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Mathematical simulation of surface heating during plasma spraying

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2017 IOP Conf. Ser.: Mater. Sci. Eng. 177 012057 (http://iopscience.iop.org/1757-899X/177/1/012057) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 191.101.65.8 This content was downloaded on 30/06/2017 at 03:21 Please note that terms and conditions apply.

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MEACS2016 IOP Conf. Series: Materials Science and Engineering 177 (2017) 012057

IOP Publishing doi:10.1088/1757-899X/177/1/012057

International Conference on Recent Trends in Physics 2016 (ICRTP2016) IOP Publishing Journal of Physics: Conference Series 755 (2016) 011001 doi:10.1088/1742-6596/755/1/011001

Mathematical simulation of surface heating during plasma spraying V I Bogdanovich and М G Giorbelidze Samara National Research University, 34, Moskovskoe Highway, Samara, 443086, Russian Federation E-mail: [email protected] Abstract. A mathematical model of temperature distribution over the flat ‘coatingsubstrate’ system section during plasma spraying, taking into account a plasma gun travel and coating buildup has been developed. It has been shown that the temperature value in the near-surface layer of the sprayed coating during the plasma gun passage can significantly exceed the temperature values in underlayers.

1. Introduction The creation of promising aviation gas turbine engines having more advanced technical characteristics is closely connected with the increase of the temperature of the gas flow at the inlet of the turbine, as a result of which increasingly high demands are made for the parts of the hot gas path of the turbine. In view of the fact that the potential of the modern refractories is virtually exhausted, one of the most efficient methods of lowering the temperature loading on the surface of the structural material is application of special coatings, possessing high indices of thermal stability, erosion resistance and providing the effective protection of the parts surface from intercrystalline high-temperature corrosion. Owing to the possibility of combination of high values of physico-mechanical and service properties during formation of the meso-ordered structure, plasma heat-resistant coatings are the best prospect. However, in the process of applying these coatings, the surface of the construction material can be subjected to considerable thermal actions on the part of the plasma jet, as a result of which the problem of temperature distribution in the system ‘coating-substrate’ is highly relevant [1-10]. 2. Statement of the problem of the mathematical model of surface heating When defining the temperature distribution during plasma spraying, the averaged influence of the heat flux on the system ‘coating-substrate’ [2] was taken into consideration. In this statement of the problem, the heat obtained by the substrate can be represented as a sum of several constituents: (1) Q = Q1 + Q2 + Q3 + Q4 + Q5 − Q6 − Q7 , where Q1 is the heat released owing to a conversion of kinetic energy of the particles into thermal energy during the impact on the substrate; Q2 is the heat obtained by the substrate from the plasma jet; Q3 is the heat released during crystallization of the molten particles; Q4 is the heat released during cooling down of the particles; Q5 is the heat released during exothermic reactions in the sprayed materials; Q6 is the heat transmitted by the surface owing to convective heat exchange; Q7 is the heat withdrawn owing to the heat conduction inside the substrate material.

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1 Published under licence by IOP Publishing Ltd



When considering some small volume of the surface of the coating and representing the process of coating in the form of the model with a continuous build-up of the layer, using the heat-balance equation, the boundary condition for the sprayed surface can be represented as: λ2

∂Т 2 =q ( t ) − α 2 (T2 − Ta ) , q ( t ) =q0 f ( t ) , when z = δ ( t ) , ∂z

(2)

where q ( t ) is the greatest heat flux on the jet axis, α 2 is the heat exchange factor, T2 is the coating temperature, Ta is the ambient temperature, z is the coordinate denoting the thickness in the ‘coating-substrate’ system, and δ ( t ) is the sprayed-surface coordinate. The heat of the plasma jet is transmitted to the part through a heating spot with diameter d , which, in general case, differs from the diameter of the spraying spot. Heat flux density q owing to uneven distribution over the heating spot (the amount of heat introduced through the spraying spot surface per time unit) can be described by means of a certain distribution law. As a rule, the density of the thermal energy, introduced by the heated gas and heated particles, is recorded with the use of different laws of distribution. However, when the plasma gun is adjusted in a certain way, so that the maximums of heat flux density coincide, the summation thermal action can be described by a single law of distribution with averaged parameters determined based on the experiments. Owing to the fact that transmission and transfer of heat in the plasma jet depend on a large number of random factors, it is possible to consider that the density of the heat flux complies with the normal law of distribution. This assumption has found its experimental verification. From the practical standpoint, an opportunity of considering the periodical influence of the plasma jet on the system ‘coating-substrate’ when calculating the temperature distribution is of special interest. The distribution of the heat flux of the plasma jet throughout the area of the heating spot in case of high capacity of the heat flux and the significant speed of the plasma gun travel can be recorded by means of the normal law of distribution: (3) = q (r ) q0 exp(− kr 2 ) , when r = ( x − Vt ) 2 + y 2 , where k is the factor of concentration of the jet heat flux, and V is the speed of the plasma gun travel. Having fixed a spot on the surface of spraying with coordinates x= y= 0 , let us write the expression for determination of the density of the heat flux during a single passage of the plasma gun through this point: (4) = q (r ) q0 exp(− kV 2t 2 ) . In the process of spraying, the plasma gun passes over this point with time interval t1 ; the duration of the influence of the plasma jet on this spot is defined by equation t2 = d x / V , where d x is the spraying spot diameter. For time t2 , the process of the coating build-up takes place at speed d δ / dt . In this case, the heat flux density has the following view: = q (t )

