0040-5795/04/3805- © 2004

MAIK “Nauka

/Interperiodica”0478

Theoretical Foundations of Chemical Engineering, Vol. 38, No. 5, 2004, pp. 478–482.Translated from Teoreticheskie Osnovy Khimicheskoi Tekhnologii, Vol. 38, No. 5, 2004, pp. 506–510.Original Russian Text Copyright © 2004 by Vyaz’min, Polyanin, Sysoev.

In theoretical consideration of chemical engineeringproblems, of particular importance are exact solutionsof unsteady-state mass-transfer equations, since theyallow one to model transient modes of operation ofchemical reactors and other equipment. Such solutionsenable one to form correct notions of the qualitativepattern of heat or mass transfer and, eventually, deter-mine the optimal time of contact between phases, whenthe process is highly nonequilibrium and the heat- andmass-transfer coefficients are large [1]. Moreover,under real industrial conditions, unsteady-state pro-cesses are much more frequent than steady-state ones[2].

Exact solutions are particularly significant for cor-rectly describing the heat or mass transfer in chemicallyreactive media. Taking into account the complex kinet-ics of chemical reactions makes these equations nonlin-ear. On the other hand, chemical reactions give rise tohigh heat or concentration gradients on the scale of theproblem, when the transfer coefficients cannot be con-sidered constant [3]. Here, there is a nonlinearity ofanother type, which is related to the nonlinear depen-dence of the transfer coefficients on temperature orconcentration. As a rule, this is the nonlinearity ofpower-law or exponential type, which allows one tosolve such problems by widely using various approxi-mate or asymptotic methods [3, 4]. However, whenconstructing and testing them, one cannot dispensewith exact solutions of the corresponding equations,which enable one to evaluate the reliability and accu-racy of the results obtained. When computer algebra isused to solve chemical engineering problems, it is nec-essary to construct exact analytical solutions of heat-and mass-transfer equations involving functional non-linearities.

Exact solutions of linear unsteady-state heat- andmass-transfer equations and methods for their deriva-tion have been quite well studied [5–7]. Nonlinear

mass-transfer equations have also recently receivedvery serious attention. Until recently, the most widelyused and efficient methods for constructing exact solu-tions of such equations have been group methods [8, 9].Methods of generalized and functional separation ofvariables enabled one to obtain a large number of newexact solutions of nonlinear heat- and mass-transfer andhydrodynamic equations [10–12]. Note also the directClarkson–Kruskal method, which allows one to con-struct noninvariant solutions [13].

NONLINEAR HEAT- AND MASS-TRANSFER EQUATION WITH AN ARBITRARY FUNCTION

Applications often involve the Kolmogorov–Petro-vskii–Piskunov equation, an unsteady-state heat- andmass-transfer equation with a nonlinear kinetic func-tion which describes the heat or mass transfer in a stag-nant medium with bulk chemical reaction:

(1)

where

C

is concentration (temperature);

t

is time;

x

is acoordinate; and

g

(

C

)

is the kinetic function of thechemical reaction, which is considered arbitrary. Previ-ously [11], a number of exact solutions of this equationat different kinetic functions were described.

If the dependence of the transfer coefficient

f

(

C

)

onconcentration (temperature) is arbitrary, then instead ofEq. (1), the following equation is used:

(2)

Since this equation is particularly significant fordescribing the heat and mass transfer in chemicallyreactive media, let us consider its exact solutions inmore detail.

∂C∂t------- ∂2C

∂x2--------- g C( ),+=

∂C∂t-------

∂∂x------ f C( )∂C

∂x------- g C( ).+=

Exact Solutions of Nonlinear Unsteady-State Equations of Mass and Heat Transfer with Bulk Reaction

A. V. Vyaz’min*, D. A. Polyanin**, and P. V. Sysoev**

* Karpov Institute of Physical Chemistry, Russian State Scientific Center,ul. Vorontsovo Pole 10, Moscow, 105064 Russia

** Moscow State University, Vorob’evy gory, Moscow, 119899 Russia

Received April 14, 2004

Abstract

—New exact solutions of some nonlinear unsteady-state mass- and heat-transfer equations areobtained by functional separation of variables. Nonlinearities related to the complex kinetics of chemical reac-tions and the dependence of the transfer coefficients on concentration or temperature are considered. Particularattention is given to the solutions of equations containing arbitrary functions.

THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING

Vol. 38

No. 5

2004

EXACT SOLUTIONS OF NONLINEAR UNSTEADY-STATE EQUATIONS OF MASS 479

Exact solutions and transformations (group classifi-cation) of Eq. (2) with functions

f

(

C

)

and

g

(

C

)

of differ-ent forms were obtained earlier [9, 14–16].

Below, we present several exact solutions of Eq. (2)with functional separation of variables of two types:

(1) Let the function

f

(

C

)

be arbitrary. Then, if thefunction

g

(

C

)

is given by the expression

where

A

1

and

A

2

are arbitrary constants,

C

as a functionof

x

and

t

is found from the implicit expression [17]

Here,

B

1

and

B

2

are arbitrary constants.

(2) Let the functions

f

(

C

)

and

g

(

C

)

be given by theexpressions

in terms of an arbitrary function

ϕ

(

C

)

and constant

A

.The dependence of

C

on

x

and

t

is given by the implicitexpression [17]

where

B

1

and

B

2

are arbitrary constants.

(3) Let us find exact solutions of Eq. (2) with func-tional separation of variables of the particular form [11]

(3)

It is necessary to find the functions

C

(

z

),

ϕ

(

t

),

and

ψ

(

t

)

and the right-hand side of the equation for

g

(

C

)

.Substitution of expression (3) into Eq. (2) with the

subsequent division of the result by yields

(4)

Let us express

x

in expression (3) through

z

and substi-tute the result into Eq. (4). This gives the functional dif-ferential equation in two variables,

t

and

z

:

(5)

Equation (5) is a functional equation of the form

(6)

where all

Φ

i

are functions of one argument, and all

Ψ

i

,of the other.

C C Z1( ), Z1 ϕ x( ) ψ t( );+= =

C C Z2( ), Z2 ϕ t( )x ψ t( ).+= =

g C( )A1

f C( )------------ A2,+=

f C( ) Cd∫ A1t12---A2x2– B1x B2.+ +=

f C( ) CϕC' C( ), g C( ) A C 2ϕ C( )ϕC' C( )---------------+

,= =

ϕ C( ) B1e2At 12---A x B2+( )2,–=

C C z( ), z ϕ t( )x ψ t( ).+= =

Cz'

ϕt'x ψt'+ ϕ2 f C( )Cz'[ ]z'

Cz'------------------------- g C( )

Cz'------------.+=

ψt'–ψϕ----ϕt'

ϕt'

ϕ----z– ϕ2 f C( )Cz'[ ]z

'

Cz'------------------------- g C( )

Cz'------------++ + 0.=

Φ1Ψ1 Φ2Ψ2 Φ3Ψ3 Φ4Ψ4+ + + 0,=

Equation (5) is reduced to Eq. (6) by taking

(7)

Equation (6) has solutions of the form

(8)

where A1, A2, A3, and A4 are arbitrary constants. Substi-tution of expressions (7) into expressions (8) yields theset of ordinary differential equations

(9)

If A4 = 0 and the function g = g(C) is arbitrary, thenf = f(C) is given by the expressions

(10)

(11)

In this case, the solution of Eq. (2) has the form

where the function C(Z) is found by inverting function(11) and B, B1, and B2 are arbitrary constants.

If A4 ≠ 0 and the function g = g(C) is arbitrary, thenf = f(C) is given by the expressions

(12)

(13)

In this case, the solution has the form

where the function C(Z) is found by inverting fun-ction (13) and

Here, B1 and B2 are arbitrary constants. In the particularcase g(C) = k = const, one can obtain f(C) = B1e2kC +(B2C + B3)ekC.

Φ1 = ψt'–ψϕ----ϕt', Φ2+ =

ϕt'

ϕ----, Φ3– = ϕ2

, Φ4 = 1,

Ψ1 = 1, Ψ2 = z, Ψ3 = f C( )Cz'[ ]z

'

Cz'-------------------------, Ψ4 =

g C( )Cz'

------------.

