Compelling Augmentation of Laminar Natural Convection from
Vertical Plates to Binary Gas Mixtures Formed with Light Helium and
Selected Heavier Gases
1ANTONIO CAMPO, 2M. MEHDI PAPARI
1Department of Mechanical Engineering, The University of Vermont Burlington, VT 05405, USA
2Department of Chemistry, Shiraz University of Technology, Shiraz, 71555‒313, IRAN
Abstract: The present study addresses a remarkable behavior of certain binary gas mixtures in connection to
laminar natural convection along a heated vertical plate at constant temperature. The binary gas mixtures are
formed with light gas helium (He) as the primary gas and selected heavier secondary gases. The heavier
secondary gases are nitrogen (N2), oxygen (O2), xenon (Xe), carbon dioxide (CO2), methane (CH4),
tetrafluoromethane (CF4) and sulfur hexafluoride (SF6). The central objective in the work is to investigate the
attributes of the set of seven He‒based binary gas mixtures in reference to heat transfer enhancement with
respect to helium (He) and air. From heat convection theory, four thermophysical properties: viscosity η, density
ρ, thermal conductivity λ, and isobaric heat capacity Cp affect the thermo‒buoyant convection of fluids. It
became necessary to construct a particular correlation equation consigned to the set of seven He‒based binary
gas mixtures, which operate in the Prandtl number closed sub‒interval [0.1, 1]. The heat transfer rate Qmix with
the seven He‒based binary gas mixtures from the vertical plate involves four thermophysical properties: density
ρmix, viscosity ηmix, thermal conductivity λmix, and isobaric heat capacity Cp, mix that vary with the molar gas
composition w. A case study is performed to elucidate the unique characteristics of the modified convective
heat transport that the set of seven He‒based binary gas mixtures bring forward as compared to the convective
heat transport of pure He and air. It was found that the He+SF6 binary gas mixture renders an absolute maximum
heat transfer enhancement ratio that is 39 times higher than the heat transfer enhancement ratio provided by
pure He and 78 times higher than air.
Key‒words: Natural convection; vertical plate; laminar boundary layer flow; light helium is the primary gas;
selected heavier gases are the secondary gases; binary gas mixtures; heat transfer enhancement ratio.
Received: May 12, 2022. Revised: October 13, 2022. Accepted: November 22, 2022. Published: December 26, 2022.
Highlights:
Laminar natural convection along a vertical plate
Momentum and thermal boundary layer flows
Binary gas mixtures formed with light helium gas and selected heavier gases
Heat transfer enhancement generated by the better helium‒based binary gas mixtures
1 Introduction
The art and science of heat transfer enhancement
has evolved into an important component of thermal
science over the last decades. State-of-the-art
review articles on heat transfer enhancement
authored by Bergles [1] and Manglik [2] have
appeared regularly in specialized journals and
handbooks on heat transfer. According to Bergles
[1], the heat transfer enhancement schemes can be
classified either as passive, which require no direct
external power, or active, which do require external
power.
The effectiveness of both schemes is strongly
dependent on the heat transfer mode, which may
range from single‒phase natural convection to two‒
phase boiling and two‒phase condensation passing
through single‒phase forced convection. A great
amount of effort has been devoted to search for heat
transfer enhancement in forced convection flows
during the past decades, but unfortunately the effort
pertinent to heat transfer
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enhancement in natural convection flows has
remained practically unnoticed.
With regards to passive schemes in solid
media, Bhavnani and Bergles [3] measured the
average heat transfer coefficients for vertical
plates owning an incrusted bundle of transverse
ribs exposed to air. The authors determined that
heat transfer enhancement ratio relative to a
similar plain vertical plate of equal projected
area was possible using transverse ribs of
certain size. Thereby, the maximum heat
transfer augmentation ratio reached 23%.
With regards to passive schemes in fluid
media, a distinction should be made between
liquids and gases.
First in the case of liquids, Kitagawa et
al. [4] carried out an experimental study to
investigate the effects of sub‒millimeter air
bubble injection on the heat transfer
characteristics of laminar natural convection of
water in contact with a heated vertical plate. The
simultaneous measurements of velocity and
temperature demonstrated that the heat transfer
enhancement ratio is directly affected by the
flow modification due to the rising air bubbles
near the heated vertical plate. The ratios of the
convection heat transfer coefficient with
injection to the convection heat transfer
coefficient without injection, intensifies with
increments in the air bubble flow rate. This
phenomenon translated into modest heat transfer
enhancement ratios that ranged from 1.35 to
1.85.
Natural convective boundary layer flows
of a nano fluid past a vertical plate was analyzed
by Kuznetsov and Nield [5] using a standard
similarity methodology of the conservation
equations followed by numerical computations.
