- Email: [email protected]
General solutions of the coupled diffusion equations
ht. J. Engng Sci., 1973. Vol. 1 I, pp. 235-241.
GENERAL
Pa’gamon Press.
SOLUTIONS
Printed inGreat Britain
OF THE COUPLED EQUATIONS
DIFFUSION
M. D. MIKHAILOV Institute for Mechanical and Electrotechnical Engineering, Sofia, Bulgaria Abstract-New finite integral transform and the corresponding infinite series are introduced, which brings the solution of the coupled dithrsion equations within the realm of integral transform theory. The formulation of this transform leads to an eigenvalue problem which is not of the conventional Strum-Liuoville type and therefore a special integral condition was derived which serves as an orthogonality relation. The solution obtained can be applied when studying diffusion in a tubular reactor, heat transfer by a turbulenty flowing fluid-solids mixture in a pipe. INTRODUCTION
IN A RECENTpaper [ 11, the author presented an analytical solution of the heat equation applicable to a large class of heat transfer problems, including unsteady heat conduction and more complicated cases, namely: heat transfer in laminar and turbulent flows of Newtonian or non-Newtonian fluids in pipes, ducts; temperature development in the entrance region of MHD channels etc. The present work complements a previous paper, developing further the method of finite integral transforms and applying it for the solution of a more general mathematical model of transfer process. The solutions presented permit the studying of many new problems. The particular problems solved in [2] and [3] are shown to be a very special case of the general results derived here. STATEMENT AND SOLUTION OF THE PROBLEM The starting point of the analysis is the simultaneous consideration differentials equations w
(M)
m
of the partial
aT?ntM.7) =divtk,(M)gradT,tM,7)l+(-I)mp(M)[a,-1TI(M,~) a7
-u,+lT,(M,7)l+P,(M,7),
m=
1,2
(11
with the initial
T,tM,O) =fitM),
m= 1,2
(2)
and boundary conditions tN)
A
w
aT,tw 7) -I-B,(N)T,(N,7) an
=qm(N,7),
m= 1,2.
It is supposed that the solutions of the problem can be represented T,W,
7) = 5 W&W i=l 235
(31
by the expansions (4)
M. D. MIKHAILOV
236
where JImi(M) (m = 1,2) are eigenfunctions to the two-region Sturm-Liouville problem
A,(N)
v
+&(Nhhn(N)
= 0
(6)
their solution being granted for known. Equations (5-6) do not belong to the conventional Sturm-Liouville family and therefore, to the determination of the coefficient Ci the well-established orthogonality cannot be used directly. Therefore it is appropriate to derive an integral condition to serve as an orthogonality relation. One begins by applying the equation (5) for two distinct eigenvalues pa and /+ M~tiplying the first by ~~(~), subtracting from the result the multip~ed with ~~j(~) second equation and inte~~g the result obtained, after taking into account the well-known Gauss Theorem, which transforms the volume integral into a surface one, one gets two expressions (for m = 1 and 2 respectively). They may be written as follows
After multiplying the equation (7) for m = 1 and 2 with cl and u2 respectively and adding the results obtained one finds
For i f j because of the boundary conditions (6) the foregoing equation (8) becomes
These equation (9) is the equivalent of the conventional Sturm-Liouville orthogonality relation for the eigenfunctions JItnl(M) and I,!J~((M).For the case of i = j equation (8) leads to
237
General solutions of the coupled diffusion equations
Now, attention may be directed to finding C*. Multiplying the equation (4) for the case m = 1 by a,~, (M)+,*(M) and the same equation for the m = 2 by a2w2(M)\j12* (M) and adding the resulting expressions and integrating into V, after taking into account the orthogonality condition (9), one finds the formula for C*. Then (4) may be written as follows m 2, cm j- w~(WJl,“*(WTtn(M, 7) dV T?n(M, 7) =
&,*(M).
c
5 o7nJ w,(W,%*(M) m=1 V
i=l
(11)
dV
To solve equations (1) at the conditions (2, 3) we define the new finite integral transform F*(T) = i
atPIJ w,(WG,*T,(M.
