Lecture 1

CHEMICALLY REACTING MIXTURE OF COMPRESSIBLE GASES.THE EQUATIONS OF BALANCE

CONTENTS

I. Equation of balance for i-th component

integrated form

differential form

continuity equation for a mixture

diffusion fluxes, mass concentration.

II. Momentum equation

integrated form

stress tensor, pressure

divergent form, momentum flow.

III. Energy equation

integrated form

full energy, full energy flow

equation of heat flow

equation of balance for private kinds of energy.

IV. Concept of entropy

the equation for entropy production

integrated and differential form of the entropy production equation

Equation of balance for i-th component

The multicomponent chemically reacting gas mixture is considered. Every component has internal degrees of freedom.

Let's allocate in a gas stream the volume which is based concerning motionless system of coordinates (Euler's point of view). Change of mass of i-th components occurs due to flowing this components through surface of volume and due to chemical reactions. The full mass of i-th

components is equal in the volume V. We shall write down balance of this mass

(1.1)

The first equality is received on the ground that volume V is motionless. The external normal to is used. Here vi is a velocity of i-th components. Using Gauss-Ostrogradskii formula

we shall receive

(1.2)

As volume V is arbitrary it is possible to pull together it up to small size and under the theorem of an average to receive

i = 1,2...... … N (1.3)

Let’s sum up (1.3) on all i. Entering definition of a average mass speed of a mixture ,

we shall receive

(or ), (1.4)

as , i.e. in chemical reactions the full mass of substance does not change.

Let's define a diffusion flux. (1.5)

as a difference of mass streams of i-th components and mixes as a whole.Substituting (1.5) in (1.3), we shall receive

(1.6)

or

(1.7)

Let's define a mass concentration i = i/ (mass of i-th components in a mass unit of a mixture). Substituting i instead of i in (1.7), we shall receive

The sum of the second and the third item in the left part according to (1.4) is equal to zero. Finally

where (1.8)

According to definition we have . Therefore among (1.8), only N 1 equation

are independent.

Momentum equation

We shall deduce a momentum equation for a mixture as a whole. Let’s allocate in a stream a liquid volume (Lagrange’s point of view). It moves under action of mass and stress forces.

(1.9)

(Change of momentum for liquid volume is equal to sum of operating forces). The equation (1.9) expresses the second the law of Newton for liquid volume.

Stress forces pn depend on orientation of the platform set by a normal n. Force pn can be expressed through stress forces on the platforms focused perpendicularly to axes of the cartesian system of coordinates

(1.10)According to a known rule of differentiation of integral on mobile volume, we have

(1.11)

As the liquid mass is fixed,

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We use Gauss-Ostrogradskii formula

(1.12)

For continuous movement on the basis of the theorem of an average we shall receive:

(1.13)

Here Р is the stress tensor, Р = рU , (U = unit tensor), is viscous stress tensor, р is hydrostatic (thermodynamic) pressure (i. e. pressure in a condition of rest at the same values of other parameters).

By means of identical transformations the equation (1.13) it is possible to lead to the divergent form

(1.14)

here vv is a diad, i. e. the tensor with components vivj, vv + P is density of momentum stream, F is "source" of momentum.

Energy equation

Let's allocate again motionless volume V and shall consider a balance of energy for it. As energy of system cannot be born spontaneously, the equation of balance of full energy of mass unit Е looks like

(1.15)

Usual transformation with use of Gauss-Ostrogradskii formula allows to reduce (1.15) to the divergent form

(1.16)

Energy of a mass unit of gas consists of kinetic energy and internal energy

(1.17)

The energy flux consists of convective flux of energy vE, works done by stress forces P · v (scalar product of tensor P on a vector v) and a non-mechanical flux q. Here

is -component of a vector (P · v). External power fields are not considered.Thus, the equation (1.16) becomes

(1.18)

The formula takes place

for any scalar variable.Using a full derivative the equation (1.18) can be written down so

(1.19)Multiplying (1.13) in scalar manner on v, we shall receive the equation of alive forces

(1.20)Here F 0.Subtracting (1.20) from (1.19), we shall receive the equation of heat inflow \

(1.21)Here h = e+p/ is the enthalpy of mixture. Equality is used here.

Similarly it is possible to receive the energy balance equations for private kinds of energy. From physics it is known that internal energy of gas consist of energy of translation motion of

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molecules, energy of rotary movement, energy of oscillatory movement, energy of electronic levels, etc. In the equations for a private kind of energy balance, it is necessary to consider source terms expressing speed of transition of one kind of energy into another. For example, at collisions of molecules, there can be a partial not elastic transformation of translation energy to energy of molecular oscillations. Speed of transformation of energy from one kind in another depends on physical (molecular) properties of considered gas.

For example, we shall consider balance of oscillatory energy evi for mass units of i-th components. We shall made again fixed volume V. We have

(1.22)Here is speed of formation of oscillatory energy inside of volume, qvi is non-mechanical flux of oscillatory energy. Making usual transformations, we shall receive

(1.23)Here Ji, i are diffusion flux and speed of formation i-th components. In perfect gas Ji = 0, i = 0 and oscillatory energy changes due to carry qv and transition from other kinds of energy v.

Concept of entropy, the equation of entropy production For the systems consisting of huge number of elements, the thermodynamics allows to

enter one more function of a condition. It is the entropy. At reversible processes in the isolated systems, entropy is constant, at irreversible it increases. A reversible process is process during which the system passes a series of equilibrium states.

For opened thermodynamic system the entropy gain consists of the contribution acting from an environment, and the contribution arising in the system.

Let's enter the entropy S of mass units of mixture. As thermodynamic system we shall consider a particle of gas, not so greater that it would be possible to consider it as homogeneous and enough greater that it contained many molecules of gas. Then it is possible to treat the entropy in the same manner as density or internal energy.

Choosing motionless volume V, we shall count up change in time the entropy of this volume, considering transfer the entropy with gas through surface of volume and its "birth" inside of volume

(1.24)Passing to small volume, for continuous movements we shall receive

(1.25)Stream Js consists of mechanical transfer (vS) and other kinds of entropy transfer: Js = vS + qs.

Certainly, use only fundamental laws of physics has not allowed us to receive the closed system of the equations for movement of the continuous media. For this purpose some additional hypotheses are necessary.

The equation (1.25) can be written down also through a full derivative

(1.26)

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