O i a wire e in c e D E- a cl a tº ice ra. - C_1 GE--------> FNGR LIBRARY JAN 23 1969 UNIV. OF WASH. IS QN) O UN O O - UNIVERSITY 1 | Of V E R M CAL | FORN 0 R || E iii. UNIVERSITY OF MICH 3 9015 09522.400 5 TID-4500 UC-34 Physics In a vurr era c e IR, a diatio ra. UN | WE R S | T W 0 F In alb coratory C A 1 ||F0 R N | A | | WE R M0 RE UCRL-50500 — A CODE FOR EQUATION-OF-STATE CALCULATIONS AT MODERATE PRESSURES GERED F. J. Rogers September 9, 1968 Contents ABSTRACT I. INTRODUCTION A. Nomenclature II. BASIC EQUATIONS . III. FORMULATION OF EQUATION A. Hugoniot Calculations Zero-Degree-Isotherm Pressure º Hugoniot Temperature E. Adiabatic Temperature and Energy and Energy Along Adiabats the Hugoniot Pressure That Intersect - - F. Isothermal Pressure and Energy Hugoniot Relations for a Second Shock IV. CODING PROCEDURE G. A. Hugoniot Calculations B. Zero-Degree-Isotherm Calculations C. Hugoniot Temperature D. Adiabatic Calculations V. GENERAL DESCRIPTION AND USE OF THE CODE A. General Description Flow Diagram B. Instructions for Setting up the Problem C. Sample Problem—Calculation of Us(Up) for APPENDIX I. Derivations of Y(Vorſ) and Y'(VoIP) APPENDIX II. Fortran Listing of the Code -iii - - - 2024 Aluminum GERED — A CODE FOR EQUATION-OF-STATE CALCULATIONS AT MODERATE PRESSURES Abstract This report describes a Fortran code which is used to convert shock wave data into equation-of-state information through the Dugdale–MacDonald, Slater, or free volume gamma relationship and the Mie-Grüneisen equation of state. The code is written to allow flexibility in the calculation of the particle velocity associated with a given free-surface velocity measured in a shock experiment. I. Introduction GERED is a versatile Fortran code 2. from shock Hugoniot data: Calculation of pressure, energy, Calculation of pressure, energy, 3. Calculation for making the following equation-of-state calculations 1. 4. and temperature along the Hugoniot. and gamma along the 0°K isotherm. of pressure, energy, and temperature along adiabats and or therms that intersect the Hugoniot. Calculation of the relationship of shock velocity to particle velocity. The report discusses the theoretical detailed instructions for its use. of what the code does may go and mathematical basis of the code and gives The reader who is interested in a general description directly A. to Section V. NOMEN CLATURE Subscript Notation. A Adiabatic H I K 0H OK iso Hugoniot Isothermal Zero degrees Kelvin Refers to the foot of the Hugoniot (initial conditions) Refers to point on the zero-degree isotherm where Pi—K = 0. Variable Definitions A. polynomial fit coefficients Us(Up) Specific CV heat at constant volume E Energy EP Specifies termination condition for iteration P Pressure t on Us(Up) Defined in text T Temperature where an adiabat intersects the Hugoniot THo Temperature V Volume VAA Volume at which an adiabat intersects the Hugoniot Minimum volume attained in the experiment Vs Volume XVOH at which gamma is Free-surface velocity known from thermal data Un Particle velocity Uf Velocity of Us rarefaction wave the Shock velocity Maximum shock velocity used in the experiment Us "Y 0D Grüneisen gamma Debye temperature 3. II. Basic Equations The following equations provide the foundation for development of the code: Hugoniot Relations U p Pºr H - EH = - U -v.) Voh V ( S Pop + OH = "sºp –F#–1-, (2) Vori 1 E0H (1) 2. + 3 (VOH - V) (PH + (3) PoH). & (E - Ek). (4) H. R. Christian, Phys. Rev. Walsh and 1544 (1955). Rice, McQueen, and Yarger, Phys. Rev. 108, 196 (1957). –2 — *Walsh, = 97, ºy P - PR M. 'J. Mie-Grüneisen Equation of State Grüneisen Gamma * Y(V) = - (#3 - # 3 -º (P dy2 \ K vº) 2 d jº (PkV ( 5) 2t/3 ) where Rarefaction t - t - t - produces the Slater gamma, 0 1 2 produces the Dugdale-MacDonald gamma, produces the free-volume gamma. Velocity Ur - V y 8P 1/2 ( #) III. This section outlines dV. (6) Formulation of Equations of the equations the code the derivation will be required to Solve. A. In a shock experiment, HUGONIOT CALCULATIONS the two velocities value of required by the Hugoniot relations Up following relationship: Urs - Up that are measured are Us and Ufs. The in Eqs. (1) - (3) is obtained from the (7) "Ur. Substitution of Eq. (6) gives U e fS = U p W. V H B. ºr)". ôV V. (8) S ZERO-DEGREE-ISOTHERM PRESSURE AND ENERGY Hugoniot data provides a reference for the Mie-Grüneisen equation which, to gether with the Y relationship of Eq. (5), produces the following differential equation: (10) (9) can now be written - E. 3V (*#)(En Fº 2te. series, - (EH thermodynamics, [from Eq. (11) (4)] becomes (13) Eq. (13) may be written - c. T. \8 (#) Porſ) PH (VoII dT. *H H V - 0. V H \ -* V av - (14) | (PH + dEH + | – | Eq. (3), = = 3 1 to According EK) dv - law H - c. dT. T. (#) 0T P.H requiring solution dS equation of thermodynamics ds. ‘‘’H dE. H (PH (12) of first Ek): 1 Ek). Along the Hugoniot, the first Using the | HUGONIOT TEMPERATURE C. T.H |# the equation T a & PH = FK + is Y expressed as | 2 = K - + K |P H x-y-K When (9) . 0K Pk dV. 2E. + = 0e 9W V - FK, Eq. (E., H Ek) + is E.K Of + gives jū In terms z = *k *H \E * E. law of thermodynamics 3 (Pa first 2. av - the *k V - v. = 0 \, 2 r 3V K\EH - Ek T At T "K., *H 2P. V P.K dV. (15) Y (16) V C V V (V) - PH - PoH |. (VOH - 2Cy + —— P' - Y(V) +++ TH V) Eqs. (15) and (16) into Eq. (14) and division by Cy dV gives |Pă TH ðT/. (#) = , Now substitution of V V =2(V) 8T (#) = Equation (4) provides the following relationship: (17) 1 0H - l V) + w) Cºw) (Voh - ŽToº = T 1 V Voh results in |r, b(V) - at Toh \ Tori | Tr(V) + Use of the initial condition - yº, ( f(V) exp = = TH form a solution of the V This equation has PH **) (18) w) # - exp = q(V) (. V Where 0H PRESSURE AND ENERGY ALONG ADIABATS THAT INTERSECT THE HUGONIOT S VOH PA “A dV E - ?'y (PA - 13 PH), (21) — (22) -ī. %). V+ º *— PA(; - Ph I' E!, # Y #)\ - Y - (;1 Pº ' = º Y . Eqs. (21) and (22) into Eq. (19) and rearrangement gives (23) FA - - V Eq. (19) gives Ş. of They are: V PH y? - - -- (20) Eq. (19) can be obtained from Eq. (20). ;Y in ' * Prº H (V)|. 8P PA Integration subtracted from the the result, after rearrangement, v Substitution EH |P(V) the Hugoniot of (# #) (#). = 8E F required The derivatives 2}= Y(V) + E.(v) H\' = E(V) ' + to If Mie-Grüneisen equation referenced Mie-Grüneisen equation referenced and P, the (19) is -(BE) 8P is (£) ôV to dV V - -- A A. an adiabatic process p. For V Wrmin § Case 1: D. H PA(V) dV. (24) Case 2; V > WOH In this region, the Hugoniot is is necessary to find a different The required reference state is not known, and it state for the Mie-Grüneisen equation. obtained from thermal data. The energy in this case is given by 'Y Po 1076 and substituting them into gives E I Eq. (19), with the approximation (9E/9P), 0, and (°E/ove, s 0 Evaluating the derivatives (25) 0 zkº, (V) [P(V) - + = E(v) - E,(v) Pol J " reference = T : the code. bP. will be TEMPERATURE dS equation of thermodynamics TA They (#). reduces to in V - Cy dTA + ſh 0 dº Rearrangement = Cº TA first (26) this region are included ADIABATIC E. Along an adiabat, the "0 in ! Several alternatives for defining discussed in the next section. Y 1 : (x 1. --= PA' -º) Y'V) dV. (27) gives * --4-(#), V V TA - *Ho when VA - VAA' 6 ."IT . E. E. - D (30) 9 c.