Thermodynamics – Interaction Studies – Solids, Liquids and Gases

410
Fig. 1. Equilibrium gaseous composition in M-F systems at total pressure of 2 kPa [7].

Thermodynamic Aspects of CVD Crystallization of Refractory Metals and Their Alloys

411
Fig. 2. Equilibrium gaseous composition in M-F-H systems at total pressure of 2 kPa and
hydrogen to highest fluoride initial ratio of 10 [31].

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

412 Fig. 3. Yield of metals (V, Nb, Ta, Mo, W, Re) from the equilibrium mixtures of their
fluorides with hydrogen (1:10) as a function of the temperature [31].
5. Equilibrium composition of solid deposit in W-M-F-H systems
A thermodynamics of alloy co-deposition is often considered as a heterogeneous

i
- mole fraction of solution
component.
The surface properties of tungsten are sharply different from the bulk properties due to
strongest chemical interatomic bonds. Therefore, there is an expedience to include the
crystallization stage in the thermodynamic consideration, because the crystallization stage
controls the tungsten growth in a large interval of deposition conditions. To determine the
enthalpy of mixing of surface atoms we use the results of the desorption of transition metals
on (100) tungsten plane presented at the Fig. 4. [35]. The crystallization energy can be
determined as the difference between the molar enthalpy of the transition metal sublimation

Thermodynamic Aspects of CVD Crystallization of Refractory Metals and Their Alloys

413
from (100) tungsten surface and sublimation energy of pure metal. These values are
presented in the table 4 in terms of polynomial’s coefficients, which were estimated in the
case of the infinite dilute solution. The peculiarity of the detail calculation of polynomial’s
coefficients is discussed in [7]. The data predict that the co- crystallization of tungsten with
Nb, V, Mo, Re will be performed more easily than the crystallization of pure tungsten. The
crystallization of W-Ta alloys has the reverse tendency. Certainly the synergetic effects will
influence on the composition of gas and solid phases.

№ М
∆H
0
m

ּ
◌ 298 К
x

solution and h
i
polynomial’s coefficients for x
i
= 0 – 0.0625 and T = 298 – 2500 K [7, 31].
Therefore the thermodynamic calculation for gas and solid composition of W-M-F-H
systems were carried out for following cases:
1. without the mutual interaction of solid components;
2. for the formation of ideal solid solution
3. for the interaction of binary solution components on the surface.
The temperature influence on the conversion of VB group metal fluorides and their addition
to the tungsten hexafluoride – hydrogen mixture is presented at the Fig.5 a,b,c. If the metal
interaction in the solid phase is not taken into account, the vanadium pentafluoride is
reduced by hydrogen only to lower-valent fluorides. It should be noted that metallic
vanadium can be deposited at temperatures above 1700 K. Equilibrium fraction of NbF
5

conversion achieves 50% at 1400 K, and of TaF
5
– at 1600 K (Fig. 5 a,b,c, curves 1).
The thermodynamic consideration of ideal solid solution shows that tungsten-vanadium
alloys may deposit at the high temperature range (T ≥ 1400 K) and metallic vanadium is
deposited in mixture with lower-valent fluorides of vanadium (Fig. 5 a, curves 2). The
beginnings of formation of W-Nb and W-Ta ideal solid solutions are shifted to lower
temperature by about 100 K (Fig. 5 b,c, curves 2) in comparison with the case (1).

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

414


) : H
2
= 10
Taking into account the interaction of component of alloys during crystallization, the
formation of W-V and W-Nb alloys possibly takes place at the temperatures above 300 K

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

416
(Fig. 5 a,b, curves 3). Temperature boundary shown at the Fig. 5 is shifted in reverse
direction for the W-Ta system (Fig. 5 c, curves 3). It should be noted, that the calculation
results performed for cases (2) and (3) (for ideal and nonideal solid solution) for the W-Ta
system are almost identical due to the small enthalpy of mixing [35].
The influence of rhenium and molibdenium on the equilibrium yield of tungsten in the M-
W-F-H systems is observed for W-Re and W-Mo alloys deposition. The ReF
6
addition to the
gas mixture with WF
6
increase insignificantly the yield of tungsten in spite of strong atom
interaction during the crystallization according to thermodynamic calculations (Fig. 6). This
effect is still smaller for the case of W-Mo co-deposition. However equilibrium yield of
metals for their co-deposition with tungsten and the energy of the interaction of metallic
components during the crystallization have the common tendency. The knowledge of
refined data of process energies will allow us to obtain a more realistic situation.
6. The application fields of the coatings
The thermodynamic background presented above is very useful for production of the
coatings based on tungsten, tungsten alloys with Re, Mo, Nb, Ta, V and tungsten
compounds (for example tungsten carbides). The tungsten coatings have found wide
application in thin-film integral circuits when preparing the Ohmic contacts in the