m=1
7) dV
(12)
V
and rearrange the expressions (11) as the inversion formulas, in the form of m*
Tm(M, 7) = i
(Ml Ti (7)
(13)
dt”
*=’j, urn[ wm@O,%,(M)
After multiplying the systems (1) and (5) respectively by ~,,,JlmiM) and a,Jm(M, 7) the result of adding the four results obtained and integrating into V as when deriving (7), one gets:
2 = cm=1
!?h.?p
$a W urn il
kr, WI T, (NV7)
s
From (3) and (6) A,(N) obtained, one gets
W,
r)
ds +
$,,arUWP,(M,d I V
db’ . I
(14)
and B,(N) are determined. After summing up the results
wm* W)
~$rn*(N)
Tm
U’,,,W, 7) an
an aTm(N,7) =
Jlmi(N)
A,(N)
-v +B,(N)
cpm(N’ ‘)-
(15)
an
Substituting this result in (14), for F*(T) is obtained an ordinary first order linear differential equation, which is easily solved using the transformed initial conditions (2) according to (12). Using this solution in the inversion formulae (13), the desired solutions of the problem are obtained as follows:
umsvd.11wa2-c
238
M. D. MIKHAIL~V
(16) If B,(N) = 0 and aocr,- aluz = 0 then p = 0 is also eigenvalue of the two-region Stun-Liouville problem and the corresponding eigenfunctions tj.~~ and ?,&must satisfy the equations flOJ/lO - crz@20
=
0
~lJIlo-wb2o
=
0.
(17)
Therefore, in this case additional terms, corresponding to the zero-eigenvalue appear in the solutions and (16) takes the form
+
ylm W,
ktN) A,(N)
7) dS +
P,(M, T) dV d7 li
If the functions (om(N, 7) , P, (M, T) are constant, polynomials and exponentials of the time, the convergence of the series in solutions (16) and (18) could be improved in thes~ewayasin~l]. ONE-DIMENSIONAL
SOLUTIONS
In connection with studying the reversible reactions and the diffusion in isothermal pipe reactors [2], the heat transfer by a turbulently flowing fluid with suspended small solid particles in a circular pipe [31,let us consider the one-dimensional case, described by the differential equations system
-~,+*~2(~,7)l+P,(x,T),
m=
1,2,
X0G X < X1, 7 b 0
(19)
General solutions of the coupled diffusion equations
239
TIn(-G0) =&I(x)
(20)
at the following initial
and boundary conditions A
aTm(xo,7)
m0
ax
+BmoTrn(~~,
7) =
Am1aTmz”)+Bm
prno(T)v
Tm(xI,~) =cpml(r). (21)
Equations (5) and (6) giving the eigenvalues and eigenfunctions become
-$[,,x) y]
+(-l)mP(~)[~m-l$h(~)-~m+lJlz(~)l+PZWm(~)&n(X)=o
Arno’&n(x~)
+BmoJlrn(xo)
=
0,
Arn~#n(x~)
(22)
+&tlrlrm(x~)
(23)
=O*
The solution ( 16) for the one-dimensional case will be Tm(x,
7) = i
t
z
$mi (x) e-&F
=
i
m=1
li
urn
UJWm(x)$ki(x)dXm=l
i=l
ax
dX
Wm(x)Jlmi(xlfm(x)
Zo
7
+
I [
e+ t(Xlhll(~)
l+!lmt(X1) -* Am1 + Bm1
$m*
(-%I) -*
-km(xO)VmO(7)
Am0 + Bmo
0
II
+
I
$mt
(xlprn(xv7) dx dr *
IO
11
The solution ( 18) valid for Bmo = B,,= 0,for the one-dimensional case becomes
7
3rm0
T~(x,T) =
i,
7
umJ12mO Wm(x)
Wm(x)fm(x)
dX
z”
zo
d7
P,(x,T)dx 30
+Ii i=l
+mf(x)e-"F wm(X)ti*(X)~ i Urn7 m=1 10
1
dX+ j 0
(24)
‘40
M. D. MIKHAILOV CONCLUSIONS