(?)", is PRESSURE AND ENERGY It ISOTHERMAL The isothermal energy can be obtained from the Debye theory. Tº yields d % ( F. (29) using the condition :ſ X e - dV. - V Y(V). Eq. (29), HT.T of - dT ...A. TA Integration (28) Eq. (16) gives O Substitution of A av. dT where V 0B The isothermal P.I = "Don a ºy V 0H *) pressure is = P, G. If ( J. exp K - E.) K’ “ (E, +% V v-I HUGONIOT RELATIONS FOR A SECOND SHOCK material under a pressure Poh and moving with a velocity Up due to a first shock wave is hit with a second shock wave, the conservation equations for the second wave may be referenced to a coordinate system moving with the first wave. The Hugoniot relations in this case are the same as Eqs. (1) - (3) except Us and Up 3.1°e Il OW in a coordinate system moving with the particle velocity of the first shock. IV. Coding Procedure This section describes how numerical solutions to the equations of Section III are obtained. A. HUGONIOT CALCULATIONS Inspection of Eqs. (23) and (26) reveals that the is entropic pressure required in the solution of Eq. (8) cannot be calculated until the Grüneisen gamma is known. To (note that calculate y from Eq. (4) it is first necessary to solve Eq. (11) for - Ek and PK Eq. (11) requires a knowledge of PH and PH which can only be PK -Ek). – In the by solving Eq. (8) for obtained from a previous knowledge of Us Up: f(U) or and iterate until the necessary guess latter case it is to make an initial for Us(Up) system produces a stationary solution for Us(Up). At low shock pressures, a good approximation to make is However, Substitution of this approximation into the Us(Urs) fit (supplied as input data) gives an initial guess for Us(Up). To provide flexibility, the code is written so that the order of accomplished the may be different than the order of the Us(Urs) fit. This is Us(Up) fit least-squares points fitting by generating 60 evenly spaced (Us, Up) and them to a polynomial. The value of on each cycle, volumes for Ur a specified These WAA. will emerge later) is calculated, (the steps leading to this value set of adiabats which intersect the Hugoniot at the depend on the Us(Up) Ur values – fit. If - fit does not correctly the then the obtained or equivalently the Ufs/ Up values Ur values f(Up), will not be the same as for the previous cycle, and the current set of (Us, Up) points al is used to generate a new set of Ai. The iterative process is terminated after specified number of iterations or when, for the adiabat that lies highest, represent 1– Us EP (;r - tº) r r ſu:- 3 0. In this analysis volume is being treated as the independent variable; thus, it is necessary to calculate Us at each Vi to obtain the Hugoniot values required in the solution of Eq. (11). The relationship between Us and V is obtained by substitution of Eq. (1) into the polynomial fit to The result, after transposing Us, is Us(Up). Ao2 U º'S ( V - ++– 1 + A., U 2 WOH 3 :( 1 is The Newton–Raphson method used on Eq. (31) involves repeated use of the formula F (31) solve for Us, each Vi. This (**) (º) F' i si si - U_. + . . . i - U n-1 - U Il y \? - S + at 1 l A, wº) = 0 = to F(U.) S correction term becomes <10-8 the process terminated. necessary guess This guess obtained from iteration U a linear approxi (32) make an initial guess each Vi: …) *i-1 – U + ( = Si Si-1 ( 1 U the - The code proceeds by using the following equation UI To start at vº) to - A2 ( - */ - is l U. . to is Eq. (31): º, mation to it is When the (33) where for convenience s1 - will be used throughout this report to - The Superscript number. U 1 0.99 - S-1 , = - n >|< U - -- - - indicate an iteration For linear fit Us is calculated directly from Eq. (33). The formula for calculating PH, from Us and V values, is obtained by eliminating Up from Eqs. (1) and (2). The result is a P.. = H P.A., OH + - .2 1 U 's Voti V - ( ) WOH (34) - ZERO-DEGREE-ISOTHERM B. CALCULATIONS Equation (11) may be reduced to three first-order differential equations in the variables R, Q, and S by means of the substitutions: dependent V R(W) = W V ! R = Q = dV Prºſv) -EK (for convenience only), = (35) 0H -E. !I - = (36) Prº, t - (37) With these substitutions and rearrangement S 1 2t --|3|# (EH + Eq. (11) produces 4t R) + - # (Pu-o] as (2 + t) 3V (EH + R) + PH - s|}/*-*. (38) The pressure and energy are obtained by application of the fourth-order Runge-Kutta method” to the three simultaneous Eqs. (36) - (38). (The code was obtained from the CIC library.) The initial conditions required R(Vok) = -Ek(Voz) Q(Vorº) = Pr(Vok) °s. ------ (38) are = 0, (39) 0, (40) (Vok) Phºvok) -? the solution of Eq. tw. (Vok) EH (Vok) |Pººvoº -?' Gill, Cambridge Phil. Soc. Proc. woºl 96 - P start 47, S(Vok) · = to (1951). -> (41) where the subscript "OK" refers to zero degrees Equation (41) was obtained by differentiating from Eq. (3). It is ' 1 = ! || – Kelvin Eq. (4). and zero pressure. The derivative of Err H is obtained - The integration is terminated when Us(V) 2. Us . When Y is expressed as a polynomial in V, the fourth-order Runge -Kutta"nethod is used directly on Eq. (12). and ^ required by Eq. (41) must be determined The values Of VOK. 7(VOR), (VOK) SO will lie near to before the integration of Eq. (38) can proceed. In general Vok that, if there are no phase transitions in this area, the following approximation pertains: *(Von) + '(Von)(V - Von). A means of approximating Y(VOH) and Y' obtained are: * At v. 2 1 Vog' - |A: I is outlined in the appendix. The results -(+++) 2A2 = (43) (44) ‘A.( 4 - 5 (*; 3 + 2 -6A,A, - *). 7(VoIP (VoIP) 1) = 3 *(v) Voh (45) V equation produces the following equation which defines the 0K the Mie-Grüneisen value of + 0 G(VOR) = Vok: Vok PH (Vok) - (46) Y(Vok) EH(Vok). of is y' 0. - to of in be n (VOR) to at of at is of Eq. (46) G(V), equally spaced points The root isolated by testing the sign Newton–Raphson method decreasing V, starting next used reduce the Voh. The 107°. uncertainty (volume) the location the root an absolute magnitude of is still undertermined. Eq. However, can now calculated from (43). (Vok) - - ! y is is Y is to - substituted for Voff, (VoIP) Since Vok near (Vok). The Grüneisen gamma calculated directly from Eq. (4). C. HUGONIOT TEMPERATURE of – is of l of 10 – in is l of of Eq. (11) has provided the set values Vi, Yi, and PH. This integration Eq. Simpson's one information utilized the (18) by application third rule, for integration of evenly spaced points. The Ph. values required in the integration are obtained by differentiation Eq. (34). The result Solution I P.. = H (47) VOH (1 where - dus * B (PH - Pori) (x + 1/3) - B) - Hu, - 1 - = A2 2 + 2A3Us3 + 3A4(Us?) •** > V VOH The code is written so that the user may elect to input a constant value for Cy or to input the molecular weight and Debye temperature and have the code calculate Cy from the following fit to the Debye equation: C, , V = –4 2.494.293 × 10_` C2 (1.0 – 3.22 X 10-4 X x 10° x* -1.7829 - 4.8772 + 3.04973 × 10^* x* -4.65022 × 10^*x” + –5 6 2.30082 X 10 ” x”) × 10^* x* (48) where molecular weight, C2 = C 1 - Debye temperature -= x = "Pon exp V - ( W Voh 7. V *) - C1/T. The error in the fit is less than 0.002% for x < 5. To start the integration of Eq. (18), the interval (V OH” Vok) is broken into 20 segments and the trapezoid rule is used to calculate Tok (=T1) and T2 using the approximation CV (T,) 2C Cy (Tori). At subsequent points, an estimate of the l temperature required to calculate Cv; (T,) is obtained from a linear extrapolation of the previously calculated temperatures D. Ti- 1 and Ti 2’ ADIABATIC CALCULATIONS Pressure along an adiabat is obtained by applying the fourth-order Runge-Kutta Eqs. method to (23) and (26) as required. FOr gamma is obtained by making a least-squares polynomial sº V S. fit to 'min Voh a specified number of the (Yi, V) points obtained from the solution of Eq. (4). The points to be used in the fit are selected according to the following formula: -11– i = Integer Part | 1 + J(M1/N2)] where M total number of points, 1 N 2 - approximate number of points to be used in the fit, J = 0, To these points are 1, 2, . . . . - (N2 1) added the point ‘Yoh. Voff), already been selected. In the region V > VOH it “M,” VM.) if it to supply the code with values of has not E,(V). polynomial fit (quadratic or less) to provide for the case when Eb(V) is not well Eð(V) determined from thermal data. To behaved (due to a phase transition), the code is written so that the region V > VOH may be divided into as many as five intervals, and a polynomial fit to Eb(V) is specified for each interval. This is done by providing is necessary and the point coefficients of The following options for defining a Y (V) in the region V > Voh are built into the code: 1. 