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Heat and Mass Transfer

, Volume 54, Issue 9, pp 2835–2844 | Cite as

Numerical analysis on interactions between fluid flow and structure deformation in plate-fin heat exchanger by Galerkin method

  • Jing-cheng Liu
  • Xiu-ting Wei
  • Zhi-yong Zhou
  • Zhen-wen Wei
Original
  • 187 Downloads

Abstract

The fluid-structure interaction performance of plate-fin heat exchanger (PFHE) with serrated fins in large scale air-separation equipment was investigated in this paper. The stress and deformation of fins were analyzed, besides, the interaction equations were deduced by Galerkin method. The governing equations of fluid flow and heat transfer in PFHE were deduced by finite volume method (FVM). The distribution of strain and stress were calculated in large scale air separation equipment and the coupling situation of serrated fins under laminar situation was analyzed. The results indicated that the interactions between fins and fluid flow in the exchanger have significant impacts on heat transfer enhancement, meanwhile, the strain and stress of fins includes dynamic pressure of the sealing head and flow impact with the increase of flow velocity. The impacts are especially significant at the conjunction of two fins because of the non-alignment fins. It can be concluded that the soldering process and channel width led to structure deformation of fins in the exchanger, and degraded heat transfer efficiency.

1 Introduction

In large scale air separation equipment, the plate-fin heat exchanger (PFHE) working a key component plays an important role for improving heat transfer performance. In general, the PFHE has some specific advantages such as high heat exchange efficiency, compact structure etc., so that it is widely used in many fields such as aerospace, petrochemical engineering, deep hypothermia, and refrigeration for heat transfer among air, oil and water [1]. In compare with other kinds of heat exchanger, PFHEs have greater heat transfer area and higher efficiency, and therefore, attract lots of scholars [2, 3] to carry out researches on a wide range of issues including material and structures improvement, turbulent flow characteristics, temperature distribution, solving and optimizing algorithm etc.

Due to its narrow channel of the exchanger, the model of the mini-channel can be used to analyze fluid flow and heat transfer characteristics approximately in order to use the heat transfer area sufficiently and improve heat transfer efficiency. Koh and Colony [4] simplified the mini-channel of the porous medium and described the fluid flow in the mini- channel by Darcy principle. Tien [5] investigated convective heat transfer boundary in mini-channel and deduced numerical solutions of velocity and temperature by conjugation analysis method. Chen and Yang [6] simplified the mini-channel heat exchanger as a porous media and then analyzed the velocity and temperature distribution by establishing double-equation model and single-equation model. Bahrami and Tamayol et al. [7] proposed an approximation model and calculated pressure distribution in the mini-channel with gliding flow, and investigated pressure variations with different geometry parameters.

Compared with the perforated fins and serrated fins, the plain fin has lower heat resistance and low heat transfer capacity. Therefore, the perforated fins and serrated fins are widely used in PFHE. For the enhanced heat transfer of rectangular cross section, Hooman [8] and Harley [9] investigated convective heat transfer and entropy variation of porous pipe with rectangular cross sections during constant wall temperature and uniform wall heat distribution. Mohammad [10] investigated heat transfer calculation method under cross section and the relationship among the local Nusselt number, average Nusselt number and aspect ratio of cross section. Furthermore, for the fluid flow of heat exchanger with different kinds of fins, reference [11, 12, 13, 14] discussed heat transfer characteristics with several kinds of fins commonly used including pin fins, herringbone fins, rectangular fins and circular fins with different parameters such as: fin space, heat exchange tube diameter and tube number etc. For the fluid-structure interaction, Jan [15] analyzed the fluid-structure interaction in cross flow heat exchanger and proposed a calculation method, finally improve calculation efficiency. Liu and Dong et al. [16] investigated 3-dimensional heat-force interaction models and proposed a new method for solving complex boundary problem. Zhou [17] proposed a simplified calculation method of strain and stress in PFHE by assuming that fins are pieces of elastic spring and solved fatigue characteristics of temperature and pressure in mini-heat exchanger of micro-chemical and thermal system.