The outcome of the analysis included velocity,
temperature and solid volume fraction of the
nanofluid in their respective boundary layers.
An enlargement in the heat transfer performance
with respect to the case of the base fluid was
found for most cases treated.
Second in the case of gases, Petri and
Bergman [6] conducted a dual numerical and
experimental investigation on natural convection
heat transfer next to a vertical plate using as a
coolant a binary gas mixture in laminar regime.
The authors found that a helium‒rich binary gas
mixture seeded with a small amount of xenon
yielded higher heat transfer rates relative to
those situations associated with pure helium. In
sum, the authors concluded that the convective
heat transfer rates are increased by a factor of
8% induced by the seeding of light helium with
heavy xenon. Arpaci [7] investigated the
coupling of natural convection and thermal
radiation from a vertical plate to a stagnant gray
gas theoretically. The study employed the
integral formulation for the two limiting thin
and thick gas approximations.
The objective of the present work is to
investigate the behavior of several binary gas
mixtures composed by light helium as the
primary gas and heavier secondary gases
associated with thermaldriven boundary layers
along a heated vertical plate at constant
temperature Tw. The central idea is to explore the
interaction between the density ρmix, viscosity
ηmix, thermal conductivity λmix and isobaric heat
capacity Cp,mix of the binary gas mixtures
composed with light He and heavier gases. The
heavier gases are seven: nitrogen (N2), oxygen
(O2) xenon (Xe), carbon dioxide (CO2), methane
(CH4), sulfur hexafluoride (SF6),
tetrafluoromethane and carbon tetrafluoride
(CF4).
Heat convection theory (Jaluria [8]
stipulates the following relation
2. Heat Transport by the Natural
Convection Mechanism
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Natural convection mechanism =
heat conduction mechanism +
fluid motion due to thermo‒
buoyant forces
. Within the heat conduction mechanism,
the heat transfer enhancement is simple because
the immobile air can be replaced with an
immobile gas having higher thermal conductivity
λ, like light helium (He). Contrarily, the heat
transfer enhancement promoted by the natural
heat convection mechanism is more complicated
because the acting thermo‒buoyancy forces need
to be stimulated as a whole. Focusing on this
aspect, a viable option strongly insinuates the
exploration of better gases. For instance, binary
gas mixtures formed with light helium (He) as the
primary gas and heavier nitrogen (N2), oxygen
(O2) xenon (Xe), carbon dioxide (CO2), methane
(CH4), sulfur hexafluoride (SF6), and
tetrafluoromethane or carbon tetrafluoride (CF4)
as secondary gases.
The heat transfer rate Q from a heated
vertical plate to a surrounding cold fluid is given
by Newton's "equation of cooling" (Arpaci [9]):
TTAhQ wS
(1)
where
h
is the average convective coefficient, As
is the surface area of the vertical plate and
is the platetofluid temperature
difference.
The estimation of the average convective
coefficient
h
for a laminar boundary layer in Eq.
(2) depends on the temperature gradient of the
fluid in contact with the vertical plate, namely
T' (0) (Jaluria [8]). For the thermal boundary
layer, the similarity variables θ and η are defined
by the ratios
(2)
where H is the height of the vertical plate.
For laminar boundary layer up‒flows
along a heated vertical plate, Ostrach [10]
calculated numerically the velocity and
temperature fields of a multitude of fluids with
Prandtl numbers ranging from Pr = 0.01 to 1,000.
Subsequently, the author developed the so‒called
Prandtl number function
)0('
= g (Pr) that led
to the comprehensive correlation equation for the
average Nusselt number
H
Nu
:
(Pr)gRaNu 4/1
H
H
(3)
which complies with the the Boussinesq
approximation (Jaluria [8]). Herein, the
thermophysical properties of the fluid are evaluated
at the film temperature
. The
coefficient of volumetric thermal expansion for
ideal gases is
where
is in absolute
degrees.Contrarily, β must be obtained
experimentally for non ideal gases and liquids
(Jaluria [8]).
As a point of reference, Bejan and Lage
[11] have demonstrated that the Grashof number
GrH (and not the Rayleigh number RaH) marks the
transition from laminartoturbulent natural
convection flows adjacent to vertical plates. The
critical Grashof number recommended stays
around GrH,cr = 109.
