Theoretical Moel of the Two-Chamber Pressure Casting Process R.G. KEANINI, K. WATANABE, an T. OKABE This article evelops a theoretical moel of the two-chamber pressure casting process. In this process, a molten metal rop, forme by arc melting a soli ingot, falls into a conical crucible attache to a gas-fille, porous cast mol. An energy-base formulation of the mol-filling process is evelope which focuses on the rop s motion within the crucible an mol cavity an on pressure evolution within the mol cavity. The moel shows that rop acceleration into the mol epens on three imensionless parameters, the Euler number, Eu, the Froue number, Fr, an the pressure loss coefficient, K, across the crucible exit. These parameters are in turn etermine by the mol s permeability to the process gas, the characteristic initial pressure ifference between the interior an exterior of the mol, the mol thickness, the process gas viscosity, an the metal ensity. Drop acceleration into the mol compresses trappe gas within the mol cavity; uner most conitions, pressure ecay ue to leakage of the trappe gas through the mol occurs at a faster rate than inertial compression. Uner these circumstances, a ownwar acting pressure force, having a magnitue etermine by the Euler number, acts on the rop. At low Froue numbers, however, gas compression occurs at a faster rate than leakage-inuce ecay an the pressure force acts upwar, again with a magnitue etermine by Eu. Scaling arguments show that friction an evaporation recoil forces are negligible in etermining rop motion, while surface tension, pressure, rop inertia, an gravity are ominant. In aition, soliification effects are shown to be negligible. I. INTRODUCTION TITANIUM has foun increasing use in ental prosthetics, principally as an alternative to allergy-proucing casting alloys. [] Although titanium is ifficult to cast using the lost-wax metho, [] technical improvements have le to various methos for preparing acceptable clinical ental prostheses. [] Dental appliance casting units can be categorize as one of two types, epening on how liqui metal is force into the cast mol: pressure ifference casting units an centrifugal casting machines. This stuy will consier the former, focusing in particular on the two-chamber pressure casting process. A typical two-chamber casting unit is shown schematically in Figure. Here, the upper an lower chambers within the casting unit are separate by an impermeable wall, which supports the cast mol. Both chambers are fille with an inert gas, typically argon, with the lower chamber pressure maintaine at approximately kpa (P l ) an the upper chamber set at approximately kpa (P u ). Due to the pressure ifference between the mol s interior an exterior, gas flows through the mol s porous bottom wall into the lower chamber. Casting is initiate when a titanium (or, more generally, a metal) ingot is arc melte within a copper crucible in the upper chamber; a molten rop eventually forms an falls into R.G. KEANINI, Associate Professor, is with the Department of Mechanical Engineering & Engineering Science, University of North Carolina at Charlotte, Charlotte, NC 83. K. WATANABE, Associate Professor, Division of Dental Biomaterials Science, Grauate School of Meical an Dental Sciences, Niigata University, Japan, 574 Gakkoucho-ori Niigata, Japan 95-854. T. OKABE, Regents Professor an Chairman, Department of Biomaterials Science, Baylor College of Dentistry, Texas A&M University System Health Science Center, Dallas, TX 7546. Manuscript submitte March 9, 4. the conical crucible positione immeiately above the mol cavity. Although the pressure within the upper chamber an that within the mol cavity are initially equal, once the rop falls into the crucible, the gas trappe within the mol cavity continues to pass through the porous mol into the lower chamber. Due to a combination of gas leakage from the mol cavity an compression ue to the rop s ownwar motion, a time-epenent pressure ifference, P u P(t), is create across the molten rop. Consiering the ynamics of the rop, it is apparent that at least two forces rive the molten metal into the mol cavity: the weight of the metal itself an the time-varying pressure ifference that evelops across the melt. Other forces that potentially play a role in cast filling ynamics inclue surface tension, impulse ue to metal evaporation, an friction. Each of these will be iscusse subsequently. With regar to the pressure force, once a molten rop falls into the conical crucible above the mol cavity, the rop s upper free surface is subject to the relatively fixe pressure, P u, within the upper chamber, while the rop s lower free surface is subject to a time varying pressure, P(t), within the mol cavity. The cavity pressure epens principally on the rate of process gas loss from the cavity, which in turn epens on the mol s permeability, k g, [5 8] the gas viscosity, g, the mol s thickness, t, the surface area within the mol cavity available for porous gas transport, A c, an the pressure ifference, P(t) P l, across the mol wall. Previous work has inappropriately ientifie the (fixe) pressure ifference between the upper an lower chambers, P u P l, as the ynamic feature riving metal flow into the mol cavity. In reality, the appropriate pressure ifference, an one of the key ynamic features examine here, is the timeepenent pressure ifference P u P(t) across the rop. [9,] Although the basic principal of operation is apparent, the physical processes unerlying the two-chamber pressure METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 36B, APRIL 5 83
