties variation near the

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pseudocritical temperature. The sharp decrease of the heat capacity ratio may result in the local reduction of the heat effectiveness, which gets alleviative in the channel with

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smaller bend angle. The local peak heat transfer behavior also benefits from relatively

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great field synergy and small heat entropy generation locally. The reverse flows mainly occur near the corner region and they could significantly

promote the local field synergy and lessen the local entropy generation. Overall field

A

synergy improvement and entropy generation reduction with the decrease in the bend angle also significantly affected by the larger reverse flows. The secondary flow gets stronger with the bend angle reduces as well. It could result in relatively even temperature distribution and great field synergy between the velocity and the 23

temperature gradient in cross sections but enlarge the entropy generation in the near wall region. The peak entropy generation at the inner side of the bend corner induced by the asymmetry secondary flow should be further optimized.

Acknowledgment

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Our work is supported by the National Key Research and Development Program-China (2017YFB0601803), National Natural Science Foundation of China

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(51676185).

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[39] A.M. Aneesh, A. Sharma, A. Srivastava, P. Chaudhuri, Thermo-Hydraulic Performance of

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Zigzag, Wavy, and Serpentine Channel Based PCHEs, in: A.K. Saha, D. Das, R. Srivastava,

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P.K. Panigrahi, K. Muralidhar (Eds.) Fluid Mechanics and Fluid Power - Contemporary Research, 2017, pp. 507-516.

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[43] E.W. Lemmon, M.L. Huber, M.O. McLinden, NIST standard reference database 23: reference fluid thermodynamic and transport properties-REFPROP, version 9.0 (2007). [44] H. Li, Y. Zhang, L. Zhang, M. Yao, A. Kruizenga, M. Anderson, PDF-based modeling on the turbulent convection heat transfer of supercritical CO2 in the printed circuit heat exchangers

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[46] L. Wei, L. Zhichun, M. Tingzhen, G. Zengyuan, Physical quantity synergy in laminar flow

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ED

M

A

4669-4672.

29

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Fig. 1. Channel configuration and meshes of numerical model.

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Figure captions

Fig. 2. Grid independence test of bulk temperature: Grid 1 (——), Grid 2 (- - - -), Grid 3

N

U

(-⸱-⸱-⸱-), Grid 4 (∙⸱∙⸱∙⸱∙⸱), wall temperature: Grid 1 (—■—), Grid 2 (- -●- -), Grid 3 (-⸱-▲-⸱-),

A

Grid 4 (∙⸱∙⸱▼∙⸱∙⸱) and pressure: Grid 1 (——), Grid 2 (- - - -), Grid 3 (-⸱-⸱-⸱-), Grid 4

M

(∙⸱∙⸱∙⸱∙⸱) for numerical model.

Fig. 3. Comparisons between simulation results under heating (—○—) and cooling (—□—)

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conditions and experimental results under heating (●) and cooling (■) conditions at 8.1 MPa, 330

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kg/(m2 s) and 20 kW/m2 [44].

Fig. 4. Schematic diagram of sub-heat exchangers.

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Fig. 5. Distributions of local (a) cp, (b) Pr and (c) Re in channels with different bend angles: hot (—●—) and cold (——) sides of 150˚, hot (- -▲- -) and cold (- - - -) sides of 130˚, hot (-⸱-▼-⸱-)

A

and cold (-⸱-⸱-⸱-) sides of 115˚. Fig. 6. Relations of (a) overall heat transfer coefficient h, (b) friction factor f, (c) heat effectiveness ε with bend angle: hot (—●—) and cold (——) sides of m =0.1 g/s, hot (- -▲- -) and cold (- - - -) sides of m =0.2 g/s. Fig. 7. Relations of (a) field synergy number Fc and synergy angle α, (b) Euler number Eu and 30

synergy angle β with bend angle: hot (—■—) and cold (- -●- -) sides of m =0.1 g/s, hot (-⸱-▲-⸱-) and cold (∙⸱∙⸱▼∙⸱∙⸱) sides of m =0.2 g/s for Fc and Eu; hot (——) and cold (- - - -) sides of

m =0.1 g/s, hot (-⸱-⸱-⸱-) and cold (∙⸱∙⸱∙⸱∙⸱) sides of m =0.2 g/s for α and β. Fig. 8. Relations of comprehensive performance criterion CPC and overall entropy generation

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number Ns1 with bend angle: hot (—●—) and cold (——) sides of m =0.1 g/s, hot (- -▲- -) and cold (- - - -) sides of m =0.2 g/s.