3. Correlation Equations Linked to
the Prandtl Number
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Focusing on the Prandtl number spectrum
0 < Pr < ∞, there are two extreme limits: one limit
deals with Pr → 0 for metallic liquids and the
other limit deals with Pr → ∞ for oils and thick
liquids. Within this vast scenario, LeFevre [12]
discovered that the average Nusselt number
H
Nu
adheres to the following two asymptotes
0PrasPrRa8.0Nu 4/1
H
H
(4a)
PrasRa671.0Nu 4/1
H
H
(4b)
The specialized heat convection literature
contains comprehensive correlation equations for
the average Nusselt number
H
Nu
= f (RaH, Pr)
covering the entire Pr spectrum 0 < Pr < ∞. For
instance, in the laminar regime restricted to
, Churchill and Chu [13] developed
the correlation equation
(4c)
with the companion Prandtl number function
(4d)
According to Martynenko and Polijakov
[14], Eqs. (4c) and (4d) can also be used for
liquid metals (Pr < 0.1) when RaH is substituted
by GrH Pr2.
It is convenient to scrutinize the individual
dependence of fluids on the Prandtl number as it
relates to the Prandtl number function g (Pr) in
the double‒valued function of Eq. (3) from
Reference [11]. More specifically, four
subgroups can be categorized with regards to
Prandtl number fluids:
(1) metallic liquids with Pr << 1;
(2) air, pure gases and vapors with Pr ≈ 1;
(3) water and light liquids with Pr around 10
(4) oils and thick liquids with Pr >> 1.
First and foremost, a specific correlation
equation for the binary gas mixtures owing
)1,1.0(Pr
has to be constructed from the
original data in the broad Prandtl number
spectrum spanning from 0.001 (liquid metals) to
1000 (oils and heavy liquids. In this sense, the Pr
data taken from Reference [10] was re‒tabulated
by Suryanarayana [15]. The Pr data is analyzed
with regression analysis using the software
TableCurve [16]. The outcome of the regression
analysis delivers the following four‒part
correlation equations:
Pr,Pr8.0
]100,1[Pr,Pr549.0
]1,01.0[Pr,Pr549.0
0Pr,Pr671.0
(Pr)g
0
044.0
171.0
4/1
(5a,b,c,d)
Herein, the correlation coefficients Rsquare are
high ~ 0.999 and the local maximum error of
3.58% occurs around Pr = 0.1.
In the definitive Pr classification
exposed in Eqs. (5), it can be seen in more detail
that Eq. (5a) handles liquid metals, Eq. (5b)
handles air, regular gases, and vapors, Eq. (5c)
handles water and light liquids and Eq. (5d)
handles oils and thick liquids.
If a new subgroup for binary gas
mixtures denoted Prmix is added to the entire Pr
spectrum, then a fifth subgroup is squeezed
inside the Pr sub‒interval [0.1,1] in Eq. (5b), but
closer to Pr = 1. Fundamentally, the Prandtl
number of binary gas mixtures Prmix occupies
the narrow subinterval [0.1, 1] as seen in
Figure 2 taken from Campo and Papari [17]. It
is observable in the figure that the minimum
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Prmix for the binary gas mixtures are: 0.1 for the
He+Xe binary gas mixture, 0.2 for the He+CF4
binary gas mixture and 0.3 for the He+SF6 binary
gas mixture.
Liu and Ahlers [18] performed an in‒
depth investigation on binary gas mixtures to
provide an explanation for the nature of
“anomalously” low values of Prandtl number.
Using a combined methodology based on
statistical‒mechanical theory and experimental
measurements of other researchers along with
their own findings, the authors have
demonstrated the possible existence of lower
Prandtl numbers for a hydrogen–xenon binary
gas mixture decreasing down to 0.16.
3.1 Specific correlation equation for the
average convective coefficient of binary gas
mixtures
Heat convection theory (Jaluria [8])
dictates that laminar boundary layer flows
promoted by thermobuoyant forces depend
upon four thermophysical properties: dynamic
viscosity η, thermal conductivity λ, density ρ,
and isobaric heat capacity Cp. In the particular
case of binary gas mixtures, the four
thermophysical properties are adequately
re‒expressed by ηmix, λmix, ρmix and Cp, mix.
Eq. (3) particularized to the proper
sub‒interval [0.1, 1] stated in Eq. (5b), along with
the corresponding Prandtl number function g (Pr)
= 0.549Pr 0.171 delivers the correlation equation
(6)
Isolating the average convective
coefficient
h
in Eq. (6), the equation
representative of the average convective
coefficient of a binary gas mixture
mix
h
turns out
to be
08.0
mix
42.0
mix,p
50.0
mix
58.0
mix
mix
C
B/h
(7)
where the subscript ‘mix’ signifies binary gas
mixture.
