İçerisinde akışkan bulunan öngerilmeli elastik tüplerde harmonik dalga yayılımı
Propagation of harmonic waves in prestressed elastic tubes containing a fluid
- Tez No: 22023
- Danışmanlar: PROF. DR. HİLMİ DEMİRAY
- Tez Türü: Doktora
- Konular: Mühendislik Bilimleri, Engineering Sciences
- Anahtar Kelimeler: Harmonik dalgalar, Öngerilmeli elastik tüpler, Harmonic waves, Prestressed elastic tubes
- Yıl: 1992
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
^ZBT Bu çalışmada, içerisinde akışkan bulunan ön gerilmeli elastik tüplerde harmonik dalga yayılımı problemi incelenmiş olup çalışma dört ana bölümden oluşmaktadır. Birinci Bölümde, konunun tarihsel gelişimi kısaca anlatılmış, konu ile ilgili genel açıklamalar ve tanımlar yapılmış, ayrıca şimdiki çalışmaya neden gerek duyulduğu ve sonuçta neler yapıldığı kısaca belirtilmiştir. ikinci Bölümde, daman ön gerilmeli elastik silindirik bir tüp, kanı da sıkışmaz Newton akışkanı varsayarak bu ortamlara ait alan denklemleri ve sınır koşulları elde edilmiştir. Akışkan denklemleri elde edilirken ön iç basıncın uniform olduğu, dolayısıyla ilk konumun dengede olduğu varsayılmıştır. Damara ait denklemler elde edilirken“Büyük şekil değiştirmeler üzerine sonsuz küçük yer değiştirmelerin süperpozisyonu teorisi”kısaca özetlenmiş ve burada incelenecek olan özel durum için yönetici diferansiyel denklemler ve sınır koşullan elde edilmiştir. Problemi daha belirgin ve çözülebilir hale getirebilmek amacıyla, damar malzemesi için üssel bir şekil değiştirme enerjisi fonksiyonu alınarak ön iç basınç ve eksenel germe altında ön gerilmeler elde edilmiş ve artımsal hareket denklemleri silindirik koordinatlarda ifade edilmiştir. Üçüncü Bölümde, alan denklemlerinin harmonik dalga tipinden çözümleri aranmıştır. Akışkanın diferansiyel denklemlerine kesin analitik çözümler vermek mümkün olduğu halde, katsayılarının değişken olması nedeniyle elastik malzeme için çözüm kuvvet serisi yöntemi kullanılarak ancak yaklaşık olarak elde edilebilmiştir. Hareketin eksenel simetrik veya simetrik olmaması hallerinde dispersiyon bağıntısı analitik veya çoğu kez sayısal olarak incelenebilmiş ve sonuçlar karşılaştırılmıştır. Tüpün bir mühendislik malzemesinden yapılması halinde çözümün nasıl bir şekil aldığı daha önce incelenip uluslararası bir dergide makale olarak yayınlandığı için bu kısma tez içinde yer verilmemiştir. Ancak tüpün mühendislik malzemesinden veya üssel fonksiyon tipindeki biyomalzemelerden yapılması hallerinde dispersiyon bağıntılarındaki açık farklılığı göstermek için makaledeki sonuçlardan yararlanılmıştır. Son olarak Dördüncü Bölümde, çalışmada elde edilen sonuçlar yorumlanmış, daha önce yapılan çalışmalar ile karşılaştırması yapılmış ve gelecekte bu konu üzerinde çalışma yapacaklar için önerilerde bulunulmuştur. iv
Özet (Çeviri)
PROPAGATION OF HARMONIC WAVES IN PRESTRESSED ELASTIC TUBES CONTAINING A FLUID SUMMARY Propagation of harmonic waves in initially stressed (or unstressed) cylindri cal elastic (or viscoelastic) tubes filled with viscous (or inviscid) fluid is a prob lem of interest since the time of Thomas Young who first studied the pulse wave speed in human arteries. Treating the artery as an elastic thin tube and the blood as an inviscid fluid, Moens-Korteweg (1809) studied the wave propaga tion in such a medium and obtained the wave mode, known in current literature as, Moens-Korteweg wave speed. Witzig (1914) is the first one who took the viscosity into account but ignored the effects of Poisson's ratio and obtained the propagation constants as a function of the frequency and viscosity. In 1954 Morgan and Kiely studied the same problem by assuming the artery as an elas tic tube and the blood as a viscous but incompressible fluid and obtained the dispersion relation in which the effect of Poisson's ratio