Compelling Heat Transfer Augmentation of Laminar Natural Convection from

Vertical Plates to Binary Gas Mixtures Formed with Light Helium and

Selected Heavier Gases

ANTONIO CAMPO

Department of Mechanical Engineering,

The University of Vermont,

Burlington, VT 05405,

USA

M. MEHDI PAPARI

Department 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 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 study is to investigate the attributes of the set of seven

He‒based binary gas mixtures for heat transfer enhancement with respect to the heat transfer with helium (He)

and air. From heat convection theory, four thermo‒physical properties: viscosity η, density ρ, thermal

conductivity λ, and isobaric heat capacity Cp affect the thermo‒buoyant convection of fluids. In this study 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 mix

Q with the

set of seven He‒based binary gas mixtures from the vertical plate involves four thermo‒physical 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 brings forward as compared to the convective heat

transport of He and air. It was found that the He+SF6 binary gas mixture renders an absolute maximum heat

transfer enhancement rate that is 39 times higher than the heat transfer rate provided by 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.

Received: May 26, 2022. Revised: January 15, 2023. Accepted: February 22, 2022. Published: March 23, 2023.

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.

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The effectiveness of the two 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

associated with heat transfer enhancement in natural

convection flows has remained practically unnoticed

in the specialized literature.

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 the 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 enhancement ratio reached

23%.

With regards to passive schemes in fluid

media, a distinction must 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 along 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 relative 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 nano

fluid in their respective boundary layers. An

enlargement in the heat transfer performance of the

nano fluid 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 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 connected to 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 theoretically the

coupling of natural convection and thermal radiation

from a vertical plate to a stagnant gray gas The study

employed the integral method based on the two

limiting approximations, i.e., thin and thick gases.

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 pertinent to thermaldriven

boundary layers adjacent to 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: nitrogen

(N2), oxygen (O2) xenon (Xe), carbon dioxide (CO2),

methane (CH4), sulfurhexafluoride (SF6),

tetrafluoro‒

methane and carbon tetrafluoride (CF4).

2 Heat Transport by the Natural

Convection Mechanism

Heat convection theory (Jaluria [8]

stipulates the following relation

Natural convection mechanism =

heat conduction mechanism +

fluid motion due to thermo‒

buoyant forces

Within the heat conduction mechanism, the heat

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transfer enhancement is simple because the immobile

air can be replaced with an immobile gas having

higher thermal conductivity λ, like for instance 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

the 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 and 󰇛TT

󰇜 is the platetofluid

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

θ 

,η  

 (2)

where H is the height of the vertical plate.

3 Correlation Equations for the Prandtl

Number

For laminar boundary layer up‒flows near 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). This finding led to the

comprehensive correlation equation for the average

Nusselt number H

Nu :

(Pr)gRaNu 4/1

H

H (3)

which complies with the Boussinesq approximation

(Jaluria, [8]). Herein, the thermo‒physical properties

of the fluid are evaluated at the film temperature

T

. The coefficient of volumetric

thermal expansion for ideal gases is β 1

T∞, where

T is in absolute degrees. In contrast, the coefficient

of volumetric thermal expansion 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

laminartoturbulent natural convection flows

adjacent to vertical plates. The critical Grashof

number recommended stays around GrH,cr = 109.

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

vscenario, 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 Gr10

, Churchill and Chu,

[13], developed the correlation equation

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Nu



0.68.