Φ1 A1Φ3 A2Φ4, Φ2+ A3Φ3 A4Φ4,+= =

Ψ3 A1Ψ1 A3Ψ2, Ψ4–– A2Ψ1– A4Ψ2,–= =

ψt'–ψϕ----ϕt'+ A1ϕ2 A2,

ϕt'

ϕ----–+ A3ϕ2 A4,+= =

f C( )Cz'[ ]z'

Cz'------------------------- A1 A3z,

g C( )Cz'

------------–– A2– A4z.–= =

f C( )A1A2C B+

g C( )--------------------------

A2A3

g C( )------------ Z C,d∫+=

Z A2Cd

g C( )------------.∫–=

C = C Z( ), Z = x± B2+

2A3t B1+---------------------------

A1

A3------

A2

3A3--------- 2A3t B1+( ),––

f C( ) 1g C( )------------ A1C A3 Z Cd∫+( ) A4

Cdg C( )------------∫– ,exp=

Z1A4------ A4

Cdg C( )------------∫–exp=

A2

A4------.–

C C Z( ), Z ϕ t( )x ψ t( ),+= =

ϕ t( ) B1e2A4t A3

A4------–

1/2–

± ,=

ψ t( ) ϕ t( ) A1 ϕ t( ) t A2td

ϕ t( )----------∫ B2+ +d∫ .–=

480

THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 38 No. 5 2004

VYAZ’MIN et al.

NONLINEAR EQUATIONSOF CONVECTIVE MASS TRANSFER

Let us consider the equation of unsteady-state two-dimensional convective mass transfer in an anisotropicmedium with bulk chemical reaction described by anarbitrary kinetic function in an axisymmetric transla-tional shear flow:

(14)

Here, a, b1, and b2 are constants, which characterize thehydrodynamic flow pattern.

This equation allows exact solutions of the form

(15)

where k1, k2, and λ are arbitrary constants and the func-tion C(z) is described by the ordinary differential equa-tion

(16)

In the particular case λ = 0, there is a steady-state solu-tion of traveling-wave form (15)–(16).

Note also that Eq. (14) also allows “two-dimen-sional exact” solutions of the form

where the function U(ξ, t) is described by the equation

UNSTEADY-STATE MASS-TRANSFER EQUATIONS WITH EXPONENTIAL

AND LOGARITHMIC NONLINEARITIESChemical engineering problems are often solved

using various extrapolations of the transfer coefficientsand the kinetic functions as functions of concentration.

(1) Let us consider the unsteady-state equation ofdiffusion with bulk chemical reaction with exponentialnonlinearity:

Direct substitution shows that this equation has theexact solution

where B1 and B2 are arbitrary constants.

∂C∂t------- ax b1+( )∂C

∂x------- ay b2+( )∂C

∂y-------+ +

= ∂

∂x------ f 1 C( )∂C

∂x------- ∂

∂y----- f 2 C( )∂C

∂y------- g C( ).–+

C C z( ), z k1x k2y λeat,+ += =

az b1k1 b2k2+ +( )C '

= k12

f 1 C( ) k22

f 2 C( )C '+( )[ ]' g C( ).–

C U ξ t,( ), ξ k1x k2y,+= =

∂U∂t------- aξ b1k1 b2k2+ +( )∂U

∂ξ-------+

= ∂

∂ξ------ k1

2 f 1 U( ) k22 f 2 U( )+( )∂U

∂ξ------- g U( ).–

∂C∂t------- a

∂∂x------ CeλC ∂C

∂x-------

be λC– .+=

C x t,( ) 1λ--- B1x

aB12

λ--------- bλ+

t B2+ +ln ,=

(2) The equation with the kinetic function of mixedtype,

has the exact solution of the form

(3) Let us now consider the equation with logarith-mic nonlinearity:

One can show that its exact solution is the function

where B1 and B2 are arbitrary constants.(4) A more complex equation,

has the exact solution

SETS OF EQUATIONS OF MASS TRANSFER WITH TWO-COMPONENT NONLINEAR

CHEMICAL REACTIONS

Among chemical engineering problems, there areoften problems with two-component chemical reac-tions, when the reaction rate depends on the concentra-tions of two reacting substances.

