ࡱ> ` c IjbjbSS l11;]\XtsZ, "\ ,hhWVLT4G  EXPERIMENTAL MODELLING OF PROTON BEAM HEATING OF THE MUON COLLIDER TARGET (Ahmed Hassanein and Kirk McDonald) Introduction. One of the main concerns in the Muon collider device is associated with the proton beam deposition in the target to produce the pions which then decay to produce the required muons. The resulting effects of the proton beam heating of the target is still unresolved due to the potential destruction of a solid or a liquid target. Different types of liquid targets are being considered because of the lifetime issues of solid targets. Low-melting temperature metals such as Ga are considered as possible primary targets for the Muon collider. A number of questions, however, are still unresolved such as the motion of liquid metal jet in a strong nonhomogeneous magnetic field of 20 T, the hydrodynamic stability of the liquid jet, thermal stresses, and shock wave effects due to the sudden deposition of the proton energy in the liquid target. It will actually be very expensive to conduct real beam-on-target experiments to study target behavior under similar conditions to those expected in the Muon collider device. In this work we propose a much cheaper and a very conveniant way to simulate the effect of sudden heat deposition on the liquid jet by using wire explosion techniques. Below is a brief summary of the actual parameters of the Muon collider as well as a brief description of the proposed experiments to simulate beam heating effects. 1. Beam parameters. Energy of protons Ep = 16 GeV, Frequency of bunches f = 15 Hz, Number of particles in one bunch= N1 = 5* 1013 The rms bunch-time length (b = 1 ns. Particles flux Sb = 1.5 * 1015 Average power Wb =Sb Eb = 4 MW Average energy in one pulse Qb = 266 kJ Beam radius R = 1 - 4 mm Target radius Ro = 10 mm Target length L = 20 cm. Below considerations are given in terms of these parameters. For other parameters, however, results can be scaled by substitution of Eb, f, N1, (b, R, and L values. 2. Target heating. Only a small part of the total energy Qb is deposited in the target and is spent in heating the target. The stopping power for the range of Eb can be estimated from the Bethe formula:  EMBED Equation.2  In table 1 the stopping range,l, of 16 GeV protons are given for different materials. Table 1. Elementrange l, cm((Q kJqL kJ / cmC15000.0133.40.17H2O27000.00741.920.096Al12600.0164.20.21Cu4900.0410.40.52Ga6500.037.90.394 where ( is the part of the proton energy spent in target heating, qL is the energy absorbed in the target per unit length. In the analysis below only Gallium target is considered with length L = 20 cm and beam size R = 1 mm. However, different parameters can be easily scaled in these equations. The absorbed energy per cm3 is  EMBED Equation.2 . Therefore, the interaction area is heated to  EMBED Equation.2  The properties of Ga used in this analysis are the following:  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2   EMBED Equation.2   EMBED Equation.2   EMBED Equation.2   EMBED Equation.2 . Therefore, since the initial target temperature is close to EMBED Equation.2 , it means that the target core is heated up to  EMBED Equation.2  (for R of about 1 mm) Let us use  EMBED Equation.2  (for any R) The deposition time (p =1 ns is much less than both the heat propagation time (( and the sound propagation time (s.  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2  where R0 is the target radius, ( is the temperature diffusivity, and cs is the sound velocity;  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2   EMBED Equation.2  Since the repetition time (F =1/f = 0.067 sec, therefore, for time (F the deposited heat does not propagate across the target as well as along the target ((k >(R). However, the sound wave can reach the target surface ((s (F). In addition, since the sound time (s is much larger than the energy deposition time, (s (p , therefore, one can consider that the energy is deposited instantly. 2.1 Shock wave formation. Therefore, for a short time, (p H"1 ns, the liquid gallium is heated up to about 3000K resulting in an increase of pressure P0 , which is given by P0= P((() + PTi + PTe , Where P((() is the cold compression pressure, PTi and PTe are the thermal pressures of ions and electrons, respectively. Under instant heating, t ( (p, one can neglect the cold pressure P((() due to the lack of target motion. Also for T< 104 K, one can neglect PTe because of the small fraction of the resulting excited electrons ((e( T / EFermi 1). Therefore, the resulting pressure can be estimated as  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2 . (Hereafter R is in mm). At this pressure the liquid Gallium is compressed to only about 2%, and during the rms deposition time, (p, the sound wave does not cross the heated area (R), since  EMBED Equation.2 . Therefore, after the interaction and beam deposition, the pressure profile resembles that of the power profile, i.e.,  EMBED Equation.2  where r is the radial distance from the center of the jet. Let us consider how the energy is propagated through the target after heating to timeYP_<T 8p YT PT   t e" (p. We are interested in time t d" L/V0 , where V0 is the velocity of the jet. This value V0 can be determined from the condition of the thermal energy removal due to the jet motion. Minimum V0 means that the heated liquid is removed from the interaction area and this area is again refilled by a fresh and colder liquid just in time for the next beam interaction. The resulting