Analysis of transverse vibration response and heat transfer characteristics of heat exchangers based on elastic structures

Abstract

To accurately explore the numerical vibration response and heat transfer characteristics of heat exchanger with fluid-solid coupling in the actual working environment. According to the finite volume method and Euler-Lagrange method, the three-dimensional model and mathematical model of coupled vibration are founded. The 0.5 m/s working condition is taken as the experimental object to verify the rationality of the coupling calculation model. Then, bidirectional and unidirectional fluid-solid coupling simulation methods are chosen to resolve the vibration acceleration and heat transfer characteristics of heat exchange tubes at different speeds, and the effects of the two methods on the vibration response of heat exchange tubes are contrasted. Results indicate that under bidirectional coupling, the heat transfer characteristics of the heat exchange tube will experience significant fluctuations due to the vibration of the tube bundle. The shell-side domain has more prominent influence on the vibration of the heat exchange tube when the water velocity is greater than 0.5 m/s. Typical digital features of vibration acceleration and vibration frequency are improved by 110.03%–140.47% and 99.71%–99.88% respectively. The research not only provides a valid analysis approach for fluid-solid coupling vibration of heat exchanger, but also gives theoretical guidance to its vibration optimization design.

Keywords

Heat exchanger unidirectional fluid-solid coupling vibration performance heat transfer characteristics bidirectional fluid-solid coupling

Introduction

Due to the widespread application and important role of heat exchangers in various industrial scenarios, people have increasingly high requirements for heat exchanger capacity and recycling in high energy consuming industries. In addition, with the increasing requirements for the safety property and thermal transfer efficiency of heat exchangers, researchers are paying increasing attention to the lateral vibration of heat exchangers. In industrial production, the shell side domain of heat exchanger design is usually closed and limited. When a heat exchanger operates under different operating conditions, the flow field around it has undergone drastic changes. These changes cause deformation in the solid domain where the heat exchange tube is located, thereby exacerbating the lateral vibration of the heat exchange tube. Therefore, as the fluid operating speed increases, the interactive coupling effect between the shell side drainage basin and the heat exchange tube would become increasingly important. The lateral vibration of heat exchange tubes (HETs) not only damages some major components of the system and reduces the lifespan of a heat exchanger, but also seriously affects the safety and stability of the heat exchanger device.^1,2 Thus, the fluid-structure interaction transverse vibration response mechanism of the heat exchanger in the real operating environment is explored.

In addition, domestic and foreign scholars have achieved certain results in the research of the transverse vibration characteristics of heat exchange tubes. Su et al.³ proposed a new type of elastic heat transfer element based on the idea of fully utilizing fluid induced lateral vibration to enhance heat transfer; at the same time, the uneven fluid impact excitation affecting the transverse vibration of the heat exchange tube is transformed into the Contact force between the fluid domain structures, and the transverse vibration dynamic model of the heat exchanger is established. Lai et al.⁴ studied the influence of periodic fluid forces on flow induced vibration of tube bundles in a rotating triangular tube array; based on the theory of solid dynamics, a dynamic model of tube bundle vibration under the influence of periodic fluid force and gap limitation was established, and numerical simulation calculations were conducted using periodic fluid force as the disturbance source. Ding et al.⁵ explored the vibration between cylindrical tubes at different staggered positions, and established the transverse vibration model of coupling between the drainage basin and the solid domain. At the same time, the effects of various parameters; for instance, the distance between heat exchange tubes and the cross angle on the dynamic characteristics of the fluid solid coupling surface is studied. Then, the influence of near wake structure and temperature distribution on the transverse vibration of heat exchange tubes is analyzed. Zhang et al.⁶ utilized the fluid kinetic energy caused by flow induced vibration, based on the unsteady RANS method and the Spalart-Allmaras turbulence model, combined with dynamic grid technology, to determine the separation characteristics of vortex induced vibration galloping coupling and the average response of composite random vibration of heat exchange tubes. Finally, the influence of parameter changes and system natural frequency on lateral vibration response was analyzed.

With the continuous improvement of water flow velocity in the fluid field of heat exchanger, the flow field and flow characteristics in the fluid field of the heat exchanger have undergone significant changes, making a larger number of complicated and grave flow problems in the heat exchanger more prominent. Meanwhile, the drastic changes in the flow field also exacerbate the lateral vibration response of the heat exchange tube in the heat exchanger. So, considering only the self-motivation interference of the solid domain within the heat exchanger system, the related scholars are unable to exactly describe the time-domain response of lateral vibration during the operation of the heat exchanger in actual environments. Researchers have begun to delve deeper into the flow field characteristics inside heat exchangers and the lateral vibration characteristics under fluid flow disturbances. Yakut and Sahin⁷ measured and recorded the amplitude and vortex shedding frequency changes along the axis under different pitch sizes of heat exchange tubes through flow induced vibration characteristics testing experiments in conical ring heat exchangers. They analyzed the influence of different heat exchange tube structural parameters on fluid flow characteristics. Wu et al.⁸ used ANSYS FLUENT software to construct a three-dimensional model of the heat exchange tube and flow field domain, and solved the motion of the heat exchange tube based on dynamic grid theory. Then, the variation process of the shell side flow field was described using pressure fluctuations and pipeline vibration, and the influence of pipeline vibration on the flow field under various flow velocities was analyzed. Anantharamu and Mahesh⁹ simulated the three-dimensional turbulence in the shell side of rectangular tube bundles at different Reynolds number through the steady incompressible Navier Stokes Equation. The simulation analyzed that the interaction between fluid and structure is unidirectional coupling, and expounded the dynamic characteristics of water flow in the shell side flow field. Carrillo Segura et al.¹⁰ constructed a three-dimensional vibration dynamic model of the interaction between an elastic tube plate and surrounding fluid, and proposed an accurate three-dimensional solution to solve the free vibration problem between the fluid and a simple support plate. Liu et al.¹¹ studied the water flow vibration response caused by lifting pipe bundles for mining ore under seawater. A nonlinear vibration control model of pipe bundles coupled in multiple flow fields under seawater mining was set up using finite element numerical simulation. At the same time, the longitudinal transverse coupling effect was considered to make use of element method, energy method, and Hamiltonian variational principle. The $Newmark ~ β$ method was used to solve the model and obtain the solution of the nonlinear vibration model, verifying the consistency between the vibration acceleration of the lifting tubes and the experimental measurement data.