is also included. In 1955 Womersley, in his pioneering work, treated the artery as a thick walled cylin drical shell and the blood as an incompressible viscous fluid and obtained the dispersion equation. In all these works either the initial stresses are neglected or the artery is assumed as a membrane. In reality the artery is subjected a mean pressure, which is about 100 mmHg and the axial stretch, which is about 1.5, under physiological condition. The initial stress (or deformation) of the arterial wall had been first taken into consideration by Atabek and Lew, in 1966, and the effects of initial stresses and tethering were numerically analyzed. Since, in these years the nonlinear the constitutive relations for arteries were not know in functional form, these initial stresses were not incorporated to initial deformations and to the incremental stresses resulting from incremental displacement which takes place in the course of pulsative blood flow. Furthermore, these initial stresses were also assumed to be constant through the thickness. The effect of initial deformation was properly taken into account by Rachev in 1980, but he simply treated artery as a membrane. However, physiological studies on arteries show that the ratio of thickness to mean radius of the artery (large blood vessels in general) changes from | to i. This means that the arterial wall is not thin enough to use the membrane theory and to take the constant initial stress distributions. Having observed the drawbacks of the previous works on this subject, in the present work, employing the governing differential equations and boundary conditions of the theory, so called“the small deformations superimposed on large initial static deformations”, the propagation of a harmonic wave in an initially stressed elastic cylindrical tube filled with an incompressible fluid is studied. We assumed the fluid is viscous for axialy symmetric motions whereas it is inviscid for nonsymmetric motions. Considering the arterial wall as an elastic, isotropic and incompressible material subjected to a large initial staticdeformation, the governing differential equations in cylindrical polar coordinates are obtained for fluid and solid body, respectively, for nonsymmetrical motions. Although a closed form solution can be obtained for equations governing the fluid body, due to the variability of coefficients of resulting differential equations of solid body, such a closed form solution is not possible; a truncated power series method is rather utilized. After employing the boundary conditions properly, the dispersion relations are obtained as a function of inner pressure, axial stretch and the thickness ratio. For symmetrical motions, two separate cases, namely, (i) the tube material is neo-Hookean, and (ii) the tube material assumes an exponential type of constitutive relations, which is suitable for soft biological tissues, are studied. For nonsymmetrical motion only the case of inviscid fluid is explored and various special cases are discusse. Governing Equations and Boundary Conditions : In the course of flow of a vis cous fluid in elastic tube, due to interaction of fluid with its container, the pulsative motion of fluid leads the wave phenomena in elastic tube as well. The governing differential equations and boundary conditions should include these interactions. Equations of Fluid : In this analysis, the blood will be treated as an incom pressible Newtonian fluid subjected to a large static presure Pi. When such an equilibrium state of the blood is disturbed, a pressure increment p(r, 0, z, t) generated by a pump or, say the left ventricle, a harmonic wave type of flow field will be developed in the blood. With vanishing initial velocity, for nonsym metrical motions, the governing differential equations in the cylindrical polar coordinates are given by _dp d^û Idû }_dH d2û û 2 dv _dû dr + ^dr2 + rdr+ r2 d<?2 + dz2 r2 r2 Öd' ~ P öt 1 dp,d2v ldv 1 ö2v d2v v 2dû.