**1. Introduction**

318 Computational Simulations and Applications

Dukler, A. E., & Hubbard, M. G., (1975), A model for gas-liquid slug flow in horizontal and

Kelly, J.E., & Kazimi, M.S., (1980), Development of the two-fluid multidimensional code

near horizontal tubes, *Ind. Eng. Chem., Fundam.*, 14, pp. 337-347.

THERMIT for LWR analysis, *AIChE Symposium Series*, pp. 149-162.

Heat transfer is the main concern in designing shell and tube heat exchangers. Flowinduced vibration (FIV) is also a major concern while designing shell and tube heat exchangers. Fluidelastic instability (FEI), turbulence, periodic instability, and acoustic resonance are the FIV mechanisms that could cause to vibrations in heat exchanger tube bundles. FEI is the most dangers mechanism, since it can cause to tube damage in short time when the flow velocity exceeds the critical flow velocity (Ucr). Therefore, intensive researches have been undergoing in the last five decades to predict and under stand this mechanism. Different theories and models are in use to predict the FEI thresholds as function of mass damping parameters (MDP). These theories and models rely on some coefficients and parameters. Experimental approaches are used to predict these parameters for some tube arrays geometries. The experimental approach is expensive and a time consumer. Computational Fluid Dynamics (CFD) is an alternative approach proposed in this study to predict these parameters. This study utilized the CFD model to simulate the unsteady flow and the resulting fluidelastic forces in a tube bundle. Numerical simulations of in-line square tube arrays with a pitch-to-diameter (*P/d*) ratio of 1.33 utilizing a 2 dimensional model are presented. In this model, a single tube was forced to oscillate within an otherwise rigid array. The numerical model solves the Reynolds-Average Navier-Stokes (RANS) equations for unsteady turbulent flow, and is cast in an Arbitrary Lagrangian-Eulerian (ALE) form to handle mesh motion associated with a moving boundary. The fluidelastic instability was predicted for both single and fully flexible tube arrays over a mass damping parameter (MDP) range of 0.1 to 200. Fluid forces acting on the oscillating tube and the surrounding tubes were estimated. The predicted forces were utilized to calculate fluid force coefficients for all tubes.

Fluidelastic instability is the most destructive excitation mechanism leading to rapid failure by fatigue or tube-to-tube clashing if the stability threshold is exceeded. Due to this potential for catastrophic failure intensive research has been ongoing for several decades on the topic of predicting and mitigating FEI effects. This has resulted in a vast amount of literature on the topic. Much of the research has been directed towards obtaining a reliable estimate of the critical flow velocity for the purpose of design. There are several models available to analyse FEI problems and the associated critical velocity. These models range from analytical approaches such as the models of Lever and Weaver (Lever & Weaver, 1982) and Païdoussis and Price (Païdoussis & Price, 1984) to the empirically-based unsteady flow

Numerical Simulations of Unsteady

**2.1 Conservation equations** 

and momentum equation:

al, 2010). The *k-*

and the


transformation between the *k-*

solution time required per time step.

where *uj* and *ui* are the fluid velocity components,

cartesian spatial coordinates, *P* is the fluid pressure,

implementation can be found in (Menter, 1994).

*u u*

to (Omar et al, 2009; Omar, 2010; Hassan et al, 2010) for more details.

Fluid Forces in Heat Exchanger Tube Bundles 321

*t x*

 

( ) <sup>0</sup> *j mj j u u*

( ) *i j mj i i eff <sup>i</sup> j ij j*

*u u uu u <sup>P</sup> <sup>S</sup> t x xx x*

laminar and turbulent contributions) , and *Si* is any additional momentum source contributions. This form allows conservative fluid flow calculations with mesh adaptation in time. The discretization process of the governing equations is similar to that applied for finite-volume/finite-element discretization (Schneider & Raw, 1987; Omar, 2010; Hassan et

> *k k ku u P k t x x x*

<sup>2</sup> ( ) *j mj*

*P t x k xx*

In these equations *Pk* is the production rate of turbulence. The SST model incorporates a

model in the near wall regions of the flow. Further details on the turbulence model

Based on the oscillation frequency and amplitude, the nodes on the moving tube surface (shaded tube designated as 1 in Figure 1) are moved to the new position and the remaining

The simulation is first run with a static tube, to obtain a quasi steady-state solution to provide initial flow conditions for the transient simulation. Following this quasi steadystate solution a transient simulation is then started but with tube 1 in motion in the lift direction and then in the drag at a specified frequency and amplitude. Reader is forwarded

For each time step convergence is achieved to within an RMS residual of 1x10-4 before moving on. The solution of the RANS equations is based on a coupled algebraic multigrid solver (Raw, 1996). Parallel computations are applied in the present case to reduce the

nodes in the interior are moved according to a set of Laplacian diffusion equations.

 

influence of turbulent mixing. The turbulence equations are cast in ALE form :

( ) \* *j mj*

based Shear Stress Transport (SST) model is used (Menter, 1994) to include the

 

(2)

(3)

(4a)

(4b)

 *is* the fluid density, *xj* and *xi* are

*eff* is the effective viscosity (includes

*k eff j j j*

*k eff*

model in the free stream regions of the flow, and the *k-*

 

*j j j*

 

The RANS equations in ALE form for mass conservation appears as:

theory models such as those of Chen (Chen, 1991) and Tanaka and Takahara (Tanaka & Takahara, 1981). These approaches provide a good estimate of the critical flow velocity, with models based on the unsteady flow theory being the more general. However, the applicability of the unsteady flow theory is restricted to the availability of the force coefficients they depend on. These force coefficients have to be measured experimentally or alternatively, as is investigated in this paper, can be generated using a comprehensive CFD model.

Several numerical techniques and studies embarked on attempting to simulate unsteady fluid forces (Weber et al, 2001; Omar et al, 2009; Omar, 2010; Hassan et al, 2010). This, in turn, could be used in conjunction with the unsteady flow model. This approach will be presented in this chapter. There has been an attempt to numerically predict the unsteady fluid forces in a tube row by Weber et al (Weber et al, 2001). They utilized a commercial CFD package (STAR-CD) to analyze a tube row with a P/d ratio of 1.35. In this work (Weber et al, 2001), the fluid forces due to the prescribed motion of one tube were predicted. Using these fluid forces, fluid force coefficients equivalent to those of Chen (Chen, 1991) were obtained.

The results obtained via this work (Weber et al, 2001)have good agreement with other experiments. Comprehensive numerical studies to simulate the unsteady fluid forces for In line square and normal triangle tube arrays were conducted by Omar et al (Omar et al, 2009); Omar (Omar, 2010); Hassan et al (Hassan et al, 2010). In these studies (Omar et al, 2009; Omar, 2010; Hassan et al, 2010), the ability to extract fluid force coefficients for the unsteady flow theory was undertaken for two tube array configurations, namely i) in-line square tube array and ii) normal triangle tube array. The ability to predict the tube array FEI from CFD derived coefficients for each configuration and for single and fully flexible tube arrays was assessed. The predicted fluid force coefficients and FEI are then compared to available experimental data. The comparison demonstrates the viability of the proposed model to provide fluid force coefficients for the unsteady flow theory. The effect of the P/d ratio and the Reynolds number on the FEI threshold was also investigated.

In the present study the accuracy of using CFD generated force coefficients in the unsteady flow theory model is tested by comparing against available experimental data for in-line square tube arrays. The base geometry, depicted in Figure 1a, and test conditions, follows the experimental study of Tanaka and Takahara (Tanaka & Takahara, 1981).
