Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 897 IX.7 ADDIIONAL INEGRAL RANSFORMS 6.7. Solutio of 3-D Heat Equatio i Cylidical Coodiates 6.7. Melli asfom 6.7.3 Legede asfom 6.7.4 Jacobi ad Gegebaue asfom Solutio of geealized heat coductio poblem 6.7.5 Laguee asfom 6.7.6 Hemite asfom 6.7.7 Hilbet ad Stiltjes asfom 6.7.8 Z asfom
898 Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 6.7. Solutio of 3-D Heat Equatio i Cylidical Coodiates u (, θ,z,t) u u u u g u + + + + = z k α t θ () ) z -vaiable a) Case of z K (Fiite Cylide) Use Fiite Fouie asfom i z -vaiable: z k { u( z) } = u () k z = K with the ivese - { } uz ( z) u k = k k k u z k ω = kuk (3) z b) Case of z (Fiite Cylide) Use the coespodig stadad, sie o cosie Fouie asfom i z -vaiable. c) Case of < z z < (Fiite Cylide) Use Fiite Itegal asfom i z -vaiable. { } u( ω) F u z = (4) u z F z = ω u ( ω) (5)
Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 899 ) θ -vaiable a) Case of θ θ θ (Piece of Cake) Bouday coditios have to be set at θ = θ ad θ = θ. Reduce to iteval θ θ ad the use Fiite Fouie asfom i θ vaiable: { u( θ )} = u (6) - u = { } Opeatioal popety: θ ( θ ) u η = u (7) θ θ = θ θ = θ b) Case of θ (full otatio) hee o bouday coditios i this case fo θ = =, but solutio has to be -peiodic: ( θ) = ( θ + ) u u Costuct itegal tasfom base o the Fouie seies o the iteval θ (see able): (,θ) ( θ) = ( θ + ) = a + a cos ( θ) + bcos ( θ) u u = whee the Fouie coefficiets ae defied by: θ a = u( θ) dθ a = u ( θ) cos ( θ) dθ b = u ( θ) si( θ) dθ Substitute them ito the Fouie seies ad eaage: u ( θ ) = u ( s) ds + u ( s) cos ( s) ds cos ( θ) + u ( s) si( s) ds si( θ) = u s ds u s cos s cos si s si ds u ( θ ) = + ( θ) + ( θ) = u s ds u s cos s ds u ( θ ) = + ( θ ) =
9 Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 he itegal tasfom ad its ivese ca be defied as: { } u θ { u ( θ )} - { } u = u s ds (8) u u s cos s ds =,,... (9) u = ( θ ) u θ = u () = Opeatioal popeties of this itegal tasfom: u θ u s = ds s u s = = s () u θ u s = cos ( θ s) ds s u s = cos ( θ s) d s u s u s = cos ( θ s) d cos ( θ s) s s = si ( θ s) d u ( s) = u si ( θ s) u ( s) d si ( θ s) + = u s cos s ds ( θ ) = u () Example:
Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 9 Applicatio of itegal tasfoms i z ad θ vaiables to the Heat Equatio (): z Fo the fiite solid cylide (combiatio of cases a ad b) the cosecutive applicatio of the itegal tasfoms () ad (8,9) togethe with the opeatioal popeties (7) ad (,) yields: u u g u k, k, k, k, + u k, ωk uk, + = k α t (4) z = K whee =,,,... k = ( ),,,... ( k =, I a case of N-N b.c. s) Equatio (4) defies the ode of the Hakel tasfom fo elimiatio of the diffeetial opeato i -vaiable (see p.388 ad p.393). heefoe, the tasfomatio of equatio (4) should be cosequetly pefomed with the Hakel tasfoms of all o-egative itege odes: H =,,,... hus, the Hakel tasfom of equatio (4) is: fo solid