Nonlinear Dynamics of Heat-Exchanger Tubes Under Crossflow: A Time-Delay Approach

The stability of DDE is investigated in the parametric space of fluid velocity and axial load. Multiple limit cycles coexist in the parametric space, which has implications for fatigue life calculations of heat exchanger tubes.

Heat Exchangers in PHWR Nuclear Reactors

Cross-flow-induced Tube Vibrations

The phase difference between different cylinders is an essential feature of this type of instability, and fluidelastic coupling between cylinders is a necessary condition for stiffness-controlled instability. 2] noted: “The existence of fluidelastic instability in cylinder banks was not discovered until the 1960s, although failures due to it had occurred earlier, but were mistakenly attributed to swirl.

Figure 1.3: Schematic of a bundle of tubes used in shell-and-tube heat exchangers of CANDU-type nuclear power plants.

Quasi-steady Model

Single Flexible Cylinder in an Array of Rigid Cylinders

Time Delay

Due to the presence of the time-delayed displacement term, DDEs are infinite-dimensional systems and their characteristic equation is a quasi-polynomial admitting infinitely many characteristic roots.

Heat-exchanger tube as Euler-Bernoulli beam

Thesis Outline

Using Galerkin approximations, the characteristic roots (spectrum) of the DDE are found and reported in the parametric space of dimensionless flow velocity and axial load. The damping present in the stable region of the parametric space is obtained from the real part of the rightmost characteristic roots.

Quasi-steady model of Price and Paidoussis

Time Delay due to Flow Retardation

Considering that the flow slows down as it approaches the cylinder, the actual time taken will be (T + ∆T) given by. According to potential flow theory, the potential function (φZ) and speed ∂φZ/∂Z for flow approaching a cylinder along the Z axis is given by.

Fluid Forces on a Downstream Cylinder

2ρLDCLU˜2 (2.13) where CD and CL are the drag and lift coefficients for cylinder i, related to the distance velocity ˜U. It can be concluded that CL0i corresponds to the flow past a stationary cylinder, and due to the symmetric nature of such flow, CL0i = 0.

Mathematical Model

Substitute the above relations into Eq. 2.28) By invoking a Galerkin approximation, z is discretized as follows. which when substituted into Eq. Thus, the DDE given by Eq. 2.22) is approximated to a system of ODEs given by Eq.

Figure 2.3: Schematic of a single flexible cylinder (solid blue color) in an array of rigid cylinders subject to cross-flow, as used in [4]

Results

Damping in the stable region

Second, it is in excellent agreement with the stability limit obtained by the method used by [4]. Only one region of alternating negative and positive damping is observed, confirming the dependence of the damping-controlled instability on µ. This suggests that the damping-controlled instability (dominant instability for low mδ [4]) depends on the flow lag parameter and hence the dimensionless time delay (τ).

Figure 2.5: Characteristic roots at mδ = 1.333, and U = 0.791, with the DDE given by ¨ y + 0.218 ˙ y + y + 1.444y(t − τ ) = 0 as obtained from Eq

Hopf bifurcation

It can be seen that the plot of the<(λr) starts from <(λr)<0, where the system is stable. By substituting these values ofωcinto Eq. 2.39), the velocities at which the corresponding λr crosses the imaginary axis in the complex plane can be obtained analytically. These velocities can also be obtained numerically by evaluating the slope of the curve in Fig.2.7 wherever it crosses the line <(λr) = 0.

Figure 2.7: Variation of the real part of the rightmost characteristic roots with reduced velocity for mδ = 1, in the vicinity of the critical points.

Chapter Summary

Additionally, we report the region of maximum damping in the parameter space, where the tube vibration response will decay the fastest. Effect of nonlinear coating stiffness on stability and hopf bifurcation of a heat exchanger tube subjected to it. Unstable regions in the parameter space of dimensionless coating stiffness and flow velocity are identified, along with the magnitude of damping in the stable region.

Mathematical modeling

A is the cross-sectional area of the cylindrical tube, X is the spatial coordinate, and T is time. The force due to the support springs is represented by ˜f, and δ(X −Xb) is the Dirac delta function. Here λ1 is the dimensionless eigenvalue of the first mode of the beam in the absence of nonlinear terms and fluid forces.