Thermodynamic Aspects of CVD Crystallization of Refractory Metals and Their Alloys

417
wear of movable units, and erosion of immobile parts of drilling bits operating underground
take special significance because their replacement is very expensive. The carbide coatings
can be deposited inside cylinders and on the outer surfaces of components of rotary or
piston oil pumps. Numerous units in the oil and gas equipment, for example, block
bearings, solution-supplying channels in drilling bits, backings directing the sludge flow,
etc. require the strengthening of their working surfaces.
Another application in this field is the coating of metal–metal gaskets in the high- and
ultrahigh-pressure stop and control valves. In addition to intense corrosion, abrasion and
erosion wear, the working surfaces of ball cocks and dampers are subject of seizing under
high pressure; W–C-coatings prevent the seizure. An important advantage of the carbide
coatings is their accessibility for the quality of surface polishing, due to the initial smooth
morphology. The examples mentioned above relate not only to oil and gas but also to
chemical industry. The W–C-coatings are promising for working in contact with hydrogen-
sulfide-rich oil, acids, molten metals, as well as chemically aggressive gases. Due to their
high wear and corrosion resistance, these coatings can be use instead of hard chromium.
The abrasion mass extrusion and the metal shape draft require expensive extrusion tools;
the product price depends on the working surface quality and life time. The extrusion tools
must often have sophisticated shape inappropriate for coating with PVD or PACVD
methods. Therefore, W–C-coating prepared by a thermal CVD-method is promising in
strengthening these tools. Strengthening of spinneret for drawing wires or complicated
section of steel, copper, matrices for aluminum extrusion, ceramic honeycomb structures for
the porous substrate of catalytic carriers may give the same effect. Also, very perspective is
the deposition of strengthening coatings onto components of equipment for the pressing of
powdered abrasion materials. One may also mention the strengthening of knife blade used
for cutting paper, cardboard, leather, polyethylene, wood, etc [38].
In addition to the surface strengthening, the W–C-coatings can function as high-temperature
glue for mounting diamond particles in a matrix when preparing diamond tools or diamond

supersaturation in these systems) increase in order: Ta, Nb, V, Mo, W, Re.
5. A lot of applications of tungsten coatings, deposited from tungsten hexafluoride and
hydrogen mixture at low temperature, as well as tungsten alloys and carbides are
reviewed in this chapter.
8. Acknowledgments
This work was supported by the Russian Foundation for Basic Research, project No. 09-08-
182.
9. Appendix 1
Description of symbols used in the text
Symbol Description
Ω Atomization energy
М Metal
Х Halid
n Valency of metal
Δ
f
Н Formation enthalpy
at Atom
φ Function
Z
m
Atomic number of metal
Z
x
Atomic number of halid
ψ Functional
Δ
s
Н Sublimation enthalpy
S Entropy

s
Δ H

Partial molar enthalpy
∆H
0
m
Standart mixing enthalpy

Thermodynamic Aspects of CVD Crystallization of Refractory Metals and Their Alloys

419
10. References
[1] Korolev Yu. M., Stolyarov V. I., Vosstanovlenie ftoridov tugoplavkikh metallov
vodorodom (Metallurguij, Moskva, 1981) 184 p. (in Russian).
[2]
Krasovskii A.I., Chuzshko R.K., Tregulov V.R., Balakhovskii O.A., Ftoridnii process
poluchenij volframa (Nauka, Moskva, 1981) 260 p. (in Russian).
[3]
Pons M., Benezech A., Huguet P., et all, J. Phys. France, Vol. 5, N. 8 (1995) pp. 1145-1160.
[4]
Lakhotkin Yu.V., Krasovskii A.I., Volfram-renievie pokritij (Nauka, Moskva, 1989) 158 p.
(in Russian).
[5]
Lakhotkin Yu.V., Protection of Metals, Vol. 44, N. 4 (2008) pp. 319-332.
[6]
Blokhinzev D.I. Osnovi kvantovoii mekhaniki (Nauka, Moskva, 1983) 664 p. (in Russian).
[7]
Malandin M.B., Lakhotkin Yu.V., Kuzmin V.P., Problemi fizicheskogo metallovedenij
(MIFI, Moskva, 1991) pp. 35-47. (in Russian).