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Fig. 9. Local heat transfer coefficient h along flow directions with different bend angles at

m =0.1 g/s: hot (—●—) and cold (——) sides of 150˚, hot (- -▲- -) and cold (- - - -) sides of 130˚,

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hot (-⸱-▼-⸱-) and cold (-⸱-⸱-⸱-) sides of 115˚

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Fig. 10. Local heat effectiveness ε in the cold side and heat capacity ratio Rc along flow direction

M

(——), 130˚ (- - - -), 115˚ (-⸱-⸱-⸱-) for Rc.

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with different bend angles at m =0.1 g/s: 150˚ (—●—), 130˚ (- -▲- -), 115˚ (-⸱-▼-⸱-) for ε; 150˚

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Fig. 11. Field synergy number Fc and entropy generation number Ns1 along flow direction in cold side at m =0.1 g/s: 150˚ (—●—), 115˚ (- -▲- -) for Fc; 150˚ (——), 115˚ (- - - -) for Ns1.

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Fig. 12. Positions of the selected (a) lines and (b) sections.

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Fig. 13. Local synergy number Fc and entropy generation number Ns1 along different lines for bend of 115° at m =0.1 g/s in the cold side: Line 1 (—●—), Line 2 (-⸱-▼-⸱-) for Fc; Line 1 (——), Line 2 (-⸱-⸱-⸱-) for Ns1.

A

Fig. 14. Velocity distributions of different positions along Z-axis for bend of 115° at m =0.1 g/s. Fig. 15. Distributions of mainstream velocity UY along different lines for different bend angles at

m =0.1 g/s: 150˚ (—●—), 130˚ (- -▲- -), 115˚ (-⸱-▼-⸱-) for Line 2; 150˚ (——), 130˚ (- - - -), 115˚ (-⸱-⸱-⸱-) for Line 1. 31

Fig. 16. The distributions of (a) radial velocity, (b) temperature gradient, (c) field synergy angle α and (d) volumetric heat transfer entropy generation on the cross-section of L=65 mm for different channels at m =0.05 g/s. Fig. 17. Contours of (a) streamlines, (b) temperature and (c) volumetric entropy generation rate in

A

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PT

ED

M

A

N

U

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twelfth wave for bend=115° at m =0.05 g/s.

32

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13.5 Pitches

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20 mm

Hot in

LX

A

tp

ED

1.8 mm

1 Pitch

0.5tw

Right

Left

Top

M

Front

LZ

N

Cold out

Bend angle

1 Wave

Bottom

A

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Fig. 1. Channel configuration and meshes of numerical model.

33

Back

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20 mm

Cold Hot Wall

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360

240

N A

320

300 40

60

ED

20

M

Temperature (K)

340

80

100

Flow direction

120

140

160

80

Pressure-8.5106Pa (Pa)

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Bend=115 Cold side m=0.1 g/s

0 160

L (mm)

A

CC E

PT

Fig. 2. Grid independence test of bulk temperature: Grid 1 (——), Grid 2 (- - - -), Grid 3 (-⸱-⸱-⸱-), Grid 4 (∙⸱∙⸱∙⸱∙⸱), wall temperature: Grid 1 (—■—), Grid 2 (- -●- -), Grid 3 (-⸱-▲-⸱-), Grid 4 (∙⸱∙⸱▼∙⸱∙⸱) and pressure: Grid 1 (——), Grid 2 (- - - -), Grid 3 (-⸱-⸱-⸱-), Grid 4 (∙⸱∙⸱∙⸱∙⸱) for numerical model.

34

IP T SC R

10

U

Maximum error: 8.5%

N

6

4

A

h (kWm-2K-1)

8

0.99

M

Maximum error: 12.8%

2

1.00

1.01

ED

Tb/Tpc

A

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PT

Fig. 3. Comparisons between simulation results under heating (—○—) and cooling (—□—) conditions and experimental results under heating (●) and cooling (■) conditions at 8.1 MPa, 330 kg/(m2 s) and 20 kW/m2 [44].

35

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1 Pitch

Tcn,in

N

Tcn,out

Thn,out

Tc1,out

Tc1,in Tc2,out

q1

Tc2,in Tcn,out

q2

Th1,out Th2,in

Th2,out Thn,in

ED

Th1,in

M

A

Thn,in

Tcn,in TcN,out

qn

qN Thn,out ThN,in

A

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PT

Fig. 4. Schematic diagram of sub-heat exchangers.