Based on the literature review, it is found that the investigations on PFHE mainly include the performance of fins, the influence on heat transfer under cross sections and heat transfer enhancement of fins. However, the investigation on fluid-structure interactions is not enough. As we know, the fluid-structure interaction in the exchanger affects heat transfer efficiency and heat transfer performance of the exchanger, especially, for large air separation equipment, slight deformation of fins that was caused by fluid impact with low temperature results in different heat transfer performance. Meanwhile, in the PFHE, the diversity of fins also causes fluid flow and heat transfer variations, therefore, the traditional calculation on fluid flow and heat transfer efficiency can result in an inaccurate solution. In this paper, the serrated fin is adopted to analyze the deformation of fins and its effects on fluid flow in PFHE of large air separation equipment by minimum potential energy principle of tiny displacement theory. Besides, the influence on fluid flow and heat transfer in mini-channel of PFHE by the deformation of fins when applying loads on fins of lateral direction is also investigated.

2 Model and flow pattern of PFHE

2.1 Model of PFHE

The core section of PFHE is composed by various kinds of fins, separating plates, sealing heads and leading flow structures. Different kinds of PFHEs were proposed and used widely in diverse applications by changing fins such as louver, serrated, perforated and corrugated fins. The leading flow structures are used to change flow direction and improve fluid uniform distribution, so that, the fluid could distribute uniformly in each channel on the same layer of the whole exchanger. Besides, the separating plates are used to isolate the cold fluid and hot fluid in different layers so that different kinds of fluids cannot mix together. The simplified model commonly used in large air separating equipment considering its high heat transfer rate with serrated fins and perforated fins is shown in Fig. 1.
Fig. 1

Simplified model of the PFHE

2.2 Flow pattern in PFHE

Generally, three kinds of flow patterns are commonly used in PFHE: co-current flow, countercurrent flow and cross flow. Temperature difference between top section and bottom section on different channels of the same layer distributes non-uniformly due to non-uniform flow distribution in inlet of the exchanger. For mass amount of fluid, the impact causes tiny deformation of fins and finally bring small influences of fluid flow and deformation. The flow variation causes slight fin deformation and finally, changes the heat transfer efficiency for each piece of fins. In order to solve the above problem, details on the mesh system were provided and finite element method (FEM) was adopted to analyze the interactions between serrated fins and fluid flow in PFHE and the maximum strain position for each piece of fins was achieved.

3 Analysis on fluid flow and fin deformation interaction

3.1 Load distribution on PFHE

In this section, the load distribution for calculating fins deformation and fluid flow characteristics at some key positions along serrated fins in PFHE were presented. In order to investigate load distribution on PFHE in large air separation equipment, the heat exchange section in a single layer of the exchanger was selected (as shown in Fig. 2). Fin arrangements in a single layer of PFHE and the optimized solution-adaptive mesh refinement are adopted in order to predict the fluid flow behavior accurately. Mesh refinement (as shown in Fig. 3) and more cells were added at some key positions where the fluid flow changed significantly. For the serrated fins, channels were compressed into an inverted trapezia zone and caused fluid velocity non-uniform, finally, decreased heat transfer efficiency.
Fig. 2

Fluid flow in channel of serrated fin

Fig. 3

Mesh generation in PFHE

3.2 Fluid-structure interaction calculation

The small deformation of fins in PFHE of large scale air separation equipment mainly originates from the fluid impact among fins. The interactions between fluid and fins based on Galerkin method and calculations on fluid domain and structure domain were analyzed respectively.
  1. (1)

    Fluid Domain

     
As we know, fluid flow in PFHE can be written as:
$$ \left\{\begin{array}{c}\frac{\partial P}{\partial x}+\rho \ddot{u}=0\\ {}\frac{\partial P}{\partial y}+\rho \ddot{v}=0\\ {}\frac{\partial P}{\partial z}+\rho \ddot{w}=0\end{array}\right. $$
(1)

In eq. (1), u, v and w represent fluid velocity along x, y and z direction respectively, ρ represents fluid density, P represents dynamic pressure.

