Flow analysis and nozzle-shape optimization for the cold-gas dynamic-spray process

Transcription

1 1 Flow analysis and nozzle-shae otimization for the cold-gas dynamic-sray rocess M Grujicic 1 *, W S DeRosset and D Helfritch 1 Deartment of Mechanical Engineering, Clemson University, Clemson, South Carolina, USA Army Research Laboratory, Processing and Proerties Branch, Aberdeen, Proving Ground, Maryland, USA Abstract: An isentroic, one-dimensional model is used to analyse the dynamics of dilute two-hase (feed owder articles lus the carrier gas) flow during the cold-sray rocess. While the hysical foundation of the model is quite straightforward, the solution for the model can be obtained only numerically. The results obtained show that there is a article-velocity-deendent, carrier-gasinvariant otimal value of the relative gas/article Mach number that maximizes the drag force acting on feed owder articles and, hence, maximizes the acceleration of the articles. Furthermore, it is found that if the cold-sray nozzle is designed in such a way that at each axial location the acceleration of the articles is maximized, a significant increase in the average velocity of the articles at the nozzle exit can be obtained. For the otimum design of the nozzle, helium as the carrier gas is found to give rise to a substantially higher exit velocity of the articles than air. All these findings are in good agreement with exerimental observations. Keywords: cold-gas dynamic sray rocess, nozzle design and otimization NOTATION A C C D D h m _m P R Re S T V x Subscrits e P nozzle cross-sectional area secific heat drag coefficient drag force convective heat transfer coefficient mass mass flowrate ressure gas constant Reynolds number molecular seed ratio temerature velocity axial distance from the nozzle throat secific heat ratio density exit quantity article quantity The MS was received on 6 November 00 and was acceted after revision for ublication on 1 August 003. *Corresonding author: Deartment of Mechanical Engineering, Clemson University, 41 Fluor Daniel Building, Clemson, SC , USA r gas/article relative quantity s shock 0 stagnation quantity 1 freestream quantity Suerscrits ot otimal quantity nozzle throat quantity 1 INTRODUCTION Thermal sray rocesses such as high-velocity oxy-fuel, detonation gun, lasma sray and arc sray are widely used to aly rotective coatings to arts and to reair worn-out arts (e.g. large-diameter shafts in turbines and ums). In these rocesses, the coating material is heated to temeratures high enough to induce melting. Consequently, the high heat inut to the art being coated/reaired accomanying these rocesses can be detrimental if the material of the art degrades when subjected to high temeratures. This roblem is generally avoided in the cold-gas dynamic-sray rocess. The cold-gas dynamic-sray rocess, often referred to as simly cold sray, is a high-rate material deosition rocess in which fine, solid owder articles (generally 1ÿ50 mm in diameter) are accelerated in a suersonic B1900 # IMechE 003 Proc. Instn Mech. Engrs Vol. 17 Part B: J. Engineering Manufacture Paer B1900

2 Reort Documentation Page Form Aroved OMB No Public reorting burden for the collection of information is estimated to average 1 hour er resonse, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and comleting and reviewing the collection of information. Send comments regarding this burden estimate or any other asect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Oerations and Reorts, 115 Jefferson Davis Highway, Suite 104, Arlington VA Resondents should be aware that notwithstanding any other rovision of law, no erson shall be subject to a enalty for failing to comly with a collection of information if it does not dislay a currently valid OMB control number. 1. REPORT DATE 003. REPORT TYPE 3. DATES COVERED to TITLE AND SUBTITLE Flow analysis and nozzle-shae otimization for the cold-gas dynamic-sray rocess 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Celmson University,Deartment of Mechanical Engineering,Clemson,SC, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) 1. DISTRIBUTION/AVAILABILITY STATEMENT Aroved for ublic release; distribution unlimited 13. SUPPLEMENTARY NOTES 11. SPONSOR/MONITOR S REPORT NUMBER(S) 14. ABSTRACT An isentroic, one-dimensional model is used to analyse the dynamics of dilute two-hase (feed owder articles lus the carrier gas) flow during the cold-sray rocess. While the hysical foundation of the model is quite straightforward, the solution for the model can be obtained only numerically. The results obtained show that there is a article-velocity-deendent, carrier-gasinvariant otimal value of the relative gas/article Mach number that maximizes the drag force acting on feed owder articles and, hence, maximizes the acceleration of the articles. Furthermore it is found that if the cold-sray nozzle is designed in such a way that at each axial location the acceleration of the articles is maximized, a significant increase in the average velocity of the articles at the nozzle exit can be obtained. For the otimum design of the nozzle, helium as the carrier gas is found to give rise to a substantially higher exit velocity of the articles than air. All these findings are in good agreement with exerimental observations. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Same as Reort (SAR) 18. NUMBER OF PAGES 11 19a. NAME OF RESPONSIBLE PERSON Standard Form 98 (Rev. 8-98) Prescribed by ANSI Std Z39-18