To increase the heat transfer effect and service life of the heat exchanger, Delgado et al.¹² and Salinas-Vázquez et al.¹³ used the heat exchange tubes with different parameters and establish a fluid-structure interaction vibration dynamic model when exploring the lateral vibration response of the heat exchange tubes under fluid disturbance. Only the unidirectional effect of fluid perpendicular to the tube bundle on lateral vibration is considered. However, in the actual operating environment of the heat exchanger, there is a bidirectional coupling effect between the transverse fluid and the heat exchange tube. Currently, many scholars are conducting research on the fluid-structure interaction of fluid and solid in the fields of ocean, natural gas, petrochemical industry, et al; for example, the hydrodynamic effect and fluid-structure interaction vibration of underwater tail fins driven by composite materials, the gas-solid flow characteristics in inclined tubes, the interaction vibration of fluid and solid in cylindrical shells based on energy method, etc.^14–16 However, currently the majority of scholars have not conducted research on the bidirectional fluid structure coupling lateral vibration control of the fluid in elastic heat exchangers. The operation of an elastic structure heat exchanger can be simplified into the fluid domain and the solid domain of the heat exchange tube, and its various operating characteristics are similar to the state of the solid at different wind speeds. Numerous researchers have conducted relatively mature research on the dynamic characteristics of gas-solid coupling.^17–20 Therefore, this paper draws lessons from the related research in the domain of air-solid coupled, and combines the characteristics of the elastic structure heat exchanger, proposes a suitable fluid-structure interaction vibration calculation method.

Heat exchanger raises the vibration of the elastic tube through fluctuating fluid, thereby improving the heat transfer efficiency of the heat exchanger. Currently, there are relatively few literatures about fluid vibration enhancing heat transfer in elastic tubes. However, the structure of the traditional elastic tube bundle makes the heat exchange area per unit volume smaller, the comprehensive heat transfer performance is lower, and the energy utilization efficiency of the heat transfer performance of the tube bundle is lower, resulting in poor overall efficiency of the heat exchanger, so it is urgent to optimize and popularize it. According to the flow conditions inside the heat exchanger, the influence of the tube bundle on the shell side condition is much greater than that on the tube side condition. Firstly, this paper addresses the problems in the study of fluid-solid coupling characteristics in the actual operating environment of heat exchangers. The mathematical model and calculation model of coupled vibration of flow field and heat exchange tube were established based on the consideration of fluid-structure interaction. Meanwhile, the numerical calculation results of this article were validated with experimental results from publicly available literature. Two solving methods were adopted sequentially to iteratively resolve different computational domains, and the flow load (impact force of water flow) and vibration acceleration of heat exchange tubes under different working conditions were obtained. At the same time, the influence of coupling phenomena of heat exchange tubes under different flow fields on their vibration response was analyzed. Finally, according to the calculated numerical results, a quantitative analysis was conducted on the heat transfer enhancement characteristics under shell-side conditions, and the heat transfer performance of the elastic tube bundle was discussed using the Performance Evaluation Criteria (PEC).

A framework of this study is as follows: The second part mainly focuses on numerical calculation methods, including geometric calculation models, flow field control, grid settings and boundary conditions, as well as verification of grid independence; By comparing the experimental results with similar literature, the correctness of the mathematical model and calculation method was verified.

Models and numerical methods

Mathematical model

Flow field control equation

When the fluid flows fully in the heat exchanger, the flow field around the heat exchange tube presents an incompressible flow state of high Reynolds number, so it can be considered that the flow field in the shell side is a three-dimensional, viscous, incompressible unsteady turbulent flow field. The numerical calculation mathematical model in this paper is based on the following assumptions^21,22 and simplifications:

(1) The working medium is an incompressible Newtonian fluid;

(2) Neglecting the coupling effect between the vibrating wall and the fluid;

(3) The heat loss generated by viscous dissipation during fluid flow is negligible;

(4) The floating force caused by the density difference and gravity of the fluid is ignored;

The motion of pipe wall vibration is harmonic motion, and its motion parameters depend on the amplitude and frequency of wall motion.

Moreover, taking into consideration the pipe wall boundary of the vibration velocity flow field in this study, on the basis of two methods (finite volume method and Euler method) and two equations model (standard $k ~ ε$ turbulence model), the control equations in the flow field inside the heat exchanger are established^23,24:

\frac{\partial}{\partial t} \int_{V} ρ ϕ d V + \int_{A} n \cdot (ρ ϕ U_{g} - Γ_{ϕ} \cdot grad ϕ) d A = \int_{V} S_{ϕ} d V

(1)

where $ϕ$ represents flow field flux, $V$ represents the volume infinitesimal of the control body, $A$ represents the surface area of the micro-element, and $S_{ϕ}$ represents a source term, $Γ_{ϕ}$ represents the diffusion coefficient, $U_{g}$ represents the speed of the heat exchange tube, which reflects the interference of vibration of the heat exchange tube to the fluid.