The above solutions from the present paper permit the easy solving of any particular case of transfer problems in the entrance region of pipes and ducts. If the substitutions (~~~(7) = qrnl (7) = 0, P,(x, 7) = 0, x0 = 0, x1 = 1, B,, = 0 and A = 1 are made, from (24) and (25) one derives a narrower class of problems to w%h belong the problems discussed in [2] and [3]. For the case x = r, r = z, T, (x, T) = c, (r, z) , w,(x) = r( 1 - 9)) k, (x) = d,r, k2 (x) = d2r, P(X) = r, fi (x) = cloI f2 (x) = czo, ~0 = (~1= 1, gz = cr3 = 5, B,, = 0, A,, = 1 the solution of [2] is obtained given also in the monograph [4]. Forthecasex=7),7=5,T,(x,7)=B(rl,~),Tz(x,7)=~~(r) [),fi(x)=fz(x)=l, WI(X) = ~vz(x) = -v_~v), k,(x) = %drl), k,(x) = 0, go = a;, = Pa,v1 = ~3 = P3,B,l = 1, Ami = 0 follows the solution of [3] given also in [5]
NOMENCLATURE
boundary coefficient functions defined on S constant boundary coefficients on x = x0 constant boundary coefficients on x = x1 initial distribution functions in I/ one-dimensional initial distribution functions in x source functions on S source functions on x = x0 source functions on x = x1 1,2,3,4 ,..., 1or2 prescribed function defined in V prescribed function defined in x point in V point on S outward normal of S internal source function per unit time per unit volume of V internal source function per unit time per unit volume of one-dimensional region S boundary of V Tm(M, 7) unsteady potentials distributions defined in equations (l), (2) and (3) TZ (M, 7) unsteady potentials distribution defined in equations (18) co-ordinate X time variable 7 prescribed functions defined in V wn(W prescribed functions defined in x wm (Xl prescribed function defined in x P(X) prescribed constant coefficients (+m-19 ~m+1 div ( ) divergence operation in M-space grad ( ) gradient vector in M-space Jlmt(M) eigenfunctions in M-space eigenfunctions in x-space Jlmi(X) CL eigenvalues
General solutions of the coupled diffusion equations
241
REFERENCES [I] M. D. MIKHAILOV, Int. J. Engng. Sci. 10,577 (1972). [2] SIBRA P., Reversible Reaction and Diffusion in an Isothermal Tubular Reactor, Master’s thesis. University of Texas (1959). [3] TIEN C. L.,J. Heat Transfer, 183 (1961). [4] R. S. SCHECHTER, The VariationalMethod in Engineering. McGraw-Hill (1967). [5] S. L. SOO, Fluid Dynamics ofMultiphase Systems. Blaisdell(l970). (Received 1 May 1972)
R&u&-La nouvelle transformation finie integrale et la serie infinie correspondante sont introduites, ce qui apporte la solution des equations de diffusion couplde dans le domaine de la theorie de transformation integrale. La formulation de cette transformation conduit a un problbme de valeur propre qui n’est pas du type conventionnel Sturm-Liouvihe et par consequent on a derive une conventionnel Sturm-Liouvihe et par consequent on a derive une condition inttgrale speciale qui sert de relation d’orthogonahtC. La solution obtenue peut Ctre appliquee lorsqu’on ttudie la diffusion dans un reacteur tubulaire, et le tcansfert de chaleur par un melange de fluide et de solides a Ccoulement turbulent dans un tuyau. Zusammenfassung- Es werden ein neues endhiches Integraltransform und die entsprechenden unendlichen Reihen eingefuhrt, was die Losung der gekuppelten Diffusionsgleichung in den Bereich der Integrahransformtheorie bringt. Die Formulierung dieses Transforms flirt zu einem Eigenwertproblem, das nicht vom konventionellen Sturm-Liouville-Typ ist, und deshalb wurde eine spezielle Integralbedingung abgeleitet, die ah Orthogonalitltsbeziehung dient. Die erhaltene Losung kann bei der Untersuchung von Diffusion in einem Rohrreaktor angewandt werden, und