DESCRIPTION OF TUNGSTEN TRANSPORT PROCESSES.IN INERT GAS INCANDESCENT LAMPS *)

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1 Philips J. Res. 38, , 1983 R 1071 DESCRIPTION OF TUNGSTEN TRANSPORT PROCESSES.IN INERT GAS INCANDESCENT LAMPS *) by E. SCHNEDLER Philips GmbH Forschungslaboratorium Aachen, 5100 Aachen, Germany Abstract This report discusses the importance of transport processes in incandescent lamps. For a model which may describe the radial and axial transports in an inert gas lamp an exact analytical solution for heat and mass transport is given, including effects of temperature dependent transport coefficients and of thermal diffusion. The influence of different inert gases upon axial and radial diffusion of tungsten in an inert gas lamp is shown. 1. Introduetion One of the most important tungsten transport processes in the vicinity of an incandescent filament is the diffusion of tungsten and its compounds 1,2,3,4). Therefore expressions for axial and radial diffusion will be derived. In connection with this the importance of thermal diffusion will be discussed. 2. Inert gas lamp 2.1. Temperature distribution As the transport coefficients are functions of the temperature T, at first the temperature distribution.inside an incandescent lamp has to be calculated. In general, this must be performed numerically, but for simple geometries an analytical solution can be derived. According to the Langmuir model of an incandescent lamp 2,3) we' assume that heat transport is based on heat conduction in a stagnant inert gas layer. This might be an appropriate approximation for inert gas lamps as well as for halogen lamps because chemical heat transport can be neglected as the concentrations of chemically reactive gases are very low. * Parts of this work have been sponsored by Bundesministerium für Forschung und Technologie under Grant No. 03E4120A. 224 Phlllps Journalof Research Vol.38 Nos 4/5 1983

2 Description of tungsten transport processes in inert gas incandescent lamps The heat transport is described by with the heat current and the heat conductivity V tl w = 0, (1) qw = -À V T (2) À=K P. (3) K is a constant independent of the temperature. The value of the exponent y lies between 0.5 and 1, depending on the kind of the inert gas 5). Eq. (1) then can be reduced to with the heat current potential T K S=j À dt= -- T1+1. o Y + 1 LIS = 0 (4) For two simple geometries the analytical solutions of (4) will be given, in order to describe the essential transport processes in incandescent lamps. 2.1.a. A hot tungsten wire of infinite length in the axis of a cylinder filled with inert gas (5) The radius of the wire is rf; rb is the radius of the diffusion zone. The two dimensional Green's function of this potential problem reads 6) G(r - r') = In Ir - r"]. (6) Considering the boundary conditions, center of the wire at r' = (0,0), T(rf) = Tf, and T(rb) = Tb, the following solution of (4) can be derived from (6) K [ TJ+1- Tr 1 r 1+1J S(r) = -- In - + If y :- 1 In rf rf rb The corresponding temperature distribution then is given by '7"'1+1- '7'1+1 J 1/(1+1).If.I b r 1+1 T(r) = [ In - + Tf. In rf rf. rb (7) (8) Philips Journolof Research Vol.38 Nos 4/

3 E. Schnedler From eq. (7) it is easy to calculate the power loss of the hot wire due to heat conduction in the inert gas E - = - f qw' cis = - f V. qw dj = f LI S dj, (9) I F F È/ I = power loss per unit length and F = area that contains the wire. Eq. (9) holds for two dimensions, together with one finds LI G(r - r') = 21t ö(r - r'), (10) E K - =21t-- I JI + 1 'T'~+I 'T'~+I.Lf -.Lb rf ln- r» If we take JI = 0, i.e. the heat conductivity is independent of the temperature, eq. (11) coincides with an equation that has been derived by Elenbaas 7). However, one finds that the temperature given by eq. (8) is somewhat higher than that given by Elenbaas. An analogue to eq. (11) has also been reported and experimentally tested by Coaton 8). 2.1.b. Two parallel tungsten wires in a stagnant inert gas atmosphere (approximation for the axial transport from one turn to another) y (11) s- 2 x The solution for the heat flow potential S again is derived from the Green's function (6) with the boundary conditions T = Tl at the surface of wire 1 and T = T2 at the surface of wire 2, 1.! _ (TrI - TrI) 2 JI + 1. ( (x - VS2 - r} + y2) ) S(x,y) = In..:..._~==;;=~;=--=--::- In (S - VS2 - r} ) (x + VS2 - r} + y2) S + VS2 - r} E: (TrI + TrI). 2 JI + 1 (12) 226 Philips Journal er Research Vol.38 Nos 4/5 1983