3 M GRUJICIC, W S DeROSSET AND D HELFRITCH Fig. 1 Schematic of a tyical cold-sray system jet of comressed (carrier) gas to velocities in a range between 500 and 1000 m/s. As the solid articles imact the target surface, they undergo lastic deformation and bond to the surface, raidly building u a layer of deosited material. Cold sray as a coating technology was initially develoed in the mid-1980s at the Institute for Theoretical and Alied Mechanics of the Siberian Division of the Russian Academy of Science in Novosibirsk [1, ]. The Russian scientists successfully deosited a wide range of ure metals, metallic alloys, olymers and comosites on to a variety of substrate materials. In addition, they demonstrated that very high coating deosition rates on the order of 5 m /min (300 ft / min) are attainable using the cold-sray rocess. A simle schematic of a tyical cold-sray device is shown in Fig. 1. Comressed gas of an inlet ressure on the order of 30 bar (500 lb/in ) enters the device and flows through a converging/diverging nozzle to attain a suersonic velocity. The solid owder articles are metered into the gas flow ustream of the converging section of the nozzle and are accelerated by the raidly exanding gas. To achieve higher gas flow velocities in the nozzle, the comressed gas is often reheated. However, while reheat temeratures as high as 900 K are sometimes used, due to the fact that the contact time of sray articles with the hot gas is quite short and that the gas raidly cools as it exands in the diverging section of the nozzle, the temerature of the articles remains substantially below the initial gas reheat temerature and, hence, below the melting temerature of the owder material. The actual mechanism by which the solid articles deform and bond during cold sray is still not well understood. The revailing theory for cold-sray bonding ostulates that, during imact, the solid articles undergo lastic deformation, disrut thin (oxide) surface films and, in turn, achieve intimate conformal contact with the target surface. The intimate conformal contact combined with high contact ressures romotes bonding. This theory is suorted by a number of exerimental findings such as: 1. A wide range of ductile (metallic and olymeric) materials can be successfully cold-srayed while non-ductile materials such as ceramics can be deosited only if they are co-cold-srayed with a ductile (matrix) material.. The mean deosition article velocity should exceed a minimum (material-deendent) critical velocity to achieve deosition, which suggests that sufficient kinetic energy must be available to lastically deform the solid material and/or disrut the surface film. 3. The article kinetic energy at imact is tyically significantly lower than the energy required to melt the article, suggesting that the deosition mechanism is rimarily, or erhas entirely, a solid-state rocess. The lack of melting is directly confirmed through micrograhic examination of the cold-srayed materials []. Because the cold-sray rocess does not normally involve the use of a high-temerature heat source, it generally offers a number of advantages over the thermal-sray material deosition technologies. Among these advantages, the most imortant aear to be: 1. The amount of heat delivered to the coated art is relatively small so microstructural changes in the substrate material are minimal or nonexistent.. Due to the absence of in-flight oxidation and other chemical reactions, thermally sensitive and oxygen sensitive deositing materials (e.g. coer or titanium) can be cold srayed without significant material degradation. 3. Nanohase, intermetallic and amorhous materials, which are not amenable to conventional thermal sray rocesses (due to a major degradation of the deositing material), can be cold srayed. 4. Formation of the embrittling hases is generally avoided. 5. Macro- and micro-segregations of the alloying elements during solidification that accomany conventional thermal sray techniques and can considerably comromise material roerties do not occur during cold sraying. Consequently, attractive roerties are retained in cold-srayed bulk materials. 6. The eening effect of the iminging solid articles can give rise to otentially beneficial comressive residual stresses in cold-sray deosited materials [3], in contrast to the highly detrimental tensile residual stresses induced by solidification shrinkage accomanying the conventional thermal-sray rocesses. 7. Cold sray of materials like coer, solder and olymeric coatings offers exciting new ossibilities for cost effective and environmentally friendly alternatives to technologies such as electrolating, soldering and ainting [4]. The solid state article-bonding mechanism discussed above suggests and exerimental observations (e.g. see reference [4]) confirm that it is desirable to maximize the velocity at which feed owder articles imact the Proc. Instn Mech. Engrs Vol. 17 Part B: J. Engineering Manufacture B1900 # IMechE 003