shock wave reaches the target surface at distance R0 after a time (R given by  EMBED Equation.2 , where Vsk is the shock wave velocity which is about the sound speed  EMBED Equation.2  since the shock wave with pressure Po close to the cold pressure at the normal density is a weak shock, i.e., the shock wave speed is close to that of the sound speed, Vsk ( cs . After a time duration of (R ( Ro / cs = 3.5 10-6 s, the shock wave reaches the target surface and the rarefied wave starts from the target surface back to the central axis of the jet. The liquid velocity behind the shock wave at time t((R can be estimated from energy conservation law:  EMBED Equation.2   EMBED Equation.2  ,  EMBED Equation.2 ,  EMBED Equation.2 . After the reflection of the shock wave at the target surface and the formation of the rarefied wave, the liquid velocity at the surface doubles, i.e, ur = 2u ( 1.64 km/s , which is comparable to the sound speed. The average pressure can then be given by  EMBED Equation.2  Thus, the deposited energy is transformed into energy of oscillation as well as a pressure energy in the liquid cylinder target. 2.2. Target instability. The oscillating liquid column in a strong and nonhomogeneous magnetic field of B ( 20 T ( EMBED Equation.2 ) could be unstable relative to MHD instabilities with azimuthal disturbances. This could also be studied experimetally in conjunction with wire explosion experiments. This can be studied in more details with state of the art numerical simulation codes that are being developed at Argonne National Laboratory. 3. Simulation experiments To model and simulate the processes of target sudden heating and jet stability, one needs experiments with intense heating of the central zone of a cylindrical liquid jet column with the following necessary conditions: the heating time (p is less than the hydrodynamic time (R , 2) the length of the liquid column,L, must be much large than the column radius R , i.e., L>> R, to exclude two-dimensional effects, and 3) the properties of the liquid media (if not the same liquid of the Muon Collider target is used) must be as identical as possible to the candidate liquid metals in terms of heat conduction and sound velocity properties. The best plan is to use the same candidate liquid metals (Ga, Sn, ...) with the same geometrical parameters, i.e., Ro=1 cm, L = 20 cm, with the same deposited energy, Qo= 266 kJ/cm3. The problem of the required intense heat source can be resolved in a similar way to that of simulating nuclear weapons effect by underwater explosion, i.e., by using a wire heated by an electric current to a high temperature (pressure) up to immediate vaporization (explosion). Deposition of QoY+@+xP+@*+L++Y+ H" 10 kJ requires that the energy of the capacitor bank, Qbank, to be of the order H" 30 kJ. A schematic illustration of the suggested facility is given in Fig.1. The liquid can be pumped from the chamber bottom to chamber top and falls freely in the from of a jet. The wire is displaced in the center of the falling liquid jet and is heated by the electric current which is supplied by a condensor battery for a time duration of (bank ( 1 microsecond. Facilities with similar parameters exist in many Institutions, for example, particularly at the TRINITI (Troitsk, Russia) or at the Kurchatov Institute (Russia). The power of discharge is given by Wbank = Qbank / (bank ( 104/10-6 = 1010 Watt, which is much less than the proton beam power Wp = Qp / (p ( 2. 105/10-9 = 2. 1014 Watt. However, the energy deposited is about the same and the time duration required does satisfy the simulation conditions, i.e., (bank ( (R H" 3 (s. Let us estimate the electric parameters, i.e., the current, I, and the voltage, U, needed for this simulation. The Ohmic heating is given by  EMBED Equation.2   EMBED Equation.2  , where Rj is the resistance and (m is the wire resistivity. This Ohmic heating should equal to the one-bunch deposited energy, i.e., Qo Y23 P2.l3D{3(3`u3D3|4$x3`33Y3|3P3|3`H" 8 kJ; therefore  EMBED Equation.2  q = 12.7 kJ/cm3. (again for R = 1 mm) The current is therefore  EMBED Equation.2  kA, where again R is in mm, (bank is in s, and L is in 20 cm length units. The bank voltage is then given by  EMBED Equation.2  Volt,  EMBED Equation.2  &!. Therefore, a facility with an electric current of a few hundreds of kA and a voltage of few tens of kV with discharge time of about 1 s is enough for accurate simulation of proton beam deposition in the muon target system. This is also sufficient for the study of various processes during proton interaction with the liquid jet. Experiments are proposed be performed in three stages: First stage. In the first stage the physical processes of the shock wave formation and its influence on the liquid target stability can be studied. For this purpose a dielectric liquid such as a purified water or organic compounds such as the Castor oil can be used. By using a transparent chamber it is possible to investigate the shock wave formation, propagation and deformations of the target surface can be studied. It will also be possible to elaborate and refine physical models and verify the computer codes. Second stage. In the second stage experiments that fully model real situations with liquid metals can be used. For this case the insulation of the conducting liquid jet from the heated wire is necessary. The wire can be insulated by a very thin film of a material with good dielectric properties such as, for example, Mylar with a thickness of a few microns. If the