Property correlations for supercritical oxygen

Why supercritical oxygen

A high-temperature superconducting (HTS) motor replaces the copper windings on its rotor with superconducting tape. The tape carries about ten times the current density with no resistive loss, so the motor is smaller and lighter for the same power. It only superconducts while it is cold, which means the motor needs a cryogenic cooling loop.

Fig. 1. The setting, redrawn from our ICEC 2014 paper. Coolant from a cryogenic system enters through the shaft and carries heat away from the superconducting windings on the rotor.

Liquid cryogens boil as they pick up heat, so the loop carries a two-phase flow, and the vapor takes far more room than the liquid: a liter of liquid nitrogen becomes about 695 liters of gas at room temperature. A supercritical fluid, held above both its critical temperature and pressure, stays in a single phase. Oxygen's critical point, 154.58 K and 5.043 MPa, sits above those of nitrogen (126.19 K, 3.396 MPa) and argon (150.69 K, 4.863 MPa), so it takes less cooling to get there.

The study was written with future superconductors in mind. The best HTS materials today lose superconductivity below about 135 K at ambient pressure, colder than oxygen's critical temperature.

Near the critical point

Just above the critical point, properties change faster than anywhere else. One kelvin above it on the critical isobar, density has fallen by 41%, thermal conductivity to a third, and specific heat by a factor of 78. Above the critical pressure the drop is gentler and moves to higher temperatures (Fig. 2).

Table 1. Oxygen at 5.043 MPa, at and 1 K above the critical temperature
Property	154.58 K (T_c)	155.58 K	Ratio
Density (kg/m³)	466.4	276.2	0.59
Viscosity (µPa·s)	33.76	21.46	0.64
Thermal conductivity (mW/(m·K))	104.4	34.98	0.34
Specific heat (kJ/(kg·K))	719.5	9.21	0.013

Fig. 2. Oxygen from the critical temperature to 210 K on 11 isobars, 5.043 to 6.043 MPa in 0.1 MPa steps, with the critical isobar highlighted. NIST reference data, as extracted for the thesis.

A thermal-hydraulic model of a cooling loop needs these properties at every temperature step. Interpolating a table this steep is slow and error-prone. A short equation removes both the table and the interpolation.

Correlations in temperature

For my M.Tech thesis I developed correlations for density, viscosity, thermal conductivity and specific heat along four isobars and validated them against NIST reference data. On the critical isobar, shown here, I took 501 points for each property from 154.58 K to 204.58 K in 0.1 K steps and fitted rational functions of temperature to them. Density and viscosity changed too sharply for one expression, so I split their ranges by trial: three pieces for density and two for viscosity. The first two points above the critical temperature were left out, which sets the lower limit of every correlation at 154.78 K.

ρ = ρ₀ + Tρ₁ + ρ₂T k = k₀ + k₁Tk₂ + T

Viscosity and specific heat take the same form as density, except for viscosity above 179.98 K, which uses the inverse of a quadratic in T. Each piece has three coefficients, tabulated in the thesis.

Table 2. Accuracy of the M.Tech correlations against NIST on the critical isobar, 154.78 to 204.58 K
Property	Pieces	Average error	Largest error
Density	3	0.24%	1.6% at 156.08 K
Viscosity	2	0.12%	2.2% at 154.78 K
Thermal conductivity	1	0.89%	4.9% at 154.78 K
Specific heat	1	5.4%	10.7% at 204.58 K

Average error here is the mean absolute relative error (AARE) over all 499 points. The largest errors sit right at the critical point, where every property is steepest, except for specific heat, which falls so far in the first kelvin that one expression cannot also follow its slow tail.

Cooling capacity

How much heat a coolant carries for a given flow and temperature rise is Q = V̇ρc_pΔT. Taking each fluid's density and specific heat at its own critical point and a 1 K rise, supercritical oxygen carries about a third more heat than nitrogen or argon at the same volumetric flow, across the 5.5 to 8 L/min range studied.