Table I. Parameter Values Fig. Schematic of the two-chamber pressure casting process. casting process are poorly unerstoo. This reflects the relative complexity of the process: casting occurs rapily, on the orer of. secons or less; the casts are small, having volumes on the orer of cm 3 ; process gas can be trappe within the mol cavity, leaing to cast porosity; [3,4] an soliification occurs rapily, possibly affecting cast quality. Progress in minimizing an controlling cast nonuniformity requires improve unerstaning of process physics, an in particular, an improve unerstaning of cast metal flow an mol filling. The purpose of this article is to theoretically investigate the two-chamber pressure casting process. We use scaling arguments to first establish the important physical features unerlying the process. A simple energy-base moel of rop motion an mol filling uring casting is then evelope. The analysis an results provie useful physical insight into the complex features unerlying rapi casting of ental prostheses an shoul provie a framework for future stuies of other ental casting processes. II. PHYSICAL FEATURES AND SCALING ARGUMENTS Prior to formulating a process moel, we use scaling analyses [] to etermine the essential physical features unerlying the casting operation. This will provie a fuller unerstaning of process physics as well as a basis for eveloping the moel. A. Heat Transfer an Drop Soliification In orer to examine the effect of soliification on the casting process, we estimate the amount of soliification that occurs at the crucible exit uring mol filling. We are particularly intereste in etermining if rapi soliification at this location can significantly slow or even shunt liqui metal flow into the mol. A straightforwar estimate, which provies an upper boun on the soliification spee, can be obtaine by assuming that the characteristic heat flux from the rop to the crucible is on the orer of the flux, q p, from Parameter Magnitue Reference h fg 8.9 6 J kg 9 h sl 3.88 5 J kg 9 H c, L c. m k g. 4 m k l, k s.9 Wm C L s.5 m P l. 4 Pa P o, P u. 5 Pa R o.5 3 m R u.5 m t 7.6 4 m T g, T o.5 C T m.945 3 C T b 3.56 3 C V 7.6 6 m U o. ms m 9. 6 m s g.5 5 Nsm m. 3 Nsm 3 m 4.54 3 Nsm.5 N m 3 the plasma torch to the ingot uring melting. Expressing the energy balance across the moving soliification front, we have k l T n r T mh sl U m k s n where k l an k s are the metal s liqui- an soli-phase thermal conuctivities, m is the metal ensity, h sl is the heat of fusion, n is the local unit normal to the propagating soliification front (irecte in the irection of front motion), an U m is the local soliification front spee. Balancing the latent heat generation term with either of the conuctive flux terms an estimating the conuctive flux as q p, we obtain q p U m r m h sl Given U m, the approximate (maximum) thickness, s, of the soliification front at the en of the cavity filling process is given by s U m fill, where fill is the time require to fill the mol. Letting fill s, [] taking q p 6 Wm, [3] an using the parameter values in Table I, we fin that s 5.7 5 m. Since s is less than 3 pct of the crucible exit raius, it is clear that negligible soliification occurs uring cavity filling an that metal flows relatively unimpee into the mol. B. Flui Friction Viscous forces between the rop an the crucible an mol walls play a negligible role in etermining rop ynamics. We show this by first estimating the characteristic shear stress,, exerte by the crucible wall on the liqui metal: t m w m n m mu o [] [] [3] 84 VOLUME 36B, APRIL 5 METALLURGICAL AND MATERIALS TRANSACTIONS B
where m is the liqui metal s ynamic viscosity, n is a coorinate normal to the crucible wall, w is the streamwise velocity, U o is the characteristic axial velocity within the crucible (taken as the characteristic velocity through the crucible exit), an is the characteristic viscous bounary layer thickness. The bounary layer thickness is estimate by balancing the ominant viscous an avective terms in the streamwise momentum equation, yieling A v m L s U o where v m is the kinematic viscosity an L s is the crucible s axial length. Thus, the approximate frictional force retaring rop motion into the cast is F f A sp, where A sp is the characteristic crucible area contacte by the rop; using parameter values liste in Table I, we fin that F f.8 N. Next, we estimate the net pressure force across the rop as (P u P c )A o, where P c is the characteristic pressure acting on the rop s lower free surface, an A o is the area of the crucible exit. (This estimate follows by first recognizing that the characteristic upwar-acting reaction force exerte on the rop by the crucible s lateral walls, P u (A u A o ), largely offsets the characteristic ownwar pressure force, P u A u, acting on the rop s upper free surface (Section III G) Secon, the characteristic ownwar reaction force exerte on the rop by the mol cavity s upper wall, P c (A c A o ), essentially offsets the characteristic upwar pressure force, P c A c, acting on the rop s lower free surface. Here, A u an A c are the cross-sectional areas at the top of the crucible an within the mol cavity. A force balance on the rop then leas to the given estimate. Thus, it is foun that friction is of negligible importance when the characteristic pressure force across the rop, (P u P c )A o, is larger than.8 N (F f ), a conition that hols essentially throughout the filling process (base on inspection of the compute results iscusse subsequently). Consiering briefly the contribution of viscous issipation to the rop s thermal energy balance, it is likewise reaily shown that frictional heating, limite to viscous bounary layers near each soli surface, is of negligible importance. C. Surface Tension Surface Potential Energy