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Research Papers: Fluid-Structure Interaction

Dynamics of Cross-Flow Heat Exchanger Tubes With Multiple Loose Supports

[+] Author and Article Information
Anwar Sadath, Harish N. Dixit

Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate,
Telangana 502205, India

C. P. Vyasarayani

Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate,
Telangana 502205, India
e-mail: vcprakash@iith.ac.in

1Corresponding author.

Contributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received August 6, 2015; final manuscript received March 10, 2016; published online April 29, 2016. Assoc. Editor: Tomomichi Nakamura.

J. Pressure Vessel Technol 138(5), 051303 (Apr 29, 2016) (10 pages) Paper No: PVT-15-1178; doi: 10.1115/1.4033091 History: Received August 06, 2015; Revised March 10, 2016

Dynamics of cross-flow heat exchanger tubes with two loose supports has been studied. An analytical model of a cantilever beam that includes time-delayed displacement term along with two restrained spring forces has been used to model the flexible tube. The model consists of one loose support placed at the free end of the tube and the other at the midspan of the tube. The critical fluid flow velocity at which the Hopf bifurcation occurs has been obtained after solving a free vibration problem. The beam equation is discretized to five second-order delay differential equations (DDEs) using Galerkin approximation and solved numerically. It has been found that for flow velocity less than the critical flow velocity, the system shows a positive damping leading to a stable response. Beyond the critical velocity, the system becomes unstable, but a further increase in the velocity leads to the formation of a positive damping which stabilizes the system at an amplified oscillatory state. For a sufficiently high flow velocity, the tube impacts on the loose supports and generates complex and chaotic vibrations. The impact loading on the loose support is modeled either as a cubic spring or a trilinear spring. The effect of spring constants and free-gap of the loose support on the dynamics of the tube has been studied.

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Figures

Grahic Jump Location
Fig. 1

(a) Heat exchanger with single baffle at the center, (b) schematic representation of a single tube as fixed–fixed beam, and (c) cross section view of the flexible tube at baffle. Note that the figures are not in same scale.

Grahic Jump Location
Fig. 2

The variation of spring restrained forces with the deflection. (a) Cubic spring (see Sec.5) with spring constant κ = 1 and (b) trilinear spring (see Sec. 5.1) for different spring constants κ (figures are not in scale).

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Fig. 3

The schematic representation of a fixed–free beam with multiple loose supports

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Fig. 4

Convergence study for bifurcation diagrams for the cantilever tube at ξ = 1 for κ1 = 1000 and κ2 = 2000 for (a) N = 3, (b) N = 5, and (c) N = 7

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Fig. 5

Bifurcation diagrams for the cantilever tube with baffles modeled as cubic spring with the following parameters: (a) ξ = 0.5, κ1 = 1500, and κ2 = 1500; (b) ξ = 1, κ1 = 1500, and κ2 = 1500; (c) ξ = 0.5, κ1 = 2000, and κ2 = 1000; and (d) ξ = 1, κ1 = 2000, and κ2 = 1000

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Fig. 6

Maximum deflection of the tube for U = 6 for different values ξ at steady state. The three different cases are adopted from Fig. 7.

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Fig. 7

Phase plots ((a), (c), (e), and (g)) and time responses ((b), (d), (f), and (h)) at the location of the tip (ξ = 1) for different velocities: ((a) and (b)) U = 3, ((c) and (d)) U = 4, ((e) and (f)) U = 5, and ((g) and (h)) U = 5.5

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Fig. 8

Bifurcation diagrams for the cantilever tube with baffles modeled as trilinear spring with the following parameters: (a) ξ = 0.5, γ1 = 0.02, and γ2 = 0.01; (b) ξ = 1, γ1 = 0.02, and γ2 = 0.01; (c) ξ = 0.5, γ1 = 0.02, and γ2 = 0.02; (d) ξ = 1, γ1 = 0.02, and γ2 = 0.02; (e) ξ = 0.5, γ1 = 0.02, and γ2 = 0.03; and (f) ξ = 1, γ1 = 0.02, and γ2 = 0.03

Grahic Jump Location
Fig. 9

Phase plots with baffle modeled as trilinear spring for parameters: (a) U = 3, ξ = 0.5; (c) U = 5, ξ = 0.5; (e) U = 3, ξ = 1; and (g) U = 5, ξ = 1. Time responses of the beam for parameters: (b) U = 3, ξ = 0.5; (d) U = 5, ξ = 0.5; (f) U = 3, ξ = 1; and (h) U = 5, ξ = 1.

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