2. A linear extrapolation according to Eq. (43). Use of thermal data, as for E,(V), to define polynomial of order two or less. required as input data.) 3. A quadratic interpolation which fits Y(VoIP), y' (VoIP), and a value of gamma > at SOme determined from thermal data. The coefficients of the Vs WOH interpolating C1 C2 = polynomial are found to be: - ~' 2 Y(VOH) * C3 VoII - Y (VOH) VoIP — .." = Y' (Vorſ) - 2C3 V OH" 3 (Vs 4. in up to five intervals as a (The coefficients of the polynomial are Y - VoIP 2 A quadratic interpolation to Vs. as in (3), then use of a fit (quadratic or less) To allow for discontinuities in Y " (V), may the Y fit (as mentioned above) be broken into as many as five intervals. However, since PA has been assumed continuous, discontinuities in Y cannot be handled directly. A discontinuity in Y can be approximated by smoothing to Y (determined from thermal data). Y-vs-V curve and taking small step sizes in No provision is made in the code to allow for the fact that the dependent. - 12 — the solution of Eq. (26). the phase lines are pressure If the adiabats are being extended to zero pressure, the solution of Eq. (26) is terminated when PA becomes <107°. The value of U, at PA = 0 is obtained from a linear interpolation (extrapolation) of the Vi) points and the (U r; V) points. (PAP l Energy and temperature along the adiabats and the velocity of the rarefaction wave are obtained by using Simpson's one-third rule to integrate Eqs. (24), (30), and (6), respectively. V. General Description and Use of the Code This section gives a general description of the code and detailed instructions for its use. A. GENERAL DESCRIPTION The flow diagram in Fig. shows the various options available within the code. Any path that leads from START to END outlines an acceptable problem. The box numbers refer to the sections labeled below which give a brief description of the operations being performed. 1 The code is constructed so that it can make calculations from any of the following gamma relationships: (a) Dugdale-MacDonald, (b) Slater, (c) free volume, having up (d) a series fit to Y to 10 coefficients. The initial values of Y and º' are 1. required in the calculations. They may be input or calculated by the code. (Normally they should be calculated within the code.) The Cy, which is used in the code to compute temperatures, may be input as a constant, or it may be calculated internally from the Debye equation. Poh may also be calculated from the Debye theory. 2. The basic problem being attacked in GERED is the determination of pressure, energy, and gamma along the zero-degree isotherm from the Mie-Grüneisen equation 3. of state and the particular definition for gamma selected in item above. This section These calculations 1 of the code will for PK, FK, PH, PH, and TH: may also be made for a second shock produced by a reflected or double shock. 4. The first law of thermodynamics and the Mie-Grüneisen equation of state output values together with PA, EA, a fit ºy, to the gamma values in 3 (just preceding) allows the calculation of TA/ To along adiabats that intersect the Hugoniot (the volume must be specified). These calculations may be made in either Hugoniot from the intersection with the as long as they remain in the range considered in 3 (otherwise the gamma fit is questionable). Pl and EI along and intersection at the direction of volume isotherms in the same volume range may be calculated from the Debye theory (this part of the code will also output 9D). equation is referenced to thermal data, 5. In the region V > Voh the Mie-Grüneisen rather than to the Hugoniot. In this region it is necessary to supply the code with – 13 GºD GENERATE A FIT FROM A U.(U.) U, U i.) 6 | DATA FIT 3 - ISOTHERM -- * W CALCULATION FOR SECOND SHOCK EXIT l AND For EXIT 2 INPUT OR CALCULATE Cy 2 SELECT GAMMA * MAKE ZERO-DEGREE ISOTHERM 3 EXIT - EXIT 4 ISOTHERMAL |_s. CALCULATIONS =Vot Yº EXIT 6 EXIT 7 | | FIT of Flow diagram 14 – Fig. – | U.(U.) W TO OBTAIN 1. 6 ITERATE YMIN | “º GERED options. EXIT 5 CONTINUE DIABATIC CALCULATION UNTIL A 4 CALCULATION TO = 0 CONTINUE ADIABATIC PA 5 | To 4 MAKE ADIABATIC CALCULATION DOWN V 4 W CALCULATION T information E,(V) and y(V). To provide the E6(V) information, coefficients for quadratic fits to Eð(V) in up to five intervals may be supplied as input data. (This gives freedom to deal with discontinuities in E,(V) due to phase transitions.) To pro vide the required information on Y (V), one may require the code to: (a) Make a linear on extrapolation of for V coefficients VoII. (b) Use up quadratic fit to gamma in > up to four separate quadratic to five intervals to define y. The for a each interval must be supplied as input data. (c) Fit a quadratic to 7 OH" won and a value of Y at some V > Voh determined from thermal data. (d) Extrapolate Y, as in (a) or (c), then use input coefficients for Y (V) to cover the rest of the necessary range. 6. The adiabatic calculations provide a means of calculating Us(Up). This involves an iterative process for which coefficients of a Us(Urs) fit must be supplied optional. The interation as input data. The order of the resultant Us(U) fit is continues for a specified number of iterations or until, for the adiabat that lies highest, - tº- fits to y than some specified value. If the problem is not completed in time, may the allotted it be restarted by using the last set of Us(Up) coefficients |U. generated. |/U. is less (These coefficients B. are punched out on cards for convenience.) INSTRUCTIONS FOR SETTING UP THE PROBLEM The code is written so that the Fortran used is acceptable to the 7030, 3600, and 6600 compilers at LRL. However, the control cards, which must accompany the Fortran deck, are not the same for the 7030 as for the 3600 and 6600 machines. In addition, because of storage considerations, for the 3600 it is necessary for the statement BANK RESULT (1) to appear directly after the COMMON statement named RESULT, which appears in the subroutine named COMGEN. (This statement–BANK RESULT (1)—is illegal on the 7030.) Except for the value of the machine-dependent variable MAC, the data cards used all three computers are identical. (MAC is used to allow for the fact that stores an array row-wise, whereas other computers store column-wise.) on The following units are assumed in the code output: Volume (cc/g), Energy (Mb-cc/g), Pressure (Mb), Velocity (cm/usec), Temperature (*K), Specific heat (Mb-cc/g-"K). – 15 — the 7030 CARD 1: MAC MAC FORMAT I5 10 on the 7030 = on the 3600 and the 6600. = 100 CARD 2: WOH VOH, EOH, TOH, USMAX, DELV = VOH VOH EOH = initial volume. = VOK (initial). initial energy. For FORMAT F10.6 a reflected shock set - If E0H = 0., the code will calculate E0H from the Debye theory. Isothermal calculations will be inconsistent if this option is EoPI = not used. TOH = USMAX = DELV = Toti = initial temperature. maximum shock velocity. volume increment used to solve Eqs. (36)–(38). It must be chosen so that at most 1000 incremental "smax 7 steps are taken. CARD 3: A(I) FORMAT F10.6 A(1) = of the Us(Up) fit. There may be up to Seven coefficients. If the Us(Up) fit coefficients are being calculated internally from a fit to Us(Ufs), coefficients this card CARD - may be left blank. CV, C1, C2, GAMOH, GAMP0H CV = specific heat at constant volume if FORMAT F10.6 4: = 0. C1 and C2 equations If CV = 0. the Debye will be used to calculate the specific heat at constant volume. In this case: C1 C2 GAMOH - GAMP0H CARD Ty OH 7 Debye temperature at V molecular weight. initial value = VOH, of gamma. initial value of gamma prime. If GAMOH = GAMP0H = 0. gamma and gamma prime will be calculated internally (this is the normal case). YOH = P0H, PK, EK, DPK, VOK P0H 0. for a single shock (this is a necessary condition 5: for start the calculation) pressure along Hugoniot for a double or initial the code to reflected shock. -1 6 FORMAT F10.6 PK = initial pressure It is EK = DPK = VOK = 0. # for on the zero-degree isotherm. a single shock. initial energy on the zero-degree isotherm. It is = 0. for a single shock. 