NUMERICAL STUDY OF THERMOVISCOUS EFFECTS IN . . . PHYSICAL REVIEW E 90, 043016 (2014)
FIG. 8. (Color online) Fifth-order polynomial fits (dashed light green lines) to IAPWS data 24–26 (full black lines) of ten thermodynamic
and transport parameters of water in the temperature range from 283 K (10 ◦C) to 323 K (50 ◦C).
the viscosity and the thermal conductivity have been obtained
using a central difference ∂η/∂ρ ≈ η(T,ρ + dρ) − η(T,ρ −
dρ)/(2dρ), with dρ = 0.001 kg m−3. The bulk viscosity is
taken from Holmes, Parker, and Povey 27, who extended the
work by Dukhin and Goetz 28. The former paper provides a
third-order polynomial fit in temperature to ηb, thus rendering
further fitting superfluous.
For each parameter, we extract 400 data points equally
spaced in the temperature range from 283 to 323 K at ambient
pressure p = 0.101 325 MPa. From these 400 values, we
only use every fourth point for the fitting of a fifth-order
polynomial, while the remaining 300 data points are used for
calculating the deviation of the fit from the data. The order
of the polynomial has been chosen as a tradeoff between low
deviation between fit and data, achieved at high polynomial
order, and low uncertainty in the polynomial coefficients,
achieved at low polynomial order. The fitted polynomial
coefficients are then truncated to a finite precision of 11
significant digits. The number of significant digits has been
chosen such that the finite precision of the fitting coefficients
does not result in larger deviations between fit and data. We
then calculated the relative deviation of the fit with respect
TABLE III. List of the coefficients Ci of the fifth-order polynomial fits to the IAPWS data for water 24–26, shown in Fig. 8. The
temperature dependence of each of the ten thermodynamic or transport parameters g with SI unit g is fitted in the temperature range from
283 K (10 ◦C) to 323 K (50 ◦C) by a polynomial of the form g(T˜ )/g = C0 + C1 T˜ + C2 T˜ 2 + C3 T˜ 3 + C4 T˜ 4 + C5 T˜ 5, with T˜ = T /(1 K). MD
and AD denote the maximum and average relative deviation, respectively, of the fit from the data. Text files with the polynomial coefficients
are provided in the Supplemental Material 37.
ρ cp cs κT αp
C0 −8.8657129122×10+3 2.9011416353×10+5 −3.1420310908×10+4 1.4812304299×10−7 −6.2142868017×10−1
C1 1.4835428749×10+2 −4.4547002259×10+3 4.4207693183×10+2 −2.2647220701×10−9 9.6088951712×10−3
C2 −8.9906362105×10−1 2.7826280608×10+1 −2.4555531077×10+0 1.3993170593×10−11 −5.9809671543×10−5
C3 2.7498750299×10−3 −8.7096216461×10−2 7.0411177557×10−3 −4.3493756936×10−14 1.8708110371×10−7
C4 −4.2441616455×10−6 1.3656418313×10−4 −1.0353739103×10−5 6.7914632960×10−17 −2.9366903765×10−10
C5 2.6348942330×10−9 −8.5786317870×10−8 6.1949139692×10−9 −4.2558992023×10−20 1.8495258230×10−13
MD 0.1×10−6 1.2×10−6 0.2×10−6 5.3×10−6 1.244×10−4
AD 0.0×10−6 0.3×10−6 0.1×10−6 1.4×10−6 0.119×10−4
γ η kth ∂η/∂ρ ∂kth/∂ρ
C0 8.6095341563×10+1 3.8568288635×10+0 −4.5378052364×10+1 −3.7043919290×10−2 3.3029688424×10−1
C1 −1.3297233662×10+0 −6.0269041999×10−2 7.0158464759×10−1 5.8520864113×10−4 −4.6998644326×10−3
C2 8.3329451753×10−3 3.7823493660×10−4 −4.3372914723×10−3 −3.7051700834×10−6 2.7065291331×10−5
C3 −2.6194143444×10−5 −1.1905791749×10−6 1.3522238266×10−5 1.1749472916×10−8 −7.8584696013×10−8
C4 4.1304638557×10−8 1.8785807901×10−9 −2.1181858883×10−8 −1.8657255651×10−11 1.1504432763×10−10
C5 −2.6113941825×10−11 −1.1881783857×10−12 1.3309059289×10−11 1.1866022487×10−14 −6.7916212133×10−14
MD 1.6×10−6 1.379×10−4 1.0×10−6 15.459×10−4 0.8×10−6
AD 0.4×10−6 0.227×10−4 0.3×10−6 3.851×10−4 0.1×10−6
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