4 Description of tungsten transport processes in inert gas incandescent lamps The temperature distribution can be calculated according to y + 1 T(x,y) = -- Sl/(Y+ll. K Like eq. (11) one can calculate the power transfer between the two wires by integrating the heat current over any surface which is surrounding one of the wires (14) È K y+1 y+1 [ rf ] -1 - = 1t -- (T2 - Tl ) In V 2 2 I y + 1 s + s - rf 2.2. Concentration distribution Before considering transport processes in halogen lamps, it is worthwhile to discuss the relevant transport processes in inert gas lamps. Especially thermal diffusion often is neglected by many authors, but as we will show, it may be one of the important transport processes which take place in incandescent lamps. In the case of an inert gas lamp the current of atomic tungsten is given by (comp. refs 5 and 9) i = density of the mass diffusion current, (! = mass density of the inert gas, c = mass concentration of tungsten, KT = thermal diffusion ratio, D = diffusion coefficient; the temperature dependence of D is given by - (13) (15) D=d TP,1.5</3<2.0, (16) where d - 1/PI takes into account the pressure dependence. For small concentrations c -e: 1 the thermal diffusion ratio reads 6,9,10) (17) In general it is possible to show, that for high temperatures a is independent of T. (This holds not only for hard sphere gases.) Generally it is V X i - VcxVT =F 0, thus there will be no potential with ;=V(/J. This (/J exists only if VcIIVT, Le. the geometry enables the solution c = c(t). As c = c(t) holds for the special symmetry (of the considered geometries) discussed in sec. 2.1.a and sec. 2.1.b an analytical solution of V ;=LI(/J=O (18) can be derived. From eq. (18) the substitution c = c(t) leads to 0= V ; = mipi d V. (TP-l dc VT+ atp-2 c(t) VT). KB dt (19). Philips Journul of Research Vol. 38 Nos 4/

5 E. Schnedler KB = Boltzmann factor, PI = inert gas pressure, mr = mass of an inert gas particle. Eq. (19) is equivalent to o = e T-l a [(P - 2) T-l (VT)2 + Ö T] + de d 2 e + dt [T-l(VT)2 (p a) + ÖT] + dt2 (VT)2. Taking into account that T is the solution of the heat conduction (eq. 1), i.e. V(PVT) = 0, one derives problem c- a. de d 2 e o = T (P y) + dt (P a - y) + T dt 2. (20) This differential equation is solved by eet) = A I Ty+2-P+ B'. T-a. (21) The corresponding diffusion potential (/J can be calculated by integration ofthe current T (/J = mipi J [TP-l ~ + atp-2 e(t)] dt + o«. KB To Inserting eq. (21) into (22) leads to the diffusion potential dt (22) I PI d mi (y P + a) Ty+l (/J =A --. rr- + (/Jo =A -- + (/Jo. (23) KB y + 1 Y + 1 As (/J and S are solutions of the same differential equation ö({j = ös = 0 for the same geometrical problem, one finds Evaluation of eqs (21)-(23) for the two problems, which have been considered in sec. 2.1, is now very simple. 2.2.a. A hot tungsten wire of infinite length in the axis of a cylinder filled with inert gas According to eq. (21) curves of constant temperature are curves of constant mass concentration. The temperature distribution of this geometry is given by eq. (8). The remaining constants A I and B' can be determined by fitting the (24) 228 Phillps Journal of Research Vol.38 Nos 4/5 1983