0. for a single shock. For a double or reflected Shock, it is the initial derivative of Pk(V). a single shock starting at VOH. for 0. For con a tinued single-shock calculation or for a reflected shock, VOK = VOK. CARD 6: NFIT, II, NY, NOIT, NIT N2, N3, N2 = = 0 if no adiabatic calculations are to be made number of points to be used in the gamma fit if adiabatic calculations are to be made. To insure sufficient accuracy, It 40. N3 FORMAT I5 this number should be at least may not exceed 98. if a Us(Up) fit is to be generated from an input U s(Ufs) fit 0 - NFIT II NY if Us(Up) fit Number of 1 a is Us(U) to be input. fit coefficients. Slater gamma is to 1 if the 2 if the Dugdale-MacDonald 4 if if gamma is to be used gamma the free-volume is to be used gamma is to be input as a series. 0 if II 3 + 4 number of coefficients II NOIT 0 = be used = if -1 4. NY if in the input gamma fit may not exceed 10. no adiabatic calculations are to be made if adiabatic calculations are to be calculations are to be made made but no Us(U) = maximum number of iterations to be allowed. iteration on Ur may be started either from a fit NIT If = Or a Us(Up) fit. This allows The Us(Ufs) the problem to be restarted at any point in the iteration. number of isotherms to be calculated. no adiabatic calculations are to be made, go to CARD 16. - NNN, NFITORDER, NX, NDP NNN 1 if the adiabatic calculations CARD 7: V = are to terminate at VOH if the adiabatic calculations are to zero pressure. 0 – 17 to be continued down FORMAT I5 NFITORDER number of coefficients fit. NX 1 if It to be used in the gamma may not exceed 10. the adiabatic calculations are not to be continued to the left of the Hugoniot (in the plane). - 2 if the adiabatic calculations are to be P-V con tinued to the left of the Hugoniot. of high pressure points, for which the value of already known, that are to be Up is NDP GO to CARD CARD 8: 10 if the number added to the set of (Us, Up) points to be generated fit. by the code from an input Us(Urs) no Us(Up) calculations are to be made. UFSMIN, UFSMAX UFSMIN = FORMAT F10.6 minimum free-surface velocity to be used in normally generating the Us(Up) fit. (It should be 0.) UFSMAX FORMAT F10.6 Us(Urs) There may coefficients. up D(I) to D(I) be 9: fit CARD maximum free surface velocity to be used in fit. generating the Us(Up) Severl. NUMADS, DELVV, EP be calculated. It number adiabats not exceed 30. may solving volume increment be used Eqs. (23) and (26). No more than 1000 incremental steps are allowed. in DELV.V FORMAT I5, 2F10.5 to NUMADS EP to 10: of CARD maximum allowed value of the ratio |U. - wº- |/U. which will terminate the iterative process. This condition must be satisfied for the adiabat that volumes at which the selected diabatics inter five cards. values. They should be listed 18 decreasing to of sect the Hugoniot. The number volumes listed require up equal may must be NUMADS. This in V.A.A.(I) FORMAT 6F10.6 NUMADS a VAA(I), I – 11. to CARD = 1, lies highest. the adiabats are not to be extended > a the end the calculated gamma and to If is the code be read will below (see > (V of 5 each segment VOH) used FORMAT 3F10.6 3*NA I=1, gamma determined from thermal of = GAMMA(I) Vof FORMAT F10.6 volume the end to express Eb(V). GAMMA(I), 14: VS the range where 14). at = VBX(I) I=1, VS the interpolated thermal data begins. use the first gamma fit CARD be made, in at volume of the thermal gamma VBX(I), FORMAT I5, F10.6 If between 2 is interpolation 13: 16. VOH. volume greater than VOH to which the gamma quadratic be extrapolated (linear). = to VS CARD CARD fit range V CARD go to NA, VS NA = number of segments into which the thermal energy derivative is to be broken in the volume 12: to CARD pressure, to zero is If coefficients There will be as many cards as there are segments called out by NA. Each card for gamma fit the successive intervals. There may in a is data. from VOH the coefficients for the first segment gamma be calculated by the code. be X(I), to is and I=1, a in GAMMA(2) GAMMA(3). FORMAT 3F10.6 3°NA coefficients the thermal energy derivative. layout The of the cards the same as for CARD except there may be no dummy coefficients. There may be 10 18. 19 gamma as a to of these. — CARD polynomial fit – 0 NDP = If of V. go to 17.) a CARD coefficients 14 FORMAT 4E20. 10 of DU(I) 1, 4, go NY to I= = DU(I), + (Note: 16: II CARD If is CX(I) -1. and leave blank spaces for GAMMA(1) of 15: = CARD this region just insert blank card. quadratic interpolation be made set = a If extrapolated is If will on each card. to Vs one to three coefficients 2 If be function CARD 17: USH(I) = UPH(I) = If NIT = CARD 18: 19: 19 I=1, There TITLE-You may punch a title, up to identify the problem. may be as many 32 spaces long, to SAMPLE PROBLEM-CALCULATION u,(U) This problem was set FOR 2024 OF ALUMINUM up as follows: 1 It was decided CARD FORMAT 7F10.6 NIT isotherm temperatures. as 20. C. CARD FORMAT 2F10.6 NDP Up values that correspond to the USH(I) values. TISO(I), = I= 1, shock velocities for which the corresponding Up values are already known (see CARD 7). CARD 0 go to TIS0(I) CARD UPH(I), USH(I), to use the 3600 machine, so MAC = 100. 2 VOH = 1/DENSITY - EOPH > H TOH ~ **TOH = 300.0 HA O = = .3592 *OH c p dT = 0.00167 - O (Determined from experimental data.) Experimentation with several different materials indicates that a step size DELV 3 0.003 VOH (=0.0012) will give at least four-place accuracy in the US MAX = .98 solution of Eq. (11). The accuracy of the solution can be determined by varying the step size. For convenicence DELV was set equal to 0.001. CARD 3 Since the CARD generated internally, a blank card was inserted. Us(Up) fit was to be 4 It was decided CV to have the code calculate = 0. C1 = 37.5.0 C2 = 27.0 GAMOH = GAMP0H 0. = 0. –20 Cy, *OH: and 'oh; therefore: CARD 5 Since all the values CARD On this card are 0., a blank card was inserted. 6 N2 = 70 N3 = 0 This is a large enough sample to give a good gamma fit. A cubic fit to Us(U) Was Selected. The Dugdale-MacDonald It has been found that the iteration. On No isotherms CARD gamma was selected. Ur converges rapidly. are required for this calculation. 7 NNN It was necessary = 0 that the adiabatic calculations be continued problem could be calculated. so that Ur PA = 0 for this high-order The fit was selected to insure a good fit to gamma. Calculations to the left of the Hugoniot were not needed for to NFITORDER NX = 1 = 10 this problem. They were obtained after the iteration was completed. NDP CARD high-pressure points were already known. Up values at four = 4 8 UFSMIN = UFSMAX CARD 0. = .75 Determined from Shock data. 9. The Us(Urs) fit coefficients The results were: D(1) CARD = were determined from 0.5282 D(2) = 0.74607 a quadratic least-squares D(3) = fit. 0.16432 10 NUMADS DELVV EP = = 21 This number was felt to be a reasonable compromise between accuracy of the Us(Up) fit—since Us is nearly a linear required machine time. function of Up —and the = .001 .001 Chosen equal to DELV for convenience. The iterative process is found to be rapidly convergent down to a level determined by machine roundoff and by interactions level, within the code. If EP is less than the noise iteration is likely to continue for NOIT cycles even –21– in the Us(Up) fit can be obtained. this reason it is a good policy to always make NOIT a though no better accuracy For small number restart and the iteration if convergence is not achieved. CARD 11 The volumes at which the adiabats intersect the Hugoniot were chosen to give a higher weight to small slope and sound velocity would be Up values so that the reproduced with good accuracy. The points used were: VAA(I) CARD = .35, .3475, .345, .3425, .34, .3475, .335, .3275, .325, .30, .2925, .285, .2775, .27, .2625, .255, .2475, .24, .315, .3075, .2325. 12 The normal melting point of aluminum is 932°K. Upon melting, the specific from 0.3796 cc/g to 0.4097 cc/g. Consequently, for V > Voh different fit to the thermal energy derivative Eð(V) was made in each of the volume changes regions 0.3592 < 0.5. In the region Y was obtained by having the code obtain a quadratic fit to V 3 0.3796, 0.3592 < V 3 0.3796, *OH' Yoh. and Y(0.3796). NA = 3 VS CARD 0.3796 × V × 0.4097, 0.4097 × V × Therefore = . 