A quality cast requires that the rop maintain its integrity against external pressure forces uring casting. This in turn requires that surface tension forces are comparable to the pressure forces. We can show that this conition generally hols by comparing characteristic surface tension forces on the rop s upper an lower free surfaces against the characteristic pressure force across the rop. The upwar-acting surface tension force within the crucible, F STu, is estimate as F STu R u cos (), where R u is the crucible s upper raius, (6.5 eg) is the conical crucible s half angle, is the surface tension coefficient, an where the contact angle between the liqui metal an crucible is assume to be approximately. Similarly, the ownwar acting force, F STl, on the lower free surface is F STl 4L c, where L c is the length of each of the square cavity s sies. Thus, we fin that surface tension forces are comparable to pressure forces when the characteristic pressure ifference across the rop satisfies P P u P c F ST /A o 64 kpa, a conition that [4] typically hols over a large portion of the filling process. (Here, F ST represents either F STu or F STl ). While surface tension forces are comparable to pressure forces, their principal effect is to maintain the rop s free surfaces against pressure, gravity, an inertial forces. In contrast, temporal variations in the rop s surface potential energy play a negligible role in the rop s energy buget. This is shown by comparing the maximum rate of surface potential energy change with the characteristic rate of work one by pressure forces: t A sa A A s Au t fill Pu na P u U u A u Base on the parameter values given in Table I, we fin that this ratio is approximately.3, showing that surface potential energy variations are of seconary importance. Here, U u is the characteristic velocity of the rop s upper free surface an U u A u U o A o (from continuity). D. Evaporation Recoil Due to the rop s relatively high temperature, the momentum flux ue to evaporation from either free surface (evaporation recoil) can be significant. The characteristic rate of metal evaporation m # is given by m # e r e v e A e Q # e /h fg, where e an e are the metal evaporate s ensity an velocity, A e is the evaporating surface area, Q # is the characteristic rate of heat transfer from the rop s interior to the evaporating surface, an h fg is the metal s latent heat of vaporization. Given the evaporate velocity, the approximate recoil force, F e, can be estimate as F e e vea e. In orer to estimate e, we note that evaporate particles thermally equilibrate with the surrouning process gas within one electron mean free path of the evaporating surface. [3] Thus, r e P o RT o where P o an T o are the ambient pressure an temperature of the process gas, an R is the gas constant for titanium. Taking Q # as being on the orer of q p A e q p A u, which again likely represents an overestimate, an using the parameter values in Table I, we fin that Fe 6 N. Since this is again several orers of magnitue smaller than the characteristic pressure force acting across the rop, (P u P c )A o, evaporation recoil plays a negligible role in cast filling ynamics. E. Mol Gas Flow an Gas Heating uring Filling The moel evelope subsequently assumes that argon within the mol cavity remains isothermal uring the filling process. This assumption is vali if the characteristic thermal iffusion time,, for gas within the cavity is much longer than the cavity fill time, fill. Although significant gas flow may be inuce as the liqui metal flows into the cavity, it is not expecte that significant convective heat transfer occurs within the gas. Specifically, gas motion will likely assume a form somewhere between one of two extremes: () for a highly permeable bottom wall, gas motion will be METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 36B, APRIL 5 85
largely uniaxial an irecte towar the bottom; an () for a slightly or a moerately permeable bottom, gas motion will be largely laminar an circulatory. In the first case, it is clear that since minimal convective mixing occurs, in-gas heat transfer will be conuction ominate. In the latter case, since the metal rop s lower free surface remains smooth (Section III G) an since the cavity walls are also smooth, no apparent mechanism exists for creating strong convective mixing within recirculating gas pockets. Thus, again, gas heat transfer is largely etermine by conuction. Comparing the iffusive time scale, given by t L c /a a, where a is argon s thermal iffusivity, with a characteristic fill time of fill (. s [] ), we fin that / fill O( ), showing that the isothermal gas approximation is vali. F. Gas Flow through Cavity Wall As molten metal flows into the mol cavity, process gas trappe within the cavity quickly leaks through the cavity s porous bottom (into the surrouning lower chamber). Assuming that the bottom s porosity is isotropic, gas transport through the wall is governe by Darcy s law: v k g m g P where v is the local gas velocity, k g is the gas permeability, g is the gas viscosity, an P is the local pressure. G. Free Surface Curvature an Rayleigh Taylor Instability The moel evelope subsequently assumes that the rop s upper an lower capillary surfaces remain flat throughout the filling process. We can estimate the characteristic curvature of the rop s free surfaces using the Young Laplace equation: P a P i s H # n where P enotes local external pressure at the upper (P u )or lower (P) interface, P i is the corresponing interfacial pressure within the rop, n is the local outwar unit normal to the interface, an H e x e x is the surface x e y y ivergence operator. The characteristic