__dv ~rd6+ ^İfr2 + r dr~ + r^W + dz2 ~ r1 + ~&W ~ P~dt dp (d2w ldw 1 d2w d2w.__dw dû û 1 dv dw,.,. dr r r dv dz where p is mass density of the fluid, p is the hydrodynamic pressure increment, (j, is the viscocity of the fluid û, v, w are, respectively, the velocity of the components of fluid in the radial, circumferential and axial directions. The components of stress tensor which we need in using the boundary con ditions are, r -, « 9û f,dû dv v. *,dû dw..“. trr = -P + 2^, ir9=K- + ---), trz =,,(- + -). (2) Equations of Solid Body : When a cylindrical tube, made of elastic material, is subjected to large static deformation, an initial stress field t0ki will be developed viin the body. The stresses under the effect of inner pressure P* and the axial stretch A, are given by in cylindrical coordinates as, DEMIRAY [44], *« = C + |f(A4 - x2)F(x), P0 = t°rr-^x2F(x), x=j Pi = ^ [' '« + j)F(t)d£, F{x) = exp «(^ + ^ + A2-3) (3) and zero for other components. In obtaining these stress components we have made use of the stress-strain relations proposed for soft tissues by DEMIRAY [43] t°kl = P°8kl + Şcllexp[a{h - 3)] (4) where P° is the hydrostatic pressure, cj^1 Finger deformation tensor, I\ the first invariant of ckl and a, f3 are two material constants to be determined from experimental measurements. The general theory of ”small deformations superimpossed on large initial static deformation“ had been explored by many scientist in continuum mechan ics (see GREEN and ZERNA [47] and ERİNGEN and SUHUBİ [46]). An outline of this theory is given in the main text of this study. The resulting incremental field equations for the constitutive relations (4) are given by Tu,k+pfi = p%jr> (5) where p is the mass density of the solid body, u is the incremental displacement vector and incremental Piola-Kirchhoff stress tensor Tkl is defined by with Tkl = skl+tki (6) skl - *fcm^',»*5 emn - n\ufn,n T V-n,m ) hi = pSki - 2P°hi + 2a(3exp[a(I1 - 3)]c~^emnc^/1. (7) The incompressibility of the elastic material imposes a further restriction on the displacement field u, i.e., uk,k = 0 (8) In order to determine the incremental field completely, equation (5)-(8) are to be supplemented with the boundary conditions which read Tklnk = tl~ e(n)t°i {onS) C(n) = iijUiUj (9) viiwhere n is the unit exterior normal vector of the surface S, tf is the initial surface traction on the cylinder and ti is the incremental surface traction. Now, we shall consider a general motion of a such a prestressed circular cylindrical shell in the cylindrical polar coordinates. For this purpose we set ui - u(r,d,z,t), i»2 = v(r, $, z,t), u$ = w(r, $, z,t) (10) where u, v and w are the incremental displacement components in the radial, circumferential and the axial direction, respectively. Introducing (10) into equa tion (6) and (7) the physical components of incremental Piola-Kirchhoff stress tensor take the following form Trr=p + 2 T09=p + 2 T T du 1, dv s dw ai- + a2~{-^ + u) + a3-£- dr r 80 dz du 1.dv. dw a2- + a4-(-+u) + a5-- dr du ryd0 dz r. dU ^ rrdr,0 1.dv. + ««;(* +-) 1.dv dw Q3d^ + a5r{dö+U) + a6Jz- + K dw ~dz~ 1 du dv V, dv r9~ P°(rd6 + dr r) + t°rrdr Or _0/l du dv v. i0 l.du. 0* where a{(i = 1,2..., 6) are defined by ttl = a/?^-F(a?) - P°, a2 = <*P^F(x), a3 = a(3x2F(x) a4 = ^F(z) - P°, a5 = a^l^ar), a8 = ap\*F(x) - P°. (12) The governing field equations in the cylindrical polar coordinates may be ex pressed as dT ldT 1,= = ÖT r öf + r (Trr ”Tee) + dz ^=P d2 u ör r 50 + ~^r* + i*r) + dT r ldT dz -P dr rz i r d9 Oz i= arZ2 r rz dz = p dt2 d2v dt2 d2w W' (13) Introducing equation (11) into (13) and utilizing the incompressibility condition we get dp -j. xd2u 1-,.du -s, s u -z, sd2u -,. 