Fig. 3. Cooling capacity of three supercritical fluids at their critical points, for 8 L/min and a 1 K temperature rise. The ranking holds at every flow rate studied, since capacity scales linearly with flow.

This compares each fluid exactly at its critical point, where specific heat peaks, so it is a best case for all three rather than a design number.

Flow in a cooling pipe

To see the coolant at work, I modeled a cooling pipe 25 mm across and 250 mm long in COMSOL Multiphysics. Oxygen enters at its critical point, 154.58 K and 5.043 MPa, and the walls are held at 160 K, about 5.4 K warmer, as the heat load from the windings. I ran inlet velocities from 1 mm/s to 0.5 m/s.

Pipe section, flow from left to right: cold oxygen only in a short wedge at the inlet; the rest of the pipe is at the wall temperature. — **Fig. 4.** Temperature in a section through the pipe, with flow from left to right, from the thesis. At low velocity the oxygen warms to the wall temperature close to the inlet. From about 5 cm/s a cold core runs the whole length and the warming is confined to a thin layer at the walls.

The cold wedge reaches about a third of the way down the pipe. — **Fig. 4.** Temperature in a section through the pipe, with flow from left to right, from the thesis. At low velocity the oxygen warms to the wall temperature close to the inlet. From about 5 cm/s a cold core runs the whole length and the warming is confined to a thin layer at the walls.

Thermal conductivity to 500 K

CFD codes evaluate fluid properties in every cell at every iteration. Reference equations of state, such as those in NIST REFPROP, are accurate but expensive to call millions of times. The M.Tech correlation for conductivity reached only 50 K above the critical point, and its errors peaked near 5% exactly where conductivity changes fastest.

My current work is a single expression for the thermal conductivity of supercritical oxygen from the critical point to 500 K. The idea is to let density carry the near-critical behavior: conductivity spikes at the critical point because density does, so a polynomial in both temperature and density can follow the spike with only six terms.

k = z₀ + aT + bρ + cT² + dρ² + fTρ

with k in mW/(m·K), T in K and ρ in kg/m³. I fitted it to 3,455 NIST points along the critical isobar, from 154.58 to 499.98 K in 0.1 K steps, with a nonlinear least-squares surface fit (Levenberg–Marquardt) in OriginPro.

Fig. 5. Thermal conductivity on the critical isobar: NIST reference data (dots) and the six-term correlation (line). At the critical point itself the correlation gives 103.3 against NIST's 104.4.

Fig. 6. Error against NIST, plotted against distance from the critical point on a log scale to spread out the near-critical region. The shaded band is ±0.5%. The M.Tech correlation covers only the first 50 K.

Accuracy
R² = 0.9998 and an average error of 0.25% over all 3,455 points. 95% of points fall within ±0.5%; the largest error, 1.5%, is half a kelvin above the critical point.
Vs. the M.Tech fit
Over the M.Tech range, 154.78 to 204.58 K, the average error halves, from 0.89% to 0.45%, and the largest error drops from 4.9% to 1.5%, with one expression instead of a range-limited one.
Independent check
A feed-forward neural network trained in MATLAB on the same data, with temperature and density as inputs and ten hidden neurons, on a random 70/15/15 split for training, validation and testing.
Scope
The correlation holds along the critical isobar, 5.043 MPa. On one isobar density is itself a function of temperature, so the fit traces a path through temperature and density rather than a full surface. Extending it across pressure is the next step.

Status

The thermal conductivity correlation is being written up. I will post its coefficients here once the paper is out.

The earlier work is my M.Tech thesis, Development of correlation for thermophysical properties of supercritical oxygen to be used in high temperature superconducting motors (Lovely Professional University, 2015), guided by Raja Sekhar Dondapati and Gaurav Vyas. A paper on the correlations, with A. Kumar, P. R. Usurumarti and R. S. Dondapati, was accepted for an oral presentation at ICEC 25–ICMC 2014.