thickness of this insulating film is in the order of (R = 3 (m thus for a time duration (bank ( 1 (s, this film will be heated by thermal conduction to T ( 3000 K. The shock wave crosses this film in a short time (t = R/cs ( 3 ns. Therefore, the existence of this thin film does not significantly influence the physical phenomena needed to be studied. Third stage. In this stage experiments from the second stage can be done in a strong magnetic field to study the effect of such field on jet stability during the heating of the jet. However, the generated pressure due to the shock wave significantly exceeds that of the magnetic pressure at least for the smaller beam sizes. Table 2. Comparison of Muon Collider and Wire Explosion simulation Parameters Parameters  Muon Collider Wire ExplosionTarget MaterialGaGa or OilTarget or Jet Size10 mm10 mmHeated Zone Size1 - 4 mm1 - 4 mmTarget Length20 cm20 cmDeposition Time1 ns100 ns 1 sPower Deposited1013 10121010 109Energy Density Deposited10 1 kJ/ cm310 1 kJ//cm3Magnetic field20 TCan be included 4. Diagnostics Diagnostics for the wire explosion simulation, as existing in the Russian facilities include measurements for the electric parameters, pressure sensors to measure the shock wave parameters, and high-speed streak and frame cameras to monitor the liquid behavior during and after the wire explosion. The pressure sensors can be placed in the liquid jet at different distances without disturbing the flowing jet. Such diagnostics have been extensively used during high-energy deposition on target materials to simulate plasma disruption in future fusion reactors. Summary and Conclusions Simulating the effects of a sudden deposition of an intense energy in a flowing liquid jet using wire explosion is a very realistic and accurate representation of the conditions encountered in the Muon Collider target. This is also indicated in Table 2 where a comparison of various parameters of the Muon Collider versus the wire explosion simulation. Such experiments can be done in a simple and controlled laboratory setup with sufficient diagnostics to monitor the liquid jet behavior due to the resulting shock wave. The estimated cost to do the initial set of these experiments is about $100K. This is much cheaper (orders-of-magnitudes) than having an energetic beam deposited in a flying liquid jet target. Valuable information can be obtained about the potential success and reliability of the current target design of the Muon Collider project. Model Experiment and Parameters of Electrical Circuit Required to Simulate Shock Heating of Liquid Targets Requirements: Energy (in pressure) ~10 kJ Time of increasing pressure wave ~1.5 s Length of exploding part of wire ~10 cm Schematic of a Model Experiment Switch Exploding wire    Oil or Liq-M C    With or without a protection tube  For ensuring a necessary energy in the channel of pressure ~10 kJ, energy in capacitor battery must be > 40 kJ (efficiency ~ 25%). When using the capacitors (3 F, 50 kV) it is necessary to have 10 banks. Each bank has an inductance ( 40 nH. Total inductance of capacitors ~ 4 nH. 3. Inductance solid switch ~ 10 nH. 4. Inductance of Load ~2(L(ln(D/d) ~ 40 nH. 5. Lead cables ~ 10 nH. Total ~ 64 nH. Time of increasing the current in such a scheme is in the order of 1-2 s. This time can also be reduced by using a plasma swith rather than the conventional switches. As far as the process of increasing the pressure it occurs approximately 4-5 times faster (peaking due to the increase of resistance). This scheme will ensure the required parameters for the pulsed pressure wave. 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sh (3) 5339 152 sh (2) 4950 686 sh (3) 6264 686 sh 384 ns (0) 2057 324 sh (195) 2345 324 sh (10) 3085 324 sh (3) 1853 858 sh (14) 2141 858 sh (10) 2702 858 sh (6) 4284 324 sh (2) 4572 324 sh (10) 4942 324 sh 384 /Times-Roman f1 (.) 2249 324 sh (.) 2045 858 sh (.) 4476 324 sh end MTsave restore dMATH*A q=q L pR 2 =0.19510 3 3.1410 -2 R 2 =6.210 3 R 2 kJcm 3 xc۔FMicrosoft Equation Editor 2.0DNQE Equation.2XܐO<t< q=q L pR 2 =0.39410 3 3.1410 -2 R 2 =12.510 3 R 2 Jcm 3 ݌ObjInfoREquation Native S/_965626555FHѽHѽOle X߼oL"/߼0l "dxpr  ", MT Extra .*" "" currentpoint ", Symbol(D,Times)T) =( "q( c +p"  (/=( 96).)2))10 (Z3 (D2).)2"8&( zJ)/)cm (PIC YLPICT [ CompObjYObjInfo3 (dJ)/()cm (3 +)kelvin))"cK(=) 2).)82( 10 (3 (R (2" ( o +K/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8032 div 1088 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 951 466 moveto 382 0 rlineto stroke 1759 466 moveto 1234 0 rlineto stroke 3121 466 moveto 2442 0 rlineto stroke 6704 466 moveto 547 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (D) 62 565 sh 384 /Times-Roman f1 (T) 297 565 sh (q) 1042 324 sh (c) 974 858 sh (J) 3843 324 sh (cm) 4225 324 sh (J) 3150 858 sh (cm) 3605 858 sh (kelvin) 4449 858 sh (R) 6768 858 sh (K) 7510 565 sh 224 ns (p) 1164 954 sh 384 /Symbol f1 (=) 633 565 sh (=) 1441 565 sh (\327) 2318 324 sh (\327) 4289 858 sh (=) 5671 565 sh 384 /Times-Roman f1 (6) 1778 324 sh (2) 2066 324 sh (10) 2436 324 sh (2) 2136 858 sh (2) 2424 858 sh (2) 5978 565 sh (82) 6266 565 sh (10) 6694 324 sh 224 ns (3) 2833 152 sh (3) 4711 152 sh (3) 4091 686 sh (3) 7091 152 sh (2) 7045 686 sh 384 /Times-Roman f1 (.) 1970 324 sh (.) 2328 858 sh (/) 4055 324 sh (/\() 3362 858 sh (\)) 5422 858 sh (.) 6170 565 sh 224 /MT-Extra f1 (o) 7400 393 sh end MTsave restore d'MATH/ DT=qc p =6.210 3 2.2J/cm 3 J/(cm 3 kelvin)=2.8210 3 R 2 , o KexlvFMicrosoft Equation Editor 2.0DNQE Equation.2OPoOl  DT=qc p =1.2510 4 2.2J/cm 3 J/(cm 3 kelvin)=56821R 2 ,  KEquation Native '_965462463FHѽHѽOle PIC LLpOd!ldxpr  !l"!l currentpoint ",Times .