6 Description of tungsten transport processes in inert gas incandescent lamps concentration to the tungsten vapour pressure at the surface of the hot wire and to the tungsten pressure at the cold bulb, i.e. C(Tf) = mw. pw(tf), mi PI mw = mass of the tungsten atom, pw(t) = tungsten vapour pressure at temperature T, and This leads and to c(t ) b = mw. pw(t b ) :::::o. mi PI A'= mwpw(t f) Tl TY+ 2 -P+a ma» f - rr':":' b B'= mwpw(tf) Tl T;+2- P +a rr=: rr=":' mipi f - b By inserting (28), (27) and (8) into eq. (21) the concentration distribution is obtained. But more important is the diffusion potential ({J, as it allows to calculate the mass loss in a simple manner. From (24), (7) and (27) one finds mwpw(tf)d Tl y f3 + a Tri - Tri r ({J(r) = In - + mo' KB T;+2- P +a- n+ 2 - P + a y rf rf"' n- rb (29) (25) (26) (27) (28) With the continuity equation a at {!c= -v i= -L1({J, (30) it is possible to calculate the mass loss of the tungsten wire according to it = at! gc dv = -!v. i dv = -!Lt({J dv. v v v Integration over a volume which contains the hot tungsten wire, gives from eq. (29) (camp. eq. (10)) the mass loss of the tungsten wire per unit length (31) M 2rc mwpw(tf) d Tl(Tr -= I KB TJ+2-p+a_ T;+2-P+a y + 1 l - Tri) y f3 + a 1 (32) This equation has already been derived by Covington and Ingold 11) in a different way. The expression for the mass loss given by Elenbaas 7), neglects the temperature dependence of the transport coefficients and thermal diffusion. Phllips Journalof Research Vol. 38 Nos 4/

7 E. Schnedler 2.2.b. Two parallel tungsten wires in an inert gas atmosphere The factors A I and B' of eq. (21) have to be determined so that the concentrations at the surfaces of the tungsten wires are determined by the tungsten vapour pressures, This leads to and _ pw(ti) mw. (T ) PI mi c I -, c(t 2 ) = Pw(T2) mw. PI mi. (33) (34) B' = mw 2_ [ (T) Ta _ pw(tl) Tt - Pw(T2) T2 a J (35) pw I I 1 ('7' / T )Y+2-P+a mi PI I Inserting (35) and (34) together with (13) into eq. (21) yields the concentration distribution. The diffusion potential then is derived from (34), (24) and (12) mw d y fj + a pw(ti) T: - Pw(T2) T; TrI - TrI qj(x,y) = 4K + 1 TY+2-P+a '7'y+2-p+a r B Y I f n s+ V S2 - (x - VS2 - rf)2 + y2 X In + qjo. (x + V S2 - rf) + y2 rj (36) The mass transfer from one wire to the other is calculated by integrating zl qj over a volume which contains one of the two wires. The mass transfer rate per unit length reads: M nmw d Y fj + a pw(ti) T: - Pw(T2) T2 a TrI - TrI -=--- I KB Y + 1 rî":": _ it":": In rf -----F=~ s+ VS2 -rf where d - l/pi expresses the dependence of the mass transport on the inert gas pressure. A similar formula but neglecting thermal diffusion and the temperature dependence of the transport coefficients has been reported in ref. 4. Eqs (32) and (37) are exact solutions for the diffusion limited mass transport rates of the corresponding geometrical problem. Eq. (32) can be considered as an approximation of the radial mass transport of an inert gas filled incandescent lamp, and eq. (37) can be used as an approximation for the axial transport from turn to turn. Obviously, not only the gradient of the tungsten vapour pressures, but also the temperatures determine the transport rates due to effects of thermal diffusion and temperature dependence of the transport coefficients of heat conduction and diffusion. The next sections show that these effects are not negligible. (37) 230 Phlllps Journal of Research Vol.38 Nos 4/5 1983