The continuity equation can be written as:
$$ \frac{\partial \dot{u}}{\partial x}+\frac{\partial \dot{v}}{\partial y}+\frac{\partial \dot{w}}{\partial z}=-\frac{\dot{P}}{K} $$
(2)
Combining with eq. (1), eq. (2) can be expressed as:
$$ \frac{\partial^2P}{\partial {x}^2}+\frac{\partial^2P}{\partial {y}^2}+\frac{\partial^2P}{\partial {z}^2}=\frac{\rho }{K}\ddot{P} $$
(3)
Equation (3) can also be expressed as:
$$ {\nabla}^2P-\frac{1}{C^2}\ddot{P}=0 $$
(4)
Where the compressing velocity of element C can be expressed as:
$$ C=\sqrt{\frac{K}{\rho }} $$
(5)
In eq. (4) and eq. (5), represents the differential operator and can be written as:
$$ {\nabla}^2=\frac{\partial^2}{\partial {x}^2}+\frac{\partial^2}{\partial {y}^2}+\frac{\partial^2}{\partial {z}^2} $$
(6)
Momentum equation:
$$ {\displaystyle \begin{array}{l}\frac{\partial \left(\rho u\varphi \right)}{\partial x}+\frac{\partial \left(\rho v\varphi \right)}{\partial y}+\frac{\partial \left(\rho w\varphi \right)}{\partial z}\\ {}=\frac{\partial }{\partial x}\left({\Gamma}_{\varphi, t}\frac{\partial \varphi }{\partial x}\right)+\frac{\partial }{\partial y}\left({\Gamma}_{\varphi, t}\frac{\partial \varphi }{\partial y}\right)+\frac{\partial }{\partial z}\left({\Gamma}_{\varphi, t}\frac{\partial \varphi }{\partial z}\right)+{S}_{\varphi}\end{array}} $$
(7)

In eq. (7), φ represents a general variable, Γφ represents generalized diffusion coefficient, Sφ represents generalized source term.

The turbulent model that can be used in high strain and mass flow conditions based on renormalization group can be written as:

Turbulent kinetic equation:
$$ \frac{\partial \left(\rho k\right)}{\partial t}+\frac{\partial \left(\rho {ku}_i\right)}{\partial {x}_i}=\frac{\partial }{\partial {x}_j}\left({\alpha}_k{\mu}_{eff}\frac{\partial k}{\partial {x}_j}\right)+{G}_k-\rho \varepsilon $$
(8)
Turbulence dissipation equation:
$$ \frac{\partial \left(\rho \varepsilon \right)}{\partial t}+\frac{\partial \left(\rho \varepsilon {u}_i\right)}{\partial {x}_i}=\frac{\partial }{\partial {x}_j}\left({\alpha}_{\varepsilon }{\mu}_{eff}\frac{\partial \varepsilon }{\partial {x}_j}\right)+{C}_{1,\varepsilon}\frac{\varepsilon }{k}{G}_k-{C}_{2,\varepsilon}\rho \frac{\varepsilon^2}{k} $$
(9)

In eq. (8) and eq. (9), k represents turbulent kinetic energy, ε represents turbulence dissipation rate.

