32
This report has been prapared foHnformation record- pmT30s?^i3Tnt''iS"'ftof fefererniad ifl ~ a ny pubiicatioiL. NATIONAL BUREAU OF STANDARDS REPORT 964 6 A REVIEW OF IKE STATUS OF FIRE PREFORMANCE PREDICTIVE METHODS Stanley P Rodak; Physicist, Heat U.S. DEPaRTMErr OF COMMERCE NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS REPORTNATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 4215122 NBSREPORT November21,1967 9646 AREVIEWOFTHESTATUSOF FIREPfiEFQRllANe-EPREDICTIVEMETHODSby StanleyP.Rodak;Physicist

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  • This report has been prapared

    foHnformation record-

    pmT30s?^i3Tnt''iS"'ftof fefererniad

    ifl~any ' pubiicatioiL.

    NATIONAL BUREAU OF STANDARDS REPORT

    964 6

    A REVIEW OF IKE STATUS OF

    FIRE PREFORMANCE PREDICTIVE METHODS

    Stanley P Rodak; Physicist, Heat

    U.S. DEPaRTMErr OF COMMERCE

    NATIONAL BUREAU OF STANDARDS

  • THE NATIONAL BUREAU OF STANDARDS

    The National Bureau of Standards^ provides measurement and technical information servicesessential to the efficiency and effectiveness of the work of the Nation’s scientists and engineers. TheBureau serves also as a focal point in the Federal Government for assuring maximum application ofthe physical and engineering sciences to the advancement of technology in industry and commerce. Toaccomplish this mission, the Bureau is organized into three institutes covering broad program areas ofresearch and services:

    THE INSTITUTE FOR BASIC STANDARDS . . . provides the central basis witbin tbe UnitedStates for a complete and consistent system of physical measurements, coordinates that system with themeasurement systems of other nations, and furnishes essential services leading to accurate and uniformphysical measurements throughout the Nation’s scientific community, industry, and commerce. ThisInstitute comprises a series of divisions, each serving a classical subject matter area:

    —Applied Mathematics—Electricity—Metrology—Mechanics—Heat—Atomic Physics—PhysicalChemistry—Radiation Physics— Laboratory Astrophysics^—Radio Standards Laboratory,^ whichincludes Radio Standards Physics and Radio Standards Engineering—Office of Standard Refer-ence Data.

    THE INSTITUTE FOR MATERIALS RESEARCH . . . conducts materials research and providesassociated materials services including mainly reference materials and data on the properties of ma-terials. Beyond its direct interest to the Nation’s scientists and engineers, this Institute yields serviceswhich are essential to the advancement of technology in industry and commerce. This Institute is or-ganized primarily by technical fields:—Analytical Chemistry—Metallurgy—Reactor Radiations—Polymers—Inorganic Materials—Cry-ogenics-—Office of Standard Reference Materials.

    THE INSTITUTE FOR APPLIED TECHNOLOGY . . . provides technical services to promote theuse of available technology and to facilitate technological innovation in industry and government. Theprincipal elements of this Institute are:

    —Building Research—Electronic Instrumentation—Technical Analysis—Center for Computer Sci-ences and Technology—Textile and Apparel Technology Center—Office of Weights and Measures—Office of Engineering Standards Services—Office of Invention and Innovation—Office of VehicleSystems Research—Clearinghouse for Federal Scientific and Technical Information^—MaterialsEvaluation Laboratory—NBS/GSA Testing Laboratory.

    Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington. D. C.,20234.

    - Located at Boulder, Colorado, 80302.

    3 Located at 5285 Port Royal Road, Springfield, Virginia 22151.

  • NATIONAL BUREAU OF STANDARDS REPORT

    NBS PROJECT

    421 5122

    NBS REPORT

    November 21, 1967 9646

    A REVIEW OF THE STATUS OF

    FIRE PfiEFQRllANe-E PREDICTIVE METHODS

    by

    Stanley P. Rodak; Physicist, Heat

    IMPORTANT NOTICE

    NATIONAL BUREAU OF STA^for use within the Government. B<

    and review. For this reason, the p

    whole or in part, is not authorize

    Bureau of Standards, Washington,

    the Report has been specifically pr

    Approved for public release by the

    director of the National Institute of

    Standards and Technology (NIST)

    on October 9, 2015

    accounting documents intend^

    bjected to additional evaluation

    sting of this Report, either in

    )ffice of the Director, National

    he Government agency for which

    ies for its own use.