cylide : gk,,m u λmuk,,m ωk uk,,m + = k α t k,,m (5) ad fo : g u λ uk, ωk uk, + = k α t k, k, (6) Now, equatio (5) is a odiay diffeetial equatio which subject to iitial coditio ca be solved by vaiatio of paamete o by the Laplace asfom to fid the tasfomed solutio: u ( t) o u ( t,λ) k,,m (7) k, he the solutio of the IBVP fo the Heat Equatio () ca be foud by the cosecutive applicatio of the ivese tasfoms: ( θ ) k,,m ( θ) ( λm) ( λ ) u,,z,t = u t, Zk z k m J m J Fouie Seies () J ( λ) u(, θ,z,t) = uk,,m ( t, θ) λdλ Z k z k J( λm) Fouie Seies () (8)
9 Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 Example: u,θ u u u + + = θ ( θ) U( θ) u, = Diichlet ( ) Apply Itegal asfom (8-9) with opeatioal popeties (-): (,θ) u u + u = =,,,... (4 ) θ = = { u(, θ) } u (, θ) U ( θ) Apply FH-I ( I ) H with opeatioal popety o p.389 (Diichlet): λ J + λ U θ λ u θ = m, m, m, m, whee m, J λ =. λ ae positive oots of asfomed Solutio: u,m ( θ) J = ( λ ) + m, λ m, U ( θ) Ivese Hakel asfom: ( θ) = U( θ) u, u, = u ( θ) ( θ) ( λ,m) ( λ,m),m m= J J Ivese Itegal asfom (Fouie seies): u(,θ ) θ ( θ) = ( θ) U ( θ ) u, u, = ( λ ) ( λ ) m, J ( λm,) J J + m, m, = m= λ = ( λ ) ( λ ) J J + m, m, U( θ ) = m= λm, J+ m, = U λ J = ( θ ) J ( λm,) ( λ ) = m= m, + m, ( λ )
Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 93 Same Example: u,θ u u u + + = θ ( θ) U( θ) u, = Diichlet ( ) Apply Itegal asfom (8-9) with opeatioal popeties (-): (,θ) u u + u = =,,,... (4 ) θ = = { u(, θ) } u (, θ) U ( θ) u u + u = =,,,... (4 ) It is a Cauchy-Eule Equatio fo =,,... with the geeal solutio: u = c + c ad u u + = with the geeal solutio: u = c + c l = (4 ) Because solutio has to be bouded: u = c u = c Apply bouday coditio: u, = U ( θ) ( θ) u = U θ u, = U ( θ) ( θ) u = U ( θ ) ( θ) = ( θ) U ( θ ) u, u, = = =
94 Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 6.7. Melli asfom u x :, p p M { u( x) } = u ( p) u ( x) x = dx (9) M { u ( p )} = u( x) i c+ i = p u p x dp () c i Examples: ) M { e x } x p e x dx = t p = e t dt p t = x Γ = ( p) p Opeatioal popeties: ) M { u ( x) } = ( p ) u ( p ) M { u ( x) } = ( p )( p ) u ( p ) ( M { u ) ( x) } d dx ( p) ( p ) Γ = u p Γ ) M x u( x) = ( ) pu ( p) 3) M u ( t) dt = ( + ) 4) Covolutios x u p p Applicatios: Examples: Melli tasfom is used fo solutio of diffeetial ad itegal equatios, evaluatio of factioal itegals ad deivatives, to summatio of ifiite seies ) (Debah, p.8) a y u u u x x y x + x + = u ( x,) = a x u ( x,) = x > x (, ), y (,) x Solutio: ( ) a u ( x, y) = x si y = ( )
Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 95 6.7.3 Legede asfom 6.7.4 Jacobi ad Gegebaue asfom Solutio of geealized heat coductio poblem 6.7.5 Laguee asfom 6.7.6 Hemite asfom 6.7.7 Hilbet ad Stiltjes asfom 6.7.8 Z asfom REFERENCES. J.V.Beck, K.D.Cole et al. Heat Coductio Usig Gee s Fuctios. Hemisphee Publishig, 99.. D.G. Duffy. Gee s Fuctios with Aplicatios, Chapma&Hall/CRC,.