∑ q (t ) − {η t − ( n − 1) t  − η t − ( n − 1) t N

n =1

n

1

1

}

− t2  ,

(5)

}

(6)

q= q0 exp {− kV 2  t − ( n − 1) t1 − 0,5t2  2 , n (t )

where η ( t ) is the asymmetric unit function; index n corresponds to the number of plasma gun passages over this point; and the concentration factor of the heat flux of the jet is related to spraying spot diameter d by relationship d = 2 ( ln 20 )1/2 k 1/2 .

The coating build-up rate and the process of its thickness increase can be presented by expressions: dδ (7) = Vz ( t ) { η t − ( n − 1) t1  − η  t − ( n − 1) t1 − t2 } , dt

2



t  − + n 1 δ ) 0 ∫ Vz (τ ) dτ , t ∈ ( n − 1) t1 , ( n − 1) t1 + t2  ( δ (t ) =  ( n −1) t1  t ∈ ( n − 1) t1 + t2 , nt1  , nδ 0 ,

(8)

t2

where δ 0 = ∫ Vz (τ ) dτ is the coating thickness obtained per single plasma gun passage, and 0

Vz ( t ) is the coating build-up rate per single plasma gun passage over the surface spot

under consideration. 3. A mathematical model of surface heating The mathematical model of the heat problem during spraying the coating onto a substrate surface of a plate-like shape has the following view in case of the averaged thermophysical parameters: ∂T2 ∂ 2T (9) = a22 22 , 0 < z ≤ δ ( t ) ; ∂t ∂z ∂T1 ∂ 2T (10) = a12 21 , − h < z ≤ 0 ; ∂t ∂z ∂T (11) λ1 2 =q ( t ) − a2 (T2 − Tc ) , z =δ ( t ) ; ∂z ∂T (12) a1 (T1 − Tc ) , z = −h ; λ1 1 = ∂z ∂T1 ∂T (13) T1 T= 0; λ1= λ2 2 , = 2, z ∂z ∂z T1 ( z ,0 ) = T0 , (14) where T1 is the substrate temperature, T0 is the initial temperature of the system, ai and λi are temperature conductivity and heat conductivity factors of the substrate (i=1) and the coating (i=2), respectively. For the convenience of solving the boundary value problem (9-14), let us introduce dimensionless variables: Vz d н Ti − T0 z a 2t δ (15) ,ζ , ζ0 , Fo , θi , i 1, 2 ; = = = = = ξ = 2 h h hV h T0 − Tc and parameters: hq0 α1 h α2h λ1 a12 , ; Bi2 ; kλ ; ka = ; β kh 2= ; Ki0 = = = = Bi1 λ1 λ2 λ2 λ2 (T0 − Tc ) a22 = Fo1

a12t1 a22t2 hVz hV , ; Fo2 ; Pez = ; Pe = = 2 h h2 a12 a12

(16)

where h is the coating thickness. Relationships (5-8) determining q (t ) and δ ( t ) in designations (15-16) are transformed as follows: N

}

{

= Ki ( Fo ) Ki0 ∑ f n ( Fo ) η  Fo − ( n − 1) Fo1  − η  Fo − ( n − 1) Fo1 − Fo2  ; n =1

 1   f n ( Fo )= exp − β Pe 2  Fo − ( n − 1) Fo1 − Fo2   ; 2   

3

(17) (18)



Fo  − + n 1 ζ ( )  0 ∫ Pez (τ ) dτ , ( n −1) Fo1   ζ ( Fo=)  ( n − 1) Fo1 ≤ Fo ≤ ( n − 1) Fo1 + Fo2  nζ 0 , ( n − 1) Fo1 + Fo2 < Fo ≤ nFo 1  