(1) Let us consider the set of two unsteady-state dif-fusion equations coupled by the kinetic function ofchemical reaction:

At f(z) = kz–m and g(z) = – kzn – m, this set describes achemical reaction of order n (of order n – m withrespect to the component C1 and of order m with respectto the component C2). The values n = 2 and m = 1 cor-

∂C∂t------- a

∂∂x------ CeλC ∂C

∂x-------

b ce λC–+ +=

C x t,( ) 1λ--- B1ebλt x

aB12

bλ2---------e

2bλtB2ebλt c

b---–+ +

.ln=

∂C∂t-------

∂∂x------ a C b+ln( )∂C

∂x------- cC.+=

C x t,( )B2 x±B1 2at–

------------------------c

3a------ B1 2at–( )– a b+

a------------– ,exp=

∂C∂t-------

∂∂x------ a C b+ln( )∂C

∂x------- cC C sC+ln+=

C x t,( ) ϕ t( ) B1 x±( ) -exp=

+ a b+( )ϕ t( ) ϕ t( ) t sϕ t( ) tdϕ t( )----------∫+d ,∫

ϕ t( ) B2e 2ct– ac---–

1/2–

.=

∂C1

∂t--------- a

∂2C1

∂x2----------- C1

k fC1

C2------

,+=

∂C2

∂t--------- b

∂2C2

∂x2----------- C2

kgC1

C2------

.+=

THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 38 No. 5 2004

EXACT SOLUTIONS OF NONLINEAR UNSTEADY-STATE EQUATIONS OF MASS 481

respond to the frequently occurring case of a second-order reaction.

This system has the self-similar solution of the form

where B1, B2, and B3 are arbitrary constants and thefunctions Y = Y(ξ) and Z = Z(ξ) are found from the setof ordinary differential equations

(2) Another similar set has the form

This set allows the self-similar solution

where the functions Y = Y(ξ) and Z = Z(ξ) are found fromthe set of ordinary differential equations

(3) Let us now consider the steady-state convectivemass transfer in a deformation shear flow in the casewhere the dependence of the kinetic functions on con-centrations is arbitrary:

The exact solution is

C1 B1t B2+( )1

1 k–-----------

Y ξ( ),=

C2 B1t B2+( )1

1 k–-----------

Z ξ( ), ξx B3+

B1t B2+------------------------,= =

aYξξ'' 12---B1ξYξ'

B1

k 1–-----------Y Yf Y /Z( )+ + + 0,=

bZξξ'' 12---B1ξZξ'

B1

k 1–-----------Z Zg Y /Z( )+ + + 0.=

∂C1

∂t--------- a

∂2C1

∂x2----------- C1

1 kn+ f C1nC2

m( ),+=

∂C2

∂t--------- b

∂2C2

∂x2----------- C2

1 km– g C1nC2

m( ).+=

C1 B1t B2+( )1

kn------–

Y ξ( ),=

C2 B1t B2+( )1

km-------

Z ξ( ), ξx B3+

B1t B2+------------------------,= =

aYξξ'' 12---B1ξYξ'

B1

kn------Y Y1 kn+ f YnZm( )+ + + 0,=

bZξξ'' 12---B1ξZξ'

B1

km-------Z Z1 km– g YnZm( )+–+ 0.=

ax∂C1

∂x--------- ay

∂C1

∂y---------+

∂2C1

∂x2-----------

∂2C1

∂y2----------- f C1 C2,( ),+ +=

ax∂C2

∂x--------- ay

∂C2

∂y---------+

∂2C2

∂x2-----------

∂2C2

∂y2----------- g C1 C2,( ).+ +=

C1 x y,( ) = U z( ), C2 x y,( ) = W z( ), z = k1x k2y,+

where k1 and k2 are arbitrary constants and the functionsU = U(z) and W = W(z) are found from the set of ordi-nary differential equations

ACKNOWLEDGMENTS

This work was supported in part by the RussianFoundation for Basic Research (project nos. 02-02-16294 and 04-02-17281).

REFERENCES

1. Protsessy i apparaty khimicheskoi tekhnologii. Yav-leniya perenosa, makrokinetika, podobie, modelirovaniei proektirovanie (Unit Operations and Equipment ofChemical Engineering: Transfer Phenomena, Macroki-netics, Similarity, Modeling, and Design), vol. 1: Osnovyteorii protsessov khimicheskoi tekhnologii (Foundationsof the Theory of Processes of Chemical Engineering),Kutepov, A.M., Ed., Moscow: Logos, 2000.

