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Unformatted text preview: 1 ME5446 Essay 15 CHEMICAL REACTIONS IN A FLOWING FLUID – CONTINUTED THE GOVERNING CONSERVATION EQUATIONS AND EQUATION OF STATE From Essay 1 4 , conservation of mass for the flowing mixture is expressed by = + dx du dx d u ρ ρ ( 1 ) For each of the components i , conservation of mass is ( ) i i i i i S dx W d dx dW dx d dx du dx d u & = − − + 2 2 D D ρ ρ ρ ρ ( 2 ) where D is the mass diffusion coefficient and i S & is either a source or a sink of i , depending on whether the reaction is producing i or depleting i . There are two more conservation laws to be considered, conservation of momentum (Newton’s Second Law) and energy conservation (First Law of Thermodynamics). First, with the aid of the diagram below, the momentum conservation law will be evaluated. In words, Newton’s Second Law for a control volume can be stated as Net xdirection forces = Rate of momentum outflow – Rate of momentum inflow (3) The forces and momenta are indicating in the diagram. The crosssectional area normal to the direction of fluid flow is A, and the area of the rim of the control volume is . Ddx π The net force in the positive x direction is 2 ( ) Ddx dx dx dp A Ddx p p A Ddx A p A p F x dx x dx x x x τπ τπ τπ − − = − − − = − − = + + (4) Next, the net rate of outflow of momentum is ( ) ( ) ( ) dx dx u m d u m u m x dx x & & & = − + ( 5 ) When Eqs. (4) and (5) are brought together to complete Newton’s Second Law, there follows ( ) dx dx u m d Ddx dx dx dp A & = − − τπ ( 6 ) With 4 2 D A π = and , Au...
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 Spring '11
 Sparrow
 Thermodynamics, Combustion, dx

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