In real life, when the heat exchanger is working, its structural components will not only improve the amount of work of simulation model, but also reduce the universality of simulation results. Due to the data of fluid-solid coupling in this paper were processed at the junction of fluid solid surfaces, the physical model of the actual operating environment of the heat exchanger was further simplified as follows:

(1) The heat exchange tube was seen as a relatively regular “hollow” thin-walled tube, neglecting the internal and external connection components of heat exchange tubes.

(2) Due to the long shell side of the heat exchanger, the fluid has a relatively small impact on the tube at the end of the shell side. So, the influence of the end of the heat exchange tube on the uniform velocity section can be ignored.

(3) In this paper, the influence of changes in physical properties was ignored, and the energy loss both the heat exchanger and the working environment was ignored.

Based on the above assumptions, the flow field control equation for this study is as follows:

Mass conservation equation:

\frac{\partial (ρ)}{\partial t} + \frac{\partial (ρ u)}{\partial x} + \frac{\partial (ρ v)}{\partial y} + \frac{\partial (ρ w)}{\partial z} = 0

(2)

where $ρ$ indicates the density of the fluid; $t$ represents time; $x, y$ , and $z$ respectively represent the three-dimensional coordinate axis; $u, v$ , and $w$ respectively represent the flow velocity in the direction.

Momentum conservation equation:

ρ \frac{d u}{d t} = ρ f_{x} - \frac{\partial p}{\partial x} + \frac{\partial}{\partial x} {μ [2 \frac{\partial u}{\partial x} - \frac{2}{3} (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z})]}

\frac{\partial}{\partial y} [μ (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}) + \frac{\partial}{\partial z} [μ (\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z})]]

(3-1)

ρ \frac{d v}{d t} = ρ f_{y} - \frac{\partial p}{\partial y} + \frac{\partial}{\partial y} {μ [2 \frac{\partial u}{\partial y} - \frac{2}{3} (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z})]}

\frac{\partial}{\partial z} [μ (\frac{\partial w}{\partial y} + \frac{\partial v}{\partial z}) + \frac{\partial}{\partial x} [μ (\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y})]]

(3-2)

\begin{matrix} ρ \frac{d w}{d t} = ρ f_{z} - \frac{\partial p}{\partial z} + \frac{\partial}{\partial z} {μ [2 \frac{\partial u}{\partial z} - \frac{2}{3} (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z})]} \\ \frac{\partial}{\partial x} [μ (\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}) + \frac{\partial}{\partial y} [μ (\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y})]] \end{matrix}

(3-3)

where $μ$ is dynamic viscosity; $p$ represents pressure; $f_{x}, f_{y}$ , and $f_{z}$ represents the component of fluid force in the $x, y$ , and $z$ directions, respectively.

Considering that the vibration of the tube bundle can generate vortex streets, using the $k - ε$ standard model, the turbulent flow energy equation is:

\frac{Dk}{Dt} = \frac{1}{ρ} \frac{\partial}{\partial y} [(\frac{μ_{t}}{σ_{k}} + μ_{t}) \frac{\partial k}{\partial y}] + \frac{μ_{t}}{ρ} (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}) \frac{\partial u}{\partial y} - ε

(4)

where $ε$ represents the dissipation rate in turbulence, then the dissipation rate equation is:

\frac{D ε}{Dt} = \frac{1}{ρ} \frac{\partial}{\partial y} [(\frac{μ_{t}}{σ_{ε}} + μ_{t}) \frac{\partial ε}{\partial y}] + \frac{C_{ε 1} μ_{t}}{ρ} (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}) \frac{\partial u}{\partial y} - C_{ε 2} \frac{ε^{2}}{k}

(5)

By coupling equations (4) and (5), the following equation can be obtained:

μ_{t} = \frac{ρ C_{μ} k^{2}}{ε}

(6)

where $μ_{l}$ and $μ_{t}$ represent the dynamic viscosity of laminar flow and turbulent flow respectively; $C_{μ}, C_{ε 1}$ , and $C_{ε 2}$ represent the constants of turbulence ( $C_{μ} = 0.09, C_{ε 1} = 1.44, C_{ε 2} = 1.92$ ), $σ_{k}$ and $σ_{ε}$ respectively represent diffusion Prandtl number ( $σ_{k} = 1.0, σ_{ε} = 1.3$ ).

The above equations were solved using simulation software ANSYS FLUENT 2021. R1 CFD. SIMPLE algorithm is selected to deal with the coupling problem of boundary pressure and water velocity in flow field. During the whole of simulation process, monitoring the residual of each operating equation and setting the convergence criteria for all variables to $1 \times 10^{- 5}$ . In simple to decrease the impact of certain parameters on the research results in this paper, special settings are made for these parameters, such as the size of the heat exchanger shell and the thickness of the tube wall. The three-dimensional simplified geometric modeling of a heat exchanger is shown in Figure 1, and the dimensions of model parameters are shown in Table 1.

Figure 1.

Three-dimensional geometric modeling of heat exchanger.

Table 1.

Parameters of geometric modeling.