/

󰇛󰇜 (4c)

with the companion Prandtl number function

(4d)

According to Martynenko and Polijakov [14], the

pair of 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 subgroups 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 reported by Ostrach [10]. In this sense, the Pr

data taken from [10] was re‒tabulated later 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)

All correlation coefficients Rsquare are high ~

0.999, while the largest 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 subgroup for binary gas mixtures

denoted Prmix is added to the entire Pr spectrum

ranging from 0.001 (liquid metals) to 1000 (oils and

heavy liquids, then a fifth subgroup 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 subinterval [0.1, 1] as seen in Figure 2

taken from Campo and Papari [17]. It is observable

in the figure that the absolute minimum 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 to provide an explanation for the nature

of the “anomalously” low values of Prandtl number

linked to binary gas mixtures. Using a combined

methodology based on statistical‒mechanical theory,

experimental measurements of other researchers as

well as 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

thermobuoyant forces depend upon four

thermo‒physical properties: dynamic viscosity η,

thermal conductivity λ, density ρ, and isobaric heat

capacity Cp. In the particular case of binary gas

mixtures, the four thermo‒physical properties are

adequately re‒expressed by ηmix, λmix, ρmix and C

p,

mix.

Eq. (3) particularized to the proper Pr sub‒interval

[0.1, 1] stated in Eq. (5b), along with the

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corresponding Prandtl number function g (Pr) =

0.549Pr 0.171 delivers the correlation equation



Nu



0.549GR

1/4Pr0.42 (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

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 T. That is

25.0

w

H

)TT(g

549.0B 











 (7a)

with units s/. Alternatively, using the

coefficient of volumetric thermal expansion β 



for ideal gases, B could be rewritten compactly as



B  0.549󰇣

󰇡

1

󰇢󰇤. (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 platetofluid temperature difference 󰇛TT󰇜

is constant, the only possible way for augmenting Q is

by enlarging the magnitude of hThereby, 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 thermo‒physical 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 the heat transfer rate mix

Qfor

laminar natural convection with binary gas mixtures

along a vertical plate, the trio of thermo‒physical

properties λmix, ρmix, and Cp,mix in the numerator of

Eq. (7) must be large and/or the single

thermo‒physical property ηmix in the denominator of

Eq. (7) must be small.

3.2 Thermo‒physical 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 C

p, mix and

inversely proportional with the 0.10 power of ηmix.

The numerator may be viewed through the optic of

the thermo‒physical property called the thermal

effusivity ε (Grigull and Sandner, [19]):

ερC/

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

ε ρC,/



Accordingly, Eq. (7) could be rewritten compactly in

terms of the thermal effusivity ε as follows



h

/B  

η

. (7c)

with minimal contribution from η.

4 Thermo‒physical Properties of

Binary Gas Mixtures

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

masses M of the participating gases are taken from

Poling et al., [20], and are listed in Table 1. In

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addition, the molar mass difference M of the eight

binary gas mixtures 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 heavier secondary gases having

the largest molar mass factors with respect to the

primary light gas He are: CF4 with a 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

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)

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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

C0p  C0v = R (13)

Herein, C

being the molar heat capacity at constant

volume of pure gas i, is quantified from





















































 k

1j

2

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,

the symbol S equates to 5/2 for linear molecules and

to 3 for nonlinear molecules as noted by McQuarrie

[22].

4.4 Viscosity

Based on the Kinetic Theory of Gases (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

0xx

xHH

 (15)

First, the element HAA along the main diagonal in the

two matrices is:























A

B

*

AB

2

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

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

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 T

∗, 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

0xx

xLL

 (16)

To save journal space, the elements LAA, HAB and LBB

in the upper and lower matrices in Eq. (16) are not

included. Instead, LAA, HAB and LBB are available in

Reference [26]. Besides, the pair xA and xB are the

mole fractions of A and B.

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5 Heat Transfer Analysis for Binary

Gas Mixtures

When the accurate formulas for the four

thermo‒physical 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

quantification of mix

h related to the seven binary gas

mixtures are carried out with fixed small steps w =

0.01 utilizing the spreadsheet software Excel [27].

The four thermo‒physical 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 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 singlevalued function B/)w(Qmix if

B/)w(Q optmix  B/)w(Qmix (17)

for all w values inside the wdomain [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 ranges of

B/)w(Qmix contained in the wdomain [0, 1].

Besides the absolute maximum, the other are called

relative maxima.