4 FLOW ANALYSIS AND NOZZLE-SHAPE OPTIMIZATION 3 target surface. While, in general, this can be accomlished by increasing the inlet ressure of the carried gas, for ractical and economic reasons it is desirable to maximize the article imact velocity at a given level of the carriergas inlet ressure by roerly selecting the tye of the carrier gas and its inlet temerature and by otimizing the shae of the converging/diverging cold-sray nozzle. A quite detailed analysis of the effect of the tye of carrier gas, the gas inlet temerature and the shae of the cold-sray nozzle on the imact velocity of the feed owder articles has been carried out by Dykhuizen and Smith [5] using an isentroic, one-dimensional gasflow model. The objective of the resent work is to extend the analysis of Dykhuizen and Smith [5] in order to include the effects of the finite article velocity and variability of the gas/article drag coefficient. The organization of the aer is as follows. A brief overview of the one-dimensional isentroic gas flow model develoed by Dykhuizen and Smith [5] and utilized in this work is resented in section. An extension of the article dynamics model originally roosed by Dykhuizen and Smith [5] to include the effect of the finite article velocity is discussed in section 3. The main results obtained in the resent work and ertaining to the determination of an otimum variation of the gas Mach number along the diverging section of the nozzle, otimization of the nozzle shae and the sensitivity of the otimum nozzle shae to variations in the drag coefficient are resented and discussed in sections 4.1, 4. and 4.3 resectively. The key conclusions resulting from the resent study are summarized in section 5. ISENTROPIC GAS FLOW MODEL In this section, a brief overview is given of the cold-sray gas-flow model develoed by Dykhuizen and Smith [5]. The model considers a tyical geometry of the coldsray converging/diverging nozzle (Fig. 1) and involves a number of assumtions and simlifications such as: 1. The gas flow is assumed to be one dimensional and isentroic (adiabatic and frictionless).. The gas is treated as a erfect (ideal) gas. 3. The constant-ressure and the constant-volume secific heats of the gas are assumed to be constant. The carrier gas flow is assumed to originate from a large chamber or duct where its velocity is zero and the ressure (referred to as the stagnation ressure) is P 0 and the temerature (referred to as the total gas temerature) is T 0. The cold-sray rocess is furthermore assumed to be controlled by the user, who can set the total temerature and the mass flowrate of the gas. The corresonding stagnation ressure can then be calculated using the following rocedure. From the basic dynamic and thermodynamic relations for the comressible fluid flow, the gas temerature T at the smallest cross-sectional area of the converging/ diverging nozzle (referred to as the nozzle throat in the following) where the Mach number (the velocity of the gas divided by the local seed of sound) is unity can be defined as T T ¼ 0 ð1þ 1 þð ÿ 1Þ= where is the ratio of the constant-ressure and the constant-volume secific heats and is tyically set to 1.66 and 1.4 for monoatomic and diatomic gases resectively. Since air is rimarily a mixture of nitrogen and oxygen, it is considered as a diatomic gas. In multi-atom molecular gases, takes on values smaller than 1.4, but such gases are rarely used in the cold-sray rocess. The gas velocity at the nozzle throat is equal to the seed of sound and is defined as V ¼ ffiffiffiffiffiffiffiffiffiffiffiffi RT ðþ where R is the gas constant (the universal gas constant divided by the gas molecular weight). It should be noted that the suerscrit is used throughout this aer to denote the quantities at the nozzle throat, i.e. the gas quantities under sonic conditions. Equation () can be used to exlain the exerimental finding that the low molecular weight (and, hence, large R), monoatomic (and, hence, large and, in turn, high T ) helium is a better carrier gas than the high molecular weight, diatomic air since for the same total gas temerature, T 0, it is associated with a higher seed of sound. From the known (user selected) mass flowrate, _m, the sonic gas density can be comuted as ¼ _m V A ð3þ where A is the (known) cross-sectional area of the nozzle throat. Next, using the ideal gas law, the gas ressure at the nozzle throat can be determined as P ¼ RT ð4þ Once the throat ressure P is comuted, the stagnation ressure P 0 can be calculated using the following isentroic relation: =ð ÿ 1Þ P 0 ¼ P T0 T ¼ P 1 þ ÿ 1 =ð ÿ 1Þ ð5þ After all of the gas-dynamics quantities (T, V, and P ) have been calculated at the nozzle throat, it is ossible to determine these quantities along the diverging section of the nozzle. Towards that end, the variation of one of these quantities or the variation of the Mach number or the variation of the nozzle cross-sectional area along the diverging section of the nozzle must be secified. Dykhuizen and Smith [5] considered the case when the variation of the nozzle cross-sectional area A B1900 # IMechE 003 Proc. Instn Mech. Engrs Vol. 17 Part B: J. Engineering Manufacture