In general, in three-dimensional flow field, it is hard to solve each kinetic equation and obtain the analytical solution under some boundary conditions. Therefore, the Galerkin method was adopted in this paper to solve the dynamic pressure distribution. Dynamic pressure distribution P(x,y,z,t) of each position can be defined as:
$$ P\left(x,y,z,t\right)={N}^T\left(x,y,z\right)P(t)=\sum \limits_{i=1}^n{N}_i\left(x,y,z\right){P}_i(t) $$
(10)
In eq. (10), NT is shape function vector, P(t) is pressure vector. NT and P(t) can be defined as:
$$ {N}^T\left(x,y,z\right)={\left[{N}_1\left(x,y,z\right),{N}_2\left(x,y,z\right),\dots, {N}_n\left(x,y,z\right)\right]}^T $$
(11)
$$ P(t)={\left[{P}_1(t),{P}_2(t),\dots, {P}_n(t)\right]}^T $$
(12)
Combining with eq. (4), the dynamic pressure distribution by Galerkin method can be expressed as:
$$ \underset{\varOmega }{\iiint }{N}_m\left({\nabla}^2{p}^{\ast }-\frac{1}{C^2}{p}^{\ast}\right) d\varOmega =0,\kern0.5em m=1,2,\dots, N $$
(13)
The eq. (13) can be solved by Green formula method as:
$$ \underset{S}{\iint }N\frac{\partial {p}^{\ast }}{\partial n} dS-\underset{\Omega}{\iiint}\nabla N\nabla {p}^{\ast }d\Omega -\frac{1}{C^2}\underset{\Omega}{\iiint }{Np}^{\ast }d\Omega =0 $$
(14)
Where, S is the area of fluid domain. Combining with eq. (11), the eq. (14) can be written as:
$$ {\displaystyle \begin{array}{l}\underset{\varOmega }{\iiint}\nabla N\bullet \nabla {N}^T Pd\varOmega +\frac{1}{C^2}\underset{\varOmega }{\iiint }{NN}^T Pd\varOmega -\underset{S_1}{\iint }N\frac{\partial P}{\partial n}{dS}_1-\\ {}\underset{S_F}{\iint }N\frac{\partial P}{\partial n}{dS}_F-\underset{S_b}{\iint }N\frac{\partial P}{\partial n}{dS}_b-\underset{S_r}{\iint }N\frac{\partial P}{\partial n}{dS}_r=0\end{array}} $$
(15)
In eq. (14) to eq. (15), S is the fluid area, S1 is the area of fluid-structure interface, SF is the area of free surface, Sb is the fixed boundary area and Sr is the area at infinity.
  1. (2)

    Boundary Conditions:

     
For the interactions of fluid and structure:
$$ \frac{\partial P}{\partial n}=-\rho \overset{\cdot \cdot }{u_n} $$
(16)
For a single piece of fin:
$$ \frac{\partial P}{\partial n}=0 $$
(17)
For the free surface:
$$ P=\rho g{\omega}_0 $$
(18)
For the infinite boundary:
$$ \frac{\partial P}{\partial \gamma }=-\frac{1}{C}\frac{\partial P}{\partial t} $$
(19)
Combining with the boundary conditions given above, the eq. (15) can be defined as:
$$ {\displaystyle \begin{array}{l}\underset{\Omega}{\iiint}\nabla N\cdot \nabla {N}^T Pd\Omega +\frac{1}{C^2}\underset{\Omega}{\iiint }N\cdot {N}^T\ddot{P}d\Omega +\underset{S_l}{\iint } N\rho {\ddot{u}}_n{dS}_l+\\ {}\frac{1}{g}\underset{S_F}{\iint }{NN}^T\ddot{P}{dS}_F+\frac{1}{C}\underset{S_r}{\iint }{NN}^T\dot{P}{dS}_r=0\end{array}} $$
(20)
In eq. (20), \( {\ddot{u}}_n \) is the accelerated velocity on the interface of fluid and \( {\ddot{u}}_n \) can be expressed as:
$$ {\ddot{u}}_n^{\ast }={N}_S^T{\ddot{u}}_n $$
(21)
In the fluid-structure coupling system, supposing \( \ddot{r} \) is structure vector and then eq. (21) can be written as:
$$ {\ddot{u}}_n=\Lambda \ddot{r} $$
(22)
Combining with eq. (21), the eq. (22) can be expressed as:
$$ {\ddot{u}}_n^{\ast }={N}_S^T\Lambda \ddot{r} $$
(23)

In eq. (22) and eq. (23), Λ is the coordinate transformation matrix.