    U.S. DEPARTMENT OF COMMERCE

    NATIONAL BUREAU OF STANDARDS

  • A REVIEW OF THE STATUS OF

    FIRE PREFORMANCE PREDICTIVE METHODS

    by

    Stanley P. Rodak; Physicist, Heat

    The problems of fire-resistance have attracted the attention ofvarious workers. This project is concerned with the theoretical aspectsof fire resistance. From a knowledge of the thermal propertities of abuilding material shaped into a particular geometry, we wish to be ableto predict the thermal fire rating**of scaled constructions of the samematerial. Ideally, we would like the calculations techniques to includevariable thermal properties and endothermic and exothermic processes.

    In order to circumvent certain numerical difficulties in using vari-able thermal properties, it was felt that sufficient information wouldbe had if the calculations are made in a manner that will allow the infor-mation to be displayed in the following fashion. Giedt (1) has suggested

    01'J

    \U

    r

    )

    Alii

    ino

    '4- =

    LD

    u4.’>

    » F F \ . r 'if

    a mean value for thermal conductivity when it varies with temperature.

    Endothermic and Exothermic processes are to be disregarded untilthe numerical techniques are sufficiently developed.

    The mathematical equations to be used are as follows:

    parabolic equation of heat flow* -Q||) T = 0 (1)

    boundary condition ,St

    ^Sn= -hT (2)

    (n is the normal surface component)

    * Nomenclature is at end of text.** The thermal fire rating will be taken as the time before the tem-

    perature in a region in the specimen under consideration reachesa value predetermined to be unsafe for fire-resistance purposes.

  • 2

    The mathematical model which we will work with is a homogeneoussolid bounded by two parallel planes. The problem will be two-dimensional.

    Over one surface, the exposed surface, we define a temperature riseby

    T(o,y,t) = f^ (t) (3)

    where f^ (t) is the ASTM time-temperature curveo

    On the other surface, the unexposed surface, the heat flux is thatfor a horizontal surface (2,3)

    h = h + h (4)n c

    h = (.174e/(T -T )) • ((T /lOO)^ - (T /lOO)^) (5)XI so s o

    1/3h = .275 T

    '

    (6)c s

    For a high, vertical surface, h c would still be proportional to Tbut the coefficient changed (3,4). The mechanism of transpirationcooling was not considered (5).

    The numerical technique we choose should be accurate enough to calcu-late temperatures for our test example (6, pg 126).

    TEST CASE:

    The region o

  • 3

    The first studied was the forward-difference explicit algorithm (8);

    pcAx*" (T. .^,3

    - .) = kAt (T^ + - 4T^ .)i^j 1-1J 1+1; J i;J-l ^^J+^ (8)

    Here, AY .. AX

    For a slab NAX thick and MAY tall,

    Tj = f (kAc) J = 1, 2, ..., m (9)j

    t lxis the temperature on the exposed surface at the k interval of time*.

    The requirement that At /Ax“ be chosen such that

    kAt ^ IpcAx^ 4 (10)

    in order to have stability, that is, the finite-difference algorithm isimmune to the accumulation of round-off errors (7), caused a survey ofother methods (metal rods were to be imbedded in the test example interior).

    The backward difference implicit algorithm (8) is defined by

    pcAx^ (T^'^JJ J

    .) = kAt (Tk+1

    i-1 ,3+ Tk+1

    i+1, j+ T

    k+1i,j-l i,j+l

    k+1- 4t7!) (11)

    This method is stable for all values of At/Ax'". Iterative and matrixreduction techniques are available to solve the resulting unknown equa-tions (9,10,11,12). For a solid of N interior points in each direction,

    equations result for the two-dimensional problem.

    The backv/ard-dif ference implicit algorithm was applied to a parti-cular problem: finite thick slab exposed on one surface to temperature

    *Ic Ic

    To meet the boundary condition (2), we replace (T . ^ . - T, , ) by

    Xj -k i

    T^ j) by 1/2 in equation 8. The unexposed surface is at i = + 1;

    i 1 is the exposed surface, we make a similar replacements in (11)and (12) to meet boundary conditions of (2).