The overall thermo‒geometric parameter
B in Eq. (7) absorbs four quantities: the
acceleration of gravity g, the height of the
vertical plate H, the plate temperature Tw and
the free‒stream temperature of the binary gas
mixture . That is
25.0
w
H
)TT(g
549.0B
(7a)
with units . Alternatively, using the
coefficient of volumetric thermal expansion
for ideal gases, B could be rewritten
compactly as
(7b)
As mentioned in the Introduction, the
main objective of the study is to explore the
possible intensification of the heat transfer rate Q
in laminar natural convection gas flows past a
vertical plate. If the surface area of the vertical
plate As in Eq. (2) remains unchanged and the
platetofluid temperature difference
is constant, the only possible way for
augmenting Q is by enlarging the magnitude of
h
Thereby, the analysis of binary gas mixtures
formed with light He and selected heavier
secondary gases will be pursued in this work.
Let us pause here momentarily to
describe the ideal values of the thermophysical
properties in the binary gas mixture should
possess in harmony with the structure of Eq. (7).
Thereby, it is evident that in order to intensify
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the heat transfer rate
mix
Q
for laminar natural
convection with binary gas mixtures along a
vertical plate, the trio of thermophysical
properties λmix, ρmix, and Cp,mix in the numerator
of Eq. (7) must be large and/or the single
thermophysical property ηmix in the denominator
of Eq. (7) must be small.
3.2 Thermophysical property called the
thermal effusivity
From an in‒depth perspective, it is
observable that
mix
h
in Eq. (7) varies directly
proportional with the square root of the product
λmix x ρmix x Cp, mix and inversely proportional
with the 0.10 power of ηmix. The numerator may
be viewed through the optic of the
thermophysical property called the thermal
effusivity ε (Grigull and Sandner [19]):
In particular, the thermal effusivity of the
He‒based binary gas mixtures related to
laminar natural convection along a vertical plate
may be expressed in an approximate manner as
Accordingly, Eq. (7) could be rewritten
compactly in terms of the thermal effusivity
as follows
(7c)
with minimal contribution from
To form the binary gas mixtures with
light He as the primary gas, the seven heavier
secondary gases are: N2, O2, Xe, CO2, CH4, CF4
and SF6. The molar mass M of the participating
gases are taken from Poling et al. [20] and are
listed in Table 1. In addition, the molar mass
difference M of the eight participating gases
are listed in Table 2.
Table 1. Molar mass of the pure gases (in
ascending order)
Gas
M (g/mol)
He
4.00
CH4
16.04
N2
28.01
O2
32.00
CO2
44.01
CF4
88.00
Xe
131.29
SF6
146.06
Table 2. Molar mass difference M (in ascending
order) of the He‒based binary gas
mixtures
Binary gas mixture M (g/mol)
He+CH4 12.04
He+N2 24.01
He+O2 28.00
He+CO2 40.01
He+CF4 84.00
He+Xe 127.29
He+SF6 142.06
Notice that the three secondary gases
having the largest molar mass factors with
respect to the primary gas He are: CF4 with a
4. Thermophysical Properties of
Binary Gas Mixtures
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factor of 20, Xe with a factor of 30 and SF6 with
a factor of 40 approximately.
4.1 Molar gas composition
The molar gas composition wi of a binary
gas mixture is defined as the mass fraction of pure
gas i
2,1ifor,
M
M
xw
mix
i
ii
(8)
where xi is the mole fraction, Mi is the molar mass
of pure gas i. Here, the molar mass of the binary gas
mixture (Poling et al. [20]) is defined by
Mmix= x1M1 + x2M2 (8a)
The molar mass of the chosen pure gases is listed in
Table 1.
4.2 Density
The density of a binary gas mixture mix at
low pressure is determined with the truncated virial
equation of state (Poling et al. [20]):
mix2
mix
B21
TR
p
Z
(9)
where Z is the compressibility factor, p is the
pressure, R is the gas constant. Herein, the second
virial coefficient B2 is evaluated with the simple
correlation attributed to Tsonopoulos [21]
)1()0(
c
c2 BB
TR
pB
(10)
where the Pitzer acentric factor , the critical
temperature Tc and the critical pressure pc for the
pure gases are taken from Poling et al. [20]. The
numerical values of B(0) and B(1) are computed from
the pair of relations:
8
r
3
r
2
rr
)0(
T
000607.0
T
0121.0
T
1385.0
T
33.0
1445.0B
(11a)
and
8
r
3
r
2
r
)1(
T
008.0
T
423.0
T
331.0
0637.0B
(11b)
where the temperature ratio Tr = T/Tc.
4.3 Isobaric heat capacity
The isobaric heat capacity of a binary gas
mixture Cp, mix at low density obeys the mixing rule
(Poling et al. [20]):
i
0
i,pimix,p CxC
(12)
where xi denotes for the mole fraction of pure gas i.