Then, the eq. (20) can be written as:
$$ {\displaystyle \begin{array}{l}P\underset{\varOmega }{\iiint}\nabla N\cdot \nabla {N}^T d\varOmega +\dot{P}\frac{1}{C}\underset{S_r}{\iint }{NN}^T{dS}_r+\\ {}\ddot{P}\left(\frac{1}{g}\underset{S_F}{\iint }{NN}^T{dS}_F+\frac{1}{C^2}\underset{\Omega}{\iiint }{NN}^Td\Omega \right)+\rho \ddot{r}\left(\underset{S_1}{\iint }{NN}_S^T{dS}_1\right)\Lambda +q=0\end{array}} $$
(24)
  1. (3)

    Structure Domain

     
For serrated fins, the channels in PFHE are compressed into an inverted trapezia zone and cause fluid velocity non-uniform distribution, finally, affect heat transfer efficiency. As shown in Fig. 4, pressure distributes on a single piece of fin along the lateral direction and flow direction.
Fig. 4

Pressure distribution on a single fin. Note: the dash line represents no deformation of fins

Due to the fluid impact, small deformation can be caused at some key positions of a single piece of fin. Combining with the finite element method, the structure domain can be expressed as:
$$ {M}_s\overset{\cdotp \cdotp }{r}+{C}_s\overset{\cdotp }{r}+{K}_sr+{f}_p+{f}_o=0 $$
(25)

In eq. (25), r is displacement vector, Ms is mass matrix, Cs is structure damping matrix, Ks is structural stiffness matrix, fp is node vector, fo is pressure vector.

On the fluid-structure boundary, the fluid pressure distribution can be expressed as:
$$ {P}^{\ast (e)}={N}_e^T{P}_e $$
(26)
In eq. (26), Pe is node dynamic pressure vector, Ne is shape function vector.
$$ \delta {u}_n^{(e)}={N}_{Se}^T{U}_n^{(e)} $$
(27)

In eq. (27), Nse represents the shape function of fins.

The virtual work of displacement on the interaction boundary can be written as:
$$ \delta {W}_e=-\underset{Se}{\iint }{P}^{\ast (e)}\delta {u}_n^{(e)} dSe=-\delta {u}_n^{(e)T}\left(\underset{Se}{\iint }{N}_{Se}{N_e}^T dSe\right){P}_e $$
(28)
Then, the kinetic equation of fins connected with fluid can be expressed as:
$$ {M}_s\overset{\cdotp \cdotp }{r}+{C}_s\overset{\cdotp }{r}+{K}_sr-{B}^TP+{f}_0=0 $$
(29)

4 Results and discussion

In this section, the interactions between serrated fins and fluid flow in PFHE under different velocities were analyzed, besides, flow characteristics and serrated fin deformation were carried out.

In order to validate the performance of the leading flow structure with specific-shaped hole in inlet of PFHE, the status detection and data processing system was established. The leading flow and enhanced heat transfer experimental facility are consisted by air loop system, heat exchange system and data processing system. The air loop system is composed by wind compressor, wind channel, dedusting and purification devices. The heat exchange system is composed by test model, moisture separator, saturated vapor producer and drainage system. The data processing system is composed by a data processing computer and several monitoring probes. The monitoring probes include the flow meters, thermocouples, pressure and differential pressure transducers.