2. Planovskii, A.N. and Nikolaev, P.I., Protsessy i apparatykhimicheskoi i neftekhimicheskoi tekhnologii (UnitOperations and Equipment of Chemical and Petrochem-ical Engineering), Moscow: Khimiya, 1987.

3. Kutepov, A.M., Polyanin, A.D., Zapryanov, Z.D.,Vyaz’min, A.V., and Kazenin, D.A., Khimicheskayagidrodinamika. Spravochnoe posobie (Chemical Hydro-dynamics: Handbook), Moscow: Kvantum, 1996.

4. Polyanin, A.D., Kutepov, A.M., Vyazmin, A.V., andKazenin, D.A., Hydrodynamics, Mass and Heat Transferin Chemical Engineering, London: Taylor & Francis,2002.

5. Lykov, A.V., Teoriya teploprovodnosti (Theory of HeatConduction), Moscow: Vysshaya Shkola, 1967.

6. Carslaw, H.S. and Jaeger, J.C., Conduction of Heat inSolids, Oxford: Clarendon, 1959. Translated under thetitle Teploprovodnost’ tverdykh tel, Moscow: Nauka,1964.

7. Polyanin, A.D., Vyaz’min, A.V., Zhurov, A.I., and Kaze-nin, D.A., Spravochnik po tochnym resheniyam uravne-nii teplo- i massoperenosa (Handbook of Exact Solu-tions of Heat and Mass Transfer Equations), Moscow:Faktorial, 1998.

8. Ovsyannikov, L.V., Gruppovoi analiz differentsial’nykhuravnenii (Group Analysis of Differential Equations),Moscow: Nauka, 1978.

9. CRC Hanbdook of Lie Group Analysis of DifferentialEquations, Ibragimov, N.H., Ed., Boca Raton: CRC,1994, vol. 1.

10. Galaktionov, V.A., Invariant Subspace and New ExplicitSolutions to Evolution Equations with Quadratic Non-linearities, Proc. Roy. Soc. Edinburgh, 1995, vol. 125A,no. 2, pp. 225–448.

11. Polyanin, A.D. and Zaitsev, V.F., Spravochnik po neli-neinym uravneniyam matematicheskoi fiziki: Tochnyeresheniya (Handbook of Nonlinear Equations of Mathe-

azU' k12 k2

2+( )U'' f U W,( ),+=

azW' k12 k2

2+( )W'' g U W,( ).+=

482

THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 38 No. 5 2004

VYAZ’MIN et al.

matical Physics: Exact Solutions), Moscow: Fizmatlit,2002.

12. Polyanin, A.D., Transformations and Exact Solutions ofBoundary Layer Equations with Arbitrary Functions,Dokl. Akad. Nauk, 2001, vol. 379, no. 3, pp. 334–339.

13. Clarkson, P.A. and Kruskal, M.D., New SimilarityReductions of the Boussinesq Equation, J. Math. Phys.,1989, vol. 30, no. 10, pp. 2201–2213.

14. Dorodnitsyn, V.A., Invariant Solutions of the NonlinearHeat Equation with a Source, Zh. Vychisl. Mat. Mat. Fiz.,1982, vol. 22, no. 6, pp. 1393–1400.

15. Dorodnitsyn, V.A. and Svirshchevskii, S.R., InvariantSolutions of the Nonlinear Heat Equation with a Source,Preprint of Inst. of Appl. Mat., USSR Acad. Sci., 1983,no. 101.

16. Galaktionov, V.A., Dorodnitsyn, V.A., Elenin, G.G.,Kurdyumov, S.P., and Samarskii, A.A., Quasi-LinearHeat Equation: Blow-up, Localization, Symmetry, ExactSolutions, Asymptotics, and Structures, in Sovremennyeproblemy matematiki (Modern Problems of Mathemat-ics), Moscow: VINITI, 1986, vol. 28, pp. 95–206.

17. Galaktionov, V.A., Quasilinear Heat Equations withFirst-Order Sign-Invariants and New Explicit Solutions,Nonlinear Anal. Theory Meth. Appl., 1994, vol. 23,pp. 1595–1621.

18. Zaitsev, V.F. and Polyanin, A.D., Spravochnik po obyk-novennym differentsial’nym uravneniyam (Handbook ofOrdinary Differential Equations), Moscow: Fizmatlit,2001.