96 Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7
Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 97 Les Modeles Globaux PP st Radiatio School Odeillo SLW- SPECRAL MODEL absoptio -lie blackbody distibutio fuctio F( C) C C C absoptio coss - sectio C=C η ( g) SLW - spectal model gay gas C κ P FC a E bη ( ) b clea gas η ( ) a =F C, =, = g b a =- a Physical meaig of SLW - paametes κ P = a κ κ R = κ Plack -mea absoptio coefficiet Rosselad-mea absoptio coefficiet η SLW- RE: I s I s = - κ I + κ I = R P b with efficietly defied gay gas paametes a, κ
98 Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 C η, cm molecule C C
Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 99 CONSRUCION OF HE EFFICIEN SLW- SPECRAL MODEL Alteatively, two paametes must be defied i the SLW - spectal model : eithe coss - sectios C ad C, o gay gas coefficiet κ ad its weight a C C η C C η κ = κ = NYC a = F( C, g,b) a = a otal et adiative flux i the isothemal plae laye at tempeatue bouded by black cold walls : SLW- { } ( ) SLW b j 3 j 3 j j= q x = I a E κ x E κ L x { } SLW- q ( x ) = I ( ) a E ( κ x) E κ ( L x) SLW b 3 3 LEAS-SQUARES OPIMIZAION M SLW m SLW m m= q x q x mi
9 Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 CONSRUCION OF HE SLW- MODEL FOR OHER GEOMERIES fid the SLW- Objective paametes a, κ with the help of the equivalet plae laye Plae-Paallel Laye -D Catesia y x M L L x 3-D Catesia z y H L x M L x
Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 9 Ifiite Cylide R Fiite Cylide z z = H R Sphee R
9 Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 SLW P - Appoximatio (equatio fo gay gases) : G j w adiative tasfe is modelled as a diffusio pocess G j G j w w G - 3κ G = -a κ I j j j j j b j =,..., -ε bouday G j( w) + κ j G j( w) = 4a j κ j I b( w) coditio ε 3 G = I,s sˆ dω j ( ˆ j ) 4 qj =- Gj 3κ qj =- Gj 3κ Qj = Gj 3κ j j j = κ jgj( ) - 4a jκ jib ( ) iadiatio (adiatio itesity itegated ove all diectios) adiative flux vecto et adiative flux divegece of adiative flux
Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 93 P- APPROXIMAION FOR PLANE LAYER L x d G j x - 3κ j G j x = -a j κ j I b dx j =,..., bouday coditio d - Gj( ) + κg j j( ) = 3 dx d G j( L ) + κg j j( L ) = 3 dx Cold black walls Aalytical Solutio : 4ajb I L q x = cosh 3κj x- j= 3 3 sih κ jl + 3cosh κ jl -4 3ajκ jb I L Q x = cosh 3κj x- j= 3 3 sih κ jl + 3cosh κ jl
94 Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 P- APPROXIMAION FOR INFINIE CYLINDER SLW - SLW - R d d d Gj 3κ jgj = -ajκ jib d j =,..., bouday coditio d G j( R ) + κg j j( R ) = 3 dx Cold black wall Aalytical Solutio : ( ) 4ajb I q = I 3κ j j= I 3κR + 3I 3κR j j ( ) ( ) whee I ad I ae the modified Bessel fuctios ( ) -a κ I Q = I 3κ j 3I 3κR +3I 3κR j jb j= j j
Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7 95 P- APPROXIMAION FOR SPHERE R SLW - SLW - L =.9 R d d Gj 3κ jgj = -ajκ jib d d j =,..., bouday coditio d G j( R ) + κg j j( R ) = 3 d Cold black wall Aalytical Solutio : ajb I R 3κ jcosh 3κ j sih 3κ j q = - j= 3 κ j - sih( 3κ jr ) + 3κ jcosh( 3κ jr) R Q = j= -6a κ I j j b R sih 3 κ j - sih 3κ jr + 3κ jcosh 3κ jr R ( 3κ j )
96 Chapte IX he Itegal asfom Methods IX.7 Additioal Itegal asfoms August 5 7