5 4 M GRUJICIC, W S DeROSSET AND D HELFRITCH is secified, which then allows determination of the corresonding Mach number M from the following equation: A 1 A ¼ 1 þ ÿ 1 =½ð ÿ 1ÞŠ M M þ 1 ð6þ Once the Mach number is determined at a given crosssectional area of the diverging section of the nozzle, the remaining corresonding gas quantities (P, T, V and ) can be calculated using the following isentroic relationshis: P ¼ P þ 1 =ð ÿ 1Þ þð ÿ 1ÞM ð7þ T 0 T ¼ 1 þ½ð ÿ 1Þ=ŠM ð8þ V ¼ M ffiffiffiffiffiffiffiffiffiffi RT ð9þ ¼ 0 ð10þ f1 þ½ð ÿ 1Þ=ŠM 1=ð ÿ 1Þ g Equations (7) to (10) can be used to comute P, T, V and at the nozzle exit, if the given exit cross-sectional area A e is substituted for A in equation (6). However, it must be noted that P, T, V and defined in this way will reflect the true conditions of the gas at the nozzle exit only if normal shock does not take lace inside the nozzle. To determine whether the normal shock will take lace inside the nozzle, the ambient ressure should be comared with the following shock ressure: P s ¼ P e þ 1 M e ÿ ÿ 1 ð11þ þ 1 where the subscrit e is used to denote the gas quantities at the nozzle exit. If the shock ressure P s is lower than the ambient ressure, a shock will occur inside the nozzle and the subsequent gas flow is subsonic. Therefore the exit ressure is not given by equation (7) but rather is equal to the ambient ressure. Under normal cold-sray oerating conditions, the shock ressure is maintained above the ambient ressure so that no shock occurs inside the nozzle and the exit ressure is defined by equation (7). At the same time, the exit gas ressure, P e, is generally lower than the ambient ressure in an effort to maximize the exit velocity of the gas (and thus the average velocity of the feed owder articles, referred to as the article velocity in the following). Under such conditions, the one-dimensional gas dynamics model develoed by Dykhuizen and Smith [5] and briefly reviewed above can be used to analyse the gas flow inside the nozzle. As the gas leaves the nozzle, it slows down as the gas ressure tries to adjust to the ambient ressure. However, due to a relatively short nozzle exit/ target surface standoff distance encountered in the coldsray rocess, the decrease in the gas velocity is not exected to be significant and the exit gas velocity can be used as a good aroximation of the gas velocity at the target surface. 3 PARTICLE DYNAMICS MODEL Dykhuizen and Smith [5] also analysed the interactions of the carrier gas with the sray articles under the aroximation of a dilute two-hase (gas lus noninteracting solid articles) flow. The article velocity V P can in this case be determined by solving the following differential equation: m P dv P dt dv ¼ m P V P P dx ¼ C DA P ðv ÿ V P Þ ð1þ where m P and A P are the average mass and crosssectional area of the articles resectively, C D is the drag coefficient, t the time and x the axial distance travelled by the article (measured from the nozzle throat). Equation (1) shows that the maximum article velocity is equal to the gas velocity. Under the condition of constant gas velocity, gas density and drag coefficient, equation (1) can be integrated to yield V ÿ VP log þ V ÿ 1 ¼ C DA P x ð13þ V V ÿ V P m P When the article velocity is very small in comarison to the gas velocity, equation (13) can be simlified as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi C V P ¼ V D A P x 3C ¼ V D x ð14þ m P D P P which is consistent with exerimental observations [6] that the sray article velocity is roortional to the square root of the ratio of the distance travelled x and the article diameter D P. In equation (14) P denotes the article material density. A simle analysis of equation (1) shows that the ultimate article velocity is equal to the gas velocity. Furthermore, examination of equations (6), (8) and (9) indicates that the gas velocity within the nozzle deends on the total gas temerature and the nozzle geometry (i.e. the cross-sectional area at a given axial distance x) but not on the gas ressure. However, equations (9), (10) and (1) indicate that the initial article acceleration (dv P =dt at V P ¼ 0) is linearly deendent on the stagnation ressure but indeendent of the total temerature. Thus, while the stagnation ressure does not affect the maximum article velocity, it has to be sufficiently high to ensure that the sray article velocity will aroach the gas velocity over a relatively short length of the diverging section of the nozzle. Proc. Instn Mech. Engrs Vol. 17 Part B: J. Engineering Manufacture B1900 # IMechE 003

6 FLOW ANALYSIS AND NOZZLE-SHAPE OPTIMIZATION 5 4 RESULTS AND DISCUSSION 4.1 Determination of the otimum Mach number Equation (1) shows that the article acceleration, dv P =dt, is influenced by two quantities of the carrier gas: the gas velocity and the gas density. Since these two conditions are mutually couled through the gas dynamics equations (and, consequently, the gas density decreases as the gas velocity increases along the diverging section of the nozzle), an otimal condition of the gas exists that maximizes the article acceleration. This condition corresonds to the maximum of the drag force acting on the article [the right-hand side of equation (1)]. Under the assumtion that the article velocity is small in comarison to the gas velocity (i.e. V P V) and for constant C D, the otimal gas condition can be obtained by differentiating the drag force, D (the righthand side of equation (1) with V P ¼ 0), with resect to the Mach number