+c +p, Symbol (=) 2).)2( GJ(0cm (?3 +kelvin (+()8)" +<D/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roPICT CompObjYObjInfoEquation Native yll sub 3456 div 1056 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1331 403 moveto 1937 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (c) 55 502 sh (J) 2226 261 sh (cm) 1479 798 sh (kelvin) 2157 798 sh 224 ns (p) 245 598 sh 384 /Symbol f1 (=) 490 502 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1552 /Symbol f3 (\() 1343 843 sh (\)) 3130 843 sh 384 /Times-Roman f1 (2) 797 502 sh (2) 1085 502 sh 224 ns (3) 1965 626 sh 384 /Times-Roman f1 (.) 989 502 sh end MTsave restore dxMATHlz c p =2.2Jcm 3 kelvin()OD|DxFMicrosoft Equation Editor 2.0DNQE Equation.2@O$z\] c p =2.2Jcm 3 K()_965387012'"FHѽHѽOle PIC !$LPICT 4 O L_ P04 uDdxpr  D"D currentpoint ", Symbol .+r) =,Times) 5).)91( 3g(,cm (;3" +/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2176 div 896 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1342 403 moveto 669 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (r) 45 502 sh 384 /Symbol f1 (=) 341 502 sh 384 /Times-Roman f1 (5) 647 502 sh (91) 935 502 sh 224 ns (3) 1851 623 sh 384 /Times-Roman f1 (.) 839 502 sh 384 /Times-Roman f1 (g) 1581 261 sh (cm) 1365 795 sh end MTsave restore dOMATHC  r=5.91gcm 3 p X VFMicrosoft Equation Editor 2.0DNQE Equation.2t ZCompObj#%YObjInfo&Equation Native __965387131)FHѽ+Hѽ n o{C r=5.91gcm 3  sOh t u}  vOL XTO  H z 6Gdxpr  G, MT Extra .* "G currentpoint ",Ole PIC (+LPICT 6CompObj*,YTimes( T +melt, Symbol ( =) 302 (6o +K/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2272 div 544 3 -1 roll exch div scale currentpoint translate 64 60 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (T) 58 324 sh (K) 1814 324 sh 224 ns (melt) 268 420 sh 384 /Symbol f1 (=) 805 324 sh 384 /Times-Roman f1 (302) 1107 324 sh 224 /MT-Extra f1 (o) 1694 152 sh end MTsave restore dPMATHD  T melt =302, o Kim FMicrosoft Equation Editor 2.0DNQE Equation.2 1 30 /XP 4D T melt =302, o K  rL L3 {ho x4 6Jdxpr  J, MT Extra ObjInfo-Equation Native `_965387210 f0F+Hѽ+HѽOle PIC /2LPICT 6CompObj13YObjInfo4 .* "J currentpoint ",Times( T +vap, Symbol ( =) 2520 (:o +K/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2368 div 576 3 -1 roll exch div scale currentpoint translate 64 60 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (T) 58 324 sh (K) 1930 324 sh 224 ns (vap) 268 420 sh 384 /Symbol f1 (=) 724 324 sh 384 /Times-Roman f1 (2520) 1031 324 sh 224 /MT-Extra f1 (o) 1810 152 sh end MTsave restore dPMATHD Z T vap =2520, o KmeFMicrosoft Equation Editor 2.0DNQE Equation.2 ` OD| D D T vap =2520, o K@ O$ \ L 0 #O $Equation Native `_9653872947F+Hѽ+HѽOle PIC 69LPICT CompObj8: YObjInfo;Equation Native x     !"#$%&'),/123456789:;<=>?@ABCDEFGHIKNORTUVWXYZ[\]^_`abcehkmnopqrstuvwxyz{|}~Vdxpr  V"V currentpoint ",Times .+Q + melt, Symbol (=) 0).)47( BkJ(>cm (M3" =/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2752 div 896 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1913 403 moveto 669 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (Q) 52 502 sh (kJ) 2079 261 sh (cm) 1936 795 sh 224 ns (melt) 351 598 sh 384 /Symbol f1 (=) 888 502 sh 384 /Times-Roman f1 (0) 1197 502 sh (47) 1485 502 sh 224 ns (3) 2422 623 sh 384 /Times-Roman f1 (.) 1389 502 sh end MTsave restore dhMATH\ Q melt =0.47kJcm 3 Q)O&& FMicrosoft Equation Editor 2.0DNQE Equation.2& ' O'P' 'P\ Q melt =0.47kJcm 3 O' (hLD 0(( O(( Ydxpr  Y"Y currentpoint ",Times .+Q + vap, Symbol (=) 21) .)71( EkJ(Acm (P3" @/MTsave save def 40 dict begin curren_9653874305_>F+Hѽ+HѽOle PIC =@LPICT tpoint 3 -1 roll sub neg 3 1 roll sub 2848 div 896 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2001 403 moveto 669 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (Q) 52 502 sh (kJ) 2167 261 sh (cm) 2024 795 sh 224 ns (vap) 351 598 sh 384 /Symbol f1 (=) 807 502 sh 384 /Times-Roman f1 (21) 1114 502 sh (71) 1594 502 sh 224 ns (3) 2510 623 sh 384 /Times-Roman f1 (.) 1498 502 sh end MTsave restore dhMATH\W Q vap =21.71kJcm 3  CompObj?A(YObjInfoB*Equation Native +x_965571028XEF+Hѽ+HѽFMicrosoft Equation Editor 2.0DNQE Equation.22H2 O2,2d2, 22 \ Q vap =21.71kJcm 3 23D3| O3(3` LF l3 3| Ole -PIC DG.LPICT 0JCompObjFHJYJdxpr  " currentpoint ", Symbol .+k,Times) T) =) 273 ( ()*))=)0).)85( pW(^cm)kelvin (Z()3)" Y7/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4768 div 992 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2813 403 moveto 1786 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (k) 52 502 sh 384 /Times-Roman f1 (T) 415 502 sh (W) 3525 261 sh (cm) 2961 795 sh (kelvin) 3488 795 sh 384 /Symbol f1 (=) 751 502 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 276 512 sh (\)) 1626 512 sh 384 /Symbol f1 (=) 1840 502 sh 384 1000 1291 /Symbol f3 (\() 2825 816 sh (\)) 4461 816 sh 384 /Times-Roman f1 (273) 1058 502 sh (0) 2101 502 sh (85) 2389 502 sh 384 /Times-Roman f1 (.) 2293 502 sh end MTsave restore dMATHw kT=273()=0.85Wcmkelvin() s<<< M< NFMicrosoft Equation Editor 2.0DNQE Equation.2<= RO<<< T==T Un kT=273()=0.85Wcm.K()== [O== \ObjInfoILEquation Native M_965626613QLF+Hѽ+HѽOle P o@@P pOL XTO@P@ t@P6Gdxpr  G, MT Extra .