Table 3. Values of the thermo‒physical 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)

Cp0

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

6 Presentation and Discussion of

Results

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 from the large

vertical plate with light He against standard air

standard air before going to the binary gas

mixtures formed with primary light He and the

secondary heavier gases N2, O2, Xe, CO2, CH4,

CF4 and SF6.. In this respect, using Eq. (7) at a

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temperature of 300K, the heat transfer enhancement

ratio Eht caused by light He with respect to air is



E 󰇡

󰇢

󰇡

󰇢 2 (18) 



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 wdomain [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

E 󰇡

󰇢

󰇡

󰇢 39 (19a)

E 󰇡

󰇢

󰇡

󰇢 78 (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

E 󰇡



󰇢HeCF4

󰇡



󰇢

24 (20a)

E 󰇡



󰇢HeCF4

󰇡



󰇢

48 (20b)

The He+Xe binary mixture supplies another

relative maximum for the relative convective

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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

E 󰇡

󰇢HeXe

󰇡

󰇢 8 (21a)

Eht 󰇡Q

B󰇢

󰇡Q

B󰇢air 16 (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0.04

is slightly smaller than

the amount of seeded CF4 at  0.064 and

much smaller than the amount of seeded Xe at

  0.215.

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 thermo‒physical 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 subgroup 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

wdomain [0, 1]. 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

wo

p

t = 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,

wo

p

t = 0.936

Thermo‒

physical

property

ratio

Density, ρ 0.16 1.51 9.44

Viscosity,

η x 105

1.99 1.89 0.95

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Table 6. He+SF6 binary gas mixture

Thermo‒

physical

property

Molar gas

composition

of He,

w = 0

Optimal

molar gas

composition,

wo

p

t = 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 related

to laminar natural convection along a heated vertical

plate that yield maximum heat transfer shares 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.

7 Concluding Remarks

The conclusions 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)

(s/)

B2 second virial coefficient (m3 mole1)

Cp mass isobaric heat capacity (J kg1 K1)

0

p

C molar isobaric heat capacity of ideal gas

(J mole1 K1)

0

v

C molar isobaric heat capacity of ideal gas

(J mole1 K1)

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Eht heat transfer enhancement ratio

g acceleration of gravity (m s 2)

Gr Grashof number, gβ

󰇛TT󰇜H

h average convective coefficient

(W m2 K1)

H height of vertical plate (m)

m molecular mass (kg)

Mi molar mass of pure gas i (kg mole1)

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 mole1 K1)

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 (K1)

ε thermal effusivity (W m2 K1 s1/2)

η viscosity ( Pa s)

 thermal conductivity (W m1 K1)

 density (kg m3)

 Pitzer acentric factor

Subscripts

mix binary gas mixture

max maximum

opt optimal

References;

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Transfer. In Handbook of Heat Transfer,

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Volume 18, 2023

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

We confirm that all Authors equally contributed in

the present research, at all stages from the

formulation of the problem to the final findings and

solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflicts of Interest

Please declare anything related to this study.

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(Attribution 4.0 International, CC BY 4.0)

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_US

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Fig. 1: Laminar natural convection from a heated vertical plate: (a) ---- Momentum

boundary layer, (b)

____

Thermal boundary layer

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Fig. 2: Variation of Prandtl number with molar gas composition w of 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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Fig. 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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Fig. 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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Fig. 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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Fig. 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.

WSEAS TRANSACTIONS on HEAT and MASS TRANSFER

DOI: 10.37394/232012.2023.18.4

Antonio Campo, M. Mehdi Papari

E-ISSN: 2224-3461

49

Volume 18, 2023

Fig. 7: Variation of the relative heat transfer rate Q

mix

/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

WSEAS TRANSACTIONS on HEAT and MASS TRANSFER

DOI: 10.37394/232012.2023.18.4

Antonio Campo, M. Mehdi Papari

E-ISSN: 2224-3461

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Volume 18, 2023