EPAPS SUPPLEMENTARY MATERIAL AT [AIP URL]
APPENDIX 2: NUSSELT NUMBER CALCULATION FOR MIXED GAS FLOWS OVER
A CANTILEVER
A Nusselt number (Nu) correlation function for presumed laminar flow over an
isothermal rectangular plate can be written in terms of the Reynolds (Re) and Prandtl (Pr)
numbers:1


Nu   F 1.42   0.742 Re L W





1
2
1
1.42 1.42

Pr 

1
3




(1)
where F is a shape factor appropriate to the cantilever aspect ratio L/W:
F

W
1 L W
L 

2
(2)
The dimensionless flow parameters Re and Pr can then be computed directly from their
definitions:
Re 
Pr 
 v LW

(3)
cp
(4)
k
where  is the mass density (kg/m3), cp is the specific heat capacity at constant pressure (J/kgK),
 is the viscosity (Pas), k is the thermal conductivity (W/m·K), and v is the freestream flow
speed (m/s). The flow speed in the direct vicinity of the cantilever array was estimated by first
dividing the measured volumetric flow rate by the external hose cross-sectional area (~2 mm2);
then, the reduced flow speed within the dilated flow cell cavity was computed using the wellknown continuity equation from fluid mechanics, and the internal dimensions of the gas cell.
The calculated values of v ranged from about 2 to 10 cm/s.
1
To estimate the quantities , cp, , and k for the various binary gas mixtures, two
approaches were utilized. The mixture values for  and cp were determined by summing the
pure component values (gas and N2 diluent) scaled by their molar fractions x1 and x2 = 1 – x1
(i.e., linear mixing):
  x1 1  x2  2
(5)
c p  x1c p1  x 2 c p 2
(6)
These mixing correlations do not account for minor, non-ideal corrections such as
compressibility or molecular non-sphericity. The pure component densities were computed via
the ideal gas law. For each pure gas constituent, the specific heat capacity cp at temperature T
was calculated using the Shomate equation, with empirical parameters obtained from the NIST
chemistry database.2
The values of  and k for the binary mixtures were modeled using a formula originally
attributed to Sutherland3 and Wassiljewa,4 respectively:
1


x 
x
   11  p 2     21  q 1
x1 
x2


1


x 
x
k   k11  p 2    k 21  q 1
x1 
x2








1
(7)
1
(8)
These formulas (and variations thereof) have been widely used to approximate the viscosity and
thermal conductivity of gas mixtures, with typical errors from the experimentally observed
values of a few percent or less (see, for example, Refs. 5-9). For simplicity, the same empirical
parameters p and q were used for both  and k, described by Sutherland as:3
2
3
8
3
8
 M1 
M 
1  M  M 2 

 p   2  q   1
4  4M 1 M 2
 M2 
 M1 
2
1
1


4
  1  2M 1  4


2M 2
1

 

 
 
2  M1  M 2  
  1  M 1  M 2 


1
8
2
(9)
where M1 and M2 represent the molar masses (kg/mol) of the pure components. The viscosity 
for each pure gas constituent at temperature T was calculated using the empirical equation:
 T

 0  T0
3
 2  T0  S 
 

 T S 
(10)
where S is the gas-specific Sutherland constant.10 A look-up table of S values was constructed
from Eq. 10 for a series of viscosities and corresponding temperatures obtained from the CRC
handbook;11 then, the Sutherland constant for each T was estimated by linear interpolation
between the neighboring temperature values in the look-up table. Finally, the values of (T)
were calculated from Eq. 10 using the reference viscosity 0 at T0 = 273 K.
For each pure gas constituent, the thermal conductivity k at temperature T was calculated
using the Eucken formula:12
k
159  5k b
128d 2
k bT
m
(11)
where  is the heat capacity ratio, kb = 1.38110-23 J/K is the Boltzmann constant, d is the
nominal molecular diameter (m), and m is the molecular mass (kg). Using values of k(T)
obtained from the CRC handbook,11 the diameter d was computed using Eq. 11; then, using this
value as a temperature-independent constant, the pure gas values k were calculated at T using the
same equation.
Solving Eqs. 1–11, the Nusselt numbers for gas flow across the cantilever were found to
be between 2.4 and 2.9 in value, which implies that convective heat transfer slightly dominates
over direct conduction. We also note that the computed values of Re from 0.01 to 1.1 support
3
the original assumption of laminar flow conditions. Another informative quantity that can be
calculated is the Grashof number (Gr), which is the ratio of buoyancy and viscous forces within
the gas:13
Gr 
g Tc  T  2  3gas
(12)
T 2
where g is the gravitational acceleration constant (9.80 m/s2), Tc is the cantilever temperature,
and lgas is the thermal boundary layer thickness over which Tc-T exists. For the gas flows in this
study, Gr values between 7.8×10-6 and 1.4×10-5 were computed. The Archimedes number Ar =
Gr/Re2 which represents the relative strength ratio of free (buoyancy driven) to forced (flow
driven) convection can then be computed, and is found to be between 1.4×10-3 and 2.2×10-3,
demonstrating that the convective heat transfer is primarily flow driven. This assessment is
consistent with thermal conduction analyses by Kim and King of heated cantilevers in quiescent
air environments, where free convection is discounted as a significant mechanism of heat
transfer.14
Therefore, since conductive and convective modes of heat dissipation are of comparable
influence, they must both be included in the model calculations for the total thermal resistance.
REFERENCES
1
M. M. Yovanovich and P. Teertstra, J. Thermophys. Heat Transf. 15, 205-211 (2001).
2
http://webbook.nist.gov/chemistry/
3
W. Sutherland, Philos. Mag. 40, 421-431 (1895).
4
A. Wassiljewa, Physik. Z. 5, 737-742 (1904).
5
E. A. Mason and S. C. Saxena, Phys. Fluids 1, 361-369 (1958).
6
D. Burnett, J. Chem. Phys. 42, 2533-2540 (1965).
4
7
A. G. Shashkov, T. N. Abramenko, N. A. Nesterov, R. K. Joshi, R. Afshar, and S. C. Saxena,
Chem. Phys. 29, 373-381 (1978).
8
F. W. Giacobbe, J. Acoust. Soc. Am. 96, 3568-3580 (1994).
9
P. T. Tsilingiris, Energy Conv. Manag. 49, 1098-1110 (2008).
10
W. Sutherland, Philos. Mag. 36, 507-531 (1893).
11
CRC Handbook of Chemistry and Physics; Vol., edited by D. R. Lide (CRC Press, Boca Raton,
FL, 2005).
12
E. H. Kennard, Kinetic Theory of Gases, With an Introduction to Statistical Mechanics, 1st ed.
(McGraw-Hill, New York, 1938).
13
F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavine, Introduction to Heat Transfer,
5th ed. (John Wiley & Sons, Inc., Hoboken, NJ, 2007).
14
K. J. Kim and W. P. King, Appl. Therm. Eng. 29, 1631-1641 (2009).
5