A brief schematic of fluid flow distribution and heat transfer experiment with leading flow structure in PFHE is shown in Fig. 5. The hot fluid is heated by an electric heater and forced into the exchanger as a hot stream and air at room temperature as cold stream is absorbed into the system by air compressor and adjusted to the required flow rate by controlling each valve. The flow rate in the passage is measured by flow meters. The dynamic pressure and temperature variations in the test model and in the flow channel were detected by pressure gauge and temperature monitoring probes. Besides, the temperature, pressure and flow rate signals were monitored and transferred into currents or voltage signals, and then these signals were collected by the status detection and data processing system.
Fig. 5

Schematic flow diagram of experimental system

4.1 Analysis on fluid flow

Considering the parameters of fluid and fins in PFHE, fluid distribution was presented. In Fig. 6, the dynamic pressure distribution along flow direction on a single channel of serrated fins was presented. As depicted in Fig. 6, the dynamic pressure distribution is in good agreement with fluid velocity. However, the dynamic pressure does not decrease linearly accompany with velocities. The maximum value of dynamic pressure reaches 614.1 Pa, meanwhile, for two array of fins nearby each other, the maximum pressure is 136 Pa at the contact positions. Figure 7 presents the dynamic pressure difference distribution in a single channel under different flow velocities. In Fig. 7, the dynamic pressure difference varies much with different flow velocities in PFHE, besides, the maximum dynamic pressure difference reaches 71.6 Pa. The velocity magnitude distribution in a single channel was shown in Fig. 8. The results show that fluid velocity magnitude decreases gradually with the inlet velocity. The maximum velocity reaches 31.5 m/s, besides, it also demonstrates the flow velocity was accelerated owing to the non-alignment arrangements of serrated fins.
Fig. 6

Dynamic pressure distribution in a single channel under different flow velocities

Fig. 7

Dynamic pressure difference in a single channel under different flow velocities

Fig. 8

Velocity magnitude distribution in a single channel under different flow velocities

In Fig. 9, the velocity magnitude difference under different inlet flow velocities was presented. It can be found that the flow velocity magnitude distribution is in good agreement with inlet velocity. Meanwhile, the velocity distribution varies much with different cases (different inlet velocities). The total energy distribution in a single channel under different flow velocities was presented in Fig. 10. Similar with the dynamic pressure distribution, the total energy distribution decrease with the increase of inlet flow velocities. Therefore, higher velocities will decrease the total energy due to some friction pressure loss and local pressure loss along with the serrated fins. The total energy difference in a single channel under different flow velocities was shown in Fig. 11, the results show that total energy difference varies much with different flow velocities. The total energy difference reveals that the dynamic velocity in each channel decreases along flow direction, meanwhile, the results also reveal that the velocity has no change on the same section when the potential energy of fluid was ignored. The main reason is the friction loss along the fins and decreases the inlet velocities and the energy of motion.
Fig. 9

Velocity magnitude difference in a single channel under different flow velocities

Fig. 10

Total energy distribution in a single channel under different flow velocities

Fig. 11

Total energy difference in a single channel under different flow velocities

4.2 Analysis on structure deformation

From the above analysis, it can be found that the serrated fins affect fluid flow characteristics owing to the non-aligned structures. For a single piece of serrated fin, the pressure distribution was investigated and then the dynamic pressure distribution on the surface of each piece of fin was obtained. By calculating the pressure difference of the two sides: side A and side B, as shown in Figs. 12 and 13, tiny deformations of fins occur because of some obstacles by serrated fins in the exchanger.
Fig. 12

Dynamic pressure distribution on surface A of a single fin

Fig. 13

Dynamic pressure distribution on surface B of a single fin

As illustrated in Figs. 12, 13, and 14, it can be found from the analysis on dynamic pressure distribution of a single piece of serrated fin that the pressure distribution is non-uniform. For the serrated fins, the structure deformation was caused by dynamic pressure difference on both sides of a single fin. In order to investigate the distributions of strain and stress on a single fin of the exchanger, as presented in Fig. 15, the structure deformation of fins was analyzed.
Fig. 14