to yield dd dm ¼ C DA P M 1 þ ÿ 1 ÿ=ð ÿ 1Þ M ÿ M 1 þ½ð ÿ 1Þ=ŠM ð15þ Since >1, equation (15) shows that the maximum acceleration of a stationary ffiffi article is achieved at the gas velocity of Mach at which dd=dm ¼ 0. The otimum value of the gas Mach number (or the relative Mach number defined as a difference between the gas and the article Mach numbers) at different values of the article velocity (or the article Mach number M P ¼ V P = ffiffiffiffiffiffiffiffiffiffi RT) are determined in the resent work, by finding (numerically) the maximum in the drag force, D, in which the V term is relaced by a ðv ÿ V P Þ term. The results of this calculation are resented in Fig.. Figures a and b show the drag force (normalized by =ðc D A P ) contour lots as a function of the article Mach number with resect to T, MP ¼ V P = ffiffiffiffiffiffiffiffiffiffiffiffi RT ) and the relative gas Mach number M r ¼ ðv ÿ V P Þ= ffiffiffiffiffiffiffiffiffiffi RT for air and helium as the carrier gases. The following (tyical) cold-sray rocessing conditions are used to generate the results resented in Figs a and b, as well as throughout the rest of this aer: the total gas temerature T 0 ¼ 600 K, the stagnation ressure P 0 ¼ bar, the mean article diameter D P ¼ 10 mm and the article material density P ¼ 8g=cm 3. The results resented in Figs a and b show that, for both air and helium, the maximum drag force occurs at a relative Mach number (referred to as the otimum relative Mach number, Mr ot ) whose value decreases with an increase in the article Mach number (i.e. with the article velocity). The variation of the otimum relative Mach number with resect to the article Mach number MP is shown in Fig. c. It is seen that the decrease in the otimum relative Mach number with an increase in the article Mach number is somewhat more ronounced in helium than in air. The variation of the otimum relative Mach number with resect to the article Mach number relative to the actual gas temerature T, M P ¼ V P = ffiffiffiffiffiffiffiffiffiffi RT, is shown in Fig. d. It is seen that, as exected, the otimum relative Mach number decreases with an increase in the article Mach number. Furthermore, it is seen that the Mr ot versus M P relationshi is indeendent of the tye of carrier gas. The results resented in Figs c and d are generally consistent with exerimental findings, which show that the sray article velocity is maximal at a Mach number around unity [6]. 4. Otimization of the nozzle shae The analysis resented in the revious section indicates that in order to maximize the sray article acceleration at each axial location along the diverging section of the nozzle, the relative Mach number should take on a value equal to the value of the otimal relative Mach number corresonding to the local value of the article Mach number. Dykhuizen and Smith [5] assumed that the otimal value of the relative Mach number is indeendent of the article Mach number and that it is equal to unity. Instead, in the resent aer, the otimal value of the relative Mach number is taken to be defined by the Mr ot versus M P relationshi, as dislayed in Fig. d. The relationshi between the gas and the article velocities can then be exressed as V ÿ V P ¼ M ot r ffiffiffiffiffiffiffiffiffiffi RT ð16þ In this section, a rocedure for otimization of the nozzle shae which maximizes the exit article velocity for the given gas stagnation ressure, the total gas temerature and the sray article roerties is resented. Differentiation of equation (16) with resect to the axial osition of the article yields dv dx ÿ dv P dx ¼ Mot r þ dmot r dm P ffiffiffiffiffiffiffiffiffiffi RT T dm P dv P dt dx dv P dx ffiffiffiffiffiffiffiffiffiffi RT ð17þ Differentiation of equation (9) gives dv dx ¼ ffiffiffiffiffiffiffiffiffiffi dm RT dx þ M ffiffiffiffiffiffiffiffiffiffi RT dt ð18þ T dx while combining equations (1) and (16) and differentiating with resect to x gives ffiffiffiffiffiffiffiffiffiffi dv P dx ¼ RTC D A P m P ðm ÿ Mr ot ð19þ Þ B1900 # IMechE 003 Proc. Instn Mech. Engrs Vol. 17 Part B: J. Engineering Manufacture

7 6 M GRUJICIC, W S DeROSSET AND D HELFRITCH Fig. (a) and (b) Contour lots of the normalized drag force relative to variations in the article Mach number and the relative gas Mach number for air and helium as the carrier gases. (c) and (d) Variations of the otimum relative gas Mach number with resect to the article Mach number relative to the nozzle throat temerature T and the actual gas temerature T resectively Also, differentiation of equation (8) gives dt dx ¼ T 0 þð ÿ 1ÞM ð0þ Combining equations (17) to (0) and using equation (10) gives dm þð ÿ 1ÞMM ot r dx þð ÿ 1ÞM ðm ÿ Mr ot Þ þð ÿ 1ÞM ÿ1=ð ÿ 1Þ ¼ ðmr ot Þ 1 þ dmot r dm P ð1þ where X ¼ xc DA P 0 ðþ m P and the stagnation density, 0, is defined by the ideal gas law as 0 ¼ P 0 ð3þ RT 0 The otimal variation of the Mach number along the length of the diverging section of the nozzle is obtained by solving equation (1). It should be noted that, since the normalized axial variable X deends on Proc. Instn Mech. Engrs Vol. 17 Part B: J. Engineering Manufacture B1900 # IMechE 003

