Transient heat transfer of superfluid ${}^4\mathrm{He}$ in nonhomogeneous geometries: Second sound, rarefaction, and thermal layer

Transient heat transfer in superfluid ${}^4\mathrm{He}$ (He II) is a complex process that involves the interplay of the unique counterflow heat-transfer mode, the emission of second-sound waves, and the creation of quantized vortices. Many past researches focused on homogeneous heat transfer of He II in a uniform channel driven by a planar heater. In this paper, we report our systematic study of He II transient heat transfer in nonhomogeneous geometries that are pertinent to emergent applications. By solving the He II two-fluid equations of motion coupled with Vinen's equation for vortex-line density, we examine and compare the characteristics of transient heat transfer from planar, cylindrical, and spherical heaters in He II. Our results show that as the heater turns on, an outgoing second-sound pulse emerges, within which the vortex-line density grows rapidly. These vortices attenuate the second sound and result in a heated He II layer in front of the heater, i.e., the thermal layer. In the planar case where the vortices are created throughout the space, the second-sound pulse is continuously attenuated, leading to a thick thermal layer that diffusely spreads following the heat pulse. On the contrary, in the cylindrical and the spherical heater cases, vortices are created mainly in a thin thermal layer near the heater surface. As the heat pulse ends, a rarefaction tail develops following the second-sound pulse, in which the temperature drops. This rarefaction tail can promptly suppress the thermal layer and take away the deposited thermal energy. The effects of the heater size, heat flux, pulse duration, and temperature on the thermal-layer dynamics are discussed. We also show how the peak heat flux for the onset of boiling in He II can be studied in our model.

Figure 1

Schematic diagrams showing the transient heat transfer in He II from (a) a planar heater, (b) a cylindrical heater, and (c) a spherical heater.

Figure 2

Experimental and simulated temporal profiles of the temperature increment $\mathrm{\Delta }T=T-T_{\infty }$ at (a) 1, 2, and 5.4 mm from the surface of a planar heater with $q_h=5$ W/${\mathrm{cm}}^2, \mathrm{\Delta }t=1$ ms, and a repetition rate of 0.2 Hz; and (b) at 1, 2, and 3 mm from the surface of a cylindrical heater with $q_h=6$ W/${\mathrm{cm}}^2, \mathrm{\Delta }t=1$ ms, $r_h=2.5$ mm, and a repetition rate of 2 Hz. The bath temperature is $T_{\infty }=1.4$ K.

Figure 3

Profiles of (1) temperature increment $\mathrm{\Delta }T$, (2) vortex-line density $L$, and (3) thermal energy density $W$ compensated by the ratio of the cross-section area $A\left(r\right)$ to the heater surface area $A_h$ in (a) planar, (b) cylindrical, and (c) spherical geometries at 1.78 K. In all cases, $q_h=23$ W/${\mathrm{cm}}^2, \mathrm{\Delta }t=0.5$ ms, and $r_h=2$ mm.

Figure 4

Profiles of the temperature increment $\mathrm{\Delta }T$ near the heater surface in (a) planar, (b) cylindrical, and (c) spherical geometries at 1.78 K. In all cases, $q_h=23$ W/${\mathrm{cm}}^2, \mathrm{\Delta }t=0.5$ ms, and $r_h=2$ mm.

Figure 5

Evolution of (a) the fraction of the heat energy $Q_{th}/Q$ deposited in the thermal layer and (b) the temperature increment $\mathrm{\Delta }T\left(r_h\right)$ at the heater surface at 1.78 K. In all three cases, $q_h=23$ W/${\mathrm{cm}}^2, \mathrm{\Delta }t=0.5$ ms, and $r_h=2$ mm.

Figure 6

Effect of the heater radius $r_h$ on the evolution of (a) $Q_{th}/Q$ and (b) $\mathrm{\Delta }T\left(r_h\right)$ in (1) the cylindrical and (2) the spherical heater geometries. In both cases, $T_{\infty }=1.78$ K, $q_h=24$ W/${\mathrm{cm}}^2$, and $\mathrm{\Delta }t=0.5$ ms.

Figure 7

Effect of the surface heat flux $q_h$ on the evolution of (a) $Q_{th}/Q$ and (b) $\mathrm{\Delta }T\left(r_h\right)$ in (1) the cylindrical and (2) the spherical heater geometries. In both cases, $T_{\infty }=1.78$ K, $\mathrm{\Delta }t=0.5$ ms, and $r_h=2$ mm.

Figure 8

Effect of the pulse duration $\mathrm{\Delta }t$ on the evolution of (a) $Q_{th}/Q$ and (b) $\mathrm{\Delta }T\left(r_h\right)$ in (1) the cylindrical and (2) the spherical heater geometries. In both cases, $T_{\infty }=1.78$ K, $q_h=22$ W/${\mathrm{cm}}^2$, and $r_h=2$ mm.

Figure 9

Effect of the bath temperature $T_{\infty }$ on the evolution of (a) $Q_{th}/Q$ and (b) $\mathrm{\Delta }T\left(r_h\right)$ in (1) the cylindrical heater geometry with $q_h=20$ W/${\mathrm{cm}}^2$ and (2) the spherical heater geometry with $q_h=22$ W/${\mathrm{cm}}^2$. In both cases, $\mathrm{\Delta }t=0.5$ ms and $r_h=2$ mm.

Figure 10

The temperature dependence of the coefficient ${\alpha }_V\left|{\mathbf{v}}_{\mathbf{ns}}\right|$ and the He II heat capacity $C_p$. The crosses mark the temperatures examined in Fig. 9.

Figure 11

The evolution of the He II state near the heater surface with a 1-m hydrostatic head pressure. In all the cases, $T=1.78$ K, $q_h=22$ W/${\mathrm{cm}}^2, \mathrm{\Delta }t=0.5$ ms, and $r_h=2$ mm. The circle and the crosses mark the start and the end time of the heat pulse, respectively.