1 d2u dr dr2 dz* r2 d62 vm-,., 1 d2v,-,,, -,,,, 1 dv &u -fe + A(r)^ + * W.(')]-âj: - ^^ + MO: '0Z2 1 /7T / x dd6(r).d2v -r. N 1 92u dr ' aö2 r drd9 1 /-Tî / s -T? / n d8Jr).du d2v a* ör2 Ör Ö02 + Mr)%£ + Mr) - Mr))£L + ±(/*T(r) - 05(r))^ =,|£ (14) where the coefficients Şi(r)(i = 1,..., 10) are defined by ?ı(r) = Kr + P°+ 2(«i - «3),. Mr) = t°99 + r^[r(ai - as)] + 2(«5 - a8) £3(r) = tie + P° + 2(a4 - <*«) + 2r^(a3 - a2), 04(r) = *L - P° ?.(r) = tî,-P°, Mr) = 2(a2 - a,), ?7(r) = *J, - ^(rP°) Mr) = *°rr-P0, )99(r) = 2(a2 - a5), ?10(r) = C + P° + 2(a« -a5). (15) These differential equations are to be supplemented with the boundary condi tions given by r=ra r=rj (16) As might be seen form the set of equations in (14), the coefficients of the dif ferential equation of the solid body are variable. Therefore, it might be very difficult to obtain a closed form solution to these equations. Solution to Field Equations : As might be seen from equations (1), a closed form of solution may be given for equations governing the fluid body. However, due to variability of the coefficients, a closed form solution cannot be given to the field equations of the solid body. We shall, therefore, present a method which utilizes the power series expansion of the field quantities. We shall seek a harmonic wave type of solution to the field equations (1) and (14). Therefore, the appropriate form of concerned variable should be as follows û = U(r)exp[i{kz -u;t)]cos(n9), v = V(r)exp[i(kz - ujt)]sin{n6) ?w = W(r)exp[i(kz - u)t)]cos(n$), p = P(r)exp[i(kz - u>t)]coa(n0). (17)ı IXu = U(r)exp[i(kz - u>t)]cos(n9), v = V(r)exp[i(kz - wt)]sin{n6) w - W(r)exp[i(kz - u;t)]cos(nd), p = P(r)exp[i(kz - u)t)]cos(nO). (17)2 where u) is the angular frequency, k the wave number, n circumferential wave number and U(r),...., P(r) are unknown amplitude functions of the field equa tions. The solution U(r), V(r), W(r) and P(r) satisfying the equations (1) may be given by Û = - - [nln(kr) - krln-i(kr)] -\ Jn_i(sr) H Jn(sr) pujr s r V = ^In(kr) - -BJn-iisr) + C[-Jn(sr) - -Jn-i(sr)} puır s r n W = ^-AIn(kr) + BJn(sr), P(r) = AIn(kr) (18) poj where A, B, C are integration constants to be determined from the boundary conditions, Jn(z) and In{z) are, respectively, the first kind modified Bessel functions of order n. In order to obtain the solution for the elastic body, we introduce equations (17)2 into (14) which result in the following ordinary differential equations + t9-^-77J~20s+^)V-O dç ^d? i + tdt r (i + £) n/36 dV n (-^ î+2~d£~ (1 + 0 rn - -x d?V d r,“...-,, 1 dV f, 2 07 (i + 02 '-n /-.. >N^6İ Ur U -= dU n (i + O2 /53+/?7 + (l+O^]^ = 0 dj, d( _ d*w £ dW r 2 2 n2 -r 2,2İTır.*W + A-jjT + jf^ + [^ - ^ - pMfV] W _ _ dU ikv - - + ikH0. - (),)-£ + ^(f), - /»”)P = 0 Because of the complex structure of the coefficient functions in (19), it is almost impossible to obtain a closed form solution to the field equations of the solid xbody. Therefore, in the present work we shall present a power series solution to the governing differential equations. For this purpose we set r-Xi + O. -±<l<± where ¥ is the mean radius, h is the thickness of the tube. Expanding the field quantities into power series of £ we have V = r(V0 + Vie + V2İ2 +..?), W = r(W0 + Wtf + W2Ç2 +...) 7i = P(Pi0+'Piit + Pi2? +.-). («“ = 1,2,...,10) (20) where the coefficients /?io,/%i, A”3 (« = 1)2,...,10), P0° and P° are known con stants to be determined from the series expansion of the corresponding initial field, while Uq,....,P2 are unknown constants to be determined from the field equations and boundary conditions. In this work since we shall keep the terms up to £2 in the expansion, the incremental Piola-Kirchhoff stress tensor should be linear £. Therefore, from (11) we have (21) where Trr0,, Trzl are unknown coefficients which can be expressed in terms of Uo,-.