36

TcN,in

ThN,out

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2.8

15

2.6

12

2.4

9

2.0

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3.0

8

6

m=0.1 g/s

U

5

A

6

60

90

L (mm)

120

3 150

1.4 30

Re 10-3

N

Pr 1.6

ED

2.0 30

4

M

2.2

6

Pr

cp (kJ/kg K)

cp (kJ/kg K)

1.8

60

90

L (mm)

120

3

4

2

2 150

1 30

60

90

120

150

L (mm)

A

CC E

PT

(a) (b) (c) Fig. 5. Distributions of local (a) cp, (b) Pr and (c) Re in channels with different bend angles: hot (—●—) and cold (——) sides of 150˚, hot (- -▲- -) and cold (- - - -) sides of 130˚, hot (-⸱-▼-⸱-) and cold (-⸱-⸱-⸱-) sides of 115˚.

37

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4

0.06

3

SC R

2

f

h (kWm-2K-1)

0.04

0.02

0 60

90

120

150

0.00 60

180

(a)

120

150

180

Bend (degree)

(b)

M

A

0.35

0.28

90

N

Bend (degree)

U

1

ED

ε

0.21

PT

0.14

0.07 60

90

120

150

180

Bend (degree)

A

CC E

(c) Fig. 6. Relations of (a) overall heat transfer coefficient h, (b) friction factor f, (c) heat effectiveness ε with bend angle: hot (—●—) and cold (——) sides of m =0.1 g/s, hot (- -▲- -) and cold (- - - -) sides of m =0.2 g/s.

38

IP T 4

80

2

75

90

120

Bend (degree)

70 180

150

4

0 60

90

90

85

80

N

8

A

0 60

SC R

12

U

85

Eu

6

M

Fc  103

16

β (degree)

90

α (degree)

8

75

120

150

70 180

Bend (degree)

A

CC E

PT

ED

(a) (b) Fig. 7. Relations of (a) field synergy number Fc and synergy angle α, (b) Euler number Eu and synergy angle β with bend angle: hot (—■—) and cold (- -●- -) sides of m =0.1 g/s, hot (-⸱-▲-⸱-) and cold (∙⸱∙⸱▼∙⸱∙⸱) sides of m =0.2 g/s for Fc and Eu; hot (——) and cold (- - - -) sides of m =0.1 g/s, hot (-⸱-⸱-⸱-) and cold (∙⸱∙⸱∙⸱∙⸱) sides of m =0.2 g/s for α and β.

39

IP T SC R

2.5

0.064

Optimum angles

Optimum angles

0.056

N

0.040

U

0.048 1.5

Ns1

(j/j0)/(f/f0)1/3

2.0

1.0

80

100

120

Bend (degree)

140

160

180

M

0.5 60

A

0.032

0.024 60

80

100

120

140

160

180

Bend (degree)

A

CC E

PT

ED

(a) (b) Fig. 8. Relations of comprehensive performance criterion CPC and overall entropy generation number Ns1 with bend angle: hot (—●—) and cold (——) sides of m =0.1 g/s, hot (- -▲- -) and cold (- - - -) sides of m =0.2 g/s.

40

IP T SC R

2.8

Cold flow direction 2.4

U N

1.6

A

h (kWm-2K-1)

Hot flow direction 2.0

1.2

M

m=0.1 g/s

0.8 30

60

90

120

150

L (mm)

A

CC E

PT

ED

Fig. 9. Local heat transfer coefficient h along flow directions with different bend angles at m =0.1 g/s: hot (—●—) and cold (——) sides of 150˚, hot (- -▲- -) and cold (- - - -) sides of 130˚, hot (-⸱-▼-⸱-) and cold (-⸱-⸱-⸱-) sides of 115˚

41

IP T SC R

0.35

0.8

U

m=0.1 g/s

60

ED

0.15 30

M

0.20

A



0.25

90

0.6

0.4

Rc

N

0.30

0.2

Cold flow direction

120

0.0 150

L (mm)

A

CC E

PT

Fig. 10. Local heat effectiveness ε in the cold side and heat capacity ratio Rc along flow direction with different bend angles at m =0.1 g/s: 150˚ (—●—), 130˚ (- -▲- -), 115˚ (-⸱-▼-⸱-) for ε; 150˚ (——), 130˚ (- - - -), 115˚ (-⸱-⸱-⸱-) for Rc.

42

IP T SC R

0.005

1.5

U

Cold side m=0.1 g/s

A

Fc

0.003

0.001

60

ED

0.000 30

M

0.002

90

1.2

0.9

Ns1

N

0.004

0.6

0.3

Cold flow direction 120

0.0 150

L (mm)

A

CC E

PT

Fig. 11. Field synergy number Fc and entropy generation number Ns1 along flow direction in cold side at m =0.1 g/s: 150˚ (—●—), 115˚ (- -▲- -) for Fc; 150˚ (——), 115˚ (- - - -) for Ns1.