8 Description of tungsten transport processes in halogen incandescent lamps 2.3. Transport coefficients Concentration diffusion and thermal diffusion coefficients are calculated according to the equations for hard sphere gases. The diffusion coefficient of a species j in an inert gas is given by 3 1 ~ (mj+mi)t KB Dj=c dj- TP = TP, 16 PI 1t m.- mi rij with ri = hard sphere radius of the inert gas particles; rj = hard sphere radius of the particle of species j, e.g. of the tungsten atom; mi = mass of the particle of species j; and rij = HrI + rj) (comp. 5,10». The hard sphere value for fj is fj = 1.5. The diffusion radii of the particles were calculated according to Dittmer 12) by an empirical formula, which fits the diffusion constants given in literature with good accuracy. The thermal diffusion constants are tedious to calculate from the rigid elastic sphere formula 5,10), but for the situation which we consider, an approximation can be given. If the concentration of the diffusing gas is very small, Cj ~ 1, the following equation for the thermal diffusion can be derived from the solution of the Boltzmann equation 5,10,11) with (38) (39) The thermal conductivity of the inert gases was calculated from a third order Sonine expansion of the solution of the Boltzmann equation 10,5) for the Lennard-Jones 6-12 potential, which fits experimental values excellently. For some commonly used gases the following constants were found for the temperature dependence À = K TY. TABLE 1 Temperature dependence of the heat conductivity Inert gas Ar Kr Xe y Kj (Joule K-y-l cm sec) Philips Journalof Research Vol. 38 Nos 4/

9 E. Schnedler 2.4. Influence of thermal diffusion on mass transport Before we consider systems with chemical reactions, the importance of thermal diffusion for the tungsten transport shall be shown. The thermal diffusion coefficient for tungsten in some commonly used inert gases is calculated by eq. (39). The collision radius of the tungsten atom is 12) rw = 1.69 Á; mw = Table 2 shows the data used for the calculation of the thermal diffusion coefficient. TABLE 2 Thermal diffusion parameter Inert gas mi ri/á a N Ar Kr Xe Defining Vrad as the ratio of the radial mass loss of a hot tungsten wire including thermal diffusion and its mass loss neglecting thermal diffusion!vi (a =F 0) T/ (y P + a) T/+ 2 - P ) - T b (y+2-p) V rad =!VI(a = 0) = T)Y+2-P+U) _ T b (Y+2-P+Q) ----=--(-y :.._p-) -, (40) one finds the values given in table 3, where the filament temperature T" resp. the bulb temperature Tb, was chosen to be T,= 3000 K, resp. Tb = 500 K. (Reactions of N2 with tungsten were neglected.) TABLE 3 Influence of thermal diffusion on radial tungsten transport Inert gas V rad N Ar 1.35 Kr 1.18 Xe 1.05 The values given in table 3 are independent of the filament or the bulb radius (or Langmuir radius). Therefore they are relevant for all inert gas incandescent lamps. 232"" Phillps Juurnal ur Research Vui. 38 Nus 4/5 1983

10 Description of tungsten transport processes in halogen incandescent lamps One finds that the thermal diffusion may not be neglected when calculating mass transports in incandescent lamps., Especially in the commonly used Ar-N 2 mixture for GLS-Iamps about of the radial mass transport is due to thermal diffusion. Similar results already have been reported elsewhere 11). An analogous consideration for the axial transport gives the results in table 4. Vax is defined as the ratio of the axial transport including thermal diffusion Max(a of 0) and the axial transport neglecting thermal diffusion Max(a = 0). Max(a of 0) Vax =.. Max(a = 0) Max was calculated from eq. 37, whereby it is assumed that one wire has a temperature of Tl = 3000 K and the other wire is at T2 = 2950 K, resp. T2 = 2500 K. Tungsten vapour pressure data are taken from ref. 13. TABLE 4 Influence of thermal diffusion on axial tungsten transport Inert gas Vax, T 2 = 2950 K Vax, T 2 = 2500 K N Ar Kr Xe Mass transport and the infiuence of the inert gas Eqs (32) and (37) show that the parameters of the inert gas determine the tungsten transport besides pw not only via the diffusion coefficient d but also via a, pand y in a complex manner. Defining VI as the ratio of the radial mass transport in the inert gas I and the radial mass transport in the inert gas Ar VI = ~rad(i), Mrad(Ar) (41) which is when neglecting the influence of convection d(i) r,ai(ty1+l _ -ny1+ l ) + 2 _ P + T1. 86 r,1.86 V; = 0 96 f f b YI I al f - b I d(ar) Tp+2-PI+aI _ T;I+2-P1+aI YI + 1 T; 573(T}"79 _ Tt 79 ), one finds for VI the values given in table 5a. The values of dl are the diffusion coefficients of tungsten in the inert gas I at unit pressure and unit temperature. PI was assumed to be the hard sphere value P = 1.5. Obviously this value may Phlllps Journalof Research Vol.38 Nos 4/