surface curvature, R s ( H n), can be estimate using the characteristic pressure ifference, P s, across either interface: H # n R s P s s Consiering first the upper interface (within the crucible), since flow within the crucible is largely invisci (iscusse subsequently), the pressure variation, P s, between the upper surface an crucible exit can be estimate via Bernoulli s equation as P s r m U o 5 3 N m (where the hyrostatic contribution, m gl s 3 N m, is of the same orer). Thus, since the upper chamber pressure P u is nearly two orers of magnitue larger, metal within the crucible remains essentially isobaric at the upper chamber pressure. [5] [6] [7] From Reference 7, it is clear then that curvature is small. (Note, since rop volumes are larger than the mol cavity volume, the rop s upper surface remains within the crucible throughout the filling process.) Consiering the lower interface, we are confronte with the problem of estimating pressure loss across the crucible exit. Large pressure losses lea to relatively flat lower free surfaces, while small pressure losses lea to large surface curvature, particularly as cavity pressure ecays. Since pressure loss is unknown, we note inirect evience that lower surface curvature remains small, i.e., significantly smaller than the inverse cavity with, L c : if the curvature was large, i.e., on the orer of L c, then the rop woul quickly impinge an cover the lower porous bounary. Process gas woul then be trappe, leaing to significant cast porosity. Since porosity is generally small, however, premature impingement appears unlikely, suggesting that the lower interface remains relatively flat. Consiering next the lower free surface s stability, since a high ensity flui (liqui metal) overlies a low ensity flui (process gas), the surface may be subject to Rayleigh Taylor capillary instability. [4] Here, isturbances on the free surface remain stable for wavelengths shorter than a critical wavelength, c, given by / s l c p c g(r m r g ) where g is the gas ensity. Using the parameters in Table I, we fin that c cm, which is significantly larger the largest (iagonal) imension within the cavity. Thus, the interface remains stable uring cast filling. H. Flow Characteristics Each metal rop is forme by melting metal ingot with a plasma torch. Due to high torch heat fluxes (on the orer of 6 Wm ), metal surface temperatures approach the metal s boiling point an thermocapillary stresses prouce by surface temperature graients become negligible. Although plasma flow over the rop rives flow within the rop, this flow is likely confine to near surface bounary layers. [5] This is ue to low liqui metal kinematic viscosity an the low viscosity of typical process plasmas. Thus, flow within the rop prior to impingement on the crucible is largely invisci. As the rop falls into the conical crucible (where the cone s interior half-angle is approximately 6.5 eg), it impinges on the sies of the crucible an on the relatively small exit hole at the bottom of the crucible (where the ratio of the crucible s exit to entrance areas is approximately /36). However, base on previous work, [6] it is expecte that generation an mixing of vorticity uring impingement will not significantly alter the invisci nature of rop flow. Thus, flow within the rop appears to remain invisci as it falls into the crucible an fills the mol. III. PROCESS MODEL Base on the physical consierations iscusse in Section III, we can now formulate a relatively simple energy-base moel of liqui metal motion within the crucible an mol [8] 86 VOLUME 36B, APRIL 5 METALLURGICAL AND MATERIALS TRANSACTIONS B
on the upper an lower free surfaces. Aing Eqs. [9] an [], we obtain the overall instantaneous mechanical energy balance on the metal rop: t r V u u V r (u u)u n S A ra P gzb u n S A r (P P )u n S Ao [] where V is the (constant) total liqui metal rop volume, A is the rop s total free surface area, an P an P are the pressures at z an z, respectively. We efine the following area-average properties P Q A Pu # n S z Q A zu # n S U Q A (u # u)u # n S Fig. Process variables an imensions. uring mol filling. A etaile sketch of the process is shown in Figure. The instantaneous mechanical energy balance for flui within the crucible is given by r t VA u u V r (u u)u n S Ao r a P gzbu n S Ao r where V s V s (t) is the volume of liqui metal within the crucible, u is the liqui metal velocity fiel, P is the associate pressure fiel, A o is the area of the crucible s exit, A s A s (t) is the liqui metal s upper free surface area (within the crucible), an n is the outwar unit normal on either A o or A s. (Note that A is locate at z o ; this allows an unambiguous erivation of the overall energy balance given subsequently. Note too that in the following, we suppress the subscript on the liqui metal ensity, m.) A similar expression can be written for liqui metal within the mol cavity: r t Vc u u V A r (u u)u n S ra P gzb u n S Ac r [9] [] where V c V c (t) is the liqui metal volume within the mol, A o A is the crucible exit area (evaluate at z o A o ), an A c is the liqui metal s lower free surface within the mol. Note that the gravitational work terms in Eqs. [9] an [] anticipate use of a one-imensional (-D) flow approximation where Q u # n S is the volumetric flow rate through A area A, an efining a loss coefficient, [7] K, as we can express [] as V r ru P P K t U V m # c P s P c g(z r s z c ) (U s U c ) K U o [] [3] where the Liebnitz rule an ivergence theorem have been use, an where U u u an U o U at the crucible exit. In orer to evaluate the volume integral in Eq. [3], we assume that the velocity istribution at any axial position z within