3796 13 VBX(1) VBX(2) VBX(3) CARD . 37.96 - .4097 - 5 14 In the mixed-phase region, Y was determined by making use of the appropriate cycle and the Clausius-Clapeyron equation. The value obtained thermodynamic was: Y = 3.126 at V = For V 0.3796. > 0.3796 Y was assumed to be a constant. Therefore GAMMA(1) GAMMA(4) GAMMA(7) CARD al = -1.0 = 3.126, GAMMA(2) GAMMA(5) = 3.126, GAMMA(8) = GAMMA(3) GAMMA(6) = = GAMMA(9) = = = .0 15 the volume range 0.3592 expression In I Eoſ V) = C < V × 0.3796, Eð(V) was calculated from the P. Vo. " –22 The following fit was obtained: = Eð(v) -10. 5766 V - 91.917.5 v2. + 63.46.67 In the volume range 0.3796 Eð(V) Was calculated from the heat of fusion. The value obtained was 0.1591. In the volume range 0.4097 × V × 0.5, again calculated from was Eð(V) CP/Va. At V = 0.4097 the value obtained was 0.229. For V > 0.4097, E,(V) was assumed to be constant. Therefore CARD CARD -10.5766, CX(2) CX(1) = CX(4) = . CX(7) = 1591, CX(5) .299, CX(8) II = = = 2, this card was = 63.4667, CX(6) CX(9) = -91.9.175, 0., = = CX(3) - 0. onitted. 17. USH (1) = 1. 195 UPH (1) = .513 USH(2) = 1.309 UPH(2) = .638 USH(3) = 1.326 UPH(3) = .673 USH(4) = 1.367 UPH(4) = .702 18 Name required since NIT CARD V 3 0.4097, 16 Since CARD × = 0. 19. The title used was: AL/D-M GAMMA/Ufs-QUAD/Up-CUBIC This problem converged in four cycles. highest three lowest and the adiabat were: The (Urs/Up)" values obtained for the Cycle (Ufs/ Up)" values for WAA value 35 0.345 34.75 of: 0.2325 1 2.00 1638 2.004448 2.001314 2.0834.66 2 2.00 1635 2.004440 2.001315 2. 09.1860 3 2.00 1636 2.004 443 2.001315 2.092 127 4 2.00 1637 2.0.04443 2. 001315 2.092 129 These results indicate that random errors occur in the sixth decimal place. However, inspection of the three low adiabats indicates that the actual accuracy of Urs/Up is approximately 0.003. (The error in Urs/Up is due to the interpolation used to obtain Ur at PA = 0. It may be reduced by reducing DELV.V.) –23– The Us(Up) Us fit obtained was = 0.52816 -- 1.4952 Up – 0.6704 2 U" O + 3 0.4081 U ’. p The fit obtained with the high pressure points included was: U-S = 0.5314 + 1.3812 U p - 0.0531 U* O - 0.3257 U". The card layout for this problem is on the next page. –24– p PROGRAMMER C-G-For COMMENT STATEMENT NUMBER PROBLEM 10 || 9 || 8 || 7 || 6 || 5 || 4 || 3 || 2 || 1 1|0|0 . I3|5||9|2 3|7|5|- E|RE |0|0|1|6||7 • ; 7 1 - - RL-389 1 || - 3|0|0|- |0 ||12||13||14 5|16||7|18|19|20121222324.25 - EXT. 35|36||37 ||38 |0|0 47|48|4 STATEM ENT 40|41|42|43 FORTRAN 5 • .. 0 || PAGE DATE 58||59[60|61||6 Appendix I. Derivations of and Y'(VoII) Y(VOH) Underlying and PK(VoH) - ! ! !I || f . . ) -> v 2 y-l PH -1 )ºv - (§ I 2t 2 ... Prºv + it 2t Pl: + V 111 - (49 OH P. * H P" V=V H H y-l PH PH + P. H 4t H * P" 2P f t \ (2 - = #)-(−4 ;-(− 3 1 -(# 4t tº (PHV ) = - t Y +; 11 (VoIP): *(vog) (Von) into Eq. (5) yields the following expressions - . of (d) Y(VoIP) and 111 P}(VoIP). the above approximations + for III P.(Vorſ) Yl Substitution 0, ~ PH(VoIP) * 3 4t (a) are the four assumptions: these derivations P' –2 V=Vo H (50) Before Y(VoIP) is - - ti it Y' equations, necessary (VoIP) can be calculated from the above III relationship and of PKCVOH) to PH(VoIP) PH (VOH) and to determine the and t to evaluate Elimination of Us from Eqs. 3 is ...) (51) obtained after rearrangement: –26– series is V(PH) a Up expressed as given by Eq. (51) yields squared, the following expression for in 1/2 = is If the resulting equation "Asº the (1) and (2) gives - of the series in A first substituting V Reversion + (AU obtained by Up 0H p - 1 - W F = PH at:2 The functional relationship between PH and Eq. (34) with the result Us(Up) fit into V Hugoniot quantities. PH. 2 V = V Arr 0H If Voh — —#: Prº H 2 * A. || 1 — —--- PrºH 2VOHA2 2 2 +| 4 A. Eq. (4) is differentiated . 2 5VoIIA3 —ºtº-t- – 2A3\ –º | 2 A. Af P. 2 -- . . . * H . (53) three times with respect to V and the approximations resulting equation, (a) through (c) are substituted in the the following relationship is obtained: ^ 111 P k (VoIP) - PH(VoIP (Vor,) OH --vi- Pk(VoH) - BH(VoIP)| Equation (5) gives the following relationship "oh! . . WOH for 7(VoIP/Voff v Ph'VoIP + (* 3V !) 2 - FHVoº) (55) Substituting Eq. (55) into Eq. (54) and taking the energy derivatives rearrangement, - 111 Pk (VOH) = |PH 2 P" - H. ſ 4P - (2 + 6V gives, after II 0P V -=V Equation (54) gives the following expressions V (54) : for (56) OH the pressure derivatives evaluated at = V2,...: OH dP dV | \, V-Vof d”P -- d°P 1 (57) ---, 4A-A* - -->2-1 —3– |x, (58) OH V-VOH - H —a dV -a-, Vori H dVT A% - H - | y, —-------. -18A*A* 2 **1 12A. A. 3 **1 (59) V V Substitution of Eqs. (57) -(59) into Eqs. (49) and (50) gives finally 7(VoIP) I "Y (VoIP) - (2 = - -( –- 1 Vof + 3 t - ) + 2 A2 (60) 2A2, + A2 ( 4 - 5t) - 3 ) –27 - 6A3A1 + 3 + (*# 2t ) - (61) Appendix II. Fortran Listing of the Code D0 till 1 is 1,50 &#5 CALL EXIT EN) ſºlºiſiºiºſº, SUBROUTINE COMCEN casesssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss" Č THIS SUBROUTINE GENERATES COMMON #ºwlā'āś tassessessesssssssssssssssssssssssssssssssssssssssssssssssssssssssssssº CLICHE tºN COM) CLICHE COM: iſfi,xx (100), SLOPE (100), X,ECH, tºn/ºsulT/PHU(1000), NOR, XOX,Rºº?0H, EHU11000),PKKT1000), EKKIIſºlilºſſ (1000). .################wiſſ CLICHE COMS º. Nºrşiţiij pirº Cºlon/TSC/URA(50), NFIT, 1. NUMADS, WAA (50), DELWW, US (100), UPP (100. 10), ENjëiíčić CLICHE COM4 COMMON/GENERAL/7, CC (5.5), ENDCLICH: CLICHE COM5 COMMON/EO/NA.CX (15), WBX [5].VS, GAMMA (15) ENDCLICH: CLICHE COM6 COMMON/PRES/P0H.DU(10), NY ENDCLICH: II ºn Rºſvºn END SUºBOUTINE GERED $88.8888.88888.88838888.8888.8888.8888.8888.88888.88888.8888.8888.8888.888 C THIS IS THE PRIMARY SUBROUTINE-IT READS IN ALL THE DATA, CALCULATES PRESSURE AND ENERGY ALONG THE ZERO DEGREE ISOT º C C C C C DEFINITIONSassºk (1) PKK (I) PHU(1) I) EHU(1) tº sº.” titfüüß s.150THERM ENERGY sis)THERM PRESSURE PRESSURE sh;0&ONIOT skuſ, ONIOT ENERGY OR ISOTHERM. ENERGY HUT 1) shºjº,0NIOT TEMPERAT ERATURE OR ISOTHERM PRESSURE GAMM sGAMMA (I) C C f US (1) s.SHOCK VELOCITY º S OR (I) si NITIAL SQR (8), USH (50), UPH (50),TISO UPP (I) spºRTICLE VELOCITY CONDITIONS EQ (11) *:::::::::::::::::::::::::iii#if: C §§ 20*.0M/TTLE/AAU, COMMON/AAZ/MAC 88.888 88.88888.8888.888 (20) AAW, AAW, AAP, ICYC,NX ICY –28 º, º żº ºś ºff ſº ############# Nºt TAPE 2,070, WCH,ECH, TH, USMAX,0tly, (All), # ºf fiš #######,” "ß, IF 670 (N2) 976 975 FORMAT (415. 1:1,71, CV, C1, if ºğir, kii o, C2 in 3,975, NNN,NFITORDER.Nx,NDP ºr wºups, DElvy, ######### 2,078,NA,ws, Vºx, tº ºff"fift (GAMMA(l), 1-1,5sNA), (CXII), 1-1.5°N (510.6)) l IF inºff fift 3,172, thut!), 1-1,NY) 6) 7% ºf twaati,i-1,Nunals, - 1A) 878 FORMAT (15,710.6/5F10.6/ (NY) 170,171, 170 Q72 P, # :FIN0:7).071,072,875 ###, (1), Isi, NIT) ºf Žižjiàº, liv.iii., ii; Mºſt (TISO C IF (CW) @30,081,380 Cººessessassssssssssssssass8888.8888.8888.8888.888388888.8888.8888.8888.88888.888 CALCULATE CW FROM EQ (49) 0.91 CW'sCWOFT (YOA) C1 CON is *:::::::::minimummim” # sº N1 sº st IF (NOIT) 997,997,996 997 NOIT 996 CONTINUE ICYC AMOH : wº s. = = CC CC CC CC CC CC +1 NINsh; 17-1 NOIT shCIT (1,1) s.66666666667 (2,1) 1.0 (5.1) s.1.3555.53553 (1,2) -1.3333333555 (2.2) 555: (5,2) s2.0 CC CC CC *0. (2,4) •.2222222222 (5,4) •.44444444444 (1.5) sº. tº $5-2.3% ##526.3% itſ?.5:1.5355335555 CCſ º C IF 000 CONTINUE (EOH) 1290, 1291, 1290 5:#y: fºtory CALC Eß frºm 1291 EJHs:THM (TOH) 1290 NNOITs),017 C*********ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss –29 t CENERATE US, UP, POINTS 27 DELL alſº SMAX/60.0 UFSaúſ SMIN-Dºll. (US, UFS) FROM * º * ####! Usitj-jtiſ-Ursa (DI2]+UFSs (D13) tufSs (DI4]+UFSs (D15)*UFS at D (6) +UFSal) (7) ))))) 42 UPP (1,2) •.5sufS caseseasassissassasassessssssssssssssssssssssssssssssssssssssssaaaaaaaas ! A LEAST (1,1) C (7) TAP; 5,610 (5) FOLLOW// (4) 1098 # ) // 1 // ºf 610 Foºlarſ//44HPOINTS USED THE FIT AND THE RESIDUALS ARE 4X.2HUP.14X.2}{US. 11K.9}{RESIDUALS 338, diſſºm"fºº 25. Usti),RRR(1), 1-1.NOPTS) 998 FORMAT (2715.6, E15.6) Riº (1) (5) /7E15.6) IN OUTPUT Fif) #Suif$ C it'ſ C C C 41 2 ! (6) (2) 23. jji Tº ºffi NFIT, UPP, US,C,888, RRR, WHN, WWV) $. C ºf" Fºrfº. 