Моделирование процесса измерения уровня жидкости акустическим методом, учитывающим факторы среды

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    2007, 10

    . .

    , , e-mail: an_gromov@rambler.ru

    , 21.01.2007, 04.05.2007

    , - . , , .

    . [112], .

    .

    , .. . ,

    . , , . , , . ,

    - . . , . , , , , . . ,

    mailto:an_gromov@rambler.ru

  • 2007, 10 _________________________________________________________________________________________

    2 (16)

    , , . . [12], .

    .

    . Saab Saab Tank Control, , . , . 10 . . . [47]. ,

    [1011]. , , , ,

    c

    12

    L c t= ,

    , , .

    L c Const= t

    , c Const= , . . , , . [3] , -, - , 5%. 1000 10.7 . , 80% . 4050

    c Const=

    . [1] ,

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    3 (16)

    . , L . , , , , .

    [ L ,0 ]

    , ( )

    2P k P = , (1)

    ( )zck

    = , .

    (1)

    ( ) ( )00 0 0P P + = , ( ) ( ) 0LP L P L + = (2) 0 , L .

    OZ( ) 0c z Const c= = , (1), (2) -

    ( 22 0k c = = ) ]. [14], , -,

    [ L ,0

    { }n ,

    =

    lim

    nn

    nL

    . (3)

    { }n , 0 0L = = , (3)

    0

    n

    n cL

    = , (4)

    ; 0c n .

    (4) [2]. { }n , ( ) 0c z Const c= = , L

    (3). ( ) Constzc (3) L , , , [1], L , . ( )c z ,

    , { }n (1), (2). (1)

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    4 (16)

    ( ) ( )00 0 0P P + = , ( ) ( ) 0LP L P L + = , (5) 0 0 0 .

    { }n (1), (5). [1] ,

    ( )c zL -

    { }n , { }n , .

    1,2,...n =

    { }n , { }n , .

    1,2,...n =

    , [1], -, . , . [17]. [13], -

    ( ) { }n , 1,2,...n = . , . .

    1.

    ( )c z c e Lz

    = 0

    , , [15]. (1), (2) . .

    { n} { }n , L . (1), (2)

    -.

    ( )=z

    tcdt

    0 , ( ) ( )

    ( )zczPY 1= , (6)

    ; ( )Y . _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    5 (16)

    ( ) YYqY ~~ =+ , [ ]0, L , (7)

    ( ) ( )

    2

    0

    0

    2

    cqL c

    =

    0

    ; . ~ = 2

    (2)

    ( ) ( ) =Y h Y0 01~

    , ( ) ( ) 0L LY HY + = , (8)

    1 0 00.5h c

    L = +

    , 0

    0.5LH c eL

    = +

    .

    L L , (7), (8),

    , (3)

    lim

    Ln

    n

    n

    =

    , (9)

    n (7), (8).

    (9), (6),

    ( )0 0

    1 1L

    tL

    L e dt pc

    = = , (10)

    0

    Lpc

    = , = e .

    [ ]0,x , (7), (8)

    ( ) [ ] + = Y q x Y Y x , 0, , (11)

    ( ) ( )2

    21 = 4 1

    L

    qq x x

    =

    ; 2

    L

    =

    .

    -. (8)

    ( )q x

    ( ) ( ) =Y h Y0 0 01 ( ) ( ) +, =Y HY 0 , (12)

    ( )

    h hp

    1 1

    1=

    ~ ,

    ( )H H

    p=

    ~ 1 ; p

    (10).

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    6 (16)

    0 0L = = , (8), (10), (12)

    h112

    =

    , H =1

    2

    . (13)

    , (1)

    01 222

    =++ PPddP

    dd

    , (14)

    z

    Lpe

    = . (14)

    ( ) ( ) ( )0 0P A J B Y = + , (15)

    . ( ) (J Y0 0 , )

    (11) (14) (6). (15), (11)

    ( )Y x AJ x BY x x=

    +

    0 0

    12

    1 1 1

    , (16)

    L

    = .

    ( ),xY (16),

    ( ) ( ) 10, 1 , 0,Y Y = = h

    )

    , (17)

    (13). 1h ( ,xY (12).

    (11), (12) ( )R

    ( ) ( ) ( ), ,R Y HY = + , (18) H , (13). ( )nxY , , .

    (16), (17), ,

    ...2,1,0=n

    BA ,

    ( ) ( ) ( )( )( ) ( )( )

    1 0

    1 0

    ,2

    ,

    Y x b Y b J b x

    J b Y b x b x

    =

    (19)

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    7 (16)

    ( ) ( )bYbJ 11 , ; a =

    1

    , b =1

    .