  • 4

    unexposed surface has constant "cooling” coefficient ho, metalrods were placed 1" from exposed face. The equations were solved foreach successive time increment (At = 1 min.. Ax = Ay = .2") by GaussSiedel Iteration*. Figs. 1 & 2 are graphs of the computer solvedproblem

    .

    Fig, 2 contains two different profiles of the distance from exposedslab surface versus temperature. One profile is taken along a line (AA

    '

    )

    normal to the slab surface and passing through one of the steel rods.The other is along a line (BB*), parallel to AA', but about 3" from thesteel rod (the steel rods imbedded in the concrete slab were on 6" centers)Temperatures at 30 min., 50 min., and 210 min., intervals along the lineAA ' from 0 to .8" have negative curvature (13). This is not the correctslope ( 14)

    .

    The Crank-Nicolson algorithm (15) is reported to be unconditionallystable (16). It is obviously based on sounder physical arguments thanthe two previous methods. The algorithm is

    . 2 /„k+lpcAx (T , ,

    J

    t; ) = (kAt/2)J

    (

    k+1 k+1 k+1 k+1^'i-1, j

    ^ \+l, j ^ j-1 ^ j+1^

    T^ + T*f . + T^ , T + T^f - 4T^"^^ - 4 t'!^ .)1-1, j 1+1, j 1, j-1 i,j+l i,j i.j( 12 )

    The same test case (metal rods imbedded in a slab as used for the backwarddifference implicit algorithm was used to test this algorithm. Graphs ofthe computer solved problem are displayed in Fig. 3 & 4 . The distance-temperature curves in Fig. 4 have the proper curvature.

    As a precursory check of the Crank-Nicolson algorithm, it was usedto calculate surface temperatures of the test example (homogeneous solidbounded by parallel planes). The resulting surface temperature calcula-tions, as well as those predicted b}’' theory, are displayed in Fig. 5.The surface temperatures differ by 307o at 100 minutes.

    In order to make the Crank-Nicolson difference approximation moreclosely approximate the differencials, equations were derived that allowthe Ax spacing to vary in an unequal manner through the slab; the spacingAy is equal. This would allow a fine grid near the unexposed surface,where the temperatures vary more slowly with time. Note 2 discusses howa grid spacing was chosen for .17, "accuracy."

    Another problem was encountered, for the above example used as acheck in the variable Ax computer program, interior temperatures werehigher than surface temperatures , k i nr. i i • i o

    (T = 100 F, k = I, . .. ; J = I, 2, .

    J

    * The equation showed diagnol-dominance, hence the choice of Gauss-Siedel Iteration techniques. See Note 1 for termination procedures.

  • 5

    1 2The aleorithm values had definitely conv®rged(T, . < T„ . ~ 139.9 F

    1,J 2, j

    after 60 iterations). The conditions under which this algorithm willconverge to values greater than the exposure surface temperatures (andhence apparently violate the first law of thermodynamics) are discussed

    2 2in Note 3. This discussion is for T

    ^ iwhere the

    2tion quess” of T. . = 0 i = 2, Xoj j = 1, 2 ...^

    j

    2 2found that similar unstable conditions (i.e. T. . < T_ )

    J • J

    2 2if the initial iteration quess was T. , = T. .or even if

    "initial itera-

    Yq. We also

    occurred even

    T^ .j were

    assigned their exact values as predicted from theory (the convergence

    of T2

    2.jwas still 140 °F).

    The present status of the project is to circumvent this problem. Itmay be that the constraints involved in keeping the Crank -Nicolson algorithmfrom violating the first law^ a At several orders of magnitude smaller willhave to be used.

  • *> .! frMr* ' ^... , .V (• /r< • t Ai^T

    M-J'A

    ofr I -

    I ,' ,

    ^

    f, t/|r»f:--.'

    ' I- < »• A'-J . *^;^^.:' VV. -.

    i - • •',*- ' U-- :-. ^ *’ '. ,'Z^-' '

    ,f.4 v.'» 1 t‘ tj .. . ' ,;.

    (t ,1 ,y^{i '-if ' '* 41% *J|J»

    • -,

    #«5'r **» i' «P

    " ft‘ '

    '_ I ,v,t:.