8 FLOW ANALYSIS AND NOZZLE-SHAPE OPTIMIZATION 7 the stagnation density 0 [and via equation (3) on the stagnation ressure P 0, the total gas temerature T 0 and the gas molecular weight], the otimal variation of the Mach number, MðxÞ, and, via equation (6), the otimal variation in the nozzle cross-section exansion ratio, A=A, along the diverging section of the nozzle are influenced by the cold-sray rocessing (stagnation) conditions and by the tye of carrier gas used. Once the otimal variation of the Mach number is found, the corresonding variation of the article velocity is obtained by combining equations (8), (9) and (10) to get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V P ¼ðMÿMr ot RT Þ 0 1 þ½ð ÿ 1Þ=ŠM ð4þ As discussed earlier, due to a short standoff distance encountered in the cold-sray rocess, the effect of article deceleration uon exiting the nozzle is not exected to be significant. Hence, if equation (4) is used to comute the article velocity at the nozzle exit, such velocity is exected to be a good aroximation for the velocity at which the feed owder articles imact the target surface. As ointed out earlier, the axial osition (X) in equation (1) is measured from the nozzle throat, so by solving this equation and using equation (6) the otimal shae (i.e. the otimal exansion ratio rofile, A=A ) of the diverging section of the nozzle is obtained. The converging section of the nozzle, on the other hand, is desirable to be as short as ossible in order to attain the required Mach 1 gas velocity at the nozzle throat quickly. However, for the isentroic flow conditions used in the model to be satisfied, a gradually converging section of the nozzle is required. Equation (1) is solved numerically in the resent work using a first-order, forward difference scheme. All the calculations are carried out for a 0.1 m long diverging section of the cold-sray nozzle and under the condition of a constant drag coefficient C D whose value is set to 1. The remaining cold-sray rocessing and feed owder conditions are set identical to the ones used in the revious section: the total gas temerature T 0 ¼ 600 K, the stagnation ressure P 0 ¼ bar, the mean article diameter D P ¼ 10 mm and the article material density P ¼ 8 g/cm 3. Once the otimal variation of the Mach number along the diverging section of the nozzle is determined, equation (4) is used to obtain the corresonding variation of the gas velocity. Next, equation (1) is used to calculate the corresonding variation of the article velocity and, finally, equation (6) is used to determine the otimal nozzle shae (i.e. the otimal variation of the otimal nozzle cross-section exansion ratio). The results of this calculation are resented in Fig. 3. For comarison, the corresonding results obtained under the aroximation used by Dykhuizen and Smith [5], that the otimal relative Mach number is 1, are also resented. The results dislayed in Fig. 3a show the variation of the velocity of air and helium along the diverging section of the nozzle. The corresonding results obtained under the assumtion used by Dykhuizen and Smith [5] are denoted as Ref. [5] in Fig. 3a, as well as in the subsequent figures. The results shown in Fig. 3a indicate that by setting the relative Mach number to its otimal value at each axial location along the diverging section of the nozzle, the gas exit velocity can be increased by 5.3 er cent for air and by.6 er cent for helium relative to the ones obtained under the condition of a relative Mach 1 number. The variation of the article velocity along the diverging section of the nozzle for the cases of air and helium as the carrier gases is dislayed in Fig. 3b. The results shown in Fig. 3b indicate that by setting the relative Mach number to its otimal value at each axial location along the diverging section of the nozzle, the gas article velocity can be increased by 3.5 er cent for air and by 17 er cent for helium as the carrier gas relative to the article velocities obtained under the condition of a relative Mach 1 number. The significantly smaller increase in the article velocity for the case of air as the carrier gas can be understood by comaring the results shown in Figs 3a and b. It is seen that, in the case of air, the article velocity is already within 5 er cent of the air velocity under the condition of a relative Mach 1 number. The variation of the otimal nozzle cross-section exansion ratios along the diverging section of the nozzle for helium and air as the carrier gases is shown in Fig. 3c. It is seen that the otimal nozzle cross-section exansion ratios for helium and air as the carrier gases are quite different from their counterarts obtained under the aroximation used by Dykhuizen and Smith [5]. The variation of the gas ressure and the corresonding shock ressure along the diverging section of the nozzle for helium and air as the carrier gases is shown in Fig. 3d. In addition, the ambient ressure of 1 atm is also indicated in the same figure. It is seen that the shock ressure for both air and helium remains above the ambient ressure until the nozzle exit, suggesting that the normal shock will not occur within the coldsray nozzle. Furthermore, it is seen that in the case of air, the gas flow is overexanded, i.e. the air exit ressure is lower than the ambient ressure. As discussed earlier, this scenario is tyically encountered in the cold-sray rocess. The results shown in Fig. 3d further indicate that there is a maximum allowable, carrier-gas deendent length of the diverging section of the nozzle which, if exceeded, would give rise to the occurrence of the normal shock inside the nozzle. If longer nozzles are needed under secific circumstances, they may be allowed, but the B1900 # IMechE 003 Proc. Instn Mech. Engrs Vol. 17 Part B: J. Engineering Manufacture

