How evaporation and condensation lead to self-oscillations in the single-branch pulsating heat pipe

The self-oscillations observed in the single-branch pulsating heat pipe (SBPHP) are explained by showing the existence of a mechanical resonator excited by a self-driving force (feedback) through linear stability analysis, with experimental validation. The SBPHP is a tube closed at one end which is initially filled with water. The closed end is heated and a vapor bubble forms and reaches an equilibrium size. From this stable state, increasing the temperature of the heater temperature beyond a threshold leads to oscillations of the liquid plug sustained over time. We would like to understand where these oscillations come from and why they do not vanish over time due to the friction. A model of this system is formulated such that it can then be linearized and solved analytically for the motion over time. The solution shows that the coupling of the spring effect of the vapor and the inertia of the liquid plug leads to a spring-mass system, which can oscillate after a small perturbation. The evaporation and condensation taking place as the oscillations occur produce a change of the vapor pressure. The resulting force on the liquid plug is positive feedback; it injects energy into the spring-mass system and makes the oscillations unstable (startup) if greater than the friction. A criterion for startup is formally provided in the form of a dimensionless instability number derived from the model. An experimental apparatus is used to validate the theoretical prediction. The predicted effects of the tube's axial temperature gradient (destabilizing), of a thin film thermal resistance (stabilizing), and of the external pressure (stabilizing) are validated. The mass of the vapor and the friction force are measured during the startup and are shown to act as feedback and dissipation, respectively, as predicted by the theory. It is concluded that the main physical mechanism behind the instability in the SBPHP is now well understood. This provides a theoretical basis for the further development of pulsating heat pipes increasingly used for thermal management of electronics and harvesting of waste heat.

Figure 1

Schematic representation of the oscillations, starting at $t_0$, in the single-branch pulsating heat pipe.

Figure 2

(a) Control volumes and (b) forces (liquid control volume) applied.

Figure 3

Wall temperature profile along the $x$ axis and the thermal resistance model for the heat transfer leading to evaporation and condensation.

Figure 4

The equilibrium state of the SBPHP [based on Eq. (12)] can be stable (small perturbation leading to either overdamped or damped oscillation) for $\mathrm{\Pi }<1$ or unstable (small perturbation leading to sustain oscillations) for $\mathrm{\Pi }>1$, with the dimensionless number $\mathrm{\Pi }=\tilde{\sigma }/{\zeta }_f$, where $\tilde{\sigma }$ is the phase-change coefficient and ${\zeta }_f$ is the friction coefficient.

Figure 5

The variables evolving over one oscillation period are shown as well as phasor representations at $\tau =\pi /4$ for ${\zeta }_f=0.04$ and $\mathrm{\Pi }=1.01$. The interest is in the phasing between variables, so only the sinusoidal part of the solution is shown. (a) Evolution of meniscus position and velocity. (b) Components $\widetilde{\mathrm{\Delta }{P}_g}$, the spring force $\widetilde{F_{\mathcal{V}}}$, which is opposite to the meniscus position, and the feedback force $\widetilde{F_m}$, which is in phase with the velocity. (c) Evaporation rate, opposite to the meniscus location, which produces the force $\widetilde{F_m}$; it can be seen that $\widetilde{F_m}$ is delayed by a quarter of a period compared to the evaporation rate.

Figure 6

Schematic of the instrumented SBPHP experimental setup, not to scale.

Figure 7

Various quantities measured during the startup. The mass of the vapor $m_g$ is in phase with the velocity, so it acts as feedback; the friction force $F_f$ is opposed to the velocity, so it acts as dissipation. The dotted lines in the phasor graphs represent the directions of $x_i$ and ${\dot{x}}_i$.

Figure 8

Measured and theoretical position $x_i$ and mass of vapor $m_g$ in the early stage of the startup, as well as friction force $F_f$, all showing close agreement. The theoretical curves displayed are based on ${\zeta }_{f\omega }=0.06$ and $\tilde{\sigma }=0.104\pm 0.014$ (the uncertainty on $\tilde{\sigma }$ is shown by the gray shaded area). On the left-hand side, each quantity is represented as a phasor in the referential $x_i-{\dot{x}}_i$ (only the angle is considered here, not the amplitude). The angles vary only slightly over the duration 0.58–0.72 s. The averages are considered in the phasor diagram.

Figure 9

Energy analysis of the startup (same experiment as in Fig. 7). (a) Works from phase change ($W_{m,\beta ,\mathrm{cycle}}$) and friction ($W_{f,\beta ,\mathrm{cycle}}$) (the value $-W_{f,\beta ,\mathrm{cycle}}$ is displayed to make the comparison of the magnitude to $W_{m,\beta ,\mathrm{cycle}}$ easier). The work $W_{m,\beta ,\mathrm{cycle}}$ is positive, so phase change is injecting energy into the system; the work $W_{f,\beta ,\mathrm{cycle}}$ is negative, so friction is dissipating energy from the system into the environment. (b) Net work on the system per cycle: $W_{m,\beta ,\mathrm{cycle}}$ is initially greater than $W_{f,\beta ,\mathrm{cycle}}$, so the net work on the system is positive; the net work eventually gets close to 0 around $1.5\mathrm{s}$. (c) Total energy of the system. The energy of the system (as well as the oscillation amplitude) increases as long as the net work is positive and saturates around $1.5\mathrm{s}$.

Figure 10

Qualitative representation of the pressure $P_g$, temperature $T_g$, and density ${\rho }_g$ distributions in the vapor bubble along $x$.

Figure 11

(a) Discriminant $d$ (with $d<0$ required for complex conjugate roots), (b) dimensionless growth rate of the real root ${\lambda }_1$ [Eq. (A27a)], (c) growth rate of the oscillations $\tilde{\alpha }$ [Eq. (A30c)], and (d) dimensionless angular frequency $\tilde{\omega }$ [Eq. (A30b)] as surfaces, along $\tilde{\sigma }$ and ${\zeta }_f$.

Figure 12

Schematic of the experimental setup for variations of $L_{g,0}$, not to scale.

Figure 13

Experimental validation of the frequency for ranges of (a) vapor length $L_{g,0}$ (fixed liquid length $L_{\ell }=14$ cm) and (b) liquid length $L_{\ell }$ (fixed vapor length $L_{g,0}=7$ cm).