44 CALL MLR(MAC, NOPTS *ITE A80WE suPP (1,2) C tº #" 45 GENERATED =1.0 J-2.NFIT UPP (I,J) supp(I,J-1) DC 43 (US, UP) POINTS TO THE ADIAB is 1,61 D0 43 UPP FIT SOUARES IN SUBROUTIN: C OR ii. tºur: 1.NFIT (AII), 1,7) is sº. 111 (I) [1] PUNCH_205, A 111 is DO C&2&38&gggggggg.8338888.88488.388.8888.8888.8888.8888.8888.88888.8888.8888.8888.8888 IF 205 FORMAT (7F10.7) (IMM-1) 106,106,107 Cºassºttasat&seassasssssssssssssssssssssssssssssssssssssssssssssssssss IV-A) (IMM-1))/URA (IMM) -EP) 109,109,106 C&#848883&438888388&#8tassssssssssssssssssssssssssssssssssssssssssssssss TEST (SECTION lºſt iš ) 40 IF [. DC 106 CONTINUE 319 319 A(1) s]. 26 (N1) 40.45, 1, (URA (IMM) -URA 14 (ABSF ( 107 IF C CONVERGENCE ifiyiii.5.1%,173 Cassassas: ºsaaaaaaaaaaaaassasssssssssssssssssssssssssssssssssssssssssss FIT I-2, Aºsſ -1 175 XUs):P&WGH NY 00 0 C CALCULATL GAMMA FROM AN INPUT 175 GAMOHsOU(1) GAMPJHaº. XP=1. GAAGHsſ, AM3H4OU 175 (I) ºxU GAMPSH-9AMP3H4ANNZ2DU(1) XPs)(U skP ºf ......ºſiſ.}}:A2). *:::::::: gº, 2 *************************************assassssssssssssssssssssssssssss #74 Alºh, FROM ºfficz, aſſºjºcºſ.; *::::::::::: 4 (45) AND (44) *minutiuniutiunium, !?!ºſa!!! ſº EOS (Aſ2-cc.(11,2)-6.sainsaſs), 47.31230,251, 230 –30– - GAMEs GAMMA(4) +W78 (GAMMA(5) +WZ8GAMMA(6)) UEX's WZ-W0H #############!/* (2) eCAMPQH-2, (5) GAMMA GAMMA tº a WQHaGAMMA sGAMOH+WQHe (WOHeGAMMA (3) -GAMP0H) 230 CONTINUE NO! TsNO!?-? IF (NOIT) 29, 109,29 29 CONTINUE Cassasssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss C WRITE DESCRIPTIVE INFORMATION TAPE 5,922, AAU, AAW, AAN, AAP OUT MRITE OUTPUT Z/) ºff foº 5X // 15 T IS D D H (5) ) D. (4) O D // D UFSMIN=F10.4,12H 2RANGE 'ºus BEEN CENERATED FROM 5X,50HTHE EIT WALID OVER UFSMAXsſ 10.4// 5X,54HTHE US-UFS FIT_CO (1) (2) (3) /4X,7F14.6 (6) D(7) / wºn 57FICIENTS FOLLOW/101 : fif"ºf"ºff".5iº gºiO, US-$ TO Uşlüfs ºf Fºrſ;Y,Zihū * 17: ºff OR OF TO OF 1D riff ºffif'ſ ºf Fºirſ; gºin ºğ'iš TO IF 922 FORMAT (1H1,4A8//) ICYCsICYC+1 WRITE OUTPUT TAPE 5,650, ICYC 658 foºirſ;Xizhcydif (NNOIT)455,455,454 456 $455.EP,NNOIT 154 455 ſhiš #ILL ITERATE OBTAIN THE CORRECTE RATIO FREE SURFACE PARTICLE VELOCITY 02HTHE 2CONDITIONS FOR TERMINATION THE ITERATIVE PROCESS ARE (UR(I)-URI //) 31-1))/UR(1) (F10.6,10H NOITs 45.7 WRITE OUTPUT TAPE 3,458, NIN, UFSMIN, UFSMAX,D TAPE 5,459 F#$$URE AND ENERGY WILL calcular:D ALONG THE 1ERO DEGREE KELY!!! ISOTHERM /5X62HFROM THE MIE-GRÜNEISEN STAT 2E AND THE HUGONIOT RELATIONS //5X11 HDEFINITIONS/ ENERGY,22X25H7OH-INITIA 510X10HWOHs INITIAL VOLUME,22X10HECH*INITIAL 455 WRITE OUTPUT formiſſãxgåHT# ſtrºńTüß/iºxiºidºſniffit"Giºli.27x26;&ºicinitial giºn Pºſº, ſix,255(W-Sºº. Hºaf"if" cºnsiſ. Wöl.77;ºjšºix. insify, ižºpºsiº Sº #"; ºilfi). is iśīšū-ji'sſfy/Iniriit ºf OF Yêiſ.’... of Mi 5 § OF EQ 2 BE 45% ##1%;##!" GO Z/5X,65HTHERMAL DATA WILL 7:RC PRESSURE //) TC 467 465 HRITE OUTPUT TAPE 5,466 : fºſſibiläijs ºft.º.º. Hºly! 468.fºº! USED 1: / WA IN IS CW a is HUGON107 # iſ fºr INITIAL THEORY //) s!.NUMADS) ſº TO THAT BE 462, NUMADS. (WAA(1), I ! IF (NNN) Iš fºſſil REIGHTss fossi..jºišićy DEBYE-MODEL/5X95HAND THE OUTPUT INPUT=0.0, HILL CALC FROM THE DEBYE INTER$fºr 465,464,465 ADIA5ATS !2H Bºřivºji IT THE EQH IS FROM IF 20ALC, 5LUE. 462is: =DEBYE-TEMPERATUREss, 18X21HC2•HOLECULAR ººt"; fºx.jsº 1/5 iſºlaſgD ALON; 13,5 Wölükş"/777f74.67; EXTEND THE ADJABATS 70 Aff';...&#5 BE Qūſpúf 865 FORMAT (10X22HC1 A9 ºf jºy. *iff /) ISOTHERM, 19x20HPH-HUGONIOT PRESSURE,20x1 70THERM/10X21HEK-ENERGY 89A.HshuGONIOT ENERGY/10X27HHUTEMPsłłUGONIOT 9US-UP FIT COEFFICIENTS, 15X21HDELW's VOLUME INCREMENT !F TO If NiššštěijūNāiſh Hijº...? IºERATRºtºrſ. išćfñº";iºnſ"; iºtiº."of" * #######, Wilſº (HY) 180,181,180 -31 - #3: (3) %4% Iºaç㺠Viljºifºliº; sº zºść intº *::::::::: //) WILL EXTEND BACK V-vºh #146; (!!N$41,317HUS-SHOCK YELOCITY,23x2]HUP-PARTICLE º; ## iºnºia is #iºts&t 10Fºić"Holidºx. TAPE 3, 102, (DU(1), MRITE OUTPUT 100 # Isl, NY, as a SERIES. THE COEFFICIENTS 5:27.70); Fºirº ######! # Tºyºtty:Riº, ºilmº-ttº, §:#### 3,105, l WQX, PK,EX,DPK Infºſſificº is ºint STARTED Aſ yolunt-F10.6, 0.4//) UTPU Fºuriš.5units/iºphyglotity-ºn/MICROSEC,208, 17MºSSºº-ºº: i Žiſtºfiyin,77) #s: º'ºï'fift'ſſists,NIT, 461 152 1522, 1321 Fºiſſºit. §§ ºff iiii 1ERATURES/15X, 10F10.2)] 20 1522 21 201 202 jjīābāºšis jjīāºis º jºiºs WRITE OUTPUT 22 WRITE OUTPUT 205 1-1,N11) THE TEMP //) FOR SLATER GAMMA FOR OUGDALE FOR FREE WOLUME GAMMA FOR INPUT GAMMA SERIES TAPE 3,202 MACDONALD //) GAMMA TAPE 5,205 ºjääitäs 909 MRITE OUTPUT º: tº 5:01 21.22.909), ºffift ITIsou), świft'ſſion:"is,%h isothºrns II išiš AT #; tº //) TAPE 3.988 //) 907 306 (6) A A A 3, A (5) OUTPUT TAPE GAMOH TOH USMIN USMAX A(3) A OH (1) USMIN, USMAX, GAMOH,CAMPOH,CW, //6F14.0//20H Aſ2) CW //2F14.0//964 MRITE 1040 WOH,EOH, WCH FORMAT (64k. CAMP0H 3 2 1 104 ICH, iitii.fi.iriſ; 104 //7F14.0// (7) (4) ) TAPE l USMIN=A (1) WRITE OUTPUT 5, C88sssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss 1040. DELV formiſſãx,&ptiv"...#iž; Ifüºdifiº,957 Riff ºffſf'fiftſ;.986.DELVy foºt (5x,&#Oftwy-f{2j fift ºff £7/5X,3561-,áž.5iº2-,E12.5//) § fºr Šiš 5ii;.COME,62 Ciſſé'ſ “ſººt, Iffwºnj995, $7.99% SECMENTS TO IMPE 3,400,NA OF RITE_QUTPUT 991 USED REPRESENT THERMAL ENERGY (NA)478,990,470 470 00, 477 Isi,NA *šion wer, QUTPUT TAPE 5,402, I Hºlſt ºf IE_3.JP'ſ TAPE 3,407, GAMOH,CAMP0H Yºhjºri.6) fºiſſéH¢iºrit.8,347 is 12//) ºf 4:3 409 NUMBER TAPE 3,409, Wiś8, WBX (1) 4:4 4:7 * 405 Riº ºf' fºliº ºffiº, ºff NT RITE_QUTPUſ ,r10.4,6H ###iviſiºn {{{WS-W8x (11) 403,404,404 (GAMAſ),J-M1, Mutz, i dºw –32– 10 IF USMAXs 1.1 sushAX WA38s WOH ,F10.4) DER: 99 "ºx, fº TAPE Fºirºtö'Pāī-fiz.6, (I) WA80s W8X NRITE TAPE 5, 101 OUTPUT Fºniſtry, ºffix,&#ima,Tx,2HPK,11x,2}{K, "Yº x-Ušºf 11X,2HPH,11X,2{H, iii. His 4?? J-MA, NHA*2) intysiſ; 12.6.3Hºws, F12.6.1H)// 3,408, (CXIJ), 406 WRITE OUTPUT iriº, Wººyº DIFs. [18X IF 1275 $5% 1./WQH s RHOOH (WOK) 894, 1275,094 sº GO TO 394 CALCULATE WOK FROM 695 CALL WOKCALC E0 C Csassssssssssssssssssssssssssssssssssssssssssssssssssaaaaaaaaaaaaaaaaaat (46) Cassassasssssssssssssssssssssssssssssssssssssssssssssaaaaaaaaaaaaaaasses 994 CONTINUE Csasssssssssssssassssssssssssssssssssssssssssssssssssssaaaaaaaaaaaaaaaat IF s s C E0 CALCULATE INITIAL CONDITIONS FOR (11) PHU (1) spHUG (WOK) EHU(1) s.5epHU(1) (WOH-WOX)+EOH DPHU(1) s-PHU(1) alſº RM (WOK) DEHUs.5s (DPHU(1) (WOH-WOK) -PHU(1)) (PK) 897,696,997 997 TO 898 896 (NY) 176, 177.176 IF :*::::: 176 SQR sº (?) "ºn-antºwn Ns? 170 ºur 1) • . + [ ] & & GO TO GAME}{8 -GAMPÖHeº 090 CONTINUE SQR (2) SQR (3) s-EK HU(1)) sº C C N=3 88.8888.88888.8888.888&68888&8888.888888388888388 SOLVE EQ Dºlys-Dºly s. 170 (11) 23WQH r CALL R&G (N.W.0%. VEND, DELW.SQR) - ? Cºë86&&&.388888&&&888&&$888&68&868&6&888&@8888.8888888.888&6888&888&ºtº888& M1 s?? WALS". 1,111 giftſ)/w011) #if:{i} G 5: DC : s 31 ) : C Qassassessesssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss NTEGRATE GAMMA (W)/W (SE: EQ (10) X-y0ſººty ...iiiºliiºliſhiii!: MNX1 EC sh: $888& 3888&8888.8888.8883 (10) WDIF's WQH-W0K tº-3AM"Hºy!HegAMP0H C: Y#Jewſkis'ſ stºº *H ) ) (1) )*PHU(1)-POH) –33 ºf C (GAMP0HsVDIF) – st!/AD7C JH PH; TC (AD (DPHU ) sºxPF RAY ; :: ; ) ) - AC7. (; M! Cº ; -1, EC C! suyº 35 JWvis (WGH/WQX'ssCOEs:XPF 53 . . 888668888.88888.888386.88836&9888.888338 NY:CRATE (1) SNTS•0.1. if (PK) 1315, 1316, 1315 1315 Hºſſ (1) aſ GH Tº sº. 3 GO TO 1317 1516 WSTEPsyDIF/26.0 & ! ###"...is ######, Pispºſº (WDCT) PD1-Pisſºl (WDCT) WNºſewºº-WDCT pºſs twäH/V5CT) as:0EaÉXPF (GAMP0HsVNET) 1314 GMTGaGMTG+YHT (WQY) Piațºjº episTºº! PD1 (WOK] gºrge-VSTEPs (CNTC+.5s (PO1s WDIF4F1)/UVN) Trisºn? G/CW 1377 CONTINUE Hºſſ (1) shºe (1.4.5&T'ſ 1/70H) GMTG2açNTC+.5e (RAT (1) +8AT (2) 1808 LV TT2sºn ſºº/CW Hºſſ (2) s?hu/ADTCſ2)s (1.4.5s,772/70H) C1 sºut!) C1 sººj (2) CW1 stwººt (HUT (1) CV2sCWOFX (HUT ) i2)] Iss, MNX? D0 36 tº-pººjſij tº; Tºsz.8%Uſ (1-1)-Hºſſ (1-2) CWSeCWQFT (TG) Hºnºran wºotºut-ºutinſºm TT2-Ty CW1scW2 CW2*CW3 aſſºwanctiºn...sºm &gggggs&sesssssssssssssssssssssssss º: Is?. 00 280 ſºrtin-wół:/yºh Is?. M! ºffij-i./iºrſ 00 2.5i Lºgº.