Shell or tube side	Parameter	Value
Shell side	Length $l_{1}$ /mm	225
	Diameter $d_{1}$ /mm	120
	Inlet diameter $d_{in}$ /mm	40
	Shell side outlet distance $d_{out}$ /mm	30
	Tube row spacing $s$ /mm	25
	Inlet and outlet pipe height $H_{h}$ /mm	15
Tube side	Diameter $d_{r}$ /mm	16
	Wall thickness $δ$ /mm	2
	Inner radius $r_{1}$ /mm	6
	Spiral length $l_{mbI}$ /mm	250
	screw diameter $l_{mbII}$ /mm	86
	Spiral turns $C$ /mm	9

Grid setting and boundary conditions

This paper used MESH CFD (ANSYS MESH CFD version 2021. R1) to grid the flow field inside the shell side. In simple to even better catch the flow field changes of heat exchange tubes and enhance the interconnection computing efficiency. Boundary layer is put up on the heat exchanger shell wall and tube wall, and grid refinement treatment should also be used here. Due to the flow distribution is relatively simple at a distance from the surface of the solid domain, it is necessary to divide the mesh here more roughly. Furthermore, to assure that the mesh density does not have a marked influence on the simulation results, this paper determined the height of mesh near the shell wall and tube wall, and set various mesh sizes and quantities to divide according to the same growth rate. Taking into account the complexity of exchanger shell, a tetrahedral mesh is used for modeling. As the structure of heat exchange tube was comparatively simple, it is mainly established by hexahedron mesh. According to Figure 2, there are eight layers of boundary layer grids near the wall of the heat exchange tube. According to this study, the standard $k ~ ε$ turbulence model, using values of $y^{+}$ between 30 and 45. At the same time, independence verification was conducted on the grid, and the verification results are shown in Figure 3. This study investigates the influence of four different grid sizes and quantities on the numerical results of heat transfer performance at an inlet speed of 0.5 m/s. For 4.53 million grids, the relative error between drop of pressure and heat transfer coefficient are less than 6.0%, respectively. Thus, the independence of the mesh is confirmed.

Figure 2.

Grid division and boundary conditions: (a) shell side, (b) heat exchange tube.

Figure 3.

Verification of grid independence.

In computer simulation, the parameters are set as follows:

(1) The inlet of shell-side fluid: water flow velocity $U_{in}$ (variables: $0.2, 0.3, 0.5, 0.7 m / s$ ); water flow temperature $T_{in} = 293.15 K$ ;

(2) Fluid outlet pressure: $P_{out} = 0 Pa$ ;

(3) Coupling interface: wall temperature $T_{F} = 333.15 K$ ; the shell wall is slip free and adiabatic.

In this study, the heat exchange tube was made of red copper, and its specific parameters are shown in Table 2.

Table 2.

Physical property parameters.

Parameter	Fluid	Copper
Density $ρ / k g^{- 1} m^{- 3}$	998.21	8900
Specific heat $c_{p} / Jk g^{- 1} K^{- 1}$	4182	385
Dynamic viscosity $μ / kg m^{- 1} s^{- 1}$	0.001005	—
Elastic modulus $E / Gpa$	—	129
Poisson’s ratio $ν$	—	0.33

This paper takes the tube in the heat exchanger as the simulation object during the time period from start to stable operation. Inspired by the wind tunnel experiment, the heat exchange tube is in a stationary state in the heat exchanger shell, and the water flow can flow to the heat exchange tube at different water flow velocities. Not only does this method affect the respective positions of water flow and pipe wall, but also the influence between water flow and pipe wall is not significant. Besides, the parameters in each domain of the simulation are shown in Figure 2. At the bottom of the heat exchanger shell, there is a velocity inlet, at the top, there is a pressure outlet, and the wall of the heat exchanger shell is an adiabatic and anti-skid moving wall. This paper selects four operating conditions: $0.2, 0.3, 0.5, and 0.7 m / s$ . In this study, two methods (spring smoothing and reconstruction) are adopted to rebuilt and renew the mesh of the boundary transformation of the heat exchange tube.

Solid motion equation

In this paper, the fluid-solid coupling calculation follows the conservation law, and the position of the contact surface between fluid and solid realizes the equality or conservation of parameters such as force, displacement, heat flow and temperature, with the following relationship:

{\begin{matrix} τ_{f} \cdot n_{f} = τ_{s} \cdot n_{f} \\ \begin{matrix} d_{f} = d_{s} \\ q_{f} = q_{s} \\ T_{f} = T_{s} \end{matrix} \end{matrix}

(7)

where $f$ represents fluid in the table below and $s$ represents solid; $τ_{f}$ and $τ_{s}$ represent stress; $n_{f}$ and $n_{s}$ represent the unit normal vector on the fluid solid contact surface; $d_{f}$ and $d_{s}$ represent displacement; $q_{f}$ and $q_{s}$ represent the heat flow rate; $T_{f}$ and $T_{s}$ represent the temperature.

During the fluid-solid coupling vibration between shell and pipe wall, due to the problem of energy loss on the fluid-solid interface, the wave equation of fluid-solid coupling whole is as follows:

[M_{f}] {\overset{\cdot\cdot}{p} (t)} + [C_{f}] {\dot{p} (t)} + [K_{f}] {p (t)} + ρ R^{T} \overset{\cdot\cdot}{d} = 0

(8)

For the solid motion, the comprehensive motion analysis of the tube wall is carried out through the arbitrary Euler theory. Heat exchange tubes vibrates under the flow fluid working medium, and after discretization treatment, the dynamic balance expression is as follows:

[M_{s}] {\overset{\cdot\cdot}{d} (t)} + [C_{s}] {\overset{\cdot}{d} (t)} + [K_{s}] d (t) = [σ_{f}] + [σ_{s}]

(9)

By combining equations (8) and (9), the following discretization expression of fluid-solid coupling whole can be obtained:

[M_{s}] {\overset{\cdot\cdot}{d} (t)} + [C_{s}] {\overset{\cdot}{d} (t)} + [K_{s}] d (t) = [σ_{f}] + [σ_{s}]

(10)

In the formula, $f$ stands for fluid and $s$ stands for solid; $M$ is the Mass matrix; $C$ represents the damping matrix; $K$ is the Stiffness matrix; $R^{T}$ represents the coupling matrix on the fluid-structure interaction contact surface.