[19]
Nuttal R.L., Kilday M.Y., Churney K.L., Natt. Bur. Stand. Rep. 73-281, (1973).
[20]
Gotkis I.S., Gusarov A.V., Pervov V.S., et al., Koordinasionnaij chimij T. 4, Vip. 5 (1978)
pp. 720-724 (in Russian).
[21]
Hildenbrand D.L. J. Chem. Phys., Vol. 62, N. 8 (1975) pp. 3074-3079.
[22]
Burgess J., Fawcett J., Peacock R.D. et al., J. Chem Soc., Dalton Trans., N. 14, (1976) pp.
1363-1364.
[23]
Politov Yu.A., Alikhanyan A.S., Butzki V.D., et al., DAN SSSR, T. 309, N. 4 (1989) pp.
897-899 (in Russian).
[24]
Politov Yu.A., Alikhanyan A.S., Butzki V.D., et al., J. Neorganicheskoii chimii, T. 32, N.
2 (1987) pp. 520-523 (in Russian).
[25]
Stout J.W., Boo W.O.J. J. Chem. Phys., Vol. 71, N. 1 (1979) pp. 1-8.
[26]
Stull D.R., Prophet H. JANAF Thermochemical Tables. NSRDS-NBS 37 US, (NBS,
Washington, DC, 1971).
[27]
Arara R., Pollard R. J. Electrochem. Soc., Vol. 138, N. 5 (1991) pp. 1523-1537.

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

420
[28] Boltalina O.V., Borzsevskii A.Ya., Sidorov L.N. J. Fiz. Chimii, T. 66, Vip. 9 (1992) pp.
2289-2309 (in Russian).
[29]

Effect of Stagnation Temperature on
Supersonic Flow Parameters with
Application for Air in Nozzles
Toufik Zebbiche
University SAAD Dahleb of Blida,
Algeria
1. Introduction
The obtained results of a supersonic perfect gas flow presented in (Anderson, 1982, 1988 &
Ryhming, 1984), are valid under some assumptions. One of the assumptions is that the gas is
regarded as a calorically perfect, i. e., the specific heats C
P
is constant and does not depend
on the temperature, which is not valid in the real case when the temperature increases
(Zebbiche & Youbi, 2005b, 2006, Zebbiche, 2010a, 2010b). The aim of this research is to
develop a mathematical model of the gas flow by adding the variation effect of C
P
and γ
with the temperature. In this case, the gas is named by calorically imperfect gas or gas at high
temperature. There are tables for air (Peterson & Hill, 1965) for example) that contain the
values of C
P
and γ versus the temperature in interval 55 K to 3550 K. We carried out a
polynomial interpolation of these values in order to find an analytical form for the function
C
P
(T).
The presented mathematical relations are valid in the general case independently of the
interpolation form and the substance, but the results are illustrated by a polynomial
interpolation of the 9
th

following differential equation (Moran, 2007 & Oosthuisen & Carscallen, 1997 & Zuker &
Bilbarz, 2002, Zebbiche, 2010a, 2010b).

0
P
CRT
dT dρ
γρ

 (1)
Using relationship between C
P
and γ [C
P
=γR/(γ-1)], the equation (1) can be written at the
following form:

[()1]
dρ
dT
ρ T T



(2)
The integration of the relation (2) gives the adiabatic equation of a perfect gas at high
temperature.
The sound velocity is (Ryhming, 1984),

2


0
P
CdT VdV

 (6)
The integration between the stagnation state (V
0
≈ 0, T
0
) and supersonic state (V, T) gives:
Effect of Stagnation Temperature on Supersonic
Flow Parameters with Application for Air in Nozzles

423

2
2 ( )VHT
(7)
Where


0
()
P
T
T
HT CTdT

(8)


0
dP
VdV
ρ

 (11)
Using the expression (3), the relationship (10), can be written as:

()
ρ
dρ
FT dT
ρ

(12)
Where

2
()
()
()
P
ρ
CT
FT
aT

(13)



424

0
00
ρ
T
P
P
ρ T









(15)
The mass conservation equation is written as (Anderson, 1988 & Moran, 2007)

tan
ρ
VA cons t

(16)
The taking logarithm and then differentiating of relation (16), and also using of the relations
(9) and (12), one can receive the following equation:


, T
*
) and the supersonic state (A,
T) gives the cross-section areas ratio:












*
T
T
dTTF
A
*
Exp
A
A
(19)
To find parameters ρ and A, the integrals of functions F
ρ
(T) and F
A

, M=1, θ=0 and A=A
*
. Therefore, the relation (20) is reduced to:

00
00

*
*
*
ρ
a
m

A ρ a
ρ
a











(21)
The determination of the velocity sound ratio is done by the relation (5). Thus,


425
3. Calculation procedure
In the first case, one presents the table of variation of C
P
and γ versus the temperature for air
(Peterson & Hill, 1965, Zebbiche 2010a, 2010b). The values are presented in the table 1.

T (K)
C
P
(J/(KgK)
γ(T) T (K)
C
P

(J/(Kg K)
γ(T) T (K)
C
P
J/(Kg K)
γ(T)
55.538 1001.104 1.402 833.316 1107.192 1.350 2111.094 1256.813 1.296
. .
. 888.872 1119.078 1.345 2222.205 1263.410 1.294
222.205 1001.101 1.402 944.427 1131.314 1.340 2333.316 1270.097 1.292
277.761 1002.885 1.401 999.983 1141.365 1.336 2444.427 1273.476 1.291
305.538 1004.675 1.400 1055.538 1151.658 1.332 2555.538 1276.877 1.290
333.316 1006.473 1.399 1111.094 1162.202 1.328 2666.650 1283.751 1.288
361.094 1008.281 1.398 1166.650 1170.280 1.325 2777.761 1287.224 1.287


I a
i
I a
i

1 1001.1058 6 3.069773 10
-12

2 0.04066128 7 -1.350935 10
-15

3 -0.000633769 8 3.472262 10
-19

4 2.747475 10
-6
9 -4.846753 10
-23

5 -4.033845 10
-9
10 2.841187 10
-27

Table 2. Coefficients of the polynomial C
P
(T).

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

and it is defined when T≤T
0
.
Substituting the relation (23) in (8) and writing the integration results in the form of Horner
scheme, the following expression for enthalpy is obtained 0
12345678910
()
[ ( ( ( ( ( ( ( ( )))))))))]
HT H -
cTcTcTcTcTcTcTcTcTc


(24)
Where

0010203040506
07 08 09 010
((((((
( ( ( ( ))))))))))
HTcTcTcTcTcTc
Tc Tc Tc Tc



(25)
and
( 1, 2, 3, , 10)

427
Taking into account the correction made to the function C
P
(T), the function H(T) has the
following form:
For
0
TT

,

0
()
P
HT C T T


For
0
TT ,we have two cases:

if : ( ) relation (24)TT HT

if : ( ) ( ) ( )
P
TT HT CTT HT
The determination of the ratios (14) and (19) require the numerical integration of F
ρ
(T) and
F

S
=6.00 (extreme supersonic) for a good representation in these ends. In
this case, we obtain T
*
=418.34 K and T
S
=61.07 K. the two functions presents a very large
derivative at temperature T
S
.
A Condensation of nodes is then necessary in the vicinity of T
S
for the two functions. The
goal of this condensation is to calculate the value of integral with a high precision in a
reduced time by minimizing the nodes number. The Simpson’s integration method
(Démidovitch & Maron, 1987 & Zebbiche & Youbi, 2006) was chosen. The chosen
condensation function has the following form (Zebbiche & Youbi, 2005a):

2
11
2
tanh (1 )
(1 ) 1
tanh( )
i
ii
bz
sb z b
b


i
in nodes i:

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

428

()
iiDG G
TsT T T

 (28)
The temperature T
D
is equal to T
0
for F
ρ
(T), and equal to T
*
for F
A
(T). The temperature T
G
is
equal to T
*
for the critical parameter, and equal to T
S
for the supersonic parameter. Taking a

3.1 Critical parameters
The stagnation state is given by M=0. Then, the critical parameters correspond to M=1.00,
for example at the throat of a supersonic nozzle, summarize by:
When M=1.00 we have T=T
*
. These conditions in the relation (10), we obtain:

2
**
2() ()0 HT a T

 (29)
The resolution of equation (29) is made by the use of the dichotomy algorithm (Démidovitch
& Maron, 1987 & Zebbiche & Youbi, 2006), with T
*
<T
0
. It can choose the interval [T
1
,T
2
]
containing T
*
by T
1
=0 K and T
2
=T
0