9 8 M GRUJICIC, W S DeROSSET AND D HELFRITCH Fig. 3 Variations of (a) the gas velocity, (b) the article velocity, (c) the otimum nozzle cross-section exansion ratio and (d) the ressure and the shock ressure along the diverging section of the nozzle under a constant drag coefficient, C D ¼ 1. See the text for details of the cold-sray rocessing conditions and feed owder roerties nozzle cross-sectional area must remain constant after the maximum allowable value of the exansion ratio is reached in order to revent the normal shock from occurring inside the nozzle. 4.3 Effect of the variable drag coefficient All the calculations erformed in the revious sections were carried out under the condition that the drag coefficient C D is constant and equal to unity. In this section, the effect of variation in the magnitude of the drag coefficient with the flow conditions on the article exit velocity and on the otimal variation in the nozzle cross-section exansion ratio is analysed. The drag coefficient correlating equations for sherical articles over a wide range of flow conditions as develoed by Henderson [7] are used in the resent work. For the subsonic flow, Henderson [7] roosed Proc. Instn Mech. Engrs Vol. 17 Part B: J. Engineering Manufacture B1900 # IMechE 003

10 FLOW ANALYSIS AND NOZZLE-SHAPE OPTIMIZATION 9 the following relation: 3:65 ÿ 1:53TP =T C D ¼ 4 Re þ S 4:33 þ 1 þ 0:353T P =T ex ÿ0:47 Re ÿ1 S þ ex ÿ 0:5M ffiffiffiffiffiffi r 4:5 þ 0:38ð0:03Re þ 0:48 Re Þ ffiffiffiffiffiffi Re 1 þ 0:03Re þ 0:48 ffiffiffiffiffiffi Re þ 0:1Mr þ 0:Mr 8 þ 1 ÿ ex ÿ M r 0:6S ð5þ Re where Re ¼ vd P = is the Reynolds number based on the article diameter, D P, the relative gas/article velocity, v, the freestream gas density,, and viscosity, ffiffiffiffiffiffiffiffi. M r is the relative Mach number and S ¼ M r =. TP in equation (5) denotes the (uniform) article temerature and T the freestream gas temerature. For the suersonic regime at Mach numbers equal to or exceeding 1.75, Henderson [7] roosed the following exression: C D ¼ 0:9 þ 0:34 M 1 M1 þ 1:86 Re 1 þ 1 þ 1:86 1= þ 1:058 TP S1 S 1 1= M1 Re 1 T 1= ÿ 1 S 4 1 ð6þ where the subscrit 1 is used to denote the freestream quantities. For the suersonic regime at Mach numbers between 1.0 and 1.75, Henderson [7] roosed the following linear interolation: C D ðm 1, Re 1 Þ¼C D ð1:0, ReÞþ 4 3 ðm 1 ÿ 1:0Þ ½C D ð1:75, Re 1 ÞÿC D ð1:0, Re 1 ÞŠ ð7þ Fig. 4 Variations of (a) the drag coefficient and (b) the change in the article temerature (and gas temerature; see the inset) along the length of the diverging section of the nozzle where C D ð1:0, ReÞ denotes the drag coefficient calculated using equation (5) at M r ¼ 1:0 and C D ð1:75, Re 1 Þ reresents the drag coefficient calculated using equation (6) with M 1 ¼ 1:75. To include the effect of variability of the drag coefficient on the article exit velocity and on the otimal shae of the diverging section of the cold-sray nozzle, equation (1) is solved numerically while the drag coefficient C D at each axial location of the nozzle is assigned a value consistent with the local flow conditions, as defined by the aroriate correlating equation [equation (5), (6) or (7)]. The results of this calculation are shown in Figs 4 and 5. The variation of the drag coefficient in the case of a nozzle with a throat diameter of 0.01 m along the length of the diverging section of the nozzle for air and helium as the carrier gases is shown in Fig. 4a. The variation of the (uniform) article temerature, T P, along the length of the diverging section of the nozzle [used for calculation of the drag coefficient in equations (5) to (7)] is obtained by numerically integrating the following heat conservation equation: m P C P dt P ¼ A P hðt ÿ T P Þ dt ¼ A P hðt ÿ T P Þ dx V P ð8þ B1900 # IMechE 003 Proc. Instn Mech. Engrs Vol. 17 Part B: J. Engineering Manufacture

11 10 M GRUJICIC, W S DeROSSET AND D HELFRITCH Fig. 5 Variations of (a) the gas velocity, (b) the article velocity, (c) the otimum nozzle cross-section exansion ratio and (d) the ressure and the shock ressure along the diverging section of the nozzle under a variable and a constant (C D ¼ 1) drag coefficient with T P ðx ¼ 0Þ ¼95 K and where the secific heat, C P ð¼ 460 J/kg K), is set to the value common for austenitic stainless steels. The convective heat transfer coefficient, h, on the other hand, is assigned a tyical natural convection value of 190 W/m K. It should be noted that, deending of the flow conditions inside the coldsray nozzle, the value of the convective heat transfer coefficient can be significantly higher than the value used. The effect of the magnitude of the convective heat transfer coefficient is not, however, studied in the resent work. The results dislayed in Fig. 4a show that for both air and helium, the drag coefficient takes on values that exceed one. Consequently, the drag force and, hence, the article exit velocities are exected to be larger than the ones obtained under the C D ¼ 1 condition. An abrut change in the drag coefficient seen in Fig. 4a is associated with the sudden change in relative Mach number, which initially increases from its value of one at the nozzle throat and, after it reaches the otimal value, starts to decrease as the otimal value of the relative Mach number decreases with the increasing article Mach number (Figs c and d). The results dislayed in Fig. 4b show that the article temerature changes by less than 0. K relative to the initial article temerature (95 K) for both air and Proc. Instn Mech. Engrs Vol. 17 Part B: J. Engineering Manufacture B1900 # IMechE 003