The isobaric molar heat capacity of the pure gas i
identified by
0
i,p
C
satisfies the equality
C0
p C0
v = R (13)
in which C0
v being the molar heat capacity at
constant volume of pure gas i, is quantified from
k
1j
2
vj
vj
2
vj
0
v
T
exp
T
exp
T
S
R
C
(14)
In this equation, the symbol j stands for the
characteristic vibrational temperature corresponding
to the vibrational degree of freedom j. In addition, S
equates to 5/2 for linear molecules and to 3 for
nonlinear molecules as noted by McQuarrie [22].
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(Hirschfelder et al. [23] and Chapman and
Cowling [24]), the viscosity of a binary gas
mixture mix is calculated with the matrix
formula:
BBAB
ABAA
BA
BBBBA
AABAA
mix
HH
HH
0xx
xHH
xHH
(15)
First, the element HAA along the main diagonal
in the two matrices is:
A
B
*
AB
2
BA
BA
AB
BA
A
2
A
AA m
m
A3
5
)mm(
mmxx2x
H
(15a)
Second, the element HBB along the main
diagonal of the two matrices is deduced from
the expression for HAA by interchanging the
subscripts A and B. Third, the elements located
off the main diagonal in the two matrices are:
1
A3
5
)mm(
mmxx2
)BA(H *
AB
2
BA
BA
AB
BA
AB
(15b)
Further, the interaction viscosity AB appearing
in the two preceding equations (15a) and (15b)
is given by the formula
)T(
1kT
mm
mm2
16
5
*
AB
)2,2*(
AB
2
AB
2/1
BA
BA
AB
. (15c)
in which the subscript A identifies the heavier
gas and the subscript B identifies the light gas,
the pair mA and mB designate the masses of A
and B and the pair xA and xB are the mole
fractions of A and B. In addition, AAB*
represents for the ratio collision integral at
,
which is defined by Bzowski et al. [25].
4.5 Thermal conductivity
For the thermal conductivity of a binary
gas mixture mix, Schreiber at al. [26] developed
the matrix formula:
BBAB
ABAA
BA
BBBAB
AABAA
mix
LL
LL
0xx
xLL
xLL
(16)
To save journal space, the elements LAA, HAB
and LBB in the upper and lower matrices in Eq.
(16) are not written, They are available in
Reference [26]. Besides, the pair xA and xB are
the mole fractions of A and B.
When the accurate formulas for the four
thermophysical properties mix (w) in Eq. (9), mix
(w) in Eq. (12), mix (w) in Eq. (15) and Cp,mix
(w) in Eq. (16) at a given pressure p and
temperature T are introduced into Eq. (6), the
average convective coefficient of a binary gas
mixture
mix
h
respond to continuous changes in
the molar gas composition w of the seven binary
gas mixtures He+N2, He+O2, He+Xe, He+CO2,
He+CF4, He+CH4, He+SF6 in the proper
w‒domain [0, 1].
The methodical algebraic calculations for
the estimation of
mix
h
related to the seven binary
gas mixtures are carried out with small steps w
= 0.01 utilizing the spreadsheet software Excel
[27].
4.4 Viscosity
Based on the Kinetic Theory of Gases
5. Heat Transfer Analysis for
Binary Gas Mixtures
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The four thermophysical properties of
The seven binary gas mixtures He+N2, He+O2,
He+Xe, He+CO2, He+CH4, He+CF4 and He+SF6
exhibit a variety of curve shapes. Parabolic
down curves λ (w) from primary light He (w =
0) to the heavier secondary gases (w = 1) are
shown in Figure 3. There are curves up and
curves down η (w) from light He (w = 0) to the
heavier gases SF6 (w = 1), CF4 (w = 1) and O2
(w = 1) are shown in Figure 4. The other binary
gas mixtures He+N2, He+Xe, He+CO2, He+CH4
follow parabolic down curves. Negative sloped
straight lines Cp (w) from light He (w = 0) to the
heavier gases (w = 1) are displayed in Figure 5.
The exponentially increasing curves ρ(w) from
light He (w = 0) to the heavier gases (w = 1) are
illustrated in Figure 6.
5.1 Maximum heat transfer rates rendered by
the light He‒based binary gas mixtures
The objective function is the relative heat
transfer rate
mix
Q
(w)/B associated with Eq. (2).