* "G currentpoint ",Times( T +melt, Symbol ( =) 302 (6o +K/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2272 div 544 3 -1 roll PIC KNQLPICT S6CompObjMOdYObjInfoPfexch div scale currentpoint translate 64 60 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (T) 58 324 sh (K) 1814 324 sh 224 ns (melt) 268 420 sh 384 /Symbol f1 (=) 805 324 sh 384 /Times-Roman f1 (302) 1107 324 sh 224 /MT-Extra f1 (o) 1694 152 sh end MTsave restore dPMATHD  T melt =302, o Kim FMicrosoft Equation Editor 2.0DNQE Equation.2GPG OG4Gl G4 GG: T melt Equation Native gV_965626634tiveSF+Hѽ+HѽS/Ole26555 FiPIC RUjL=302KOIJ IJDJ| OJ(J` LZ"XJ J| dxpr  , MT Extra .* " currentpoint ",Times(T, Symbol) =) T +melt (.+) D)T) =) 302)+( r2)PICT lCompObjTVYObjInfoWEquation Native .)85) )10 (3 (R (2"q+ (=) 3122 ( o +K/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6816 div 960 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3572 466 moveto 1414 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (T) 58 565 sh (T) 706 565 sh (T) 1973 565 sh (R) 4070 858 sh (K) 6379 565 sh 224 ns (melt) 916 661 sh 384 /Symbol f1 (=) 394 565 sh (+) 1438 565 sh (=) 2309 565 sh (+) 3270 565 sh (\327) 4311 324 sh (=) 5094 565 sh 384 /Symbol f1 (D) 1738 565 sh 384 /Times-Roman f1 (302) 2611 565 sh (2) 3593 324 sh (85) 3881 324 sh (10) 4429 324 sh (3122) 5396 565 sh 224 ns (3) 4826 152 sh (2) 4347 686 sh 384 /Times-Roman f1 (.) 3785 324 sh 224 /MT-Extra f1 (o) 6269 393 sh end MTsave restore dMATH] T=T melt +DT=302+2.8510 3 R 2 =3122, o KEd6XUV 7UUFMicrosoft Equation Editor 2.0DNQE Equation.2/VVTVp <0 =V > T=T melt +DT=302+56821R 2 =5984KVWP ErVWPW GWlL", X_965626663JZF+Hѽ+HѽYOlenfo PICion Native Y\LPICT7131 F ]0YZ Z( ^%Ydxpr  , MT Extra .* " currentpoint ",Times(T, Symbol) =( 2).)82))10 (?3 ++) 302))R (q2 (@R (I2"] ( o +K)/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4576 div 960 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 712 466 moveto 3014 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (T) 58 565 sh (R) 3279 324 sh (R) 2010 858 sh (K) 4113 565 sh 384 /Symbol f1 (=) 394 565 sh (\327) 1465 324 sh (+) 2201 324 sh (\327) 3123 324 sh 384 /Times-Roman f1 (2) 733 324 sh (82) 1021 324 sh (10) 1583 324 sh (302) 2487 324 sh 224 ns (3) 1980 152 sh (2) 3556 152 sh (2) 2287 686 sh 384 /Times-Roman f1 (.) 925 324 sh 224 /MT-Extra f1 (o) 4003 393 sh end MTsave restore dMATH_ T=2.8210 3 +302R 2 R 2 , o K16 _ FMicrosoft Equation Editor 2.0DNQE Equation.2` O`` ` aaL Or T=5682+302R 2 R 2 ,  Kaa Oaa CompObj[]YObjInfo^Equation Native _965390771aF+HѽHIѽYOle PIC `cLPICTbj CompObjbdYff8 Oef LWi f8 f .dxpr  ." . currentpoint ", Symbol .+t +k (=,Times( R + o(%2 ( c"o/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1472 div 1024 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 828 466 moveto 475 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (t) 61 565 sh (c) 963 858 sh 224 ns (k) 246 661 sh 384 /Symbol f1 (=) 510 565 sh 384 /Times-Roman f1 (R) 856 324 sh 224 ns (o) 1133 420 sh 224 /Times-Roman f1 (2) 1133 152 sh end MTsave restore dNMATHB t k =R o2 c 5kx Ok$k\ FMicrosoft Equation Editor 2.0DNQE Equation.2kx kObjInfoeEquation Native ^_965390850<hFHIѽHIѽOleion Native Omm m B t k =R o2 cnn Onln no L4Fl OooL !o-dxpr  -"- currentpoint ", Symbol .+t,TPIC gjLPICT CompObjikYObjInfolimes +s (=( R +o (c +s" S/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1440 div 992 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 777 403 moveto 474 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (t) 61 502 sh 224 /Times-Roman f1 (s) 242 598 sh (o) 1082 357 sh (s) 1045 891 sh 384 ns (R) 805 261 sh (c) 866 795 sh 384 /Symbol f1 (=) 459 502 sh end MTsave restore dWMATHK t s =R o c s 2uv ^3u _FMicrosoft Equation Editor 2.0DNQE Equation.24 csD dosD e%X fK t s =R o c s wP j9wwP lwL`i Equation Native Fg_965390853oFHIѽHIѽOle PIC nqL z zD Oy3 'dxpr  '" ' currentpoint ", Symbol .+c) =( k,Times(c +p"  4/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1248 div 1024 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 686 403 moveto 382 0 rlineto stroke /cat { dup lePICT  3 CompObj&'(+-./0pr56789Y@ObjInfoEFGHIJKLMNOsUVW`Equation Nativeijklmnop uvwxyMngth 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (c) 60 502 sh (k) 764 261 sh 384 /Symbol f1 (=) 368 502 sh 384 /Times-Roman f1 (c) 709 795 sh 224 ns (p) 899 891 sh end MTsave restore d=MATH12 c=kc p O4FMicrosoft Equation Editor 2.0DNQE Equation.2  0 O 1 c=kc p 0    _965544075    {vFHIѽHIѽ Ole#$%&'),/ 56789@PICCDEFGHIKNO uxUVWXYL`PICTehkmnop uvwxyk      !$&'()*+,-./0123456789;>ACDEFGHIJKLMNOPQRSTUVWXY[^_bdefghijklmnopqrstuvwxyz{|}~L  + cm @( -k"dxpr  """ currentpoint ", Symbol .+c) =( k,Times(c +p"  (&=( 10).)63(42).)3"0(I=) 0).)273( pcm (2 (rsec"o/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4384 div 1088 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 686 466 moveto 382 0 rlineto stroke 1494 466 moveto 700 0 rlineto stroke 3512 466 moveto 682 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (c) 60 565 sh (k) 764 324 sh 384 /Symbol f1 (=) 368 565 sh (=) 1176 565 sh (=) 2302 565 sh 384 /Times-Roman f1 (c) 709 858 sh (cm) 3535 324 sh 224 ns (p) 899 954 sh 384 /Times-Roman f1 (0) 1517 324 sh (63) 1805 324 sh (2) 1612 858 sh (3) 1900 858 sh (0) 2611 565 sh (273) 2899 565 sh 224 ns (2) 4024 152 sh 384 /Times-Roman f1 (.) 1709 324 sh (.) 1804 858 sh (.) 2803 565 sh (sec) 3605 858 sh end MTsave restore dMATH c=kc p =0.632.3=0.273cm 2 sec 3 mFMicrosoft Equation Editor 2.0DNQE Equation.2 L R     " " #Z $ c=kc CompObjwyYObjInfozEquation Native _965544088}FHIѽHIѽfp =0.632.3=0.273cm 2 sprotocol&StdFileEditingL; 0 InsertableT T`dxpr  `"` currentpoint ",Times .