Model calculation and analysis method

This paper mainly studies the coupling effect between fluid and heat exchange tube in a heat exchanger. In simple to more accurately calculate the true coupling between the fluid and solid domains in the heat exchanger, this paper adopts a bidirectional coupling analysis method. This method simultaneously solves the fluid domain control equation in the heat exchanger and the transient equation of coupled vibration in the solid domain of heat exchange tubes. It considers the real-time interaction between the fluid flow in the heat exchanger and the vibration posture change of the heat exchange tube. The heat exchange tube is constantly in different vibration posture changes, and different vibration postures correspond to different fluid loads on the heat exchange tube, different fluid loads will react on the heat exchange tube and disturb its vibration posture. The specific bidirectional coupling analysis method flow is shown in Figure 4.

Figure 4.

Flow chart of bidirectional coupling analysis method.

In the simulation calculation of this paper, in each time step, data exchange and iteration were carried out between the heat exchanger flow field calculation domain and the heat exchanger pipe calculation domain. Using the interpolation algorithm of coupling module to finish data conversion of fluid impact force and deformation displacement at nodes, and using the bidirectional coupling method to quantity the coupling effect of fluid and solid in the actual operating environment of the heat exchanger. According to Figure 5, the iterative coupling starts at a certain time point $t_{n}$ and solves equation (1) through time steps $t$ to obtain the pressure on the wall of the heat exchange tube during time period $(t_{n} + Δ t)$ . Using fully conservation interpolation algorithm at the interface of fluid-solid coupling, the pressure transfer at the node of coupling surface is completed. Iteratively solve equation (8) through time step $t$ to obtain the displacement of each node in the solid domain of the heat exchange tube at time $(t_{n} + Δ t)$ . Applying profile preserving interpolation algorithm to exchange displacement data on fluid-solid coupling nodes, and using time step $t$ to resolve the computational model when the time transitions to $(t_{n} + Δ t)$ , the pipe wall pressure at time $(t_{n} + Δ 2 t)$ after mesh updating is obtained. By analogy, by iteratively solving the computational model for the fluid and solid domains of a three-dimensional heat exchange tube, and the operation is not completed until the set time. Figure 5 shows that the fluid calculation domain and the heat exchange tube physical calculation domain undergo mutual iterative calculations and exchange data within a set time step. Therefore, the two parts need to maintain consistency in time step settings. In the “coupling” module of the numerical simulation calculation, set the “coupling” time step to 0.003 s based on the physical inherent characteristics of the heat exchange tube. For satisfying the convergence criterion in the computational domain, the number of iterations for per time step is 15. A total time step for calculating coupling is 1.2 s. The calculation under different working conditions is carried out on the workstation Inter(R) Xeon(R) Platinum 8179M, with 22 parallel CPUs, and the calculation time under each working condition is about 17 h.

Figure 5.

Bidirectional coupling interaction iterative calculation process.

Verification of calculation results

For verifying and testing the accuracy and reliability of the numerical method proposed in this paper, the numerical results were compared with those in relevant literature.^21,22 The mathematical model with the same parameters was established based on the cone spiral tube bundle vibration test bench in reference²¹ and the single row elastic tube bundle vibration test bench in reference.²² During the experiment, the acceleration amplitude and vibration frequency at diverse inlet flow rates were measured by an acceleration sensor. Adopting the grid setting and numerical simulation method described in this article, the experimental results were used to compare with the numerical results, as shown in Figure 6(a) and (b). From the following two figures, it can be seen that the numerical results are basically consistent with the experimental results in relevant literature. The maximum relative error between the numerical results and the experimental results is 5.79%, which meets the requirements for subsequent vibration calculation analysis in this paper.

Figure 6.

Comparison chart of simulation results and experimental results under different flow velocities: (a) comparison chart of simulation results and experimental results of frequencies at different flow rates, (b) comparison between simulation results and experimental results of acceleration amplitude at different flow velocities.

Based on the numerical calculation method of this study and using the same physical parameters, the numerical results were compared with the experimental results in reference.²⁵ The elastic tube model constructed in this paper is the same as the heat transfer tube used in the experiment in reference²⁵ in terms of tube structure size, material, and fluid flow. Using the numerical calculation method and geometric model in this article, the heat transfer of shell side water flow velocity in the range of 0.019–0.136 m³/s was simulated. According to reference,²⁵ different Nu values can be obtained at different Reynolds numbers Re, and are shown in Figure 7(a) with a 10% error bar. The error between the Nu gained from numerical simulation and experimental data is within the range of 3.48%–8.42%. These errors all originate from the simplification of measurement and geometric models, which ensures the accuracy of numerical results and lays the foundation for the follow-up. Moreover, this paper also compared it with the simulation in reference.²⁶ Parameters that are the same as those in reference²⁶ are adopted to simulate various working conditions. The average heat transfer coefficient of tube bundles with different Reynolds numbers is calculated and compared with the numerical results in reference,²⁶ as shown in Figure 7(b). Comparing with the simulation results in this paper, the numerical results in the literature are represented by a 10% error bar. The error of the heat transfer coefficient is within the range of 3.48%–8.42%, which further shows that the numerical calculation method in this paper can meet the subsequent analysis of heat transfer data.

Figure 7.

Comparison of heat transfer numerical calculation results: (a) comparison chart with experimental results, (b) comparison chart with simulation results.