43

IP T A2

A4 A1

A5

A2 A1

N

A5

A3

U

Left side A4

SC R

A3

A

Flow direction Line 1 Line 2

ED

M

Right side

Z1=R-0.1 mm Left side

Z2=2/3 R Z3=1/3 R Z4=0.1 mm

A

CC E

PT

(a) (b) Fig. 12. Positions of the selected (a) lines and (b) sections.

44

IP T A A5

A4

A3

0.003

0.002

M ED

0.25

0.00

SC R

0.50

U

0.75

Bend=115° m=0.1 g/s Cold side

N

0.004

Ns1

Fc

1.00

0.001

A2

A1

0.000

A

CC E

PT

Fig. 13. Local synergy number Fc and entropy generation number Ns1 along different lines for bend of 115° at m =0.1 g/s in the cold side: Line 1 (—●—), Line 2 (-⸱-▼-⸱-) for Fc; Line 1 (——), Line 2 (-⸱-⸱-⸱-) for Ns1.

45

IP T SC R

Cold side

Flow direction

A

N

Reverse flows

U

Reverse flows

Z2

M

Z1

Z3

Z4

A

CC E

PT

ED

Fig. 14. Velocity distributions of different positions along Z-axis for bend of 115° at m =0.1 g/s.

46

IP T SC R

0.06 Cold side m=0.1 g/s

U

0.02

N

Velocity UY (m/s)

0.04

-0.02

A5

M

A

0.00

A4

A3

Reverse flows

A2

A1

A

CC E

PT

(-⸱-⸱-⸱-) for Line 1.

ED

Fig. 15. Distributions of mainstream velocity UY along different lines for different bend angles at m =0.1 g/s: 150˚ (—●—), 130˚ (- -▲- -), 115˚ (-⸱-▼-⸱-) for Line 2; 150˚ (——), 130˚ (- - - -), 115˚

47

z

Bend=115 degree

IP T

Bend=150 degree

SC R

Bend=180 degree

U

(a)

M

A

N

(b)

ED

(c)

z (d)

x

A

CC E

PT

Fig. 16. The distributions of (a) radial velocity, (b) temperature gradient, (c) field synergy angle α and (d) volumetric heat transfer entropy generation on the cross-section of L=65 mm for different channels at m =0.05 g/s.

48

SC R

A2

Left

IP T

A2

A1

Right

A3

A3

A4

ED

N

M

A

(a)

U

A5

A

CC E

PT

(b) (c) Fig. 17. Contours of (a) streamlines, (b) temperature and (c) volumetric entropy generation rate in twelfth wave for bend=115° at m =0.05 g/s.

49

Table captions

Table 1 Geometric parameters and inlet mass flow rates for different channels.

IP T

Table 2 Mesh details of numerical model for bend of 115°.

A

CC E

PT

ED

M

A

N

U

SC R

Table 3 Comparison between numerical results and Ishizuka’s experimental results [5].

50

Table 1 Geometric parameters and inlet mass flow rates for different channels. Pitch length /mm

Total length /mm

Mass flow rate /g s-1

180

10.6712

184.06

0.05,0.1,0.2

150

10.3076

179.15

0.05,0.1,0.2

130

9.6714

170.56

0.1,0.2

115

9

161.5

0.05,0.1,0.2

100

8.1746

150.36

0.1,0.2

85

7.2094

137.33

0.1,0.2

70

6.1207

122.63

0.1,0.2

A

CC E

PT

ED

M

A

N

U

SC R

IP T

Bend angle /°

51

Table 2 Mesh details of numerical model for bend of 115°. Element number

y0/mm

y+

Grid 1

2,324,814

0.008

1.53625

Grid 2

2,663,232

0.004

0.857752

Grid 3

3,222,336

0.002

0.445403

Grid 4

4,090,944

0.001

0.198212

A

CC E

PT

ED

M

A

N

U

SC R

IP T

Case

52

\ Table 3 Comparison between numerical results and Ishizuka’s experimental results [5]. Temperature difference/K

Pressure drop/Pa

Cold side

Hot side

Cold side

Experimental result

169.6

140.38

24180

73220

Numerical result

170.58

137.286

26152.8

76000.5

Relative Error/%

0.58

2.20

8.16

3.80

A

CC E

PT

ED

M

A

N

U

SC R

IP T

Hot side

53