-,P2. If (20) is substituded into (19) and the coefficients of the same power of £ are set equal to zero we obtain a set of algebraic equations relating these unknown coefficients. Furthermore, if (21) is introduced into the boundary conditions (16), we get another set of algebraic equations. The number of unknown coefficients is the same with the number of homogeneous algebraic equations. Even for large arteries the mean radius ¥ is very small as compared to the wave lenght, that is |£| <C 1. In this case these homogeneous algebraic equations may be given by ^0 + foCFi + n%0 - fi8 + rj2Ş40) + T2)U0 + n[fc(ri + £50) + T2] V“ + it}[idT3 + r4] Wo + UnPoVi + iiltdPgW! = 0 UnPQ + n(P0° + {dr5)Uo + [P0° + ( - ir}nÇdr7W0 + (To + &r8)Vi = 0 -itfCdPo + M-PS +(r0 + &rn)Wi=o,-o 2P0 + r12U0 + nT12V0 + ir}T13W0 - 2(/i - n -* n2 - r}2 + - )A0 xi £dnP0 + n(P0° + tdT5)Uo + [P0° + MS? + £50 - Po + V2fa + nare)]V0 - iritdP0 + n?(-P0° + ^r9)C/0 + irjnUr9V0 + &(-$f + n2£50 + 772r10)Wo- 2rj(n - 1 - f2)B0 - 2t}[İQ - (1 + n)f2]C0 = 0 2nP£Uo + 2P°V0 + 2r0 Vi - 2n{fx - 1 - n)A0 - rj[2(l - n) + f2]B0 - [2İÜ + f2(iQ - 2n(l + n))]C0 = 0 2tjP°U0 + 2iT0W1 - 2??(/1 - n)A0 - (İÜ - rç2 - nf2)B0 + vnf2C0 = 0 «W-./.M.+^ + nAC-o in - y0 + nA0 - rjBo + (nf2 - İÜ)C0 = 0 Î2 Wo + v AQ + f2B0=0 (22) 92 where w - ZZ2”> Î1 - => A* - Î2^2 Jl ~~ TFT“”' -/2 _2~,,x _, r,<_2,x Jn(|7),,« Jn_i(7) pupr' spr pr' and T,- (i = 1,2,..., 13) are known constants depending on the initial field. In order to obtain a non-zero solution for unknown coefficients the determi nant of the coefficient matrix must vanish. This vanishing determinant gives the dispersion relation between u>, k and the initial field. In general, it is very difficult to analyse this relation by analytical means. Before studying the gen eral case it might be pertinent to analyse some special cases, particularly the symmetric case. Axially Symmetric Case : The symmetrical case may be obtained from the gen eral case by simply setting n = 0 in equations (22). The special case has been studied for two different elastic materials; (i) Tube material is a neo-Hookean, (ii) Tube material admits an exponential type of strain energy function which is suitable for soft biological tissues. The first case has been worked out separately and published as a scientific paper (DEMİRAY and ERCENGİZ [45]). For the exponential type of constitutive relation, the dispersion relation for the sym metrical case is still too complicated for analytical treatment. Before studying the general case it might be instructive to study some special cases. Longwave Approximation : Generally the wavelenght is very large as compared to mean radius of arteries. Therefore, in this case |?;| <C 1 and the function /i = \if. Hence the dispersion relation takes the following form 2fi4{^(25l/2r0 - rn) - 2r0} + n\2[2q1^d[-r0r1(2qif2 +4?1 - 2) + 2r0r3/2 + rnCTi - 2r3)] + «ı&[r0(2r12gı/2 + 4Çlrı2 - 4?1r2/2 - 8Sir2 - 4r10/2 + 4r4/2 - 4r3 + 4r9 + 4r2 + 2^) + rn(4P0° + 2r13 - 4r4 - r12 + 2r12)] + gir0(2r13 + 2r2 - 4r4 - r12)} xii+ 2r?V&<(/2 + 2){<?1ar0r1(2r10 - r13) + r0r3(r12 + 2r9) + 2r3r11p0°] + gi[-r0r13(r2 + r9) + r0r10(2r2 - r12) + r0r4(2r9 + r12) + p0°ru(2r4-r13)]} = o. (24) In addition, if the viscosity of the fluid is quite small, then /2 - ? oo and the dispersion equation becomes 4(^) qıUTo + (^) {4ro9ı^(r3-rı) + 2gı^r0(çır12 - 2?1r2 - 2r10 + 2r4)} + 2q1u{qiU^oT1(2r10 - r13) + r0r3(r12 + 2r9) + 2r3rnp0°] + Çl[-r0r13(r2 + r9) + r0r10(2r2 - r12) + r0r4(2r9 + r12) + rnp0o(2r4 - r13)]} = o (25) where.