Results and discussion

Flow field response of heat exchange tube vibration

For understanding the vibration characteristics of heat exchange tubes, the shell side drainage basin is first analyzed. Figure 8 shows the velocity vector diagram of the shell side at an inlet velocity of 0.5 m/s. At this point, the fluid flow inside the shell has reached a stable state and the flow field has fully developed. In Figure 8(a) and (b), the fluid flows around the heat exchange tube, leading to a great deal of vortices in the gap between adjacent tube spacing. Moreover, for the bottom of the heat exchange tube, it is more susceptible to fluid impact. According to Figure 8(a1)–(a5), it can be seen that after the fluid enters the shell of the heat exchanger, the water flow will impact the inner wall of the shell, and then the direction of the water flow will change, causing the water flow to impact the inner cavity of the heat exchanger shell again. Due to the reflux phenomenon of the fluid, a large number of eddies will be generated. As we know, the fluid flow impacting the tube bundle and the inner wall will produce vortex, and the vortex has a certain scouring effect on its surface, which not only reduces the thickness of the Boundary layer of the velocity field on the outer surface of the tube wall, but also improves the vibration effect of the tube.

Figure 8.

Shell side velocity vector diagram: (a) z = 0, (b) y = 0. (a1) x = 22.5, (a2) x = 67.5, (a3) x = 112.5, (a4) x = 157.5, (a5) x = 202.5.

Time domain response of heat exchange tube vibration

When the heat exchanger operates in a real environment, it will generate forward and backward vibrations and lateral vibrations in the vertical and parallel directions to the heat exchange tube. From the previous study, the vertical vibration of heat transfer tube bundles is not only similar to the up-and-down vibration in the dynamic model, but also close to their inherent properties.^27,28 According to Section “Flow field response of heat exchange tube vibration” of this paper, we know that due to the asymmetric arrangement of the heat exchange tube structure at the axis, asymmetric flow loads are generated when subjected to fluid action, making it easier to exacerbate the lateral vibration of the heat exchange tube. Therefore, this section simulates the transverse vibration acceleration of the solid region of the heat exchange tube under four diverse working conditions: single direction coupling, two-way coupling and without other fluid disturbance (only the frequency disturbance of the heat exchange tube itself). The results are shown in Figures 9 –Figures 12. The typical numerical characteristics of lateral vibration acceleration are taken as root mean square, mean, minimum, and maximum values. Table 3 shows that the typical values without other fluid disturbances are taken as evaluation indicators or comparative analysis of coupled vibration effects. During this paper, the calculated lateral vibration acceleration refers to the up-and-down vibration.

Figure 9.

Time domain response of vibration displacement in the x, y, and z directions of the heat exchange tube: (a) x-direction, (b) y-direction, (c) z-direction.

Figure 10.

Time domain response of vibration acceleration of the heat exchange tube.

Figure 11.

Frequency domain response of vibration acceleration of the heat exchange tube.

Figure 12.

Horizontal vibration response of heat exchange tube: (a) 0.2 m/s, (b) 0.3 m/s, (c) 0.5 m/s, (d) 0.7 m/s.

Table 3.

Comparison of the representative numerical features of the vibration acceleration under different velocity conditions.

Running speed no disturbance (m/s)	Methods	Vibration acceleration/ $mm \cdot s^{- 2}$
Running speed no disturbance (m/s)	Methods	RMS1134.50	Mean688.27	Max4153.70	Min61.87
0.2	Unidirectional coupling	738.86	187.54	3902.30	1.35
0.2	Bidirectional coupling	1050.17	403.43	4165.90	20.66
0.3	Unidirectional coupling	1054.80	443.10	4162.40	28.82
0.3	Bidirectional coupling	1070.17	511.47	4159.50	47.36
0.5	Unidirectional coupling	1163.60	748.02	4151.40	71.81
0.5	Bidirectional coupling	1248.29	899.43	4148.90	90.96
0.7	Unidirectional coupling	1527.72	1279.24	4143.10	101.49
0.7	Bidirectional coupling	1593.56	1366.70	4141.60	89.90

From Figure 10 and Table 3, we can see that in relative to the horizontal vibration velocity which ignores the fluid disturbance, when the water velocity is less than 0.5 m/s, the representative numerical features of one-way coupling and two-way coupling have varied by 65.13%–92.97% and 92.57%–94.33% separately, and the vibration amplitude has also changed by 93.95%–100.21% and 93.93% respectively. By combining the time-domain response diagrams of the heat exchange tube in different directions again, we can see that when the water velocity is less than 0.5 m/s, the fluctuation caused by the fluid load on the heat exchange tube and itself vibration is relatively small. Thus, the interconnection influence between the fluid and the heat exchange tube has a reversely weak impact on vibrations in different directions, while the self-excitation of the heat exchange tube is the main source of disturbance.

Combined with Figures 11 and 12, we can see that when the water flow velocity is greater than 0.5 m/s, the difference in water pressure characteristics around the heat exchange tube becomes increasingly significant as the water flow velocity in the shell side increases. Combined with Figure 9, it can be seen that when the water flow velocity is greater than 0.5 m/s, as the water flow velocity on the shell side increases, the difference in water flow pressure characteristics around the heat exchange tube becomes greatly obvious. This causes a corresponding increase in the flow load on the heat exchange tube, and the horizontal vibration response of the heat exchange tube ultimately increases. In addition, from Table 3, we can see that when the water velocity is not less than 0.5 m/s, the representative numerical features of one-way coupling and two-way coupling have improved to 102.57%–134.67% and 110.03%–140.47%, and the vibration amplitude has also increased by 99.74%–99.94% and 99.71%–99.88%, respectively. According to Figure 12(a)–(d), we can see that at each iteration time step, compared with the one-way coupling between the shell side and the water flowing outside the heat exchange tube, the horizontal vibration acceleration under the two-way coupling is obviously increased, and it is more prominent with the increase of water velocity. From Figures 9 to 10, it can be seen that in each iteration time step, the fluid on both sides of the heat exchange tube is more asynchronous because the moving state of the heat exchange tube changes the flow field around it, which leads to significant fluctuations in the “equilibrium” flow load. In addition, fluctuating flow loads and external random excitations also play a role, ultimately leading to a significant increase in the horizontal vibration acceleration of the heat exchange tube, making the structural safety of the heat exchanger more important.