+c +s, Symbol (=) 2).)873))1Oleion Native "PIC26634tive |F#LPICT6555 F%CompObj~:Y0 ( F5 ( Lcm+s" L/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3072 div 896 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2376 403 moveto 518 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (c) 55 502 sh (cm) 2399 261 sh (s) 2559 795 sh 224 ns (s) 234 598 sh 384 /Symbol f1 (=) 451 502 sh (\327) 1665 502 sh 384 /Times-Roman f1 (2) 758 502 sh (873) 1046 502 sh (10) 1783 502 sh 224 ns (5) 2182 330 sh 384 /Times-Roman f1 (.) 950 502 sh end MTsave restore dhMATH\s c s =2.87310 5 cmsexMicrosoft Word Picture. FMicrosoft Equation Editor 2.0DNQE Equation.20, 26 61 &Open, 0, 26 61 &Ob c s =2.87310 5 cms e2 2ObjInfo<Equation Native =~_965543949FHIѽHIѽOleion Native ?PIC26663 F@LPICTfo BCompObjNativeZYObjInfo1F\L" XZ Z&&{00020970-0000-0000-C000-00000dxpr  " currentpoint ", Symbol .+t +k (=,Times( )R + o(22 (2))0).)273"((G=) 1).)83) R) (sec)7/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4416 div 960 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 828 466 moveto 1293 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (t) 61 565 sh 224 ns (k) 246 661 sh 384 /Symbol f1 (=) 510 565 sh (\327) 1101 858 sh (=) 2229 565 sh 384 /Times-Roman f1 (R) 1265 324 sh (R) 3210 565 sh 224 ns (o) 1542 420 sh 224 /Times-Roman f1 (2) 1542 152 sh 384 ns (2) 849 858 sh (0) 1252 858 sh (273) 1540 858 sh (1) 2505 565 sh (83) 2793 565 sh 384 /Times-Roman f1 (.) 1444 858 sh (.) 2697 565 sh (\(sec\)) 3507 565 sh end MTsave restore dMATH{ t k =R o2 20.273=1.83R(sec)h HD:Microsoft Office 98:FMicrosoft Equation Editor 2.0DNQE Equation.2f d> Macintosh HD Macintosx t k =R o2 20.273=1.83R(s)HELPDIR!"Macintosh Equation Native ]_965543991FHIѽHIѽLOlebj `PICObj aLosoft Office 98$~/Macintosh HD:MicL 01i$~Microsoft Word=Tdxpr  " currentpoint ", Symbol .+t,Times +s (=( ,R + o (2).)873))10 (G5" 1 (O=) 3).)48) )10 ( -)6 +PICTfo cCompObjNativeYObjInfo5FEquation Native  R +o ((sec))=) 3).)48)R +o (m)s /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7616 div 896 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 777 403 moveto 1607 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (t) 61 502 sh (m) 7076 502 sh 224 /Times-Roman f1 (s) 242 598 sh (o) 1648 357 sh (o) 4630 598 sh (o) 6934 598 sh 384 ns (R) 1371 261 sh (R) 4353 502 sh (R) 6657 502 sh (s) 7297 502 sh 384 /Symbol f1 (=) 459 502 sh (\327) 1705 795 sh (=) 2492 502 sh (\327) 3513 502 sh (=) 5646 502 sh 224 ns (-) 4035 330 sh 384 /Times-Roman f1 (2) 798 795 sh (873) 1086 795 sh (10) 1823 795 sh (3) 2794 502 sh (48) 3082 502 sh (10) 3631 502 sh (3) 5948 502 sh (48) 6236 502 sh 224 ns (5) 2222 623 sh (6) 4157 330 sh 384 /Times-Roman f1 (.) 990 795 sh (.) 2986 502 sh (\(sec\)) 4813 502 sh (.) 6140 502 sh end MTsave restore dMATH& t s =R o 2.87310 5 =3.4810 -6 R, o (sec)=3.48R, o ms0000046}Bookmark\ \FMicrosoft Equation Editor 2.0DNQE Equation.2rTypeLib&&{0905-0000-0000-C000-000000000046}Version8.0 t s =R o 2.87310 5 =3.4810 -6 R, o s()=3.48R, o ms()Version8.0V V&&{000-000000000046}Variable\LF l00000000046}r "rTypeLibdxpr  " currentpoint ",Times .+P +0, Symbol (=) P +Ti (+=_965396983mFHIѽHIѽOle PIC LPICT ) 3)n +0 (FkT)=) 3( kr(gm + a" g (xkT/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4416 div 992 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3235 403 moveto 508 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (P) 60 502 sh (P) 807 502 sh (n) 1823 502 sh (kT) 2176 502 sh (m) 3263 795 sh (kT) 3807 502 sh 224 ns (Ti) 995 598 sh (a) 3580 891 sh 224 /Times-Roman f1 (0) 247 598 sh (0) 2038 598 sh 384 ns (3) 1631 502 sh (3) 3006 502 sh 384 /Symbol f1 (=) 493 502 sh (=) 1329 502 sh (=) 2704 502 sh 384 /Symbol f1 (r) 3386 261 sh end MTsave restore dMATHy P 0 =P Ti =3n 0 kT=3rm a kTf 6\FMicrosoft Equation Editor 2.0DNQE Equation.2rTypeLib&&{0905-0000-0000-C000-000000000046}Version8.0y P 0 =PCompObj Y ObjInfo&'(+-./056789@Equation NativeIJKLMNO UVW`_965552089tiveijklmnoptFHIѽXIѽM Ti =3n 0 kT=3rm a kT\ ?\ProxyStubClsid4-0000-0000-C000-000000000046}r @rL.FlVersion8.0X AX&&{000dxpr  " currentpoint ",Times .+n +0, Symbol (=( r(m + a" Oleion Native PIC26634tive FLPICT6555 FCompObjY ( Q6).)1(-1).)67))10 (T-)24 + )69) .)72" .S(=) 5).)24))10 ( 22 + cm ( -)31/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7008 div 992 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 840 403 moveto 508 0 rlineto stroke 1412 403 moveto 2708 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (n) 61 502 sh (m) 868 795 sh (cm) 6088 502 sh 224 ns (a) 1185 891 sh 224 /Times-Roman f1 (0) 276 598 sh (24) 2768 623 sh (22) 5783 330 sh (3) 6703 330 sh 384 ns (6) 2540 261 sh (1) 2828 261 sh (1) 1402 795 sh (67) 1690 795 sh (10) 2242 795 sh (69) 3234 795 sh (72) 3714 795 sh (5) 4534 502 sh (24) 4822 502 sh (10) 5383 502 sh 384 /Symbol f1 (=) 522 502 sh (\327) 2124 795 sh (\327) 3087 795 sh (=) 4228 502 sh (\327) 5265 502 sh 224 ns (-) 2646 623 sh (-) 6581 330 sh 384 /Symbol f1 (r) 991 261 sh 384 /Times-Roman f1 (.) 2732 261 sh (.) 1594 795 sh (.) 3618 795 sh (.) 4726 502 sh end MTsave restore dMATH#3 n 0 =rm a 6.11.6710 -24 69.72=5.2410 22 cm -3 &{00020905-0000-000FMicrosoft Equation Editor 2.0DNQE Equation.200-C000-000000000046}Drp\ \\ProxyStubClsid&&{000 n 0 =ObjInfo3FEquation Native _965552120tiveFXIѽXIѽZYOlenfo1 Frm a =6.11.6710 -24 69.72=5.2410 22 cm -3 {00020953-0000046}_ParagraphFormat\ L" X000000046}r frTypeLibJdxpr  " currentpoint ",Times .