der Warmeiibertragung durch eine turbulent fliessende Fliissigkeitsfeststoffmischung in einem Rohr. Sommario-Se introducono nuove serie finite di trasformazione integrale e le corrispondenti serie infinite, il the porta alla soluzione delle equazioni di diffisione accoppiata nel campo della teoria di trasformazione integrale. La formulazione di questa trasformazione porta a un problema di eigenvalore the non t del tipo di Sturm-Liouville convenzionale e percio si ricava una condizione integrale speciale the serve da rapport0 di ortogonalit~ La soluzione ricavata puo venire applicata quando si studia la dhfusione in un reattore tubolare, il trasferimento di calore mediante una miscela di solidi e fluidi the defluiscono in turbolenza in un tubo. Atkqnwr-
Baeneno HOBOS. nHTerpanbHoe npeo6pa3oIwuie H cooTseTcrBymI.&4 6ecKonewioti pm ero, 'IT0 06eClIe'IHTpelIIeHHe CHCTeMbl CBR3aIiHbIX ypaBHeHHZf ,4E44y3HH B o6nacrrr TeOpHH HHTe~palIbHOrO npeo6pasoaauna. @op~ynamia 31oro npeo6pa30~a~1fn npsr~%onrr~K npo6neMe co6craeunoro 3naYenmi, a OHaOTnHYHBOT o6bmnoro THTI~~T~~M-~OBHJ~JI~,II~~TOM~B~IBOD~TCII oco6oe HHTerpaJIbHOeyCJIOBHe, KOTOpOe CJIyWiT COOTHOIIIeHHeM OpTOrOHaJIbHOCTE. nOny'feHIiOe peIJIeHEe BO3MOHMO IIpHMeIiHTb IIpH H3ySeHHH ,I&l4y3HH B Tpy6YaTOM mope, TeIInOO6MeHa ITpK Typ6yneHTHOM IIOTOKe CMeCH XWJUCOCTH HTBepmIXTenBTpy6C.
GENERAL
Pa’gamon Press.
SOLUTIONS
Printed inGreat Britain
OF THE COUPLED EQUATIONS
DIFFUSION
M. D. MIKHAILOV Institute for Mechanical and Electrotechnical Engineering, Sofia, Bulgaria Abstract-New finite integral transform and the corresponding infinite series are introduced, which brings the solution of the coupled dithrsion equations within the realm of integral transform theory. The formulation of this transform leads to an eigenvalue problem which is not of the conventional Strum-Liuoville type and therefore a special integral condition was derived which serves as an orthogonality relation. The solution obtained can be applied when studying diffusion in a tubular reactor, heat transfer by a turbulenty flowing fluid-solids mixture in a pipe. INTRODUCTION
IN A RECENTpaper [ 11, the author presented an analytical solution of the heat equation applicable to a large class of heat transfer problems, including unsteady heat conduction and more complicated cases, namely: heat transfer in laminar and turbulent flows of Newtonian or non-Newtonian fluids in pipes, ducts; temperature development in the entrance region of MHD channels etc. The present work complements a previous paper, developing further the method of finite integral transforms and applying it for the solution of a more general mathematical model of transfer process. The solutions presented permit the studying of many new problems. The particular problems solved in [2] and [3] are shown to be a very special case of the general results derived here. STATEMENT AND SOLUTION OF THE PROBLEM The starting point of the analysis is the simultaneous consideration differentials equations w
(M)
m
of the partial
aT?ntM.7) =divtk,(M)gradT,tM,7)l+(-I)mp(M)[a,-1TI(M,~) a7
-u,+lT,(M,7)l+P,(M,7),
m=
1,2
(11
with the initial
T,tM,O) =fitM),
m= 1,2
(2)
and boundary conditions tN)
A
w
aT,tw 7) -I-B,(N)T,(N,7) an
=qm(N,7),
m= 1,2.