    (18),(19)

    ( ) ( ) ( )[ ( ) ( )]aYbJaJbYabR 11112 = . (20) (20)

    { } , 0n n

    =

    = 0 , ( ) ( ) ( ) ( )Y b J a J b Y a1 1 1 1 0 = , (21) , (19). ba, (19) = 0

    ( ) 21

    0,

    =

    bxbxY , (22)

    = =0 0 (11),(12). [13],

    ( )20

    ,n nY x dx

    = , (23)

    ( ), nY x (11),(12). (19), (21), (23),

    :

    ( ) ( ) ( ) ( )( ) n n n n n nb a J a Y b J b Y a=

    2

    12

    22

    0 1 1 0

    2n, = 1 2 3, , .... (24)

    0 ,

    (22) (23)

    ( )100

    12

    b b x dx += = . (25)

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    8 (16)

    2.

    085.0= . 10=L ,

    50.

    / 4.29

    { }4000n n

    =

    ,

    , (21), (24), (25). 1.

    { }4000n n =

    n n .

    1.

    n

    n

    n

    n

    n

    n

    0 0 3,01359204925 396 6,93642535e-07 1,57079633207

    1 0,27434330e-03 1,57162227253 397 6,91895073e-07 1,57079633205

    2 0,13729871e-03 1,57100315690 398 6,90156582e-07 1,57079633202

    3 9,15482070e-05 1,57088827968 399 6,88426474e-07 1,57079633199

    4 6,86652869e-05 1,57084805589 400 6,86705333e-07 1,57079633196

    nF

    Lnc

    fn 20= ,

    (4). n , 1,

    (11) . , 2

    nn

    L

    F

    = . nnn fF = .

    1n = 7.0111 = fF . 1 .

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    9 (16)

    0 100 200 300 4000

    100

    200

    300

    Hz n

    n

    . 1.

    , ,

    .

    { }4000n n

    =

    , { }4000n n = ,

    . L ( )xq , .. -. , [1],

    ( )

    q~4

    324

    2=

    , (26)

    ( )2

    L L

    q q

    =

    .

    ( )Lz L = . LL z= . ( )xq , [13],

    ( ) ( ) ( ) ( ) (K x y F x y K x t F t y dt y xx

    , , , ,+ + = 0

    0 0 ) , (27)

    ( ),F x y ,

    ( ) ( ) ( )F x y x y D u D vnn

    nn

    , cos cos= + =

    +

    =

    1 10

    0 01 1

    + , (28)

    ( )D u u nunn

    n =

    12

    1

    cos cos , ( )D v v nvnn

    n+ =

    12

    1

    cos cos ,

    u x y= , 0 u , v x y= + , 0 2 v .

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    10 (16)

    (27) (

    x( )K x y, ,

    ) .

    x

    ( ) xq

    ( ) ( )q x ddx

    K x x = 2 , , (29)

    (27). ( yxK , ) (28), (27), .

    , ( )yxF , . , . [13],

    :

    ( )q x

    313 ...n

    kknn n

    = + + + , ...~~

    2 44

    22 +++=

    nd

    nd

    n , (30)

    n , n (11), (12).

    , , , (30) , 1=n , (27).

    2. 1 3 5 7, , ,k k k k 2 4 6, , ,d d d d8

    2.

    n

    n2/1

    1k 2,74682402284552 e-04 2d 1.67743442374836e-04

    3k 3,40132335254744e-07 4d 4.61927618275191e-007

    5k 9,44690448318419e-10 6d 2.1410365069608e-009

    7k 8,58707037946961e-11 8d 1.65673220678243e-011 [13] , . 1k Tk

    ( )10

    1 12T

    k h H q t d

    = + +

    t , (31)

    , 1h H (13).

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    11 (16)

    (31) . 1k

    71 100% 1.8 10 %TT

    k kk

    .

    . ,

    , . 1k

    ( )uDn (. 28). ( )vDn+ . vu

    3 513 5 ...n

    k kkn n n

    = + + , (32)

    (30), , .

    nn

    uncos , (28),

    cos cos cos sin sinn n nu u nu u= nu . (33)

    (32), cos sin nu , nu n

    ( )cos n

    ii

    iu

    c un

    = +=

    1 2 21

    , ( )

    sin ni

    ii

    us un

    = =

    2 12 11

    , (34)

    , , . ic2 12 is

    (34)

    ( ) 22

    12 2

    uk

    uc = , ( ) 4412314 241 ukukkuc += ,

    ( ) 6614331223516 7201

    61

    21 ukukkukkkuc +

    += ,

    , ( ) ukus 11 = ( ) 33133 61 ukukus = , ( ) 551332155 120

    121 ukukkukus += ,

    ( ) 7715341323152177 50401

    241

    21

    21 ukukkukkkkukus +

    += ,

    (35)

    (32). 7531 ,,, kkkk

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    12 (16)

    (33), (34) n2/1 ,

    ( ) ( ) ( )

    ==

    =

    =

    +

    +

    Nn

    kkn

    kkn

    n rnnuuM

    nnuuMuD cos sin=

    122

    1k 11212

    1, (36)

    Nr1ni

    ; ( )uM i

    u . : ( )uM i

    ( )

    +++= 1223221212 ...