    P t - ^ :[» 5s.->i. ;_—. . -y.h- K -f'^’. 'm

    ): ••:. Xi

    ec '«.•'•

    • •“• '^3

  • h

    hc

    hn

    k

    ]

    T

    To

    Ts

    .

    0,1

    t

    y

    d

    e

    P

    Ax, Ay

    At

    NOMENCLATURE*

    surface film coefficient

    convection heat-transfer film coefficient

    radiation heat transfer film coefficient

    conductivity

    thickness

    temperature

    unexposed surface temperature

    exposed surface temperature

    finite-difference notation for temperature at time

    increment, x coordinate of iAx = Ax and y coordinate

    jAy .. jAx.

    t ime

    space coordinates

    dif fersivity

    surface emissivity

    density

    finite division of x or y coordinate

    finite division of time

    *Units are English

  • REFERENCES

    1. "Principles of Engineering Heat Transfer^" W. Giedt, van Nostrand Co.,Inc., 1957, p 2.

    2. "The Transmission of Heat by Radiation and Convection," E. Griffiths

    & A„ H. Davis, Food Investigation Special Report No. 9, Dept, ofScientific and Industrial Research, London, England, 1931.

    3. "Heat Transfer," M. Jakob, John Wiley & Sons, Inc., Vol I, 1949,

    PP 526-527, 532„

    4. "Elements of Heat Transfer and Insulation, " M. Jakob and G. Hawkins,John Wiley and Sons, Inc., 1950, p 99.

    5. "Heat Transfer," Mo Jakob, John Wiley & Sons, Inc. 1957, Vol II, p 304.

    6. "Conduction of Heat in Solids," Carslaw & Jaeger, 2nd Edition, Oxfordat the Clarendon Press, 1959.

    7. "Numerical Methods in Fortran," J. McCormick & M. Salvador!, Prentice-Hall, Inc., 1964, PP 21, 79, 208-209.

    8. "The Numerical Solution of Parabolic and Elliptic Differential Equations,"D. Peaceman and H. Rachford, Jr., Industrial and Applied Mathematics,Vol 3, 1955. PP 28-41.

    9. "Introduction to Numerical Analysis," F. Hildebrand. McGraw-Hill BookCo., Inc., 1956.

    10. "Numerical Methods for Scientists and Engineers," R. W. Hamming, McGraw-Hill Book Co., Inc., 1962.

    11. "Elements of Numerical Analysis," P. Henrice, John-Wiley & Sons, Inc.,1964.

    12. "Iterative Methods for the Solution of Equations," J. Traul, Prentics-Hall, Inc., 1964.

    13. "Advanced Calculus," R. Buck, McGraw-Hill Book Co., 1956. p 255.

    14. "Tests of the Fire Resistance and Thermal Properties of Solid ConcreteSlabs and Their Significance, " C. Menzel, ASTM, Vol 43, 1043, pp 1136-1145.

    15. "A Practical Method for Numerical Evaluation of Solutions of PartialDifferential Equations of Heat-Conductive Type," J. Crank & P. Nicolson,Proc. Comb. Phil. Soc., 43, 1947, pp 50-67.

  • References (continued)

    16. "A Stable, Explicit Numerical Solution of the Conduction Equation forMulti-Dimensional Non-Homogeneous Media, " Allada & Quon, AIChEPreprint 27, pp 1-16.

    17. "A Treatis on Theoretical Fire Endurance Rating," T. Z. Harmathy,ASTM Special Technical Publication No. 301, 1961, pp . 10-35.

    18. "Thermal Radiation Properties Survey." Gubareff, Honeywell ResearchCenter, Minn-Honeywell Regulator Co., Minn., Minnesota, 1966, p 197.

    19. "An Error Analysis of Numerical Solutions of the Transient Heat Con-Duction Equation," D, B. Mitchell, Thesis, Air Force Institute ofTechnology Air University, 1969, pp 2-3.

    20. "The Effect of Latent Heat on Numerical Solutions of the Heat FlowEquation," P. Price & M, Slack, Brit. J. Appl. Phy., Vol 5, 1954,pp. 285-287.