It is supposed that the solutions of the problem can be represented T,W,
7) = 5 W&W i=l 235
(31
by the expansions (4)
M. D. MIKHAILOV
236
where JImi(M) (m = 1,2) are eigenfunctions to the two-region Sturm-Liouville problem
A,(N)
v
+&(Nhhn(N)
= 0
(6)
their solution being granted for known. Equations (5-6) do not belong to the conventional Sturm-Liouville family and therefore, to the determination of the coefficient Ci the well-established orthogonality cannot be used directly. Therefore it is appropriate to derive an integral condition to serve as an orthogonality relation. One begins by applying the equation (5) for two distinct eigenvalues pa and /+ M~tiplying the first by ~~(~), subtracting from the result the multip~ed with ~~j(~) second equation and inte~~g the result obtained, after taking into account the well-known Gauss Theorem, which transforms the volume integral into a surface one, one gets two expressions (for m = 1 and 2 respectively). They may be written as follows
After multiplying the equation (7) for m = 1 and 2 with cl and u2 respectively and adding the results obtained one finds
For i f j because of the boundary conditions (6) the foregoing equation (8) becomes
These equation (9) is the equivalent of the conventional Sturm-Liouville orthogonality relation for the eigenfunctions JItnl(M) and I,!J~((M).For the case of i = j equation (8) leads to
237
General solutions of the coupled diffusion equations
Now, attention may be directed to finding C*. Multiplying the equation (4) for the case m = 1 by a,~, (M)+,*(M) and the same equation for the m = 2 by a2w2(M)\j12* (M) and adding the resulting expressions and integrating into V, after taking into account the orthogonality condition (9), one finds the formula for C*. Then (4) may be written as follows m 2, cm j- w~(WJl,“*(WTtn(M, 7) dV T?n(M, 7) =
&,*(M).
c
5 o7nJ w,(W,%*(M) m=1 V
i=l
(11)
dV
To solve equations (1) at the conditions (2, 3) we define the new finite integral transform F*(T) = i
atPIJ w,(WG,*T,(M.
m=1
7) dV
(12)
V
and rearrange the expressions (11) as the inversion formulas, in the form of m*
Tm(M, 7) = i
(Ml Ti (7)
(13)
dt”
*=’j, urn[ wm@O,%,(M)
After multiplying the systems (1) and (5) respectively by ~,,,JlmiM) and a,Jm(M, 7) the result of adding the four results obtained and integrating into V as when deriving (7), one gets:
2 = cm=1
!?h.?p
$a W urn il
kr, WI T, (NV7)
s
From (3) and (6) A,(N) obtained, one gets
W,
r)
ds +
$,,arUWP,(M,d I V
db’ . I
(14)
and B,(N) are determined. After summing up the results
wm* W)
~$rn*(N)
Tm
U’,,,W, 7) an
an aTm(N,7) =
Jlmi(N)
A,(N)
-v +B,(N)
cpm(N’ ‘)-
(15)
an
Substituting this result in (14), for F*(T) is obtained an ordinary first order linear differential equation, which is easily solved using the transformed initial conditions (2) according to (12). Using this solution in the inversion formulae (13), the desired solutions of the problem are obtained as follows:
umsvd.11wa2-c
238
M. D. MIKHAIL~V
(16) If B,(N) = 0 and aocr,- aluz = 0 then p = 0 is also eigenvalue of the two-region Stun-Liouville problem and the corresponding eigenfunctions tj.~~ and ?,&must satisfy the equations flOJ/lO - crz@20
=
0
~lJIlo-wb2o
=
0.