The goal of the sub‒section is to search
for the absolute maximum of the objective
function. Conceptually, a point w = wopt is an
absolute maximum of a singlevalued function
B/)w(Qmix
if
B/)w(Q optmix
B/)w(Qmix
(17)
for all w values inside the wdomain [0, 1]
(Sioshansi and Conejo [28]). In other words,
the location of the optimal molar gas
composition wopt corresponds to the w value that
renders an absolute maximum
B/Q maxabs,mix
for
all computed ranges of
B/)w(Qmix
contained in
the wdomain [0, 1]. Besides the absolute
maximum, the other maxima are called relative
maxima.
Table 3. Values of the thermophysical properties
at T = 300K and p = 1 atm. (The highest and
lowest values are highlighted)
Gas
M
g/mol
ρ_
kg/m3
_
Pa.s
mW/(m.K)
Cp
0
J/(kg.K)
He
4.00
0.1624
19.92
155.70
5199.11
CH4
16.04
0.6553
11.19
34.89
2230.47
N2
28.01
1.1379
17.96
25.88
1039.66
O2
32.00
1.3004
20.78
26.64
918.21
CO2
44.01
1.7964
15.08
16.79
848.40
CF4
88.00
3.5410
17.34
15.16
699.36
Xe
131.29
5.3610
23.20
5.52
158.49
SF6
146.06
5.8585
29.70
13.20
671.39
The presentation and discussion of results
will be conveniently divided in two parts.
6.1 Natural convection heat transfer using
light helium
First, it was deemed appropriate to
contrast the natural convection heat removal
with light He against standard air before going
to the binary gas mixtures formed with primary
light He and the secondary heavier gases. In
this respect, using Eq. (7) at a temperature of
300K, the heat transfer enhancement ratio Eht
caused by light He with respect to air is
(18)
6. Presentation and Discussion
of Results
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The physical explanation for this behavior is
that the thermal conductivity of He is eight
times larger than the thermal conductivity of air,
the viscosities of He and air are about the same
and the isobaric heat capacities of He and air are
very close.
6.2 Natural convection heat transfer using
the He‒based binary gas mixtures
Second, the natural convection heat
removal by the seven He‒based binary gas
mixtures is discussed next. At this point, it is
instructive to highlight the characteristics of
the four thermophysical properties for He. In
comparison with the seven secondary gases, it
is observable in Table 3 that He possesses the
highest thermal conductivity He = 155.70
mW/m.K, the highest isobaric heat capacity
Cp, He = 5199.11 J/kg.K, but the lowest density
ρHe = 0.1624 kg/m3 and an intermediate
viscosity He = 19.92 Pa.s. Therefore, in
reference to Eq. (7a) the lowest ρHe value needs
to be compensated with a high ρ value coming
from the heavier secondary gases N2, O2, Xe,
CO2, CH4, CF4 and SF6. In addition, the
intermediate He value needs to be compensated
with a low value coming from the heavier
secondary gas.
An additional figure has been prepared
to illustrate how the target parameter for the
seven He based binary gas mixtures, i.e., the
relative heat transfer rate Qmix/B varies with the
molar gas composition w in the proper
wdomain [0, 1]. In the figure format, the
abscissa associates the light primary gas He
with the left extreme w = 0, whereas the right
extreme w = 1 represents each of the seven
heavier secondary gases N2, O2, Xe, CO2, CH4,
CF4 and SF6. In this regard, Figure 7 displays
the family of seven curves for the relative heat
transfer rate Qmix/B varying with the molar gas
composition w at the film temperature T =
300K. Obviously, the point of reference is the
primary light He that owns a relative heat
transfer rate QHe/B = 12. Among the binary gas
mixtures examined, it is seen that three He+SF6,
He+CF4 and He+Xe binary gas mixtures exhibit
maxima relative heat transfer rates Qmix,max/B.
Further, the optimal molar gas compositions
wopt for these three binary gas mixtures are
located near the right extreme w = 1 occupied
by the heavier secondary gases SF6, CF4 and Xe.
First, the He+SF6 binary gas mixture produces
the absolute maximum relative heat transfer rate
Qmix,max/B = 16.70 that happens at the optimal
molar gas composition wopt = 0.960. As
compared to (Q/B)He, and (Q/B)air, the He+SF6
binary gas mixture generates remarkable heat
transfer enhancement ratios
(19a)
(19b)
Another indicator is that among the seven
binary gas mixtures under scrutiny, the He+SF6
binary gas mixture has the largest molar gas
difference ΔM = 146.06.