    21. "Heat Conduction in a Melting Solid," H. Landau, Quarterly of AppliedMathematics, Vol. 8, 1950, pp 81-151.

    22. Private conversation with Dr. Robert J. Arms at the NBSo

  • Note 1

    A DISCUSSION OF WHEN TO TERMINATE CALCULATIONS

    k+1OF T . BY GAUSS-SIEDEL ITERATION*

    J

    Consider the equations

    (4 + ^7^) . + T^^J . + 1 + T^"*"! . + .kAt 1,.] 1-1,1 i+l,J i,J+l J-1 i,j ( 1 )

    + Ax ^ + 2) T^"-^ = ^ (T^-"! . + .kAt k i,j 1-1, J 2 i,J-l i,J+l i,J( 2 )

    where a°i,j = T^^ . + t; . + t. + T‘f . , - 4T,' 1-1, J 1+1, J

    ^ + T^ - 4T^iJ+1 iJ-1 i,j

    a" i, j = TV . . + ^ (T^^ . , + T;' ) + (' 1-1, J 2 i,j-l 1, J+1

    pc .,h

    -Ax f -2)T? .kAt k 1, j

    Equation 1 is the Crank-Nicolson (CN) algorithm for finding interiortemperatures of a solid bounded by two infinite parallel planes. Equa-

    tion 2 is the CN algorithm for the unexposed surface temperatures. (The

    solid is NAx thick and MAy tall (Ax = Ay)). Since the equations arediagonally dominate, Gauss-Siedel Iteration techniques are used to solvethe N'M unknown „k+l .. m.i , too • , •T^

    ^

    (i = 2,3, ..., N+1; j = 1,2, 3 M; i = 1 is

    the unexposed surface).

    k k+1All T. , and T. . are known.

    i,J

    k+1As an initial start to the iteration, all T. , are set equal to

    ' 1. J

    2 2zero. Equation 1 is used to calculate T„ ^ . This new value of T„

    z, 1 2,12

    is used now in the calculation of T„ The iteration process of

    k+1replacing the old values of T^ . by the newly calculated values of

    k+1T. . continues methodically by working through the j = 1,2, ... M for

    J

    * The method of determining the termination of the calculationsis mainly from (21) . See also (9)

    .

  • Note 1

    - 2 -

    k+1each i index until a new set of T

    , ,have been iterated.

    J

    Let

    n k+l lc+1AT

    , , = (T (nth iteration) - T. . (n- 1 iteration)

    (In the following discussion^ it is convenient to consider only valuesof n > 3 . )

    For our diagonally dominate matrix. AT? . should become small

    after a finite number of iterations. That is^ we assume

    lim at: . = 0J

    n-e eo

    i = 2,3, . . .,N

    j = 1,2,...,M

    In practice, this process must be terminated after a finitenumber of iterations (Iq)* This is called truncating. Let S. .

    J

    be the '"exact" temperature value at [(i+l)Ax, JAy]. The radius of

    k+1convergence is given by (T . . in this formula assumes the value of the

    io iteratioii)

    :

    r.k+1T. . - S. .

    J iJ< 1>

    1-LAT^ .

    ^,2

    where* L = supn=3

    supi =

    j = 1, 2. . o,Mi = 2,3. . .,N

    jat’^'^^:

    1

    —I

    AT^' J

    That is, for the (i+1) iteration, we find the maximum value \t of the

    -T+1 i

    ratioATu.

    AtJ . 1J

    A._ = sup

    i = 2,3,

    j - 1.2, M

    AT+1

    AT. .

    ATT .J

    reference 13, pg 9

  • Note 1

    - 3 -

    From the set of (3 < T <

  • Note 2

    GRID SIZE DETERMINATION

    The following houristic method of arriving at an equation todetermine the proper grid size to use in one-dimensional finite dif-ference equations was described by Dr. Arms (21). We can extend theresults to two-dimensional problems. Let f. ((i+1) Ax) be the itera-tived temperature value for grid size Ax. Call f((i+l) Ax) the "exact"value. Then^ in an approximate manner^

    £Ax ((i+l)^x) = f((i+l) Ax) + Ax^.A

    A can be thought of as an operator. (The series expansion was terminatedon the first term). Taking a grid size of Ax/2

    'Ax/

    2

    = f +Ax^

    orf . - f Ax/2Ax

    A + 2nd order terms.