(17)
Therefore, in this case additional terms, corresponding to the zero-eigenvalue appear in the solutions and (16) takes the form
+
ylm W,
ktN) A,(N)
7) dS +
P,(M, T) dV d7 li
If the functions (om(N, 7) , P, (M, T) are constant, polynomials and exponentials of the time, the convergence of the series in solutions (16) and (18) could be improved in thes~ewayasin~l]. ONE-DIMENSIONAL
SOLUTIONS
In connection with studying the reversible reactions and the diffusion in isothermal pipe reactors [2], the heat transfer by a turbulently flowing fluid with suspended small solid particles in a circular pipe [31,let us consider the one-dimensional case, described by the differential equations system
-~,+*~2(~,7)l+P,(x,T),
m=
1,2,
X0G X < X1, 7 b 0
(19)
General solutions of the coupled diffusion equations
239
TIn(-G0) =&I(x)
(20)
at the following initial
and boundary conditions A
aTm(xo,7)
m0
ax
+BmoTrn(~~,
7) =
Am1aTmz”)+Bm
prno(T)v
Tm(xI,~) =cpml(r). (21)
Equations (5) and (6) giving the eigenvalues and eigenfunctions become
-$[,,x) y]
+(-l)mP(~)[~m-l$h(~)-~m+lJlz(~)l+PZWm(~)&n(X)=o
Arno’&n(x~)
+BmoJlrn(xo)
=
0,
Arn~#n(x~)
(22)
+&tlrlrm(x~)
(23)
=O*
The solution ( 16) for the one-dimensional case will be Tm(x,
7) = i
t
z
$mi (x) e-&F
=
i
m=1
li
urn
UJWm(x)$ki(x)dXm=l
i=l
ax
dX
Wm(x)Jlmi(xlfm(x)
Zo
7
+
I [
e+ t(Xlhll(~)
l+!lmt(X1) -* Am1 + Bm1
$m*
(-%I) -*
-km(xO)VmO(7)
Am0 + Bmo
0
II
+
I
$mt
(xlprn(xv7) dx dr *
IO
11
The solution ( 18) valid for Bmo = B,,= 0,for the one-dimensional case becomes
7
3rm0
T~(x,T) =
i,
7
umJ12mO Wm(x)
Wm(x)fm(x)
dX
z”
zo
d7
P,(x,T)dx 30
+Ii i=l
+mf(x)e-"F wm(X)ti*(X)~ i Urn7 m=1 10
1
dX+ j 0
(24)
‘40
M. D. MIKHAILOV CONCLUSIONS
The above solutions from the present paper permit the easy solving of any particular case of transfer problems in the entrance region of pipes and ducts. If the substitutions (~~~(7) = qrnl (7) = 0, P,(x, 7) = 0, x0 = 0, x1 = 1, B,, = 0 and A = 1 are made, from (24) and (25) one derives a narrower class of problems to w%h belong the problems discussed in [2] and [3]. For the case x = r, r = z, T, (x, T) = c, (r, z) , w,(x) = r( 1 - 9)) k, (x) = d,r, k2 (x) = d2r, P(X) = r, fi (x) = cloI f2 (x) = czo, ~0 = (~1= 1, gz = cr3 = 5, B,, = 0, A,, = 1 the solution of [2] is obtained given also in the monograph [4]. Forthecasex=7),7=5,T,(x,7)=B(rl,~),Tz(x,7)=~~(r) [),fi(x)=fz(x)=l, WI(X) = ~vz(x) = -v_~v), k,(x) = %drl), k,(x) = 0, go = a;, = Pa,v1 = ~3 = P3,B,l = 1, Ami = 0 follows the solution of [3] given also in [5]
NOMENCLATURE
boundary coefficient functions defined on S constant boundary coefficients on x = x0 constant boundary coefficients on x = x1 initial distribution functions in I/ one-dimensional initial distribution functions in x source functions on S source functions on x = x0 source functions on x = x1 1,2,3,4 ,..., 1or2 prescribed function defined in V prescribed function defined in x point in V point on S outward normal of S internal source function per unit time per unit volume of V internal source function per unit time per unit volume of one-dimensional region S boundary of V Tm(M, 7) unsteady potentials distributions defined in equations (l), (2) and (3) TZ (M, 7) unsteady potentials distribution defined in equations (18) co-ordinate X time variable 7 prescribed functions defined in V wn(W prescribed functions defined in x wm (Xl prescribed function defined in x P(X) prescribed constant coefficients (+m-19 ~m+1 div ( ) divergence operation in M-space grad ( ) gradient vector in M-space Jlmt(M) eigenfunctions in M-space eigenfunctions in x-space Jlmi(X) CL eigenvalues