One relative maximum for the relative
convective coefficient Qmix,max/B = 14.89 is
provided by the He+CF4 binary gas mixture,
which takes place at the optimal molar gas
composition wopt = 0.936. Herewith, the heat
transfer enhancement ratio for the He+CF4
binary gas mixture referred to He and air have a
significant magnitude
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(20a)
(20b)
The He+Xe binary mixture supplies
another relative maximum for the relative
convective coefficient Qmix,max/B = 13 operating
at wopt = 0.785. Here, the heat transfer
enhancement ratio provided by the He+Xe
binary gas mixture when paired against QHe/B
descends to the modest values
(21a)
(21b)
Attention is now turned to the seeded
heavier secondary gases in the He‒based binary
gas mixtures under study. Thereby, the amount
of seeded at is slightly smaller
than the amount of seeded CF4 at
and much smaller than the amount of
seeded Xe at
Furthermore, it can be observed in
Figure 7 that as Qmix,max/B decreases from the
upper He+SF6 binary gas mixture to the
intermediate He+CF4 binary gas mixture ending
in the lower He+Xe binary gas mixture, so that
the optimal molar composition wopt shifts
gradually toward the left extreme on the w
abscissa.
The physical explanation for the heat
transfer enhancement delivered by the trio of
He+SF6, He+CF4 and He+Xe binary gas
mixtures that own the largest molar mass
difference M may be explained as follows. The
curves in the pair of Figures 3 and 5 reveal that
the thermal conductivity mix and the isobaric
heat capacity Cp,mix of the seven He‒based
binary gas mixtures decrease with the molar gas
composition w. Therefore, this behavior means
that the two thermophysical properties mix and
Cp,mix do not contribute to the heat transfer
enhancement. Consequently, the heat transfer
enhancement will depend solely on the interplay
between the density ρmix and the viscosity mix.
The numbers listed in the three Tables 4, 5 and
6 reveal that the density ρmix is the major
contributor to the heat transfer enhancement.
First, the density ρmix of the He+SF6 binary gas
mixture at the optimal molar gas composition
wopt = 0.96 is 15.25 times higher than the
density of pure He. Second, the density ρmix of
He+CF4 binary gas mixture at the optimal molar
gas composition, wopt = 0.936 is 9.44 times
higher than the density of pure He. Third, the
density ρmix of the He+Xe binary gas mixture at
the optimal molar gas composition, wopt = 0.785
is 4.06 times higher than the density of pure He.
In contrast, for the three best binary gas
mixtures He+SF6, He+CF4 and He+Xe, the
corresponding ratios for the viscosity between
wopt and w = 0 for He range between 0.95 and
1.61, so that the variability is not that
significant.
The remaining four Qmix/B vs. w curves
related to the subgroup of He+N2, He+O2,
He+CO2 and He+CH4 binary gas mixtures form
a tight cluster, all exhibiting monotonic
decreasing trends with increments in the molar
gas composition in the wdomain [0, 1].
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Obviously, for this particular trio, the primary
light He is preferred as the coolant, instead of
the four binary gas mixtures He+N2, He+O2,
He+CO2 and He+CH4.
Table 4. He+Xe binary gas mixture
Thermo‒
physical
property
Molar gas
composition
of He,
w = 0
Optimal
molar gas
composition
wopt = 0.785
Thermo‒
physical
property
ratio
Density, ρ
0.16
0.65
4.06
Viscosity,
η x 105
1.99
2.51
1.26
Table 5. He+CF4 binary gas mixture
Thermo‒
physical
property
Molar gas
composition
of He,
w = 0
Optimal
molar gas
composition,
wopt = 0.936
Thermo‒
physical
property
ratio
Density, ρ
0.16
1.51
9.44
Viscosity,
η x 105
1.99
1.89
0.95
Table 6. He+SF6 binary gas mixture
Thermo‒
physical
property
Molar gas
composition
of He,
w = 0
Optimal
molar gas
composition,
wopt = 0.96
Thermo‒
physical
property
ratio
Density, ρ
0.16
2.44
15.25
Viscosity,
η x 105
1.99
3.21
1.61
6.3 Beneficial characteristics of the best
He‒based binary gas mixtures
The three best He‒based binary gas
mixtures that yield maximum heat transfer for
laminar natural convection along a heated
vertical plate share the following characteristics.
1) As indicated in the last three lines of
Table 2, the three largest molar mass
difference M are: He+CF4 with M =
84.00, He+Xe with M = 127.29 and
He+SF6 with M = 142.06.
2) The three lowest Prandtl numbers are:
He+Xe with Prmix = 0.1, He+CF4 with
Prmix = 0.2 and He+SF6 with Prmix =
0.3. These three numbers can be
observable in the lower right part of
Figure 2.
3) The absolute maximum relative heat
transfer rate Qmix,max/B = 16.70 is
delivered by the He+SF6 binary gas at
the optimal molar gas composition wopt
= 0.960. One relative maximum for the
relative convective coefficient Qmix,max/B
= 14.89 is provided by the He+CF4
binary gas mixture at wopt = 0.936. The
He+Xe binary mixture supplies another
relative maximum for the relative
convective coefficient Qmix,max/B = 13
operating at wopt = 0.785. These numbers
can be observable in the upper right part
of Figure 7.