+P +0PICion Native LPICT2463 FJCompObjYObjInfo (3122)K, Symbol (()&))=( D6).)77))10 (k4 (SR (\2"D+ (ratm/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4352 div 960 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2114 466 moveto 1432 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (P) 60 565 sh (K) 1339 565 sh (R) 2621 858 sh (atm) 3596 565 sh 224 /Times-Roman f1 (0) 247 661 sh (4) 3377 152 sh (2) 2898 686 sh 384 ns (3122) 524 565 sh (6) 2133 324 sh (77) 2421 324 sh (10) 2973 324 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1291 /Symbol f3 (\() 395 586 sh (\)) 1630 586 sh 384 /Symbol f1 (=) 1796 565 sh (\327) 2855 324 sh 384 /Times-Roman f1 (.) 2325 324 sh end MTsave restore dMATH P 0 3122K()=6.7710 4 R 2 atm002094B-0000-0000-C000-000000000046}CoFMicrosoft Equation Editor 2.0DNQE Equation.200000046}r ~rTypeLi&&{00020905-0000-0000-C000-000000000046} P 0 5984K()=1.3.10 5 R 2 atm000-000000000TypeLib&&{00020905-00L(F lEquation Native F_965628060FXIѽXIѽOle PIC L0020905-0000-0000-C000-000000000046}Xdxpr  " currentpoint ", Symbol .+t,Times +r (=( R + o (c +s"  (,) 0).)3))10 ( V-)6 +m)s)>>)t +pa/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4448 div 992 3 -1 roll exch div scale cuPICTbj  X CompObj&'(+-./056789Y@ObjInfoNativeIJKLMNOUVW`Equation Nativeijklmnop F      !"#$%&'()*+,-./0123456789:;<=>@CDGIJKLMNOPQRSTUVWXY[^acdefghijklmnpuwxyz{|}~rrentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 786 403 moveto 474 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (t) 61 502 sh (m) 3027 502 sh (t) 3955 502 sh 224 /Times-Roman f1 (r) 253 598 sh (o) 1091 357 sh (s) 1054 891 sh (p) 4147 598 sh 384 ns (R) 814 261 sh (c) 875 795 sh (s) 3248 502 sh 384 /Symbol f1 (=) 468 502 sh (\273) 1367 502 sh (\327) 2195 502 sh (>>) 3436 502 sh 224 ns (-) 2717 330 sh 384 /Times-Roman f1 (0) 1672 502 sh (3) 1960 502 sh (10) 2313 502 sh 224 ns (6) 2839 330 sh 384 /Times-Roman f1 (.) 1864 502 sh end MTsave restore dMATH t r =R o c s 0.310 -6 ms>>t p 905-0000-0000-C000-000000000046}FMicrosoft Equation Editor 2.0DNQE Equation.2Endnote\ \oxyStubClsid&&{00020424-0000-0000-C000-0000000 t r =R  c s 0.310 -6 s>>t p 00-000000000046}t\ \ProxyStubClsidL\peLib&&{00020905-0000-0000-C000-00000"kdxpr  "k""k currentpoint ",Times .+W)r, Symbol (() ))) W + 0 (9e ( ?-(Er (I2 (DR (J_965553302tiveFXIѽXIѽ`Ole87294 FPIC LPICT 2" D  +([2+p"W#"[#"V/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3424 div 1088 3 -1 roll exch div scale currentpoint translate 64 38 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th 2119 294 moveto 316 0 rlineto stroke /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .34 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .34 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 492 380 384 2729 998 16 sqr 16 th 2697 567 moveto 556 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (W) 61 666 sh (r) 574 666 sh (W) 1245 666 sh (e) 1777 666 sh 224 ns (r) 2172 208 sh (R) 2135 542 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 430 676 sh (\)) 721 676 sh 384 /Symbol f1 (\273) 934 666 sh (\327) 2537 666 sh 224 ns (-) 1968 351 sh 224 /Times-Roman f1 (0) 1583 762 sh 160 ns (2) 2275 108 sh (2) 2309 442 sh 384 ns (2) 2879 425 sh 384 /Symbol f1 (p) 2986 959 sh end MTsave restore dMATH Wr()W 0 e -r 2 R 2 2 p  liA4-0000-0000-C000-000000000046}FMicrosoft Equation Editor 2.0CompObj0F?YObjInfoAEquation Native B_965568864FXIѽXIѽDNQE Equation.2ield\ \ProxyStubC&&{00020424-0000-0000-C000-000000000046}r  Wr()2 p  W 0 e -r 2 R 2 ,BrowseProxyStubClsid&&{0002042L`4 @00-000000000046}Styles6}OleObj EPICnfo FLPICTon Native HpCompObj1FZYp'dxpr  '"' currentpoint ", Symbol .+ t,Times +R ( = (R +0 (D" O/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1248 div 512 3 -1 roll exch div scale currentpoint translate 64 35 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 791 250 moveto 345 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (t) -3 349 sh 224 /Times-Roman f1 (R) 187 445 sh (R) 821 152 sh (D) 884 442 sh 384 /Symbol f1 (=) 473 349 sh 160 /Times-Roman f1 (0) 996 208 sh end MTsave restore dEMATH9 t R = R 0 D2400-000000000046}FMicrosoft Equation Editor 2.0DNQE Equation.2Version8.0X X&&{00020927-0000-0000-C000-000000000046} N t R =R e0 V esk  rTy&{00020905-0000-0000-C000-000000000046}      !$9"#%(&')*+.,-0/312456;78:M<B=>?@ACDEJFGHIKLNSOPQRVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ObjInfoNative\Equation Native F]j_965398918FXIѽXIѽYOlenfo _PIC44075     F`L PICT$%&'),/ 56789b8@CompObjEFGHIKNOUVWXYoY`ObjInfoehkmnopuvwxyqL,ProxyStubClsid&&{00020424-0000-0000-8 dxpr   " currentpoint ",Times .+ c +s/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 352 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (c) -9 261 sh 224 ns (s) 170 357 sh end MTsave restore d$MATHA c s\FMicrosoft Equation Editor 2.0DNQE Equation.2bClsid&&{00020424-0000-0C000-000000000046}  c sL 4x@"Fdxpr  F"F currentpoint ",Times .+ E + kEquation Native r4_965454423FXIѽXIѽOleion Native sPIC44088 FtLPICTfo v"CompObjNativeYObjInfo9FEquation Native oin, Symbol ( ) E +  potential ( B,/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2240 div 512 3 -1 roll exch div scale currentpoint translate 64 32 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (E) -4 261 sh (E) 989 261 sh 224 ns (kin) 257 357 sh (potential) 1250 357 sh 384 /Symbol f1 (\273) 679 261 sh 384 /Times-Roman f1 (,) 2066 261 sh end MTsave restore d_MATHSA E kin E potential ,TiFMicrosoft Equation Editor 2.0DNQE Equation.2S E kin E potential ,LN$,5dxpr  5"5_965454509 FXIѽyJѽY Olenfo&'(+-./0 56789@PICion NativeIJKLMNO UVWL`PICT2089tiveijklmnop F currentpoint ",Times .