    1 sdsdsuM kkkk , ( ) kkkk dcdcuM 222222 ...

    1+++=

    , (37)

    ,..., 42 dd n2/1 .

    [16],

    =+

    112

    sin

    nknnu ,

    =12

    cos

    nknnu

    . (36)

    ( ) ( ) ( ) ( ) =

    +

    =

    +

    Nkkk

    k

    nn ruM

    uBk

    uD 2!2

    21-= 1k

    12

    1 , (38)

    . ( )uBk (38), 7=k ,

    ( ) =

    =

    + Nin

    n rueuD14

    0ii

    1= , (39)

    (37). 14,...,0, =iei . 3

    (

    ).

    ie

    10 14,...,e e

    3. ie

    0e -0.000378324724177147 5e 5.29329003909441e-013

    1e 0.00017204179135448 6e 1.42645001113616e-012

    2e 2.60571505254777e-006 7e -1.54491479236586e-013

    3e 3.4514805373702e-008 8e 6.98866109936978e-015

    4e 4.58803525031662e-010 9e -2.26764572914655e-019

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    13 (16)

    (39) ( )vDn+ yx, ,

    ( ) ( ) ( ) ryxyxF ii

    i += =

    8

    1, , (40)

    r , (30). ( )xi , ( )xi

    ( ) =

    + =

    14

    2ki

    2i

    21 2

    kikik xeCx , ( ) kk yx 21 =+ , 7,...,1,0=k , (41)

    . knC

    (40) r , , (27) 8 , .

    ( ) ( ) (=

    =8

    1,

    iii yxfyxK ) , (42)

    ,

    .

    ( )xfi

    , ( )xfi , :

    ( )( ) ( )=

    =+8

    1ikikiki xxfx ( ) , (43)

    ki ; . ( ) ( ) ( ) ki k ix

    x y y= 0

    dy

    (41), ( ) ( )yy ik ( )xki x . (43) (41),(42) ( )xxK , , (29) . ( )xq ( ),TK x x ( )xxK , , (29)

    (11)

    ( )1

    1 3,8 1 8T

    K x x x 1

    = , (44)

    , (10).

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    14 (16)

    (44) . . 2

    ( ) ( ) ( )( ), ,

    ,T

    T

    K x x K x xx

    K x x

    =

    ( )xxK , .

    0 1 2 3 45 .10 4

    0

    5 .10 4

    0.001

    0.0015

    x j( )

    x j

    . 2. (27)

    . ( ) %106.20 7

    ( ) ( ) %1.0max = x . , (40) " . (29), (26).

    4 10 . ( )xq

    4. (26)

    kz

    ( )kz

    kz

    ( )kz

    9.894 0.028668361 10.064 0.029140157 9.928 0.028762775 10.098 0.029234434 9.962 0.028857162 10.132 0.029328684 9.996 0.028951521 10.166 0.029422907 10.03 0.029045853 10.2 0.029517102

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    15 (16)

    4 , ( ) 028951521.0996.9 = , ,

    . , 4 (

    ).

    ( ) 21 2.8963112 10L p = = L 996.9 L

    % 04.0

    , - . , . .

    , . -. ,

    - [17].

    1. . . , . C XI , 2001, . 2, . 262265.

    2. . ., . ., . ., .., - . . . . ., 2000, . 46, 3, . 320324.

    3. . ., . ., . . -. . ., 2001, . 47, 4, . 564566.

    4. . ., . ., . . . . ., 1999, . 45, 6, . 825831.

    5. . . . 2064666, 1996.

    6. Nyce D. S., Togneri M. G., Bulkowski R. S. Magnetostrictive position sensing probe with waveguide referenced to tip for determining fluid level in a container. US Patent 5848549, 1998.

    _________________________________________________________________________________________ . . ,

  • 2007, 10 _________________________________________________________________________________________

    16 (16)

    7. . ., . . . XI , 2001, . 2, . 265274.

    8. . . , .: , 1969. 9. . ., . . , .: ,

    1977. 10. . . -

    . .: - -, 1989, 249 . 11. Stapleton et al. Ultrasonic liquid measuring device for use in storage tanks containing

    liquids having a non-uniform vapor density. US Patent 5085077, 1992. 12. . ., . . . , , .

    , 2006, 8. 13. . . -. .: , 1984. 14. . . . .: , 1984. 15. . ., . . . .: , 1989. 16. . ., . ., . . . .: ,

    1981. 17. . ., . ., . ., . ., . .

    - Fieldbus Foundation. , 2001, 11, . 1014.

    _________________________________________________________________________________________ . . ,

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