Table 3.1: Values of nondimensional parameters α 1 to α 7 used in the nonlinear DDE given by Eq

Stability and bifurcation analysis

Linear stability

Therefore, to obtain the stability limit, we substitute λ = jωcr and k1 = kcr1 in Eq. For a given value of the dimensionless critical linear stiffness (k1cr), Eqs. 3.20a) and (3.20b) can be solved numerically to determine the values of U and ωcr along the stability limit. To counter this limitation, a Galerkin method is used to obtain information on the stability in each region (I−VI) and the amount of damping associated with the stable regions.

Spectrum

The initial value problem (Eq. 3.23)) is then converted to an initial boundary value problem and Eq. 3.24)) with the boundary conditions given by Eq. When we increase the number of terms (N) in the series solution given by Eq. 3.34) converge to the characteristic roots of Eq. Therefore, the stability of the DDE is given by Eq. 3.21a) can be studied by determining the eigenvalues of Eq. Shifted Legendre polynomials are used as basis functions in Eq. 3.26), as the literature shows that they result in relatively faster convergence to the characteristic roots [49]. Figure 3.5 shows the 12 characteristic roots on the far right in Eq. 3.21a), corresponding to points P1 to P5, which lie on the stability limit.

Figure 3.4: Stability chart in the [U, k 1 ] plane obtained from the Galerkin ap- ap-proximation, with color contours representing the damping present in the rightmost

Hopf bifurcation

This indicates the possibility of Hopf bifurcation, which is explored in the next section. Furthermore, it should be noted that in the points P2 and P4, Γ > 0 (Table 3.2) indicates that the rightmost roots cross from left to right in the complex plane, i.e. the system goes from stable to unstable. Similarly, in the points P1, P3 and P5, Γ < 0, indicating the crossing of purely imaginary roots from right to left on the complex plane, i.e. the system goes from unstable to stable.

Method of multiple scales

If we substitute the values of kcr1 and ωcr at point P1 (see table 3.2) in Eq. 3.56b) Figure 3.7(a) shows the local bifurcation diagram (supercritical Hopf bifurcation) obtained from Eq. First, with initial conditions R(0) and θ(0), the set of ODEs given by Eqs. 3.56) (red line) and the system response obtained by integrating Eq. Then we substitute the values of kcr1 and ωcr at point P3 (see table 3.2) in the equation.

Figure 3.7: (a) Local bifurcation diagram at point P 1 . System response at P 1 for (b) ∆ = −1 with initial conditions for Eqs

Subcritical Hopf bifurcation at points P 2 and P 4

3.7, 3.8 and 3.9 that both the transient and steady-state solutions of the normal form equation obtained using MMS closely match the results of direct numerical integration. At this point, the response of the system in the unstable regions (white areas in Figure 3.4) is still unknown. Furthermore, the MMS only provides information about the system behavior in the vicinity of the split points.

Figure 3.10(a) shows the local bifurcation diagram (subcritical Hopf bifurcation) obtained from Eq

Global bifurcation analysis

If it vibrates with an initial amplitude greater than 0.4, it will settle into the stable limit cycle (solid red line in Fig. 3.12) originating from point P5 and will vibrate with the corresponding amplitude. For an initial amplitude of approx. 0.3 fork1 = 50 (Fig. 3.13(c)), the oscillations will die out, while fork1 = 5 (Fig. 3.13(a)), the system will settle into periodic motion associated with a stable limit cycle (solid blue line in Fig. Finally, for an initial amplitude greater than or equal to 0.7, the system settles into periodicity.

Figure 3.13: Limit cycles for U = 7 with (a) k 1 = 5, (b) k 1 = 25 and (c) k 1 = 50.

Chapter Summary

Furthermore, it is clear that k2 = 50 is a stronger choice for our analysis, as it corresponds to larger limit cycle amplitudes, i.e. worse case between the two. When there are multiple periodic solutions, fatigue life calculations should be based on the worst possible scenario, i.e. the maximum amplitude of the limit cycle. From the global bifurcation analysis (Section 4.5), the coexistence of several stable and unstable periodic solutions in the parametric space of flow velocity and axial load is shown.