º.º.º. XYAs?HU (#1) wº 6) i.i. º riºtzhai. (I), CAMM(I),PKK (I), EKKII),PHU(1), • ‘w 9 - tilt QUTPUT º Hºlſ. 88.8888.88888.88888.8888.8888.8888.8888.88888.8888.888 Rºsul TS TAPE_3,102, (RAT AND PLO'ſ tº r QUT . CHPRESSURE ########. kºtº'ºs-isomon *ºtºqf kºi AAU. AAW. AAW. AAP ºf Pºisſtäki. stipchº.º.º.º.5%) Cºll_201N1 XYAsºſ !!!,3; A ; Zºli ºf 4, J., XYA,0,10HPRESSUREIMEGABARS),3) (MEGABARS),3 1,0,5HV/WWH, *###..º.º.º.º.w.... A8 Ç C 251 • 250 S8888.88888.8888.8888 M1 (1) $6 fººl,M1,1,1) (1H8, AD7C,PKK.M1,1,1) (Mi) ºRID (14,9,1,1,3,5HV/WQH,4,0., CALL SETCH (30, ,5%. XYA,0,23HENERGY (MEGABARS-CC/GRA ,0,0,1,0) -34 931 , ICYC, AAU, AAY, Ali,Mº ºf'." §º.º.º.º. # jºin ################# I4/4A0/ ºwrºat 1GY /ins-isºmon be tº jºinićtiº,M1,1,1] CALL SETPCH(0,0,1,0,500) # # #; S: CH! * ,0,5HGAMMA,4) - jūīºf riffitºšicyc, C e AAU, AAV, AAM,AAP §:##1.” “...fºllºw-unminuºus" #: ! 1509 70 till tº išoikºsti,TIsott)) IF (N2) 70,70,71 RETURN POINTS Ks () C TO BE USED TO MAKE A LEAST SQUARES FIT TO CAMMA AM1 shi AN2*M2 UNUMsAM1/AN2 WXY's 1.-UNUM DO 58 Kºkº'ſ is 1, N2 WXY sy)(Y+UNUM Js WXY WAD (K,2) =WO (J) 58 GAMAD (K) • GAMM (J) IF (M1-J) 141,141,140 141 NFITPTSsk 140 WAD (K+1,2) =WO(M1) GAWAD (K+1} sº AMM(M1) GO TO 66 NF: TPTS=k*1 GO TO 66 65 DO 39 l =1, M1 WAD 39 (I.2) swo (I) ºffirginiſt1) NFITPTS-M1 66 NFITPTS=NFITPTS+1 WAD (NFITPTS,2) syſ)H twº ſºiffsj-tingh Cºssass8888.888.888.888.888.8888.888388.88888.8888.8888.8888.8888.8888.88888.88888.8888 Dºlys-DELV IF (NNOIT) 20,109,20 Cºssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss C MAXE LEAST SQUARES 28 CALL GAMF. T FIT TO GAMMA Cassssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss C MAKE ADJA8ATIC CALCULATIOS CALL ADI A8 C88388.8888.8888.888sssssssssssssssssssssssssssssssssssssssssssssssssssssss 09: IF (NNOI7+1) tº 1,109,091 IMMs (MM+! URA (IMM) sºu (M1) USMAXs 1.7.1.1 sushAX GC 70 44 :29 F (NDP) $65,865,864 864 CC 063 s!,NDP Ksh;CPYS+: JS K sºjSH!!) -35 — SUCROUTINE WOXCALC C&CCCCCCC866888.808888&6888.8888.8888.8888.8888.8888.88888.8888.88888.8888.8888.8888 CALCULATES WOK BY SOLVING EQ. (46) C THIS SUBRIUTINE , C8&8888888.8888.88888.88888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888 BY ITERATIVE SOLUTI THIS SUBROUTINE CALCULATES VOX, MHERE PK • EX • (W) s () OF EQUATION: Wsph (W) - GAMMA (W) C C sº COM! USE WW2 s WOH # "...wº-tº Dºls. 01&WOH D0 5 K-1, 15 WW1swV2 WW2sWW2-DEL G1s02 C2•C (WW2) IF (C1862) $5,21,3 5 CONTINUE 35 sI º !º-ºwn-wn/iº-ºn CALL EXIT 50 CCsC (WOt.) IF (G1sqG) 55,21,34 55 WW2-vOL MºsVW1-WW2 MºsVW2-WW1 G! •CC 56 IF (ABSF (G2) -. 57 WT =WOL4.0000 IF (G(WT) scG) 21 WOKswCL RETURN END *iºliiliº * > THIS 38888.8888.8888.8888.88888.888ssssssssssssssssssssssssssss SUBROUTINE CALCULATES US 9W SOLVING EQ (31) :****************assassssssssssssssssssssssssssssssssssssssssssssssssss USHIN, USMAX - ESTIMATED RANCE OF SHOCK VELOCITY WILUES US: COM! USE COM5 –36– ºh".” XX (1) axºDIF XX (2) a 1.00018XX (1) USMs, 958USMIN , Fºirºtºzºro 10 ! 2. OF US, WALUE FUNCfióN CHECK ºnzº.”.” AT THIS BY HAND, WALUE MAY NEED OF B DOES NOT WIELD UNIQUE LOWER USMAX) RETURN 20 35 IF (NFIT-2)22,25,22 Y. Tij} (7:it:35 RETURN 22 Z1 =7ERO (8, XX (1)) Z2-7:RO (8, XX (2) ) iſ 5 T-2.5| inckxx-??sixx (1)-xx(1-1))/(22-21) XX (1+1) IF (ABSF syx ( ) -8NCRXX (3NCRXX) -.00000001 31 21 =72 1 fºurtigh impf ºff ºffijfND'Souffon 5 Z2*7:R0 (8, XXII +1).) 16; 5. ) 50, 50,51 101 fo ZEROtx) = 0 WITHIN 50 ITERATIONS) **** CALL EXIT 50 DIFs). X (1+1)-X Xs)(X (I +1) RETURN END SUBROUTINE RKGIN, XS, XF.H, Y) Cºasts&sasagaaaaaass8888.888&ssssssssssssssssssssssssssssssssssssssssssss *:::::: CTHIS SU880UTINE ! USE COM2 USE COM6 gº USES THE RUNGA KUTTA METHOD TO SOLVE EQ. CºON/RUNGA/E (4), F(4), R(4) disj.Yū), Yºſë Eſ 2) s. 29.20952.1081345 Eſ 5) s 1.707106781 18655 Eſ 4) s. 16666666666666666666 Fſ 1) s2. F F F ( 2) s?. 3) s?. ( 4) s2. Rſ 1) a .5 R (2) s. 29.20952.108.1545 R{5} =1.707106781 18655 R(4) s.5 MsABSF M! st ( [XC-XF}/H]+.5 #:#; {5}*.5s (Y (4) +POH) s (WOH-A1)+ECH +: DC 5 K-1, M M’ sh;14 PHU (K}sy (4) !? (Y-.9suSMAX)21,21,24 –37 – (11) OR (12) 24 RETURN 21 WO(X) •A" -- (PHU(K)-P0H) alſº RM (A1) sy (5) Gºitº) sais (PHUIK) -Y (2)) / (ERUIK) *Y (3)] Pºº (Y) sy (2) Exx{K} =-Y (3) 00 5 Ja 1,4 Dººjtk) ºutk) tº fo tiš,11,15, 11), 11 A1*A1+A2 J DERIV (A1, Y.YP) 15 CALL D0 5 1 =1, N (Jia (YP (I) sh-F (J) sq (1)) YII) syll) +TEMP TEMPs: 5 Qii) sqt!)+3.87 ERP-R(J) syp (I) sh Mºſuº SUCROUTINE DERIW (WOL,Y,DERSOR) Cººeeeeeeeeeeeeeeeeeeeeeeººººººººººººººº. 88.8888.08888.8888.88888.8888.8888.8888 C THIS SUCROUTINE CALCULATES S PRIME (E0. 50) Caeseºeeeeeeeeeeeeeeeeeeeges&8888.888.8888.8888.8888.8888.8888.88888.8888.8888.888 ),DERSOR(0 ) DIMENSION US: COM! Y( Pººjº (WOL) E}{s.5s (Płłłł'OH)2 (WOH-WOL) Y (4) Y (5) sº H sº tº OH Zsſ. WWOL IF (NY)4,5,4 4 XPs.I/WOL CAMWs).0 D0 6_1 =1. NY YUs)(Psw(\L CAWs. AMW+}\}{1}a}{U 6 MPs)(U tº Rſ1} Y (3) s-Y (1) *-ºsº (1) =-PH-SAMWs (Y (1)-EH) * #. UºN 5 km ſººty (5) jaz DERSCR (1) -- (Y (2) sza (CCIII.4) sºlº CC (11,5]s [PH-Y (2))) ! +2.sy (1) s (CCIII.1) shºpH-Y (2))) / (EH-Y (3) ) Q? (2) sy (1) (3) sy (2) EN) SUCROUTINE *:::::::::::::::::::: 3.15.5ºlº. SIMPSONS 1/3 RuLE *:::::: C*******assasssssssssssssssssssssssssssssssssssssssssssssssssssssssssss 88.8888.88888.88888.88888 WSES, D! ###!! 10N.W.A. (1000),Y (1000) }/H; +1.5 ####..s. *WALS+.56% (Y (1)) +Y (2) ) is 3.M ' Wiſſi-Wittſ-24.333333333333styū-21-4.syſi-1},ytinsh 50 ſ –38– Rºſvºn EN) SUCROUTINE GAMFIT 86888088888.880880888.888 S8888.8888.8888.8888.8888.8888.8888 C8888.8888.08868880888 THIS SUCROUTINE MAKES A LEAST SQUARES FIT TO CAMMA Ceeeeeeeeeeeeeeeeeessessesssssssssssssssssssssssssssst C USE &8888.8888.8888.88888 COM! COMMON/AAZ/MAC l l D0 = 1, NFITPTS WAD (1,1) =1.0 = 2, NFITORDER D0 1 Pºitº'ºtrišf shºts fir'fö ºn foºiriſhi. tuft RRR,M,www. / his BEEN MADE 1NT ºf stºirs TAP; 5,101, IWAD (1,2), GAMADI1), RRR(1), Isl, NEITPTS) ºff ºfficifºr; i2ha 1 HPITE_OUTPUT fºrf7/4&pººfs jśń"in l 101 WAD, GAMMA ######! ºf º'fit º'ºsińuits (2715.6, RESIDUAL // 100 J Jºjº Wapſi,z) ºitºitR, GAMA),C,Dee, ºf ſºirºfs, jºičijj.I.T.ſºſſº Wipii, Jy-Winfi, W 1 7744h E15.6].) RETURN END SUCROUTINE ADIMB Cººessessessssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss THIS SUBROUTINE COMPUTES PRESSURE, ENERGY, PARTICLE VELOCITY. AND "Alon; DIABATIC ENERGY DEFINITIONSassº(X(1) s. Ajºſs' INTºº-Wºliº. C C C Tºfu: PKXII) sho! ABATIC PRESSURE PHU(1) shºjº,0NIOT PRESSURE EHU(1) shuCONIOT ENERGY i USE RATIOslº A/70 ), WL (0 888&8888.8888.8888.8888.8888.8888.8888.8888.8888.8888.88888.888 Y ſºlº (I) JºA (I) slºpg|RATURE sº G AMM ADTC (2) ,DELLL (2) COME5 CºN/TITLE/AAU, AAW, AAW, AAP, ICYC,NX WL = 1./WOH R}00H WL (1) syſ)H (2) s/MOTC (M1) syſ)H lºſs ADTC (M1) XYAs?HU (M1) º;; stºß,6; OUTPUT Tip: FORMAT (12HCYCLE XYA,0,1 CHPRE SSUREIMEGABARS (MEGABARS),5) , w ºil MºITE AAW AAM.AAP 1, 7, • ! # i. #####, ** 00 | ššuº'Along Apiagars 4/4A0/2 $ºsºvº'ſſong 0,0) CALL SEICRI (ADTC (1),PHU(1) i.i) CALL POINTsūjić, pºſſ, 98.031.W.W NUMADS WAsRHOOH —39 – - 8096 1. WA*WAA (1) s 15 WG ######$!!...vii.º.º. W/WOH, 4,0. ) IFINA) 47,47,40 47 WGZs 1.05 GO TO 49 * tºº) Xsa (1) / (1,-A(2) XGUESsuSMIN+.98 (USMAX-USMIN) If tº-XGUES)46,46,45 45 XaXGUES 46 XXYY ex DIFs, § #. 0.18X USSTART-SQRTF (WJHePHUG (WA)/8088) T USSTART, MRITE su & G i##" ########### 102 UPSTART NUMBER,15/4X,5HUSs, F15.6/4X,5HUP-,F15.6) Dºl\}(X=0&LW Dºlwe0&LWW D0 13 IZ-1, NX Y (1) s?HA2 Ne! Xs):YYY Ceeeeeeeeeeeessessesssssssssssssssssssssssssssssssssssssssssssssssssssss C INTEGRATION OF EQ. (25) CALL RECAIN, WA, WL (IZ ),00LLL (17), Y) Ceeeeeeessesssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss *italiº-waw".