Example

Transcription

1 hapte Exaple A single hllw fibe is placed within a vey lage glass tube. he hllw fibe is 0 c in length and has a diaete f 400 icns. he flw ate f a liquid cntaining a peeable slute thugh the hllw fibe is l/in. It is fund that the cncentatin f the peeable slute exiting the hllw fibe is 0% f the cncentatin f this slute when enteing the hllw fibe. Estiate the peeability f the hllw fibe ebane f this slute. Slutin Hllw fibe =0 f We will cnside a siplified del f slute tanspt in the hllw fibe. We assue that the flw ate in the shell space is uch highe than that f the fibe s that the slute cncentatin in the shell space is e and the esistance t the slute within the fibe is negligible s that P. he steady state slute balance n the cntl vlue π gives Vπ f Vπ f + = π ( 0) (E-) whee is the veall ass tansfe cefficient between the slute in the hllw fibe and the slute in the shell space next t the hllw fibe wall, and is the slute cncentatin in the fibe. he slute cncentatin in the suunding shell space is e. he veall ass tansfe cefficient is elated t the fil ass tansfe cefficient and the peeability by the expessin = + k P 5 Funie,. L., Basic anspt Phenena in Biedical Engineeing, ayl & Fancis, 007, p

2 whee,, and ae the ttal ass tansfe esistance f slute f the fibe t the k P utside suface f the fibe, the ass tansfe esistance in the fibe side, and the ass tansfe esistance thugh the fibe wall, espectively. Since << k P P P Dividing equatin (E-) by π f and taking the liit as 0, we btain d V = d f P (E-) d π f = d Vπ f P = π Q f P (E-3) We have eplaced the liquid velcity V with the vluetic flw ate Q. Equatin (E-3) can be integated ( ) O d π = Q f P d 0 ( ) = π f Q P At the exit f the hllw fibe, = 0 c and ( ) = 0.. heefe P = (0.) Q π (0) f.303 (/ 60) π (00 0 )(0) = 4 = c/s.7 Slute Peeability high N S lw Bulk slutin Bulk slutin Mebane x=0 x= Figue 4.5b- Slute tanspt acss a ebane. -5

3 Fick s law f the diffusin f slute acss a ebane is epesented by J S = D A P τ d = DAP dx ω τ d dx (.7-) his equatin is integated acss the ebane t btain J S = DA P ω high τ t lw (.7-) he slute tanspt can als be expessed in tes f the peeability P f the ebane J S = P S( high lw ) (4.5b-) paing equatin (4.5b-) with equatin (4.5b-) we have a elatin between peeability P and diffusivity D P = D A p S τω In se cases, it is difficult t estiate the suface aeas A P f the pes the capillaies invlved. Equatin (4.5b-) is then e cnvenient t use since yu can wk with the pduct P S, athe than the specific values f P S. he fllwing celatin ay be used t estiate capillay P S values f a given slute f adius a P S = 0.084a -.3 P S = 0.087a -.9 a < n a > n In these expessins, the unit f a is n and the units f P S ae c 3 /sec/00g f tissue..8 Slute anspt between apillay and issue Space he lecules equied f tissues ae caied in the bld vessels t the capillaies whee they diffuse thugh the capillay wall t the tissue space. develp the del f slute tanspt f the capillay t the suunding tissues, we will use a shell balance and the gh tissue cylinde del. he gh tissue cylinde del is a siplified del f the tissue suunding the capillay. It assues a cylindical laye f tissue suunding each capillay with the slute tansfeed nly f that capillay. he capillay is assued t be cylindical and f cnstant adius. he gh tissue cylinde is shwn gaphically in Figue.8-. apillay issue cylinde Figue.8- he gh tissue cylinde -53

4 As the slute ve ves ag the capillay, its cncentatin deceases because f slute tanspt thugh the capillay wall. We can ake a slute balance n the cntl vlue π shwn in Figue.8- assuing that the bld flws thugh the capillay with an aveage velcity V. c+t apillay wall ntl vlue in the tissue space = π c Figue.8- ntl vlues f the capillay and the tissue space. he steady state slute balance n the cntl vlue π gives Vπ Vπ + = π ( c + ) (.8-) whee is the veall ass tansfe cefficient between the slute in the capillay and the slute in the tissue space next t the capillay wall, is the slute cncentatin in the capillay, and is the slute cncentatin in the suunding tissue space. Dividing equatin (.8-) by π and taking the liit as 0, we btain d V = d ( c + ) (.8-) hee ae tw dependent vaiables and c + in this equatin. heefe, we need e infatin befe we can slve f (). Making a steady state shell balance aund cntl vlue π shwn in Figue.8- f slute in the tissue space, we have D (π ) d d + D (π ) d d + = ( )π In this equatin, ( ) is the cnsuptin ate pe unit vlue f slute in the tissue space. We will assue that the eactin ate is e-de in the slute cncentatin, ( ) = = -54

5 cnstant. Dividing the equatin by the cntl vlue π and taking the liit as the cntl vlue appaches e, we btain D d d d = (.8-3) d With the assuptin f e-de eactin, the slute cncentatin () ag the axial psitin can be btained f the balance ve pat f the capillay f the entance t tissue space capillay Figue.8-3 Slute leaving in the fist pat f the capillay. Vπ Vπ () = π[ ( + ) ] his equatin states the fact that the change f the slute within the bld is equal t the slute cnsuptin in the suunding tissue space. he axial slute cncentatin is then Equatin (.8-3) D cnditins () = [ ( + ) ] (.8-4) V d d d = can be integated with the fllwing tw bunday d = +, = d c + and =, d = 0 whee d c + is btained f equatin (.8-): V t = d ( c + ) c + = d + V = d () [ ( + ) ] c + = () [ ( + ) ] -55