^-2 (±Y-tL c2_£ The dispersion relation is further simplified when the initial deformation van ishes, i.e, x = 1, A = 1 and the dispersion relation becomes (-)* - 2(-)2[2 - Ml - 2<?i)] + 12qiU = 0. (26) For small values of £,*, the roots of equation (26), in terms of real physical quantities, c? = f + Ol(W,cI = § (27) which correspond to Lamb and Young (Moens-Korteweg) modes. The more general case is studied numerically and the result is discussed in the main text. Non - axisysmmetric Case : The dispersion equation for the most general non symmetrical case may be obtained by setting determinant of the coefficients matrix of equations (22) equal to zero AialqfTf + BialqjT? + daoqilf + Dx = 0 (28) where the coefficients A\, B\, C\ and D\ are obtained analytically by using the“reduce”program in computer and the result is given in the appendix of the main text. In general it is quite difficult to analyse the general case, even by numerical means. Therefore, in the present work we shall neglect the effect of viscosity of the fluid and assume a special value for transverse wave number n, namely xinn = 1. Before studying this general case of inviscid fluid by numerical means, it might be instructive to study a special case. Elastic Tube Without Initial Stress : This limiting case may be obtained by set ting x - * 1 and A - * 1, and dispersion equation becomes tfpqiUft ~ 1)1 + nV[io«i&(i - /i) + 2fc - 5] + 18gi&(l - h) + 4U - 5} + 2ftV[4<Zı&*(/ı - 1) + 2 - &] + 772[24Çled(/ı - 1) + 4 - 3£rf] + 8ft6,(/, - 1) + 2(1 - &)} + 24gı^4(l-/ı)=0. (29) This shows that there are three waves all of which are dispersive. Normally, the wavelength is very large as compared to midradius of the artery; therefore, the condition |?/| «C 1 is. satisfied. In this case as 77 - > 0, /i - > 2, and dispersion equation takes the following form (2gi& + l)lf + (-18îi& + 4£d - 5)0* + 2(8<&& + 2 - 2td)T? = 0. (30) This cubic equation in Û has three roots which we call them cut-off frequencies and are given by a; îöşe^ + i) ^2 = 0, ^ By expanding the frequency into power series of the nondimensionalized wave number, it is possible to obtain the frequencies analytically for small wave num bers, but we will not give them here. The general dispersion relation for inviscid fluid is studied by numerical means and the results are depicted on some graps. The variations of wave speeds with inner pressure and stretch ratio are similar to those of the symmet rical waves. Therefore, it is meaningless to repeat all those words here. The striking differences between the symmetrical and nonsymmetrical cases are that (i) in nonsymmetrical case the wave has a finite cut-off frequency (ii) the non symmetrical waves are more dispertive than those of axially symmetric waves. Other details of nonsymmetrical wave are given in the main text. xiv
Benzer Tezler
- İçerisinde akışkan bulunan viskoelastik ve elastik tüplerde nonlineer dalga modülasyonu
nonlinear wave modulation in viscoelastic and elastic thin tubes filled with an inviscid fluid
GÜLER AKGÜN
- Ön gerilmeli elastik tüp içerisinde pulsatif akımın incelenmesi
An investigation of pulsatile flow in a prestressed thin elastic tube
İLKAY BAKIRTAŞ
Yüksek Lisans
Türkçe
1997
Mühendislik Bilimleriİstanbul Teknik ÜniversitesiMekanik Ana Bilim Dalı
PROF. DR. HİLMİ DEMİRAY
- İçerisinde akışkan bulunan öngerilmeli ince elastik tüplerde nonlineer dalga yayılması
Nonlinear wave propagation in a prestressed fluid-filled thin elastic tabes
NALAN ANTAR
- Akışkan ortam içerisinde parçacıkların çökelme hareketinin sayısal ve deneysel incelenmesi
Başlık çevirisi yok
FUAT DERVİŞ SOLAK
Yüksek Lisans
Türkçe
1993
Makine MühendisliğiKaradeniz Teknik ÜniversitesiMakine Mühendisliği Ana Bilim Dalı
YRD. DOÇ. DR. BURHAN ÇUHADAROĞLU
- Formaldehyde and iso-propanol treatment in an anaerobic fluidized bed reactor and kinetic evaluations
Havasız akışkan yataklı bir reaktörde formaldehit ve iso-propanol arıtımı ve kinetik değerlendirmeler
MOİZ ELNEKAVE