Mathematical modeling

The coexistence of multiple limit cycles in the parameter space is found and has implications for fatigue life calculations of heat exchanger tubes. Additionally, as explained later in this chapter, tensile axial loads can be induced to control the dynamic response of the pipe. The axial loads P0 in the pipe are a consequence of the tensile axial load PA due to prestressing and thermal expansion.

Figure 4.1: (a) Schematic of the heat-exchanger tube bundle with its isometric view along with the coordinate axes, (b) the cross-sectional view of the tube bundle, and (c) idealized model of the heat-exchanger tube as a simply supported beam

Linear stability

It should be noted that in both (a) and (b) only the first few rightmost roots of the infinite spectrum are present. DDE given by Eq.For different values of axial load pcr, Eqs. 4.16a) and (4.16b) can be solved numerically to determine the variables U and ωcr along the stability limit. To determine the critical curve along which static bifurcation can occur, we substitute ωcr = 0 in Eq.

In Fig.4.4(a), although the system is unstable, the second pair of eigenvalues lie exactly on the imaginary axis and satisfy the analytical conditions imposed on the stability limit. The distribution of the characteristic roots (spectrum) of Eq. 4.13) will also be studied for different flow rates and axial loads. It should be noted that in both (a) and (b), only the first few rightmost roots of the infinite DDE spectrum given by .

Figure 4.3: Critical curves for Hopf and static bifurcation for equilibrium points q ¯ 1 = 0 and ¯q 2

Spectrum

The system of ODEs given by Eq. 4.20), and the eigenvalues (ˆλi) of G approximate the characteristic roots of Eq. As N increases, the eigenvalues of G converge to the rightmost characteristic roots of Eq. 4.14)), obtained by substituting the eigenvalues of G into Eq. If we consider N terms in the series solution given by Eq. 4.25), approximately N/2 eigenvalues of matrixG converge to the rightmost roots of the characteristic polynomial (Eq.

Figure 4.5: Stability chart in the [U, p 0 ] plane, generated using the Galerkin approximation method, with N = 100

Hopf bifurcation

Hopf bifurcation at point P 1

Dropping the non-linear term in Eq. 4.40)) into the characteristic equation for the linearized problem, we have: where λ is an implicit function of p0. At the Hopf bifurcation point, the transient solution of Eq. 4.50a) decays with time, since the roots of the characteristics lie on the left half of the complex plane (see fig. For a given ∆, the amplitude of unstable periodic solutions is obtained from equation 4.47) for increasing values of constant history function and trace the equilibrium solution.

Figure 4.9: (a) Local bifurcation diagram at point P 1 . System response at local bifurcation point P 1 for (b) ∆ = −0.1 with initial conditions for Eqs

Next, the normal form equations for Eq. 4.38), around the Hopf bifurcation points for the equilibrium at ¯q= ¯q2, are derived. Since there is a quadratic nonlinearity in Eq. 4.38) to take the normal form near the Hopf bifurcation point. The approximate DDE solution presented in Eq. 4.72), accurate toO(0), can now be written using the normal form equations (Eq.

Global bifurcation analysis

Of these, two stable limit cycles arise from the supercritical Hopf bifurcation arising at points P1 (red solid line) and P3 (brown solid line), and two unstable limit cycles arise from the subcritical Hopf bifurcation arising at points P2 (red dashed line ) and P5 (brown dotted line). However, if the operational conditions push the system into multiple limit cycle regions, the fatigue life calculations must be based on the worst case scenario of the limit cycle amplitudes. The inner and outer stable limit cycles have remarkably different amplitudes; this makes the behavior of the system in the vicinity of the larger unstable limit cycle highly unpredictable.

Figure 4.16: (a) Limit-cycles for U = 1 with (a) p 0 = 1, (b) p 0 = 13 and (c) p 0 = 20.

Chapter Summary

Weaver, “A theoretical model for fluid-elastic instability in heat exchanger tube bundles,” Journal of Pressure Vessel Technology, vol. Meskell, “Estimation of the time delay associated with the damping of a controlled fluidelastic instability in a normal triangular tube array,” Journal of Pressure Vessel Technology, vol. Li, “Cross-flow induced chaotic vibrations of heat exchanger tubes affecting loose supports,” Journal of Sound and Vibration , vol.