the crucible an cavity is -D an is irecte in the axial irection. As iscusse previously, flow within the crucible is largely invisci. Since it is expecte that the velocity fiel within the crucible approaches three-imensional potential flow into a mass sink (locate at the crucible s apparent vertex), an estimate of the error associate with the -D flow assumption can be obtaine by calculating the ratio,, of the kinetic energy flux ue to potential sink flow against that ue to -D parallel flow: a (pr (z)) au r u z [4] where u an u z are the raial an axial velocity components for sink flow an where the integration is carrie out over the crucible s cross-sectional area at any position z. The calculation shows that.9. Thus, ue to the relative steepness of the crucible s conical wall, the -D flow assumption is reasonable. Within the mol cavity, a short perio of lateral spreaing occurs when the metal first enters the cavity an spreas to meet the mol s vertical walls. After this, cavity flow is preominantly -D in the axial irection. Base on the flat free surface assumption escribe previously, the uniform flow approximation also appears to be reasonable within the mol cavity. U b r sin fufr METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 36B, APRIL 5 87
The volume integral in Eq. [3] thus becomes r U V V t [5] where the first integral is taken over the volume of liqui metal within the mol cavity, V c, while the secon is over the liqui metal volume within the crucible (V s ). The argument in the first integral follows by application of continuity between the metal rop s upper free surface an any axial position z occupie by liqui metal within the cavity: z # cl, where L c z # spr s c A c is the cavity s cross-sectional area, an R s is the upper free surface s cross-sectional area (evaluate at z z s ). Note that R s R s (z s ) mz s R o, where m (.5) is the conical crucible s slope an R o is the raius at the crucible exit. The secon integral follows in a similar manner. (For clarity, we have roppe overbars on the average positions, z s an z c, of the upper an lower free surfaces.) Carrying out the integrations in Eq. [5] an rearranging then leas to the equation governing rop motion: V r where A s pr s. t U V rz A. Mol Gas Pressure r A c Vc r Vs t [z# s(mz s R o ) p] z t [z# s(mz s R o ) p] z # s A s z c A c rz# s z s A s mp [6] We assume that the process gas is argon an that it behaves as an ieal gas. The rate of gas loss from the mol cavity is given by m # g RT g t (PV ) [7] where R is the gas constant for argon, T g is the constant gas temperature, an P P(t) an V V(t) are the timeepenent gas pressure an volume within the cavity, respectively. Since gas loss occurs ue to flow through the cavity s porous bottom wall, we can also express the rate of mass loss as m # g Ac r g v g n A Ac r g k g m g P n A [8] Neglecting small spatial pressure variations ue to gas flow within both the cavity an lower chamber an assuming that the bottom wall is of constant thickness, t, then P n P P l t Cz # sa s pr s mz s D CR o R s D [9] where again P l is the lower chamber pressure. Note, since pressures on both sies of the bottom wall are spatially uniform, the pressure fiel within the wall epens only on the axial coorinate z an is governe by P/z. Integrating this expression shows that the preceing erivative is exact. Now, since P is spatially uniform within the cavity, the integral in Eq. [8] can be evaluate as kg r g P n A k g A c P (P P S m l ) g m g t RT g [] where the ieal gas law has been use to express g in terms of P an T g. Setting Eqs. [7] an [8] equal to one another an substituting Eq. [] for [8] yiels [] where P # P an V # V. t t The time rate of change of gas volume within the cavity is relate to the instantaneous spee of the molten rop s lower bounary, z # c, as follows: [] where V(t) (H c z c )A c, an H c is the cavity s height. (Note that z c.) This proves convenient when expressing z c in terms of the rop s upper free surface position, z s, using V A c z c p 3m [(mz s R o ) 3 R o 3 ] [3] where again V is the rop s fixe volume. Thus, substituting Eq. [] into [], we obtain the equation escribing pressure evolution within the mol cavity: P # VP # PV # k ga c m g t P (P P l ) V # A c z # c c kg P (P P H c z c m g t l ) Pz # c [4] where z c is relate to z s through Eq. [3]. Notice from Eq. [4] that mol pressure evolution is etermine by two competing processes: pressure ecay ue to gas leakage from the bottom of the mol (represente by the first term on the right) an compression ue to the pistonlike motion of the liqui metal into the cavity (represente by the secon term). Thus, if the magnitue of liqui metal s ownwar (negative) velocity z # c is large enough to overcome ecay ue to leakage, trappe gas can be compresse, leaing to increasing pressure, P. B. Nonimensional equations Equations [3] an [4] represent two couple equations governing rop motion uring mol filling an cavity pressure evolution. In orer to obtain a fuller qualitative unerstaning of the process as well as to facilitate the numerical solution, Eqs. [4] an [3] are nonimensionalize as follows: () all lengths are scale using the cavity with L c (which is on the orer of both the cavity height H c an the crucible length L s ); () pressures are scale using the pressure ifference P between the upper an lower chambers, P P u P l ; (3) the time scale, o, is chosen as the characteristic mol fill time, o V m /Q o, where V m H c L c is the cavity volume an Q o is the metal s characteristic volumetric flow rate into the cavity. In orer to estimate Q o, we 88 VOLUME 36B, APRIL 5 METALLURGICAL AND MATERIALS TRANSACTIONS B