General solutions of the coupled diffusion equations
241
REFERENCES [I] M. D. MIKHAILOV, Int. J. Engng. Sci. 10,577 (1972). [2] SIBRA P., Reversible Reaction and Diffusion in an Isothermal Tubular Reactor, Master’s thesis. University of Texas (1959). [3] TIEN C. L.,J. Heat Transfer, 183 (1961). [4] R. S. SCHECHTER, The VariationalMethod in Engineering. McGraw-Hill (1967). [5] S. L. SOO, Fluid Dynamics ofMultiphase Systems. Blaisdell(l970). (Received 1 May 1972)
R&u&-La nouvelle transformation finie integrale et la serie infinie correspondante sont introduites, ce qui apporte la solution des equations de diffusion couplde dans le domaine de la theorie de transformation integrale. La formulation de cette transformation conduit a un problbme de valeur propre qui n’est pas du type conventionnel Sturm-Liouvihe et par consequent on a derive une conventionnel Sturm-Liouvihe et par consequent on a derive une condition inttgrale speciale qui sert de relation d’orthogonahtC. La solution obtenue peut Ctre appliquee lorsqu’on ttudie la diffusion dans un reacteur tubulaire, et le tcansfert de chaleur par un melange de fluide et de solides a Ccoulement turbulent dans un tuyau. Zusammenfassung- Es werden ein neues endhiches Integraltransform und die entsprechenden unendlichen Reihen eingefuhrt, was die Losung der gekuppelten Diffusionsgleichung in den Bereich der Integrahransformtheorie bringt. Die Formulierung dieses Transforms flirt zu einem Eigenwertproblem, das nicht vom konventionellen Sturm-Liouville-Typ ist, und deshalb wurde eine spezielle Integralbedingung abgeleitet, die ah Orthogonalitltsbeziehung dient. Die erhaltene Losung kann bei der Untersuchung von Diffusion in einem Rohrreaktor angewandt werden, und der Warmeiibertragung durch eine turbulent fliessende Fliissigkeitsfeststoffmischung in einem Rohr. Sommario-Se introducono nuove serie finite di trasformazione integrale e le corrispondenti serie infinite, il the porta alla soluzione delle equazioni di diffisione accoppiata nel campo della teoria di trasformazione integrale. La formulazione di questa trasformazione porta a un problema di eigenvalore the non t del tipo di Sturm-Liouville convenzionale e percio si ricava una condizione integrale speciale the serve da rapport0 di ortogonalit~ La soluzione ricavata puo venire applicata quando si studia la dhfusione in un reattore tubolare, il trasferimento di calore mediante una miscela di solidi e fluidi the defluiscono in turbolenza in un tubo. Atkqnwr-
Baeneno HOBOS. nHTerpanbHoe npeo6pa3oIwuie H cooTseTcrBymI.&4 6ecKonewioti pm ero, 'IT0 06eClIe'IHTpelIIeHHe CHCTeMbl CBR3aIiHbIX ypaBHeHHZf ,4E44y3HH B o6nacrrr TeOpHH HHTe~palIbHOrO npeo6pasoaauna. @op~ynamia 31oro npeo6pa30~a~1fn npsr~%onrr~K npo6neMe co6craeunoro 3naYenmi, a OHaOTnHYHBOT o6bmnoro THTI~~T~~M-~OBHJ~JI~,II~~TOM~B~IBOD~TCII oco6oe HHTerpaJIbHOeyCJIOBHe, KOTOpOe CJIyWiT COOTHOIIIeHHeM OpTOrOHaJIbHOCTE. nOny'feHIiOe peIJIeHEe BO3MOHMO IIpHMeIiHTb IIpH H3ySeHHH ,I&l4y3HH B Tpy6YaTOM mope, TeIInOO6MeHa ITpK Typ6yneHTHOM IIOTOKe CMeCH XWJUCOCTH HTBepmIXTenBTpy6C.