6 issue cylinde d /d = 0 () apillay d Figue.8-4 At the gh cylinde adius d = 0 d he secnd bunday cnditin =, d = 0 is the esult f the gh tissue del whee the tissue cylinde adius is at the psitin between the capillaies. Integating equatin (.8-3) nce we btain D d = d + A he cnstant A can be slved with the bunday cnditin at = t give A = D d = d D d = d d Integating the equatin ve the liit = + t yields D ( + ) = c [ ( + ) ] 4 + (, ) = c + () + 4D [ ( + ) ] D + Since c + = () [ ( + ) ] -56

7 (, ) = () [ ( + ) ] + 4D [ ( + ) ] D + (.8-5) Hence, equatins (.8-4) () = [ ( + ) ] and equatin (.8-5) descibe the V slute cncentatin in the capillay and in the tissue space espectively. cit tissue space with n slute capillay cit Figue.8-5 egin f tissue space with n slute. Unde se cnditins when the aunt f slute supplied t the tissue space is less than the aunt equied, se egins f the tissue will have n slute as shwn in Figue.8-5. We can define a citical adius cit, the adial distance beynd which n slute is pesent in the tissue. he bunday cnditin at the ute edge f the tissue cylinde beces d = cit, d = 0 and = 0 Beynd the axial distance cit shwn in Figue.8-5 whee the citical adius begins t exist the slute cncentatin in the capillay and in the tissue space wuld still be given by equatins (.8-4) and (.8-5) espectively; hweve, the gh tissue cylinde adius is eplaced by cit () = [ cit ( + ) ] (.8-6) V (, ) = () [ cit ( + ) ] + 4D [ ( + ) ] D + (.8-7) We need t deteine the citical adius befe equatins (.8-6) and (.8-7) can be used t descibe the slute cncentatin in the capillay and in the tissue space. At = cit, = 0, theefe 0 = [ cit ( + ) ] V -57 [ cit ( + ) ]

8 + [ cit ( + ) ] 4D D cit + t 4D Multiplying the equatin by ( + t ) yields 4D 0 = ( + t ) 4D [ cit V ] + D [ cit + ] + cit + cit + cit + t Let = cit +, the equatin beces 4D 0 = ( + t ) 4D V ( ) D ( ) + ( ) Since =, the abve equatin can be eaanged t 4D = ( + t ) + ( D 4D )( ) V 4D Let A =, B = ( + t ) D 4, and D = V D, we have finally = A + (B D)( ) (.8-8) We can nw find the citical axial lcatin cit whee tissue cylinde adius is devid f slute. At = cit, = cit =, theefe = cit + t = + t = knwn value cit is then slved f equatin (.8-8) cit = [B ( A)] (.8-9) D Beynd the citical axial lcatin cit, the citical adius ust be slved nueically f equatin (.8-8). Let x =, at any lcatin > cit -58

9 B D = E = cnstant he nnlinea equatin t be slved is f(x) = x (x) E(x ) A = 0 f (x) = (x) + E x can be btained using Newtn s ethd whee x = x f ( x) f ' ( x) Since x = cit +, the citical adius is then cit = ( + )x / Exaple a) Plt the slute cncentatins in the capillay (), at the utside wall f the capillay c +, and at the gh cylinde adius (, ) using the data in able.8-. able.8- apillay chaacteistics Value Ppeties Inside diaete (D c ) Length (L) Wall thickness ( ) Aveage bld velcity (V) Enteing glucse cncentatin ( ) issue glucse cnsuptin ate ( ) gh tissue cylinde adius ( ) Glucse tissue diffusivity (D ) Oveall ass tansfe cefficient ( ) 0.00 c 0. c c 0.05 c/sec 5 µl/c 3 0.0µl/c 3 sec c c /sec c/sec b) epeat (a) with the glucse tissue diffusivity changed t c /sec. c) epeat (a) with the aveage bld velcity changed t 0.005c/s. d) Plt the citical tissue adius as a functin f the axial distance ag the capillay. Slutin a) he slute cncentatins in the capillay (), at the utside wall f the capillay c +, and at the gh cylinde adius (, ) ae pltted in Figue.8-6 (the tp left cne). he equatins used f the plt ae () = [ ( + ) ] (.8-4) V -59

10 c + = () [ ( + ) ] (, ) = c + + [ ( + ) ] 4D D + t hee is little diffeence between the glucse cncentatins at the ute suface f the capillay wall and at the gh cylinde adius. his indicates that the pcess is eactin liited that is the ass tansfe ate is uch faste than the eactin ate. b) When the diffusivity f glucse in the tissue space is educed by a fact f 0 we can see the diffeence between the glucse cncentatins at the ute suface f the capillay wall and at the gh cylinde adius. he gaph is at the tp ight cne. 5 V=.05 c/s, D=8e-6 c/s 5 V=.05 c/s, D=8e-7 c/s (Mic-l/c3) 4 3 w (Mic-l/c3) 4 3 w (c) (c) 5 V=.005 c/s, D=8e-6 c/s 4 x 0-3 itical adius calculatin (Mic-l/c3) 4 3 w citi (c) (c) (c) Figue.8-6 ncentatins and citical adius f exaple