Frequency domain characteristics of heat exchange tube vibration

The research takes into account the idea of normalizing random variables, which transforms the horizontal vibration acceleration signal of heat exchange tubes randomly distributed in time domain into the vibration acceleration power density spectrum in frequency domain. According to Figure 13, this paper not only discussed the change rule of transvers random vibration of heat exchange tubes, but also gained the frequency domain distribution of random vibration of heat exchange tubes under one-way coupling, two-way coupling, and no other interference factors. For more clearly comparing the influence of “one-way coupling, two-way coupling, and no other interference factors” on the transvers vibration of heat exchange tubes, the linear scale of the maximum amplitude of transvers vibration acceleration in the frequency domain is transformed into a scale $lo g_{10}$ . It can be seen from Figure 10 that under the action of one-way coupling, two-way coupling and no other fluid disturbance, the maximum amplitude of power spectrum density diagram of horizontal vibration acceleration of the heat exchange tube is mainly concentrated in the range of 20–32 Hz.

Figure 13.

Power density spectrum of fluid-structure interaction horizontal vibration acceleration of heat exchange tube at different speeds: (a) 0.2 m/s, (b) 0.3 m/s, (c) 0.5 m/s, (d) 0.7 m/s.

As can be seen from Figure 13(a) and (d), the power spectral density values of horizontal vibration acceleration of heat exchange tubes are all close to $0.015 m m^{2} / s^{3}$ , and the maximum difference of power spectral density diagrams is within 0.15% under the conditions of one-way coupling, two-way coupling and no other fluid interference. Furthermore, it can be seen that when the fluid velocity is less than 0.5 m/s, the coupling effect between axial flow and heat exchange tube has a significant impact on horizontal vibration. As can be seen from Figure 13(c) and (d), when the flow velocity in the fluid domain is greater than 0.5 m/s, compared with the power spectrum density map value of vibration acceleration without other fluid disturbances, the value of effect of its coupling increases in the wake of the increase of operating velocity, and the augment of power spectrum density values of two-way coupled influence is more obvious.

According to the discuss in Section “Time domain response of heat exchange tube vibration,” when the heat exchanger runs in actual environment, the mutual coupled between the fluid in the heat exchanger shell and the heat exchange tube makes the horizontal vibration of the heat exchange tube more intense.

Analysis of heat transfer performance of heat exchanger

Kinetic energy distribution in fluid domain

For exploring the heat transfer performance of the heat exchanger, the turbulent kinetic energy diagrams of the shell side of the heat exchanger were studied on the cross-section at various inlet velocities. Figure 14(a)–(d) show that the diameter of the heat exchange tube is 110 mm. It is not difficult to see from Figure 14 that as the inlet velocity increases, the turbulent kinetic energy in the computational domain significantly increases. The greater the turbulent kinetic energy, the stronger the turbulent characteristics of the fluid, and the better the heat transfer effect obtained. This is also the reason why the heat transfer efficiency inside the tube increases with the increase of inlet velocity.

Figure 14.

Turbulent kinetic energy distribution diagram in various inlet velocities in the shell side domain: (a) 0.2 m/s, (b) 0.3 m/s, (c) 0.5 m/s, (d) 0.7 m/s.

Heat transfer analysis of heat exchanger

In simple to measure the comprehensive enhanced heat transfer performance of heat exchangers, $PEC (Performance Evaluation criteria)$ evaluation criteria can be used.²⁹ When the $PEC$ value is greater than 1, it indicates that the heat exchanger has achieved an improvement in heat transfer performance, and its calculation formula is:

PEC = (N u_{v} / Nu) / (f_{v} / f)

(11)

where $N u_{v}$ is the Nusselt number under vibration; $Nu$ is the Nusselt number without vibration; $f_{v}$ is the fluid drag coefficient under vibration; $f$ is the fluid drag coefficient without vibration.

Nu = \frac{h D_{T}}{k}

(12)

where $h$ is the Heat transfer coefficient $W / (m^{2} \cdot K)$ ; $K$ is the thermal conductivity coefficient of the fluid $W / (m \cdot K)$ ; $D_{T}$ is the outer diameter of the pipe, $m$ .

In the shell side basin, the calculation expression for the fluid resistance coefficient is³⁰:

f = \frac{2 Δ P D_{T}}{H ρ u_{m}^{2}}

(13)

where $H$ is the height of the heat exchanger, $m$ ; $ρ$ is the flow density, $kg / m^{3}$ ; $u_{m}$ is average velocity of the fluid, $m / s$ .

According to the analysis in Section “Frequency domain characteristics of heat exchange tube vibration,” the larger the flow velocity of the water flow, the greater the vibration amplitude of the heat exchange tube. At the same time, the vibration performance of the heat exchange tube also affects the heat transfer performance of the heat exchanger. Figure 15 shows the heat transfer coefficients of the shell-side of a heat exchanger at several inlet velocities $U_{in}$ , where $C = 9$ . Meanwhile, it can be clearly seen that as the inlet velocity $U_{in}$ increases, the heat transfer coefficient of the heat exchange tube gradually increases. The main reason is that as the inlet flow rate increases, the turbulence intensity of water flow in the shell side basin of the heat exchanger also increases, and it can significantly reduce the thickness of the thermal boundary layer near the wall of the heat exchange tube.

Figure 15.

Heat transfer capacity of heat exchange tubes at various inlet flow rates.