ºut. XFe2.swC (M1) Y (1) spºk (M1) IF (NNN)63,64,65 ifiºſºziºs 64 C INTEGRATION 88.8888.8888.8888.88.8888.8888.8888.88888.8888.8888.888 OF EQ. (26) CALL ADIAS2IN, W0 (M1), XF, DELV,Y) 21 *::::: Jºl,M flºwntºw º D0_50 30 Cºasessessessessesssssssssssssssssssssssssssssssssssssssssssssssssssssss C INTEGRATION OF CAMMA (W)/W (SEE EO 30) CALL SIMP (ADTC, RAT, WALS,08:LLL (17), WA, W0 (M)) 88.888883&SC38888.8888.8888.8888.8888.8888.8888.8888.8888.8888 D0_52 Jel,M 32 ADTC (J) ºf:YPF (-AOTC (J) ) Cassasses asssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss C INTEGRATION OF EQ (6) CALL SIMP (EMU,0PHU, WALS,DELLL (12), WA, W0 (M)) 838888.8888.8888.8888.8888.8888.88888.0888.8888.8888.88.08888.8888.8888 WALSs:0}}+.5s (WOH-WA) s?HA2 Caesaaeedassssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss C INTEGRATION OF EQ (24) CALL SIMP(EKK,PKK, WALS, DEL, WA, W0 (M)) - 88.8888.8888.8888.8888.88888.8888.8888.888&sssssssssssssssssssssssssss IF iſ Z-1)22 25,22 WS-Upsilºſºſin): tººlſ M-1)) 1 s (EHU(M) RATIOisufs 22 00_41, 41 ..?] 115 ... :75 ARY Js 1, *ºft.§:%mº ºf Wiſ (J) syO(J)/WOH 00 42 (PKKIM-1)/IPKKIM-1)-PKKIM))-1.0) s] . If [Z-1126,27.26 J fºr º.º.º. 3.; 3,113,5,110 fºirſ/74%. Tºš/Wºlfº.6% fºlſ (63,5M€7A,70X,2HPA, 11X,2RPH,11x,2HEA,9x,5HGAMMA, –40 – 25 Fº łºśījj, 5x,5HTA/70,7x 10X,5HW/WQH,9X, IHW//) Riº Olimpiji "ijtjäſuºjičij. Vijjīn) ºf º, º: fºiſt Ž; TAPE 3,29 ºwn.º.º. fêtioWING IS AN EXTENSION US 1, iFúz.;; OF THE ADIASAT.10 THE BY THE KNOHN VALUE.0F.GAMMA //) 4,545.6, F12.6,F15.6) 1) J), º: fiº.3, roºfiñº, fiº. 25 26104 1High PRESSURE LIHIT/6x,35HALLOWED WRITE , Jº ! ADTC (J), EHU (J CALL POINTS (HUT, PKK, M1, ºut.J),EKKt.J), GAHºtJ), 3.35.1% (i+1) =D(i)+UFSs (DI2]+UFSs (DI3]+UFSs +UFSA (DI5]+UFS s 1 () (6) +UFSs) (7) ))))) UPP (1+1,2) suPSTART (4) GO TO 28 26 WRITE OUTPUT ID ! US (1) (1) = 0 15 tonfiniſt UPP[1,2) =0.0 NOPTS=NUMADS41 UPP (1,1) 1, NOPTS = l = 40 DO DELW's DELWXX 1.0 J-2, NFIT 40 D0 ºft.*.**i.” *:::::* 40 RETURN END C TO Y) SUBROUTINE REGA (N, X0, XF.H, C888&séssassssssssssssssssssssssssssssssssssssssssssssssssssssssst THIS SUBROUTINE USES THE RUNGE KUTTA METHOD SOLVE EQ. 25 (4), F(4), R(4) Qiqi, Yiği, Yºtº COMMON/RUNGA/E jºnsſøn ſ 0. = H ( (I) = 0 8 MsABSF [XO-XF)/H]+1.000001 H2s.58 1,N D0.8 sy M sy K-1, GAMM (K) PHU (K) W0 (K)s A1 5 DO Als)(0 CALL DERIWA (A1, Y.YP) 24!} RETURH ) :: ) - =0 Y (I) sh-F (J) sq.(1)) +TEMP (!) +3.87EMP-R (J) syp (1) END DERIWA (WOL, OPA) -41 - SUBROJTINE Y, 5 Y!: J 1,N TEMP-E {J}s (YP sh L = 15, ºf 70 5 !! 15 DC 5 OC (YP (1)) 1.2.2 5PHÚſkj-5057f7-YP (1)) Ja 1.4 11), (+5.11, Als th2 CALL DERIWA (A1, Y.YP) ºf 2 If (2) (3) PKK (K) sy (1) 288.888 C888.888.888&8888.8888.888.8888.888.8888.8888.8888.8888.8888.8888.8888.88888.888 : THIS SUBROUTINE CALCULATES THE ADIABATIC sessess PRESSURE DERIVATIVETEO. 23) C88888.8888.888&8888.8888.8888.8888.8888.8888.88888.8888.8883.38888.8888.8888.88888.888 USE COM! DIMENSION Y (6 ), DPA (8 ) WA*WOL PAsPHUG (WA) Y (3) spa GAs CAMºU (WA) MPA (2) “GA GAMART sºlº U(WA)/GA DPH--PAs TERM (WA) * DEHe .5s (DPH's (WOH-WA) -PA) -GAWs):H&DPH+Y DPA (1) spas (RHO-GAMART) (?)s SUCROUTINE ADIAS2IN, X0, XF,H,\) Cºººººººººººººººººººººººººººººººººººººººººººººººº C THIS SUCROUTINE USES THE RUNGA-KUTTA METHOD (CAMART-RHO-GAW) &8888.888.8888.8888.8888.888 TO SOLVE EQ. (26) Cºººººººººººººººººººººººººººººººº.388.8888.8888.8888.8888.8888.8888.8888.8888.8888 US: US: COM2 COME5 (4), F (4), R(4) COMMON/RUNGA/E jñºsión disj. Yiği, whº Fºsſtºj}#}.5 H2s.5sh Y (7) sº Aº (M1) Y (0) skſ Y (2) sy (T) YP (1) =-1.080pHUIM1) D05 K-1.M DPH1 (M1) =SQRTF =0.0 sp?HU(M1) (-YP (1)) PH.U.M1+1) Pºk (H1) sy (1) W0 (Mi) sº? GAMM(M1) sy (2) If [Al-W8x (NA))33,35,1 35 IFIPKK (M1)-.000001) 131 ºf TURN 31 is 1,151,132 132 M1 shit! 00 5 Ja 1.4 § 70 tiš,11,15, 11), J 11 A1*A1+H2 15 CALL SERIVB (A1, 1,N D0 5 is TEMPs: (Jia s (YP Y.YP) (I) sh-F (J) sq.(1)) ######prº *Q (I) +5.87EMP-R #" ......Sººº...!!NE, (J) syP (1) I sh DERIVB W, Y, YP) *************sassasssssssssssssssssssssssssssssssssssssssssssssssssssss -42– {}; THIS SUCROUTINE CALCULATES THE ADI iLSö täLCULATESTGAMMA FOR WoWOHISECTION if Č C SSUAE DERIWATIVE (EQ 26) IV-0) COMS Y DIMENSION YP10 ) COM! ), USE USE (0 Casaaaaaaaaaaaaaaaaaaaaaaaassssssssssssssssssstaaaaaaaaaaaaaaaaaaaaaaaas WA*W R}{0s!. IF WWA 1,1,2 50'5"ri.Ni (WA-W8X (1)) N=1 +58 (I-1) (I-1) GAA=GAMMA(N)*WAs (GAMMA (N+1)*CAMMA CAMPsGAMMA (N+1)+2.8 WA86AMMA (Nº.2) (2) sº AM gº” /GAM) syſ 1}-GAMER}{08EOPFNC (2) acAM (GAM+1.-W.As (N+2) aWA) GAMP0H/CAM) sy RETURN CAMs) (7) +GAMP0H2 (WA-W (0) Y 1 6 ºw”,” Ne4+58 Y 6 3 GO TO 4,4,5 (WA) ) 4 IF 2 (WA-WS) (1) •GAMs?!!08:0PFNC (WA) SUCROUTINE ISOTHERMSIN0,7) Cºººººººººººººººººººººººººººººººººººººººº & 8888.8888.8888.88888.8888.88888.8888 THIS SUBROUTINE CALCULATES PRESSURE C AND ENERGY ALONG ISOTHERMS C86888.8888&8888.888888.8888.88888.8888.8888.8888.8888.888.8888.8888.8888.8888.8888.888 © USE COM! US: COM2 101 MRITE_OUTPUT TAPE 3,101,NO.7 formiſſiºi, 20x, ºf$öß'Number 12,52x,12HTEMPERATURE-FM.2/ 120Y. 10H------------------ ,52X,21H--------------------//) Mºſt jumpur TAPE 5,102 vol.,5x, ºpressure, 10x,&ENERGY 102 fºurt;,&Völunt, ižºisºlative ſījiàº'àº.77) 1.727.5%iºi, pā’īf.º.º.2 ti-ſphuiſ) EDIFs:THM (T) (I) •PKK (I) * +EDIF +GAMM(I) /W0 (I) agolf (!), ADTC TAPE 5,105, (1),WT (I),EMU(1), CAMM(I), I HUſ W0 'ºn stºk (1) E}{U(1) 105 FCºAI (F11.5,0X, F10.5,0x,F11.6,7X,F11.6,0x,FC.4,10x.F9.2) FUNCTION PHUG (WOLUME) E0 PH Cºassassssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss THIS FUNCTION CALCULATES FROM (54) *:::::::: *intinuuuuuuuuuuuuuuu WQL*WO-UM: 80s!.-WOLs?HOOH CALL JSCALC (88) PHUGsks KsKHOOHs38+P0H RETRI l ! US: COM6 ENC -43 *::::::::::::::inuuuuuuuuuuuunrººf": tº ſºlº...? FUNCTION CALGULATES THE SPECIFIC HEAT FROM THE Essassassessssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss DIMENSION US: DD (10) COM) IFIC!)6,7,6 7 CWOFT's CW Rºſvºn 6 CONTINUE DD (1) = 1.0 s-5.223-4 s-4.0772E-2 DD (4) e-1.70293-5 DD (5) sy.04975:-5 DD (6) s-4.650222-4 DD (7) s2.300022-5 DD (2) DD (3) XODeC1/TEMP FODs 1.0 Us)(0) D0 1 1s2.7 fön-fúDºğütl)su 1. Usue)(OD - CWOFTsz4.94.295:-58FOD/C2 RETURN END “rºutinium" THM (T) FUNCTION Cººººººººººººººººººººººººººs C 88.8888.8888.8888.8888.8888.888.888.8888.8888.8888.8888 THIS FUNCTION CALCULATES THE THERMAL ENERGY YsC1/T AL00sL00F (X) ETMMs. 00024.94295/C2s (T +C1s (-.574002+5.22E-48ALOG 1 +Xs (4.0772:-24x8 (.091453-54xa (-1.016502-54x8 (1.16256E-4-.46016&-58)()))))) FUNCTION G (WOLUME) Cººeeeeessesssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss CTHIS FUNCTION CALCULATES G (W)-DEFINED *:::::::::COM! BY EQ. (46) 8888.88888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888 WOLs.WOLUM: CAMsCAM0%GAMP0Hs (WOL) (WOL-WOH) PłłłsºuG 0, 0 E}}{s.58PHHs (WOH-W ôLjith glºw-tº tº ###" ! 'ºuï', w #(ºst WQX LIES IE QUTSIDE SIDE SUSROUTINE WOKCALC) CALL EXIT EM) –44 THE W WOLUME R RANGE OF º FUNCTION EOPFNC (WDUM) Cassaaaaaaaaaaaaaaassassassssssssstaaaaaaaaaaaaaaaaaaaaaaaaaaaaassessess CTHIS FUNCTION CALCULATES THE THERMAL ENERGY DERIVATIVE (SECTION IV-D) Cassassassssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss USE COM! USE COM5 Nº. 1 IF (NA-1)5,4,5 5 DO 5 Isl, NA-1 IF (WDUM-W8X(1)).2.2.5 2 N=1 +38 (I-1) GO TO 4 3 Ne4+58 (I-1) 4 EV-FNC's CX [N]+WDUMs (CX (N+1)+CX (N+2) a WDUM) #!" º ãº"minimummim” FUNCTION TERM (WOLUME) # C8888.88.888&8888.8888.888.8888.8888.8888.8888.8888.8888.888.888.8888.8888.8888.8888.8888 CALCULATES THE HUGONIOT PRESSURE DERIWATIVE/PRESSURE COM! USE COM3 WAs WOLUM: DUS's A (2) 8 is 1.-WA/WOH UPs)(88). ZXsUP IF (NFIT-2)3,4,5 5 50'ſ i-2, 8X8s I ºff: DUS*DUS+8)(387Xs 1 ZX-ZXs UP 4 TERMs RETURN A (1 +1) (DUS+1./8H) / (WOHs (1.-Bºis DUS)] END º: FUNCTION ZERO (B.DDD) Cºssaaaaassasssssssssssssssssssssssssssssssssssssssssssssssssssssssssss CTHIS CALCULATES F (US)-DEFINED BY EQ. 31 USE COMS 7:R0s A (1) 9Cs5800D XDUMsgC DC : ! 1s2,NFIT ŽRQ-ZER}+\{1} (DUMs), DUM28C axDUM ZEROszERO-ODD RETURN ENC fiſh C7: Oh GAMFU (WDUM) **************ssessesssssssssssssssssssssssssssssssssssssssssssssssssss –45 – : THIS FUNCT! ON CALCULATES GAMMA(V) FROM THE LEAST SQUARES FIT TO GAMMA Catecasaasaaaaaaaaassessessessesses&8888.8888.88888.8888.8888.8888.8888.8888.888 XDUM*WDUM CQ 1 i = 2, NFITORDER GAMFU's SAMFU+C ºxDUM (I) ! XDUMexDUMeWDUM RETURN END FUNCTION C8888&0&0&0&0&0 CAMPFU (WOUM) &0&0&0&0&0&0&08&888888.8888.88888.8888.8888.8888.8888.888.8888.888 CTHIS FUNCTION DIFFERENTIATES THE LEAST SQUARES FIT TO CAMMA C88888&688&6888.88888.888.8888.8888.8888.888.8888.8888.8888.888.8888.8888.8888.8888.888USE COM! CAMPFUsC (2) XOUMsWDUM # } jºritain CAMPFUsGAMPFU+GIs C (I) syDUM ! XDUMs)(DUMs WDUM Rºſſuſºn END -46