    By doing the same problem for two different grid sizes^ A can bedetermined (we neglect the 2^^^ order terms), and hence the propervariable grid spacing for the desired degree of accuracy.

  • - s*.-

    .

  • Note 3

    A DISCUSSION OF THE CONSTRAINTS

    IMPOSED ON CRANK-NICOLSON FINITE -DIFFERENCE

    ALGORITHM BY GAUSS -SIEDEL ITERATION

    Consider a solid (i thick) bounded by two parallel planes. Onone surface^ there is a sudden temperature rise Ts for t>0. The ini-tial temperature of the solid is zero. Using Crank-Nicolson differenceequations^ we wish to find the interior temperatures at time At. Gauss-Siedel Iteration will be used to solve for T^ • . As a first quess setall T? . equal to zero. The equations for a irariable Ax and Ay are:

    J

    i=l

    exposedsurface

    for t>o)

    inside solid:

    (9sAll. + ^ + AiL + 1)^kAt Ax^ (Axj^+Ax^) Ax2(Ax^+Ax 2> i, j

    ^Ax^(Ax^ + AX2)

    k+1 Av^ k+1 1i-l,j Ax2(Axj^ + Ax^) i+1, j 2 J-1

    +

    Tk+1i, j+1

    ) ( 1 )

    _Ay^ k Ay^ k+1 1 .k+1

    ^i,j“ Axj^CAy^ + Ax2> Ax2(Ax^ + Ax2> i+1, j 2 ^ i, j-1

    )4f£gAyl ^ k

    i^ j+2^ \kAt Axj^(Ax^ + Ax^) Ax2(Ax^ + Ax^) / ^i

  • Note 3

    - 2 -

    at surface 'k = i

    i+l,j

    Axj k^

    kA: ^ ^i,J

    /AZ N2 ™k+l'^Ax/

    , 1 /„k+l , „k+l . ,i-lj 2 ^i,j+P * \ j

    ^kAt- (Ax,

    Ay hAx k

    ' 1) T

    let Ax-i = Ax, Ay i = 2^ • *1 1

    m.

    Equation (1) then becomes (noting for our case = 0);

    i, j^

    "i+1^ ^

    ^^i,j-l i,j+r

    Where 0 = (2pcAy'‘

    k At

    -1+ 2(gr +2) ^ < 1

    Let us introduce some symmetry into the boundry condition:

    _k k^i, 1 - ^i,3

    Then the first iteration is:

  • Note 3

    - 3 -

    T? . = gu)T (1 + P) =Zm

    y^ 8 ^y J-

    T24

    = ^“^8 ^

    2 Q ^ /I I Q . q 2 . , Qin-2. Q „T» = P(A)T (1 + P + P + .00 + P ) = PodT (t

    Q )

    2, m s s i-p

    2 2 2t:^ „ = pVt3, 2 8

    „ = pVt (1 + 2 P)3, 3 s

    „ = pVt (1 + 2p + 3P^)J j s

    T? = P^OD^ n + 2P + 3P^ +3,m s

    + (m-1)l-p”~^-(m-l)p"^

    - (1 - P)^

    etc

    ,

    second iteration:

    - = P [u)T + P^U)\ + 2Puyr (1 + p)12, 2 s s s

    = PtuT (+ 2P + 2P^ + P^oo^) < 1s

    For the various heat-conclusion problems under study^ this typeof analysis was used to understand the constraints imposed on the iterationtechniques. It was found that when the constraints were violated, therewas convergence of the (all equations had diagonal dominance), but

    J

    k+1 kthe

    jwere larger than for several Ax^ from the exposed surface ^).

  • 3^' &

    m

    lU',

    * .•• 4 k/' ^ J. -

    . ‘ti •. ;i I .W.-rV' , , Cc>5ij»

  • FKsi.1 - BACKWARD OiFFEREMtfe IJrvPUUT CALCOLATIOM^

    MWS: ceiMCWt : ICS ps ^ C:.l3: K.i 2fc.t

    ,jS»a 4^0^ Ci .113

    AR.» At«. 1 nui*^«s ,9

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    uor^: CoMtPtTf. KSJU.Z5

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