The vibration of the heat exchange tube improves its own heat transfer performance and plays a positive role in the heat exchanger. From Figure 15, it can be seen that the inlet flow rate is within the range of $U_{in} = 0.2 m / s ~ 0.7 m / s$ , compared with the case without vibration, the vibration of the heat exchange tube increased its heat transfer efficiency by 0.53%, 2.88%, 4.68%, and 14.67%, respectively. In addition, it can be seen that as the inlet flow rate $U_{in}$ increases, the corresponding average heat transfer coefficient also significantly increases, fully proving that vibration can elevate the heat transfer effect of heat exchange tubes itself. All in all, the average heat transfer coefficient on the surface of an elastic heat exchange tube will increase with the increase of flow velocity, and it can also improve the heat transfer effect of flow induced vibration enhancement in the elastic heat exchange tube.

As shown in Figure 16, it represents the comprehensive heat transfer and enhanced heat transfer performance of a heat exchanger at different water flow rates $U_{in}$ , where $C = 9$ . Obviously, as can be seen from Figure 16, the unit pressure drop heat transfer coefficient will decrease with the increase of inlet velocity. Compared with the non-vibration working condition, when $U_{in} = 0.2 m / s$ , its own vibration increases the heat transfer effect per unit pressure drop by 0.53%, while when $U_{in} = 0.7 m / s$ , it increases by 14.35%. In addition, as the inlet flow rate $U_{in}$ increases, the PEC value of the heat exchanger also gradually increases. Within the range of $U_{in} = 0.2 m / s ~ 0.7 m / s$ , their $PEC$ values are 1.005, 1.029, 1.047, and 1.145, respectively. Through calculation and analysis, when the inlet velocity increases from $U_{in} = 0.2 m / s$ to $U_{in} = 0.7 m / s$ , the $PEC$ value increases by 13.93%. This further indicates that the overall heat transfer effect of the heat exchanger will be more significant when the water flow speed increases. During industrial chemical production, the vibration effect of elastic heat exchange tubes and the overall heat transfer efficiency of heat exchangers can be enhanced by increasing the flow velocity in the basin, and the overall economic applicability should also be considered comprehensively.

Figure 16.

Overall heat transfer performance of heat exchanger at various inlet flow velocities.

Conclusions

The coupling calculation model of fluid and solid domains was built on the basis of finite volume method ( $FVM$ ) and Euler Lagrange method ( $ELL$ ). Inspired by time discretization, the cyclic calculation and data conversion between fluid and solid are realized by using system coupling module. The coupling relationship between the fluid in the heat exchanger and the solid in the heat exchange tube was explored with four different operating speeds as the research objects. The following conclusions were drawn:

The correctness and rationality of the three-dimensional calculation model for the coupled vibration between the shell side drainage basin and the heat exchange tube were verified through numerical simulation of the liquid-solid coupling of a 0.5 m/s heat exchanger. The vibration acceleration calculated by bidirectional fluid-solid coupling simulation not only has a smaller error with the real value, but also can express the fluid-solid interaction coupling of the heat exchanger more correctly in the actual working scene.

Compared with the situation where the influence of fluid on the heat exchange tube is not considered, as the flow rate increases, the difference in flow characteristics on both sides of heat exchange tubes becomes more significant, and the fluid load on heat exchange tubes increases significantly. In addition, the self-excitation of heat exchange tubes intensifies the vertical vibration of heat exchange tubes.

Maximum amplitude of the lateral vibration acceleration of heat exchange tubes is mainly concentrated in the range of 20–32 Hz. Moreover, when the water velocity is higher than 0.5 m/s, power density spectrum value of two-way coupling influence between the fluid in the heat exchanger and heat exchange tubes increases greatly, indicating that the horizontal vibration response of the heat exchange tube is the most obvious for this coupling effect.

Due to the two-way coupling influence between axial flow and heat exchange tube, the flow load not only fluctuates greatly with the variation of the motion state of the heat exchange tube, but also the peak of the wave is directly proportional to the second power of water flow velocity. During the whole operation, as the fluid attitude changes, pressure also acts on the heat exchange tube, making the transverse vibration of heat exchange tubes more serious. The representative numerical features of vibration acceleration increase by 110.03%–140.47%.

In summary, this method not only improves the shortcomings of numerical simulation vibration research on heat exchangers, but also provides a kind of valid research idea for seeking the fluid structure coupling vibration response of heat exchangers in actual working environments, and gives some theoretical supports to develop new optimal vibration reduction devices for heat exchangers.

This study is part of the basic research on fluid-induced vibration to enhance heat transfer. Due to the constraints of experimental conditions and time, there are still many problems to be further discussed, including:

(1) Lack of systematic research on passive (fluid-induced)-active (mechanical-excited) coupled vibration and heat transfer of elastic tube with the participation of mechanical vibration. On the basis of the research in this paper, mechanical vibration can be effectively used to quickly adjust the vibration intensity and frequency. The fluid induced vibration of the heat exchanger itself is inevitable. Considering these two main factors, a coupled vibration numerical calculation model is established to study the vibration mechanism of the elastic tube of the heat exchanger under passive active coupling.

(2) Effective control of elastic heat exchange tubes, comprehensive heat transfer of heat exchangers and accurate control of heat exchange tubes cannot be achieved. We can develop a vortex generator, which can not only meet the vibration control and prediction of the tubes in the heat exchanger, but also improve the vibration consistency between tubes.

Footnotes

Handling Editor: Dr Sharmili Pandian

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the support of National Natural Science Foundation of China, scientific problems of energy efficiency and safety coordination of heat exchange equipment, through grant number 52275154.

ORCID iD

Luyun Li

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