Equations of Mass Transfer in Quiescent or Moving Media with Chemical Reactions

2.5.1. Equations of Mass Transfer in Quiescent or Moving Media with Chemical Reactions

∂w ∂ 2 w

2 – ∂t f (w). ∂x ∂y ∂z This equation describes unsteady mass or heat transfer with a volume reaction in a quiescent medium.

The equation admits translations in any of the variables x, y, z, t.

1 ◦ . There is a traveling-wave solution, w = w(k 1 x+k 2 y+k 3 z + λt).

following forms in cylindrical and spherical coordinates, respectively:

3 ◦ . “Three-dimensional” solution:

2 2 2 w = u(ξ, η, t), 2 ξ=y+ , η = (C − 1) x −2 Cxy + C z ,

where

C is an arbitrary constant (C ≠ 0), and the function u = u(ξ, η, t) is determined by the equation

2 2 ∂u 2 1 u ∂ u

2 ∂u

+4 C ( ξ + η) 2 + 2(2 C − 1) − f (u). ∂t

∂η Remark. The solution specified in Item 3 ◦ can be used to obtain other “three-dimensional”

solutions by means of the cyclic permutations of the space variables.

4 ◦ . “Three-dimensional” solution:

w = u(ξ, η, t), 2 ξ = Ax + By + Cz, η= ( Bx − Ay) +( Cy − Bz) 2 +( Az − Cx) 2 , where

A, B, and C are arbitrary constants and the function u = u(ξ, η, t) is determined by the equation

− f (u).

2 ∂t + ∂x ∂y ∂z 2 + f (t)w ln w + g(t)w.

1 ◦ . There is a functional separable solution of the form

X w(x, y, z, t) = exp X ϕ

nm ( t)x n x m +

ψ n ( t)x n + χ(t) ,

x 1 = x, x 2 = y, x 3 = z.

n,m=1

n=1

2 ◦ . There is a incomplete separable solution of the form

w(x, y, z, t) = Φ 1 ( x, t)Φ 2 ( y, t)Φ 3 ( z, t).

3 ◦ . For f (t) = b = const, the equation also has a multiplicative separable solution of the form w(x, y, z, t) = ϕ(t)Θ(x, y, z),

where ϕ(t) is given by

ϕ(t) = exp Ae bt + e bt

e bt g(t) dt ,

A is an arbitrary constant, and Θ(x, y, z) is a solution of the stationary equation

– f (w).

This equation describes unsteady mass transfer with a volume chemical reaction in a steady trans- lational fluid flow.

The transformation

ζ=z−a 3 t leads to a simpler equation of the form 2.5.1.1:

w = U (ξ, η, ζ, t),

ξ=x−a 1 t,

η=y−a 2 t,

2 2 ∂U 2 ∂ U ∂ U ∂ U

− f (U ).

∂t

2 ∂y + ∂z 2 – g(w). This equation describes unsteady mass transfer with a volume chemical reaction in an unsteady

translational fluid flow. The transformation

Z w = U (ξ, η, t),

f 3 ( t) dt leads to a simpler equation of the form 2.5.1.1:

2 ∂z + ∂y 2 ∂x + ∂z 2 – f (w). This equation describes unsteady mass transfer with a volume chemical reaction in a three-

+( a 3 x+b 3 y+c 3 z+d 3 )

dimensional steady translational-shear fluid flow.

1 ◦ . Let λ be a root of the cubic equation

b 3 = 0,

and let the constants A 1 , A 2 , and A 3 solve the degenerate system of linear algebraic equations

One of these equations can be omitted, since it is a consequence of the other two. Suppose w(x, y, z, t) is a solution of the equation in question. Then the function

w 1 = wx+A 1 Ce λt , y+A 2 Ce λt , z+A 3 Ce λt , t

where

C is an arbitrary constant, λ is a root of the cubic equation (1), and A 1 , A 2 , and A 3 are the corresponding solution of the algebraic system (2), is also a solution of the equation.

2 ◦ . Solution:

w = w(ξ), ξ=A 1 x+A 2 y+A 3 z + Ce λt ,

where

C is an arbitrary constant, λ is a root of the cubic equation (1), A 1 , A 2 , and A 3 are the corresponding solution of the algebraic system (2), and the function w(ξ) is determined by the ordinary differential equation

( λξ + A 1 d 1 + A 2 d 2 2 2 + 2 A 3 d 3 ) w ′ ξ =( A 1 + A 2 + A 3 ) w ′′ ξξ − f (w).

3 ◦ . Let λ be a root of the cubic equation (1) and let A 1 , A 2 , and A 3 be the corresponding solution of the algebraic system (2). “Two-dimensional” solution:

w = U (ζ, t),

ζ=A 1 x+A 2 y+A 3 z,

where the function U (ζ, t) is determined by the differential equation ∂U 2 ∂U

− f (U ). Remark. In the case of an incompressible fluid, the equation coefficients must satisfy the

condition a 1 + b 2 + c 3 = 0.

∂y ∂z ∂z This equation describes unsteady anisotropic mass or heat transfer in a three-dimensional steady

translational-shear fluid flow.

1 ◦ . Let λ be a root of the cubic equation

b 3 = 0,

and the constants A 1 , A 2 , and A 3 solve the degenerate system of linear algebraic equations

One of these equations can be omitted, since it is a consequence of the other two. Suppose w(x, y, z, t) is a solution of the equation in question. Then the function

w 1 = wx+A 1 Ce λt , y+A 2 Ce λt , z+A 3 Ce λt , t

where

C is an arbitrary constant, λ is a root of the cubic equation (1), and A 1 , A 2 , and A 3 are the corresponding solution of the algebraic system (2), is also a solution of the equation.

2 ◦ . Solution:

(3) where

w = w(ξ), ξ=A 1 x+A 2 y+A 3 z + Ce λt ,

C is an arbitrary constant, λ is a root of the cubic equation (1), and A 1 , A 2 , and A 3 are the corresponding solution of the algebraic system (2), and the function w(ξ) is determined by the ordinary differential equation

( λξ + A 1 d 1 + A 2 d 2 + A 3 d 3 ) w ′ ξ =[ ϕ(w)w ′ ξ ] ′ ξ ,

2 2 ϕ(w) = A 2

1 f 1 ( w) + A 2 f 2 ( w) + A 3 f 3 ( w).

3 ◦ . Let λ be a root of the cubic equation (1) and let A 1 , A 2 , and A 3 be the corresponding solution of the algebraic system (2). “Two-dimensional” solution:

(4) where the function U (ζ, t) is determined by the differential equation

1 f 1 ( U)+A 2 f 2 ( U)+A 3 f 3 ( U ).

Remark 1. A more general equation, with an additional term g(w) on the right-hand side, where

g is an arbitrary function, also has solutions of the forms (3) and (4). Remark 2. In the case of an incompressible fluid, the equation coefficients must satisfy the

condition a 1 + b 2 + c 3 = 0.