Dynamic pressure difference between surface A and surface B of a single fin

Fig. 15

Structure deformation of fins

As shown in Fig. 15, the structure deformation of fins shows that the non-uniform load can be generated due to the disturbing effect of fins. Analyzing the stress distribution, we can find that the maximum value of net stress is at the corner position of a single fin. The maximum value of net stress that happens reaches up to 31.2 N and the maximum of strain reaches by 0.6 mm. Considering the variations of stress and strain with different inlet velocities, the stress and strain distributions under different inlet velocities were analyzed. As shown in Fig. 17, the maximum stress and strain increased with fluid velocity. According to the dynamic pressure difference on a single fin, the stress and strain varies accompany with flow velocities. As can be observed from Figs. 16 and 17, the increment of maximum value of stress is less than that of strain with the increase of fluid velocity. The maximum net value of stress increment is at the position of minimum value of stress with fluid velocity reaches by 1 m/s. Besides, the increment of the net stress is up to 19.39 times of the minimum value of stress, meanwhile, the maximum net strain is up to 7.78 times of the minimum of the minimum value. Therefore, the PFHE, a multi-layer structure with many channels in it, causes the loads along with the fins distributed non-uniformly and results in non-uniform deformation of fins.
Fig. 16

Stress distribution on fins with different Reynolds number

Fig. 17

Strain distribution on fins with different velocities considering fluid structure interaction

5 Conclusion

The interactions between serrated fins and fluid flow are extensively appeared in PFHE. The stress and strain distributions under three-dimensional predictions are carried out to investigate the effects on velocity magnitude, dynamic pressure and total energy in the exchanger. The governing equations for solving the interactions between fins and fluid in the exchanger were deduced by FVM. In this paper, the interactions of fluid flow and fins in serrated fin heat exchanger were investigated based on Galerkin method. The following conclusions were drawn:
  • Firstly, the dynamic pressure on both sides of a single fin distributes non-uniformly. In PFHE, the dynamic pressure on the two sides of the exchanger is larger than that on the middle part of the fin. The results show that the disturbing effect at the two sides of fins results in the dynamic pressure is larger than that at the middle part.

  • Secondly, the fluid flow in the exchanger was influenced by obstacles coming from the fins because of certain thickness and the non-alignment of fins. Besides, the fluid flow in PFHE shows that the velocity and dynamic pressure vary periodically normal to the direction of fluid flow. Therefore, the dynamic pressure distributed non-uniformly and brought net difference of dynamic pressure.

  • Thirdly, the net pressure difference between the two sides of fins in the exchanger causes tiny deformation of fins. The analysis on the net value of strain and stress distribution on the two sides of fins show that the maximum value of strain and stress on fins is on the rise with fluid velocity, meanwhile, the minimum value of strain and stress of fins decreased.

Notes

Acknowledgements

This paper is supported by the National Natural Science Foundation of China (No. 51705297) and Natural Science Foundation of Shandong Province, China (Grant No. ZR2016EEP09). Besides, we would also like to show our appreciation to Mr. WEI Zhenwen, the chairman of Doright Corporation, for his kindly help during the leading flow and enhanced heat transfer experiment period.

Compliance with ethical standards

Conflict of interest

The authors have no financial or other relationship that might be perceived as leading to a conflict of interest that could affect authors’ objectivity. Besides, this manuscript has not been published elsewhere and it has not been submitted simultaneously for published elsewhere.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Jing-cheng Liu
    • 1
  • Xiu-ting Wei
    • 1
  • Zhi-yong Zhou
    • 2
  • Zhen-wen Wei
    • 1
    • 3
  1. 1.Shandong Provincial Key Laboratory of Precision Manufacturing and Non-traditional MachiningShandong University of TechnologyZiboChina
  2. 2.Hangyang Co., Ltd.Designing InstituteHangzhouPeople’s Republic of China
  3. 3.Doright Energy Saving Equipment Co., Ltd.QingdaoPeople’s Republic of China

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