11 E. Schnedler be somewhat too small, as for hot gases the particles are repelled according to a power law interaction potential cp - rï:. This potential results in a value for p = 3/2 + 2/(v - 1),10). For hot gases the Lennard-Jones value v = 12 may be a useful approximation. The resulting small correction of 2/11 in the expression for P has been neglected for the sake of simplicity. Tt was assumed to be T f = 3000 K and Tb = 500 K. TABLE Sa Influence of the inert gas parameter on tungsten transport neglecting convection Inert gas di/arb. units ex 'Y di/d(ar) VI N *) Ar Kr Xe *) Value adopted from ref. 11. The deviation of the mass transport ratios VI from the ratios of the diffusion coefficients is of the order of some percent, and especially not negligible for heavier noble gases. Including effects of convection via different Langmuir radii 7) one finds for VI the values given in table Sb, for which a filament radius of mm, an inert gas pressure of 1 bar, a filament temperature of 3000 K, and a bulb temperature of 500 K was assumed. TABLE Sb Influence of the inert gas parameters on tungsten transport including convection via the Langmuir radius Inert gas rb/rf VI N Ar 38 1 Kr Xe The Langmuir radius was calculated according to ref. 7 and the viscosities of the inert gases according to ref. 5 with the tabulated.values for the Lennard- Jones 6-12 potential. 234 Philips Jourooi of Research Vol.38 Nos4/5 1983

12 Description of tungsten transport processes in halogen incandescent lamps 2.6. Mass transport and geometrical parameters The axial tungsten transport between neighbouring turns is determined by geometrical parameters as filament radius rf and distance 2s of the wires via the factor [ln(rf/(s + VS2 - rj))]-i. Considering the ratio Vs of the axial transport at single and double distance one finds for M(s) -.--= M(2s) In (rf/(2s + V 4s 2 - rj)) In (rf/(s + VS2 - rj)) rf = cm; s = cm; Vs = 1.7. Vs, (42) I.e. the axial transport can be reduced by a factor of 1.7, if the distance of the two wires is doubled, and the temperature of the wires is kept at the same level. In a following paper the transport processes in halogen incandescent lamps shall be discussed. REFERENCES 1) J. R. Langmuir, Phys. Rev. 34, 401 (1912). 2) O. R. Fonda, Phys. Rev. 21, 343 (1923). 3) O. R. Fonda, Phys. Rev. 31, 260 (1928). 4) H. Hörster, E. Kauer and W. Lechner, Philips Techn. Rundschau 32, 165 (1971). 6) J. O. Hirsch felder, C. S. Curtis and R. B. Bird, Molecular Theory of Oases and Liquids, J. Wiley, New York, ) P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McOraw Hill, New York, ) W. Elenbaas, Philips Res. Repts 18, 147, ) J. R. Coaton, Lighting Research and Technology 9,25, ) L. D. Landau and E. M. Lifschitz, Lehrbuch der Theoretischen Physik, Bd. VI, Akademie- Verlag Berlin, ) S. Chapman and T. O. Cowling, The Mathematical Theory of Non-uniform Oases, 3rd edition, Cambridge, ) E. J. Covington and J. H. Ingold, Journalof les 4, 198, ) O. Dittmer, Private communication. 13) E. R. Plante and A. B. Sessoms, Journ. Res. NBS 77 A, 237, Philips Journalof Research Vol. 38 Nos 4/

DESCRIPTION 'OF TUNGSTEN TRANSPORT PROCESSES IN HALOGE~ INCANDESCENT LAMPS *)

Philips J. Res. 38, 236-247, 1983 R 1072 DESCRIPTION 'OF TUNGSTEN TRANSPORT PROCESSES IN HALOGE~ INCANDESCENT LAMPS *) by E. SCHNEDLER Philips GmbH Forschungslaboratorium Aachen, 5100 Aachen, Germany Abstract