6.4 Comparison of numerical calculations with
experimental data
The magnitude of the heat transfer
enhancement ratio Eht = 8 for the He+Xe binary
gas mixture in Eq. (21) coincides with the heat
transfer enhancement ratio obtained numerically
and experimentally by Petri and Bergman [6],
but at a slightly different molar gas composition
w.
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The concluding remarks that may be
drawn from the present study are enumerated
next.
The first conclusion is that the He+SF6
binary gas mixture yields a remarkable absolute
heat transfer enhancement ratio Eht = 39% at the
optimal molar gas composition wopt = 0.960
compared to the primary light He. The
corresponding Prmix = 0.3.
The second conclusion is that the
He+CF4 binary gas mixture at the optimal molar
gas composition wopt = 0.960 delivers a
significant relative heat transfer enhancement
ratio Eh = 24% at the optimal molar gas
composition wopt = 0.936 compared to the
primary light He. The corresponding Prmix = 0.2.
The third conclusion is that the He+Xe
binary gas mixture provides a modest relative
heat transfer enhancement ratio Eht = 8% at the
optimal molar gas composition wopt = 0.785
compared to the primary light He. The
corresponding Prmix = 0.1.
In the global picture, it could be pointed
out that usage of exotic binary gas mixtures like
the trio He+CF4, He+SF6 and He+Xe formed
with light He and heavier gases CF4, SF6 and Xe
may be envisioned for special engineering tasks
that demand high heat transfer rates in a multitude
of industries over the world.
Nomenclature
As surface area of vertical plate (m2)
B thermogeometric parameter in Eq. (7a)
()
B2 second virial coefficient (m3 mole1)
Cp mass isobaric heat capacity (J kg1 K1)
0
p
C
molar isobaric heat capacity of ideal gas
(J mole1 K1)
0
v
C
molar isobaric heat capacity of ideal gas
(J mole1 K1)
Eht heat transfer enhancement ratio
g acceleration of gravity (m s 2)
Grashof number,
h
average convective coefficient
(W m2 K1)
H height of vertical plate (m)
m molecular mass (kg)
Mi molar mass of pure gas i (kg mole1)
H
Nu
average Nusselt number,
Hh
p pressure (bar)
pc critical pressure (bar)
Pr Prandtl number,
p
C
Q heat transfer rate (W)
R gas constant (J mole1 K1)
RaH Rayleigh number, GrH Pr
T temperature (K)
Tc critical temperature (K)
Tw plate temperature (K)
T free‒stream temperature (K)
w molar gas composition of a binary gas
mixture
wopt optimal molar gas composition of a binary
gas mixture
x mole fraction
Z compressibility factor
Greek letters
β coefficient of volumetric thermal
expansion (K1)
ε thermal effusivity (W m2 K1 s1/2)
η viscosity ( Pa s)
thermal conductivity (W m1 K1)
density (kg m3)
7. Concluding Remarks
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Pitzer acentric factor
Subscripts
mix binary gas mixture
max maximum
opt optimal
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Commons Attribution License 4.0
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Figure 1. Laminar natural convection from a heated vertical plate: (a) ---- Momentum
boundary layer, (b) ____ Thermal boundary layer
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Figure 2. Variation of Prandtl number with molar gas composition w in the
seven He‒based binary gas mixtures at T = 300 K, p = 1 atm.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w
Pr
He+N2
He+O2
He+CO2
He+CH4
He+Xe
He+SF6
He+CF4
T=300 K
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Figure 3. Variation of the thermal conductivity with molar gas composition w of the seven
He‒based binary gas mixtures at T = 300 K, p = 1 atm.
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Figure 4. Variation of the viscosity with molar gas composition w of the seven
He‒based binary gas mixtures at T = 300 K, p = 1 atm.
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Figure 5. Variation of the isobaric heat capacity heat capacity with molar gas composition w
of the seven He‒based binary gas mixtures at T = 300 K, p = 1 atm.
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Figure 6. Variation of the density with molar gas composition w of the seven
He‒based binary gas mixtures at T = 300 K, p = 1 atm.
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Figure 7. Variation of the relative heat transfer rate Qmix/B with the molar gas composition w
of the seven He‒based binary gas mixtures at T = 300 K, p = 1 atm.
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w
Qmix/B
He-N2
He-O2
He-CO2
He-CH4
He-Xe
He-SF6
He-CF4
T=300K
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