12 FLOW ANALYSIS AND NOZZLE-SHAPE OPTIMIZATION 11 helium as the carrier gases. Initially, as seen in the inset of Fig. 4b, the carrier gas has a higher temerature than the articles and, consequently, the article temerature rises. However, as the carrier gas raidly exands downstream in the diverging section of the nozzle, its temerature decreases below the article temerature, causing the article temerature to begin to decrease. In any case, the average article resident time Ð l 0 dx=v P (l ¼ 0:1 m is the length of the nozzle) is quite short (0. ms for air and 0.1 ms for helium) so the overall change in the article temerature is quite small. It should be noted that the magnitude of the article temerature change is greatly affected by the choice of the convective heat transfer coefficient, h. The variations of the gas velocity, the article velocity, the otimal nozzle cross-section exansion ratio and the ressure/shock ressure along the diverging section of the nozzle are shown in Figs 5a to d resectively. For comarison, the variations in the same quantities under the C D ¼ 1 condition are dislayed in Figs 5a to c and denoted using dashed lines and a C D ¼ 1 label. The results dislayed in Figs 5a to c show that the variability of the drag coefficient has a relatively small effect on the variation of the gas velocity, the article velocity and the otimal shae of the diverging section of the cold-sray nozzle. This finding is consistent with the results dislayed in Fig. 4a which show that, while the drag coefficient varies along the length of the diverging section of the nozzle, its value never dearts more than 10 er cent from C D ¼ 1:0. As redicted, since C D takes on values larger than 1, the article velocities are larger than the ones obtained under the C D ¼ 1:0 condition. However, this effect is relatively small. The results dislayed in Fig. 5d show that the inclusion of the deendence of the drag coefficient on the flow conditions does not cause normal shock to occur inside the nozzle and that the air flow is again overexanded. 5 CONCLUSIONS Based on the results obtained in the resent work, the following main conclusions can be drawn: 1. Due to its monoatomic character and a lower molecular weight, helium is a more efficient carrier gas, giving rise to roughly twice the average article exit velocity as that obtained in the case of air as the carrier gas under the otherwise identical coldsray rocessing conditions.. For the given cold-sray rocessing conditions and a fixed article size, the average exit article velocity can be significantly increased by ensuring (via the use of a roerly designed nozzle) that the gas/ article relative Mach number takes on a articlevelocity-deendent, carrier-gas-invariant otimal value. 3. For both air and helium as the carrier gases, the gas/ article drag coefficient C D under the tyical coldsray rocessing conditions does not deart significantly from unity. Consequently, the frequently used C D ¼ 1 simlification aears justified. ACKNOWLEDGEMENTS The material resented in this aer is based on work suorted by the US Army Grant DAAD The authors are indebted to Drs Walter Roy and Fred Stanton of the ARL for their suort and a continuing interest in the resent work. The authors also acknowledge the suort of the Office of High Performance Comuting Facilities at Clemson University. REFERENCES 1 Alkhimov, A. P., Payrin, A. N., Dosarev, V. F., Nestorovich, N. I. and Shusanov, M. M. Gas dynamic sraying method for alying a coating. US Pat. 5,30,414, 1 Aril Tokarev, A. O. Structure of aluminum owder coatings reared by cold gas dynamic sraying. Metal Sci., Heat. Treat., 1996, 35(3 4), McCune, R. C., Payrin, A. N., Hall, J. N., Riggs, W. L. and Zajchowski, P. H. An exloration of the cold gas dynamic sray method for several material systems. In Thermal Sray Science and Technology (Eds C. C. Berndt and S. Samath), 1995,. 1 5 (ASM International, Ohio). 4 Bisho, C. V. and Loar, G. W. Practical ollution abatement method for metal finishing. Plate Surf. Finish, 1993, 80(), Dykhuizen, R. C. and Smith, M. F. Gas dynamic rinciles of cold sray. J. Therm. Sray Technol., 1998, 7(), Neiser, R. A., Brodeman, J. E., Ohern, T. J., Smith, M. F., Dykhuizen, R. C., Roemer, T. J. and Teets, R. E. Wire melting and drolet atomization in a high velocity oxy-fuel jet. In Thermal Sray Science and Technology (Eds C. C. Berndt and S. Samath), 1995, (ASM International, Ohio). 7 Henderson, C. B. Drag coefficients of sheres in continuum and rarefied flows. Am. Inst. Aeronaut. Astronaut. J., 1976, 14(1), B1900 # IMechE 003 Proc. Instn Mech. Engrs Vol. 17 Part B: J. Engineering Manufacture