recognize that this quantity is etermine by the gas s characteristic rate of flow through the mol, Q o k g PL c m g t, Thus, o is given by [5] The resulting nonimensional forms of Eqs. [4] an [] are given by an z & & z cz & s p sm s A & c t o t H c m g Pk g & P z & [P & (P & P & l) z & c ] c (z & & s z ) c Fr & ( z sz & c) Eu(P & u P & ) K & a Rs 4 & b Ro ccr & R & s A& & s m p zc A & s A & c [6] [7] where, an A & R & P & s R s L c, & PP, & z s R o R o L c, A & z s L c, & z c z c L c, P & l PlP, s A s L c c A c L c. Three imensionless parameters, the Euler number, Eu, the Froue number, Fr, an the loss coefficient K, appear in Eq. [7] an are efine as follows: Eu t o P t m g H c L c r m Pk g r m L c Fr L c t o g P k g L c t H c m g g [8] [9] where the efinition of o has been use. The Euler number inicates the relative importance of pressure forces to inertia, while the Froue number inicates the relative importance of inertia to gravity forces. In interpreting the results presente in the next section, it proves useful to recognize the physical meaning of the last three terms in Eq. [7]. The term Fr (z & s & z c) etermines & $ the rop s ownwar acceleration (emboie in zs) ue to gravity. From the efinition of Fr in Eq. [9], it is seen that rop inertia is etermine by the pressure ifference, P, mol permeability an thickness, gas viscosity, an since H c L c, mol epth. If gravity is significantly larger than inertia, Fr, an as shown in Section V, ownwar acceleration can be rapi enough for compression to overtake leakage-inuce ecay, causing cavity pressure to increase. The term Eu(P u P) etermines rop acceleration ue to the pressure ifference across the rop s upper an lower free surfaces. Uner most circumstances, when compression lags, pressure ecay, P u P, is positive an the rop is accelerate ownwar. However, when acceleration is rapi enough for compression to ominate ecay, P u P is negative an the pressure force acts upwar, retaring ownwar acceleration. Finally, the term K(R s /R o ) 4 always acts to retar the rop s ownwar acceleration, with a magnitue that epens on K. IV. RESULTS AND DISCUSSION In this section, we examine the effects of the Euler an Froue numbers an the loss coefficient an mol crosssectional area on mol filling ynamics an mol pressure evolution. The imensionless Eqs. [6] an [7] are solve using fourth-orer Runge Kutta integration. Initial conitions are efine as follows. The metal rop is assume to fill the crucible, so that & z u().5. In aition, it is assume that & the rop s upper surface is initially stationary, or zs (). Assuming that the rop falls a istance h o (. m) into the crucible after melting from the ingot, the approximate initial spee of liqui through the crucible exit is gh o (.4 ms, neglecting the retaring effects of surface tension an gas compression within the mol). Thus, by continuity, () zs, inicating the valiity of the stationary & initial conition. Finally, the initial cavity pressure is set equal to the upper chamber pressure: P & () P & u.. Reference values for the Euler an Froue numbers are etermine using the nominal parameter values in Table I, giving Eu 59 an Fr.54. A reference value for the loss coefficient can be estimate by balancing mass, linear momentum, an mechanical energy between the crucible s exit plane (z ) an an arbitrary ownstream location within the mol. [7] The result is given by K ( A o A c ) [3] Since the ratio of crucible exit area to mol cross-sectional area A o /A c.96, the reference value for K was taken as K.646. Finally, a reference value of the mol s crosssectional area was chosen as A c 4 m. Mol filling ynamics will be characterize by the imensional time require to fill the mol, t f (which we will refer to as the mol fill time), while mol pressure evolution will be characterize by the final imensionless mol pressure, P & f, i.e., the imensionless pressure extant at the instant when liqui metal completely fills the mol. Process behavior will be investigate over the following ranges of Fr, Eu, K, an A & c: Fr, Eu 3, K 4, an à c 7. The effect of varying Euler an Froue numbers on final mol pressure an casting time are shown in Figures 3 an 4, respectively. At low Froue numbers (Fr / ), P f is actually higher than the initial mol pressure of.. As mentione, from the energy an pressure evolution, Eqs. [7] an [6], when Fr is large, ownwar rop acceleration, represente by z &s, becomes large enough for compression to overtake pressure ecay ue to leakage; in this case, pressure increases. Similarly, at fixe Fr less than /, final mol pressure increases with ecreasing Eu. In this case, since P u P is negative, the pressure force acts upwar an resistance to ownwar motion becomes smaller with ecreasing Eu; compression an associate final mol pressure thus increase. When Fr is greater than approximately /, the gravitational acceleration term becomes comparable to (an, at larger Fr, smaller than) the pressure-riven acceleration term. Since all terms on the right sie of Eq. [6] are initially small, initial ownwar acceleration is likewise small, an inspection of the time-epenent mol pressure shows that compression oes not overcome ecay ue to gas leakage. METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 36B, APRIL 5 89