(19)

Solving the boundary value problem using the differential series method [3, 4] and neglecting the terms of order of smallness ζ k , k ≥ 2 , let us write the following conditions on stationary boundary ζ = 0 : ∂θ ∂ξ

= Ψ ( Fo ) −

k ∂θ Bi 1 + θ ( 0, Fo ) ) − ζ ( kλ kλ ∂Fo

a 2 2 1 1 ξ 0= ξ 0 =

;

(20)

Bi2ζ Ki 1 Ψ ( Fo ) = −  Ki − Bi2 (1 + θ1 ( 0, Fo ) )  . (21) kλ kλ 1 + Bi2ζ The solution of the boundary value problem of thermal conductivity intended for the substrate (having stationary boundaries ζ = −1 , ζ = 0 ) differs from the solution obtained in [3] by the fact that in this approach, owing to supplementary terms of the expression, the heat losses and heating of the lower layers of the coating are taken into consideration. This approach is the most important one when solving the problems with rapidly moving heat sources. Using a series of the subsequent transformations, including the Laplace transformation by variable Fo, and applying the expansion theorem, let us write the expression for relatively excessive temperature: ∞ A cos µnξ + Bn sin µnξ exp ( − µn2 Fo ) + θ1 (ξ , Fo ) =−1 + 2∑ n D n =1 n . (22) Fo ∞ An cos µnξ + Bn sin µnξ 2 exp ( − µnτ )dτ + 2 ∫ Ψ ( Fo − τ )∑ Dn n =1 0 Substituting solution (21) in (22) for Ψ ( Fo ) , we will obtain:

Fo Ki ( Fo − τ ) kλ −1 + Bi2 Bi3 ( Fo − τ ) An cos µnξ + Bn sin µnξ ⋅ exp ( − µn2 Fo ) + 2 ∫ 1 + Bi2ζ ( Fo − τ ) Dn n =1 0 ∞

θ1 (ξ , Fo ) =−1 + 2∑

Fo Bi2ζ ( Fo − τ ) An cos µnξ + Bn sin µnξ ⋅ exp ( − µn2τ ) dτ + 2 Bi3 ∫ θ 2 ( 0, Fo − τ ) ⋅ 1 + Bi2ζ ( Fo − τ ) Dn n =1 0 ∞

⋅∑

(23)

An cos µnξ + Bn sin µnξ exp ( − µn2 Fo ) dτ . Dn n =1 ∞

⋅∑

tg µ = −µ

ka µ 2 − Bi1 − Bi3

µ 2 (1 + ka Bi1 ) − Bi1 Bi3

, µ > 0, n ≥ 1 ;

An = µn ( Bi1 + Bi3 cos µn ) + Bi1 Bi3 sin µn

;

= Bn µn ( Bi3 cos µn + Bi1ka µn ) + Bi1 Bi3 ( cos µn − 1) ;

(24) (25) (26)

An µn2 ( µn cos µn + Bi1 sin µn ) ; =

(27)

= Bn µn2 ( Bi1 cos µn + µn sin µn ) ;

(28)

(1 + Bi1ka + 2ka ) µn3 − Bi1 Bi3  cos µn − Dn =   , −  ka µn − (1 + Bi1ka + Bi1 + Bi3 ) µn2 − Bi1 Bi3  sin µn , where µ is the root of characteristic equation= (24); Bi3 Bi = k a kλ . 2 kλ ; k a

4

(29)



Having inserted ξ = 0 in (23), we will obtain the Volterra integral equation for determining θ1 ( 0, Fo ) : Fo Ki ( Fo − τ ) kλ −1 + Bi2 Bi3ζ ( Fo − τ ) An ⋅ exp ( − µn2 Fo ) + 2 ∫ 1 + Bi2ζ ( Fo − τ ) n =1 Dn 0 ∞

f ( Fo ) =θ1 ( 0, Fo ) =−1 + 2∑

(30) Fo Bi2ζ ( Fo − τ ) ∞ An An 2 2 ⋅∑ exp ( − µn Fτ ) dτ + 2 Bi3 ∫ f ( Fo − τ ) ⋅ ∑ exp ( −µn Fτ ) dτ 1 + Bi2ζ ( Fo − τ ) n 1 Dn Dn n 1= 0 The Volterra integral equation of the second kind (30) is solved using the method of successive approximations taking into consideration the smallness of the parameter: Bi2ζ ( Fo )