Effect of Inhomogeneous Heat Flow on the Enhancement of Heat Capacity in Helium-II by Counterflow near Tλ

Abstract

Abstract. In 2000 Harter et al. reported the first measurements of the enhancement of the heat capacity ΔC Q ≡C(Q)-C(Q=0) of helium-II transporting a heat flux density Q near T λ . Surprisingly, their measured ΔC Q was ~7-12 times larger than predicted, depending on which theory was assumed. In this report we present a candidate explanation for this discrepancy: unintended heat flux inhomogeneity. Because C(Q) should diverge at a critical heat flux density Q c , homogeneous heat flow is required for an accurate measurement. We present results from numerical analysis of the heat flow in the Harter et al. cell indicating that substantial inhomogeneity occurred. We determine the effect of the inhomogeneity on ΔC Q and find rough agreement with the observed disparity between prediction and measurement.Keywords: helium, superfluid, heat capacity, lambda transition. PACS: 67.40. Kh, 67.40.Pm, 64.60.Ht In order to evaluate the idea that unintended inhomogeneity of the heat flow in the Harter et al. [1] experiment might account for the discrepancy between measurement and predictions [2,3] of ΔC Q , we must estimate the heat flow field Q(r) in the helium-II. It is not difficult to show [4] that thermal counterflow in helium-II can be solved simultaneously with the diffusive heat flow in the enclosing experimental cell using a standard finite-element solver [5], if the helium-II is nondissipative, nonvortical, nearly isothermal, and free of net mass flow (J=0). These conditions should have been well-approximated in the Harter et al. experiment. Such a numerical model has been constructed and solved for the Harter et al. cell. The model geometry is shown in Fig. 1. Not visible at this scale is the model for the Kapitza boundary resistance R K : an artificial thin envelope of thickness δ=25 μm and thermal conductivity κ RK =δ/R K interposed between the helium and the cell walls.For best accuracy, the helium-II diffusion coefficient should be modeled as κ He =α(ρ s /ρ n ), where α is a large constant required to reduce the variation of the scalar superfluid velocity potential function [4]. However, Harter et al. had a very short cell (0.64 mm) and did not approach closer to T λ than ~0.5×10 -6 K, limiting the maximum variation of ρ s /ρ n over the height of their cell to ~10%. To reduce the number of required computations we have approximated ρ s /ρ n as constant and set κ He =10 6 W/cmK. Test reductions of κ He to 10 5 W/cmK changed calculated enhancements by only ~0.01%, verifying that κ He is sufficiently large.To within their measurement noise, Harter et al. found that t -α ΔC Q was linear in (Q/Q c ) 2 , where t=(T λ -T)/T λ is the reduced temperature. Keeping only the Cell model geometry with heat flow streamlines.The model is axisymmetric about r=0. Streamlines show heat flowing from the primary heater to the cooled surface. The principal cause of inhomogeneity of the heat flow in the helium is the "well" cut into the upper endplate to accommodate the diaphragm valve.