5.5 Final Mol Pressure 4 3 Eu = - - Log Fr Fig. 3 Effect of Froue an Euler numbers on final imensionless mol pressure. K.646, A c 4 m. Final Mol Pressure.5 Eu = Eu = Eu= Eu= 3 4 5 6 7 Fig. 5 Effect of imensionless mol cross-sectional area on final imensionless mol pressure. K.646, Fr.54. Mol Fill Time (s).5..5..5 Eu = - - log Fr Fig. 4 Effect of Froue an Euler numbers on imensional mol fill time. K.646, A c 4 m. In contrast to conitions extant uner small Froue numbers (Fr / ), the pressure force thus acts ownwar, with a magnitue that increases with increasing Euler number. Thus, at fixe Fr an increasing Eu, pressure ecay is increasingly offset by compression, so that final pressure increases with increasing Eu. Trens in mol fill time, shown in Figure 4, are consistent with trens in pressure evolution iscusse previously. Specifically, uner conitions where compression ominates leakage-inuce ecay, (Fr / ), the upwar acting pressure force slows the rop s ownwar motion, leaing to increase fill time. Thus, conitions causing cavity pressure to rise above the upper chamber pressure also lengthen fill times (compare Figures 3 an 4). In contrast, when final pressure is less than P u, inspection of the associate time varying mol pressure (not shown) shows that the pressure force is ownwar. Since the ownwar force is proportional to Eu, ownwar acceleration increases with increasing Eu, leaing to increasing compression an ecreasing mol fill time. Note too that preicte fill times are consis- tent with those observe by Watanabe et al. in an earlier stuy of plate mol casting. [8] Consiering the effect of mol cross-sectional area on mol filling, Figure 5 shows that final mol pressure is less than P u at all Euler numbers an areas consiere. Since P is less P u, then, as before, an for any given area, the pressure force acts ownwar with a magnitue that increases with increasing Eu. Thus, compression increasingly offsets pressure ecay, resulting in higher final pressures. (Although not shown, compression ominates ecay when Eu 4, leaing to final pressures higher than P u.) At any given Eu, we observe the intuitively reasonable result that final pressure ecreases with increasing cross-sectional area. From Reference 7, it is seen that since z c, both the enominator an numerator ecrease with increasing A c ; however, comparing magnitues of the two terms involving A c, the term in the numerator is foun to be larger. Thus, ownwar acceleration ecreases with increasing A & c, implying less compression relative to ecay an lower final pressures. Physically, this is a continuity effect, since gravity an pressure forces o not vary significantly, i.e., by less than an orer of magnitue; then, as area increases, given vertical isplacements within the crucible prouce smaller vertical isplacements within the mol. Mol fill times, shown in Figure 6, are again consistent with trens in mol pressure evolution. In particular, since the pressure force is ownwar uner all conitions shown in Figure 5, an since the force magnitue is again proportional to Eu, then for any fixe area, ownwar acceleration increases, an thus fill time ecreases, with increasing Eu. The near-linear increase in fill time with increasing à c (at fixe Eu) reflects mass conservation. Since forces are relatively fixe, it is foun that following a short initial perio, the volumetric flow rate from the crucible, Q, remains essentially constant an inepenent of A c. Thus, t f Q A c. Consiering finally the effect of loss coefficient, K, on mol filling, we fin that trens in final mol pressure an fill time are qualitatively similar to those observe for varying cross-sectional areas. Referring to Figure 7, for example, we see that as K increases at any given Euler number, 9 VOLUME 36B, APRIL 5 METALLURGICAL AND MATERIALS TRANSACTIONS B
.8 8 Mol Fill Time (s).6.4. Eu= Eu= Eu = Eu = 3 4 5 6 7 Fig. 6 Effect of imensionless mol cross-sectional area on imensional mol fill time. K.646, Fr.54. Mol Fill Time (s) 6 4 Eu= 3 4 Log K Fig. 8 Effect of pressure loss coefficient on imensional mol fill time. Fr.54, A c 4 m. Final Mol Pressure.5.5 Eu = 3 4 Log K Fig. 7 Effect of pressure loss coefficient on final imensionless mol pressure. Fr.54, A c 4 m. the final pressure ecreases; see Figure 5 for comparison. Similarly, at any given K, final pressure increases with increasing Eu. We can use Eq. [7] an arguments similar to those previously given to explain these results. Likewise, as shown in Figure 8, at fixe Eu, fill time increases with increasing K, similar to the behavior observe for increasing cross-sectional area; refer to Figure 6. Physically, as K increases at any given Euler number, resistance to flow through the crucible s exit also increases. Thus, ownwar acceleration an gas compression ecrease, leaing to lower final mol pressures an longer fill times (Figure 8). V. CONCLUSIONS Scaling an physical arguments inicate that minimal soliification occurs uring mol filling, that frictional effects play a negligible role in rop ynamics, that process gas within the mol remains essentially isothermal, that evaporation recoil is negligible, an that the rop s free surfaces remain relatively flat uring filling. Base on these arguments, a simple energy-base moel of the two-chamber pressure casting process is evelope. The moel is use to examine the effects of various process parameters on casting ynamics. The most important results an observations are as follows.. Two competing processes, pressure ecay ue to gas leakage an compression ue to rop motion into the mol, etermine mol pressure evolution an consequent mol fill times.. When Fr /, gravitational forces on the rop are large enough to compress gas within the mol. In this case, the net pressure force on the rop acts upwar so that as the Euler number increases at any given Fr, the pressure force also increases, reucing both the rop s ownwar acceleration an consequent gas compression within the mol. Corresponing mol fill times increase while final mol pressures ecrease. 