Simpson’s formula with condensation of nodes towards the left end, the critical density ratio
is obtained.
The critical ratios of the pressures and the sound velocity can be calculated by using the
relations (15) and (22) respectively, by replacing T=T
*
, ρ=ρ
*
, P=P
*
and a=a
*
,
3.2 Parameters for a supersonic Mach number

For a given supersonic cross-section, the parameters ρ=ρ
S
, P=P
S
, A=A
S
, and T=T
S
can be
determined according to the Mach number M=M
S
. Replacing T=T
S
and M=M
S
in relation

and ρ=ρ
S
in relation (14) and integrating the function F
ρ
(T) by using the
Simpson’s method with condensation of nodes towards the left end, the density ratio can be
obtained.
Effect of Stagnation Temperature on Supersonic
Flow Parameters with Application for Air in Nozzles

429
The ratios of pressures, speed of sound and the sections corresponding to M=M
S
can be
calculated respectively by using the relations (15), (22) and (19) by replacing T=T
S
, ρ=ρ
S
,
P=P
S
, a=a
S
and A=A
S
.
The integration results of the ratios ρ
*
/ ρ
0


 (33)
The introduction of relations (21), (22) into (32) gives as the following relation:


**
0
0*0

E
FE
aa
CTM
aa



 

 


 

 
 (34)
The design of the nozzle is made on the basis of its application. For rockets and missiles
applications, the design is made to obtain nozzles having largest possible exit Mach number,
which gives largest thrust coefficient, and smallest possible length, which give smallest
possible mass of structure.

(35)
The letter y in the expression (35) can represent all above-mentioned parameters. As a rule
for the aerodynamic applications, the error should be lower than 5%.

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

430
4. Application
The design of a supersonic propulsion nozzle can be considered as example. The use of the
obtained dimensioned nozzle shape based on the application of the PG model given a
supersonic uniform Mach number M
S
at the exit section of rockets, degrades the desired
performances (exit Mach number, pressure force), especially if the temperature T
0
of the
combustion chamber is higher. We recall here that the form of the nozzle structure does not
change, except the thermodynamic behaviour of the air which changes with T
0
. Two
situations can be presented.
The first situation presented is that, if we wants to preserve the same variation of the Mach
number throughout the nozzle, and consequently, the same exit Mach number M
E
, is
necessary to determine by the application of our model, the ray of each section and in
particular the ray of the exit section, which will give the same variation of the Mach number,
and consequently another shape of the nozzle will be obtained.

() ()

AA
HT e PG
AA






(38)
The relation (36) indicates that the Mach number of the PG model is preserved for each
section in our calculation. Initially, we determine the temperature at each section; witch
presents the solution of equation (37). To determine the ratio of the sections, we use the
relation (38). The ratio of the section obtained by our model will be superior that that
determined by the PG model as present equation (38). Then the shape of the nozzle obtained
by PG model is included in the nozzle obtained by our model. The temperature T
0
presented
in equation (38) is that correspond to the temperature T
0
for our model.
The second situation consists to preserving the shape of the nozzle dimensioned on the basis
of PG model for the aeronautical applications considered the HT model.

**
() ()
SS
AA
HT PG
AA

0
increases, we can
see the difference between these values and it influences on the thermodynamic parameters
of the flow.

0 1000200030004000
Stagnation Temperature (K)
950
1000
1050
1100
1150
1200
1250
1300
1350

Fig. 4. Variation of the specific heat for constant pressure versus stagnation temperature T
0
.

0 1000 2000 3000 4000
Stagnation Temperature (K)
1.24
1.28
1.32
1.36
1.40
1.44

0 1000 2000 3000 4000
Stagnation Temperature (K)
0.82
0.83
0.84
0.85
0.86
0.87
0.88
0.89

Fig. 6. Variation of T
*
/T
0
versus T
0
. 0 1000 2000 3000 4000
Stagnation temperature (K)
0.62
0
0.624
0.628
0.632
0.636

Fig. 8. Variation of P
*
/P
0
versus T
0
.
Figure 9 shows that mass flow rate through the critical cross section given by the perfect gas
theory is lower than it is at the HT model, especially for values of T
0
.
0 1000 2000 3000 4000
Stagnation Temperature (K)
0.576
0.578
0.580
0.582
0.584
0.586
0.588Fig. 9. Variation of the non-dimensional critical mass flow rate with T
0