APPLICATION OF ORDINARY DIFFERENTIAL
EQUATIONS IN SOLVING DIFFUSIVE PROBLEM
Tkachuk A.M., Mazur O.K.
Ukraine, Kyiv, National University of
Abstract: It is investigated functional
dependence, which expresses changing’s of concentration of
expanded fluid in thickness of diffusive stratum. A solution of the given
problem has been received as a solution of an ordinary second-order
differential equation i.e. as a solution of Cauchy problem.
Keywords: ordinary second-order differential equation,
diffusion, chemical reaction, general solution, Cauchy problem.
функціональну залежність, яка виражає зміну концентрації розчиненого газу по
товщині дифузійного шару. Отримано розв’язок даної задачі
у вигляді розв’язку звичайного диференціального рівняння другого порядку, а
саме розв’язку задачі Коші.
диференціальне рівняння другого порядку, дифузія, хімічна реакція, загальний
розв’язок, задача Коші.
Mathematical models of many engineering and technological processes can
be equated to a single differential equation or to their system. For instance,
studying kinetics of ion exchange in processes of flowing extraction, filtering
of liquid, analyzing settling of solid particles the significant results were
brought by wide application of differential equations. Therefore
qualitative theory of differential equations is an effective mathematical
instrument for describing of phenomenon in many fields. The usage of
differential equations defines their practical value. Owing to their usage
it is possible to set up a connection between a basic physical or
chemical law and often a whole group of variables, which have major
significance studying the food processes.
This article is dedicated to the construction of the mathematical model
of the diffusion problem which is accompanied by a chemical reaction. The diffusion’s
speed in the liquid is proportional to the concentration gradient.
We consider a diffusion layer of liquid which bounds
with an interphase bound fluid - liquid. It is necessary
to find the functional dependency which expresses a change of concentration of soluble
fluid in thickness of the diffusion stratum.
Conditions of the process are the same in any plane that is perpendicular
to the direction of diffusion. Let us separate in the bounded layer an element with
its thickness . This element is bounded by
planes that are parallel to the plane of distinction of phases and they have
been constructed at the distance and from this
plane. We should make a material balance for this element. The area of this
element is assumed to be equal to the unity.
The diffusion rate at points of the plane that is distant from the
plane of distinction of phases will be equal to
where is the diffusion coefficient, is
concentration of fluid in liquid at the depth of Since
concentration decreases in the direction of the diffusion flow then the diffusion
coefficient should be taken with the negative sign.
Quantity of passing fluid through this plane during the time is
Likewise, quantity of passing fluid
through an opposite bound of the elementary layer that is distant from the
plane of distinction of phases will equal:
since the concentration gradient in this plane
Fluid interacts with liquid in time of the diffusion
through an elementary volume. The rate of this interaction is proportional to
its quantity which this layer contains. Note the volume of the considered
element equals , it implies the quantity of the
passing substance through the elementary volume will be obtained by multiplying
of this volume by concentration . Nevertheless the rate of the
chemical reaction is proportional to concentration, therefore it is equal to ,
where is a constant of the rate of the
Thus, quantity of passing fluid that enters into the chemical reaction in the elementary volume for
the time becomes equal to the product
Therefore an equation of the material balance will
have the following form:
After simplifying we will obtain .
For solving of this
equation we substitute for then it will be
got the following form
Equation (1) is an ordinary second-order
differential equation with constant coefficients. Whereas the look of roots of
the characteristic equation, we will receive a general solution of equation (1):
Assume the concentration of fluid in the bounded layer and in the layer that
from the bounded one be known. In other words, . Solving the system of
constants and we will get
Therefore, the solution of equation (2) will have the following form
Thus, a Cauchy
problem has been solved.
Conclusion: The functional dependence has been found, which shows a
change of concentration of soluble gas in thickness of the diffusion layer in
the form of a second-order differential equation with constant coefficients. In
case, concentration of gas in the boundary layer and in the layer at distance from the
boundary one is known, then a solution of Cauchy problem has been received.