3. When Fr /, gravitational forces are no longer large enough for compression to ominate leakageinuce ecay. Here, mol pressure rops an remains below its initial magnitue, proucing a ownwar acting pressure force; as the Euler number increases at fixe Fr, the pressure force also increases, increasing both the rop s ownwar acceleration an gas compression within the mol. Corresponing mol fill times ecrease while final pressures increase. 4. Uner the reference conitions chosen, an over the range of mol cross-sectional areas examine, A & c 7, final mol pressure is in every instance less than the upper chamber pressure. Since the pressure force is always ownwar an is proportional to the Euler number, at any given A & c, rop acceleration an associate gas compression both increase with increasing Eu. Thus, corresponing final mol pressures increase while mol fill times ecrease. Since the forces acting on the rop are largely fixe, it is foun that the volumetric flow rate from the crucible is likewise fixe. This leas to an essentially linear relationship between mol fill time an A & c. 5. Drop acceleration an associate gas compression ecrease with increasing loss coefficient at the crucible METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 36B, APRIL 5 9
exit. Thus, at fixe Euler number, corresponing final mol pressures ecrease while fill times increase. ACKNOWLEDGMENTS The authors acknowlege support for this work from NIH/NIDCR Grant No. DE787. They are also grateful for the eitorial assistance with this article from Mrs. Jeanne Santa Cruz. LIST OF SYMBOLS A c cross-sectional area of mol cavity A o cross-sectional area of crucible exit A s rop s upper free surface area A sp characteristic area of contact between rop an crucible A u cross-sectional area of crucible h fg latent heat of vaporization h sl latent heat of fusion H c, L c height an with (epth) of mol cavity k g gas permeability K pressure loss coefficient at crucible exit k l,, k s liqui an soli phase thermal conuctivities L s crucible height m slope of crucible lateral wall rh g gas mass flow rate through bottom of mol n, n unit normal vector an normal irection P time-epenent mol pressure P c characteristic pressure on the rop s lower surface P o ambient upper chamber pressure (P u ) P u, P l upper an lower chamber pressures q p characteristic rop surface heat flux; characteristic plasma torch flux Q volumetric flow rate R gas constant for metal evaporate R o crucible exit raius R s characteristic free surface raius of curvature R u crucible top raius T temperature t thickness of mol s bottom wall t f mol fill time T g, T o process gas temperature T m metal melting temperature T b liqui metal boiling temperature u liqui metal velocity U o liqui metal spee at crucible exit; characteristic axial velocity of crucible U s, U c velocity of metal rop s upper an lower free surfaces U u characteristic velocity of rop s upper free surface v g gas velocity within lower mol wall V time-epenent gas volume within mol cavity V metal rop volume z s, z c axial positions of the rop s upper an lower free surfaces Greek letters m metal thermal iffusivity characteristic viscous bounary layer thickness s characteristic soliification thickness P pressure ifference between upper an lower surfaces g process gas viscosity m liqui metal viscosity v m liqui metal kinematic viscosity m liqui metal ensity crucible half angle liqui metal surface tension characteristic shear strain at crucible wall fill characteristic observe mol fill time theoretical characteristic mol fill time o REFERENCES. S. Canay, N. Hersek, A. Culha, an S. Bilgic: J. Oral Rehabil., 998, vol. 5, pp. 759-64.. T. Okabe, C. Ohkubo, I. Watanabe, O. Okuno, an Y. Takaa: J. Met., 998, vol. 5, pp. 4-9. 3. H. Her, M. Syveru, an M. Waarli: Dent. Mater., 993, vol. 9, pp. 5-8. 4. H.S. Al-Mesmar, S.M. Morgano, an L.E. Mark: J. Prosthet. Dent., 999, vol. 8, pp. 5-. 5. T. Togaya an K. Ia: Trans. 3r Worl Biomater. Congr., 988, pp. 5p-34, 575. 6. M. Syveru an H. Her: Dent. Mater., 995, vol., pp. 4-8. 7. Y. Inoue: J. Jpn. Dent. Mater., 995, vol. 4, pp. 3-. 8. I. Watanabe, J.H. Watkins, H. Nakajima, M. Atsuta, an T. Okabe: J. Dent. Res., 997, vol. 76, pp. 773-79. 9. K. Watanabe, S. Okawa, O. Miyakawa, S. Nakano, H. Honnma, N. Shiokawa, an M. Kobayashi: J. Jpn. Dent. Mater., 993, vol., pp. 496-55.. K. Watanabe, S. Okawa, M. Kanatani, T. Okabe, an O. Miyakawa: J. Jpn. Dent. Mater., 999, vol. 8 (Spec. Iss.), p. 34.. S. Mahajan: Ph.D. Dissertation, California Institute of Technology, Pasaena, CA, 998, pp. -89.. T. Togaya, S. Tsutsumi, Y. Tani, Y. Yabugami, H. Hiroshima, S. Iwaki, S. Ohyagi, an S. Shimakawa: Proc. 7th Dental Titanium Conf., 994, pp. -. 3. R.G. Keanini: Phys. Rev. E, 995, vol. 5, pp. 457-75. 4. P.G. Drazin an W.H. Rei: Hyroynamic Stability, Cambrige University Press, New York, NY, 98, pp. 9-. 5. S. Ostrach: Annual Rev. Flui Mech., Annual Reviews, Palo Alto, CA, 98, pp. 33-45. 6. X. Zhang an O.A. Basaran: J. Coll. Interface Sci., 997, vol. 87, pp. 77-78. 7. B. Munson, D. Young, an T. Okiishi: Funamentals of Flui Mechanics, 3r e., Wiley, New York, NY, 998, pp. 5-. 8. K. Watanabe, S. Okawa, O. Miykawa, S. Nakano, N. Shiokawa, an M. Kobayashi: J. Jpn. Dental Mater. Dev., 99, vol., pp. 77-96. 9. ASM Hanbook, vol., Properties an Selection: Nonferrous Alloys an Special-Purpose Materials, ASM INTERNATIONAL, Materials Park, OH, 99, p. 35.. K. Watanabe: Niigata University, Niigata, Japan, private communication,.. F.P. Incropera an D.P. DeWitt: Funamentals of Heat an Mass Transfer, 4th e., Wiley, New York, NY, 996, p. 83.. Physics of Weling, J.F. Lancaster, e., Pergamon Press, New York, NY, 984, p.. 3. R.G. Keanini an B. Rubinsky: Int. J. Heat Mass Transfer, 993, vol. 36, pp. 383-98. 9 VOLUME 36B, APRIL 5 METALLURGICAL AND MATERIALS TRANSACTIONS B