+ 2)E +kin, Symbol ( ) Q + 0/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1696 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (2) -11 261 sh 224 ns (0) 1478 357 sh 384 /Times-Roman f1 (E) 197 261 sh (Q) 1182 261 sh 224 ns (kin) 458 357 sh 384 /Symbol f1 (\273) 880 261 sh end MTsave restore d@MATH4 2E kin Q 027 \FMicrosoft Equation Editor 2.0CompObj3FYObjInfoNativeEquation Native FP_965566846FyJѽyJѽDNQE Equation.2StubClsid&&{00020424-00000-C000-000000000046}r rTypeLib4 2E kin Q 0Z&&{00020916-0000-0000-C000-00000 CustomLabels\ \ProxL |TypeLib&&{00020905-0000-0Gdxpr  G"GOle PIC LPICT CompObjY currentpoint ",Times .+2, Symbol ( r)u (2 (2"  (p)R ('2 +=) Q + 0/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2272 div 608 3 -1 roll exch div scale currentpoint translate 64 50 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 235 299 moveto 418 0 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TextColumns\ 5\dxpr  " currentpoint ",Times .+u, Symbol) = (!Q +0 (rp)R (*2" "#"# (3=PICion Native LPICT7294 FCompObjYObjInfo) 0).)58) )10 ( c5 +cm)/)s)) 0).)6)km)/)s/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5984 div 768 3 -1 roll exch div scale currentpoint translate 64 51 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 866 330 moveto 579 0 rlineto stroke /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .34 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .34 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 871 661 384 606 661 16 sqr /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (u) -3 429 sh (cm) 3281 429 sh (s) 3972 429 sh (km) 5014 429 sh (s) 5727 429 sh 224 ns (Q) 1005 232 sh (R) 1131 578 sh 384 /Symbol f1 (=) 288 429 sh (=) 1585 429 sh (\327) 2613 429 sh (\273) 4212 429 sh 160 /Times-Roman f1 (0) 1191 288 sh (2) 1305 478 sh 384 ns (0) 1894 429 sh (58) 2182 429 sh (10) 2731 429 sh (0) 4517 429 sh (6) 4805 429 sh 224 ns (5) 3130 257 sh 224 /Symbol f1 (rp) 887 578 sh 384 /Times-Roman f1 (.) 2086 429 sh (/) 3812 429 sh (.) 4709 429 sh (/) 5567 429 sh end MTsave restore dMATH2 u=  Q 0 rpR 2  =0.5810 5 cm/s0.6km/sch&&{00020424-0000-0000-C000FMicrosoft Equation Editor 2.0DNQE 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Fran CooseAH0000-C000-000000000046}P,VerbC000-000000000046}86 NotInsertable>>  Ole1Class WordDocument::ProgID WordDocument\\ AutoConvertTo&&{00020906-0000-0000-C000-000000000046}86 NotInsertableTTCLSID&&{00030   , 8 DPX`h'Introduction.ontr Fran CoosenranNormaloAHm21mMicrosoft Word 8.0d@v#@ۅѽ@lѽ@h0xҽ C1O,d ՜.+,D՜.+,8 hp|  'ALdi<: Introduction. Title 6> _PID_GUID'AN{66BFF401-3B3F-11D2-9CFB-080007BF7EBB}@xO$\ FMicrosoft Word DocumentNB6WWord.Document.8xDNQE Equation.2MSGraph.Application\ P\ AutoConvertTo&&{00020803-0000-0000-C000 R j =r em LpR e  e2 =1.64.10 -2 LR e  e2 r em r eGa () MiscStatus1H U$  Oh+'0pc IjbjbSS 11;]rrrr\tw2Z###t1v1v1v1v1v1v1,35\1#! ###1g+vRg+g+g+#t1,hh#t1g+g+0V1Tt1B4e rr(h1 EXPERIMENTAL MODELLING OF PROTON BEAM HEATING OF THE MUON COLLIDER TARGET (Ahmed Hassanein and Kirk McDonald) Introduction. One of the main concerns in the Muon collider device is associated with the proton beam deposition in the target to produce the pions which then decay to produce the required muons. The resulting effects of the proton beam heating of the target is still unresolved due to the potential destruction of a solid or a liquid target. Different types of liquid targets are being considered because of the lifetime issues of solid targets. Low-melting temperature metals such as Ga are considered as possible primary targets for the Muon collider. A number of questions, however, are still unresolved such as the motion of liquid metal jet in a strong nonhomogeneous magnetic field of 20 T, the hydrodynamic stability of the liquid jet, thermal stresses, and shock wave effects due to the sudden deposition of the proton energy in the liquid target. It will actually be very expensive to conduct real beam-on-target experiments to study target behavior under similar conditions to those expected in the Muon collider device. In this work we propose a much cheaper and a very conveniant way to simulate the effect of sudden heat deposition on the liquid jet by using wire explosion techniques. Below is a brief summary of the actual parameters of the Muon collider as well as a brief description of the proposed experiments to simulate beam heating effects. 1. Beam parameters. Energy of protons Ep = 16 GeV, Frequency of bunches f = 15 Hz, Number of particles in one bunch= N1 = 5* 1013 The rms bunch-time length (b = 1 ns. Particles flux Sb = 1.5 * 1015 Average power Wb =Sb Eb = 4 MW Average energy in one pulse Qb = 266 kJ Beam radius R = 1 - 4 mm Target radius Ro = 10 mm Target length L = 20 cm. Below considerations are given in terms of these parameters. For other parameters, however, results can be scaled by substitution of Eb, f, N1, (b, R, and L values. 2. Target heating. Only a small part of the total energy Qb is deposited in the target and is spent in heating the target. The stopping power for the range of Eb can be estimated from the Bethe formula:  EMBED Equation.2  In table 1 the stopping range,l, of 16 GeV protons are given for different materials. Table 1. Elementrange l, cm((Q kJqL kJ / cmC15000.0133.40.17H2O27000.00741.920.096Al12600.0164.20.21Cu4900.0410.40.52Ga6500.037.90.394 where ( is the part of the proton energy spent in target heating, qL is the energy absorbed in the target per unit length. In the analysis below only Gallium target is considered with length L = 20 cm and beam size R = 1 mm. However, different parameters can be easily scaled in these equations. The absorbed energy per cm3 is  EMBED Equation.2 . 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