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Unformatted text preview: 4A8: Environmental Fluid Mechanics Mixing and Reactions in Turbulent Flows 17 3. Statistical description of turbulent mixing In this Chapter, we will derive the governing equation for a reacting scalar in a turbulent flow and we will demonstrate why the turbulence affects the mean reaction rate. We will also present concepts from probability theory that are useful for understanding why the concentration fluctuations are important in environmental pollution and for providing measures to describe these. The material here is needed background for understanding the practical Air Quality Modelling techniques introduced in later Chapters. 3.1 Governing equation for a reacting scalar Conservation of mass Consider an infinitesimal control volume ∆ V (Fig. 3.1). Inside the CV we have a uniform mixture of species undergoing chemical reactions. Mass may cross the surfaces of the CV. For simplicity of presentation we assume a one-dimensional geometry. Then, the principle of mass conservation of each species i reads: [Rate of accumulation] = [Rate at which species comes in] – [Rate at which species leaves] + [Rate of generation due to reaction] In mathematical terms, V w z y m m z y m t Y m i i i i i V ∆ + ∆ ∆ ′ ′ ∆ + ′ ′- ∆ ∆ ′ ′ = ∂ ∂ ) ( ) ( (3.1) with the following definitions: m V (kg) total mass of mixture inside the control volume, m V = ρ ∆ x ∆ y ∆ z Y i (-) mass fraction of i ρ (kg m-3 ) mixture density i m ′ ′ (kg m-2 s-1 ) mass flow of species i per unit time per unit surface, the mass flux i w (kg m-3 s-1 ) mass of species generated per unit volume per unit time due to chemical reactions T,Y,u m " i dV=dxdydz ρ Chemical reaction T+dT Y + dY u+du m"i+dm"i + d ρ ρ Figure 3.1 Control volume for derivation of species conservation equation. 4A8: Environmental Fluid Mechanics Mixing and Reactions in Turbulent Flows 18 Letting ∆ x go to zero, we obtain the species conservation equation : i i i w x m t Y + ∂ ′ ′ ∂- = ∂ ∂ ) ( ρ (3.2) Equation (3.2) is a partial differential equation (in time and space) and to be in a position to solve it, we need expressions for the mass flux and the rate of generation due to chemistry. The latter was covered in Section 2.1, while the former is discussed next. Mass flux, mass transfer and Fick’s Law of diffusion The mass flux i m ′ ′ for each species that appears in the species conservation equation is composed of two parts: an advective and a diffusive part. This result is given here without proof, as it can be proven from the Kinetic Theory of Gases (4A9, Part IIB). DIFF i ADV i i m m m , , ′ ′ + ′ ′ = ′ ′ (3.3) The advective mass flux is due to the bulk fluid motion and is given by: u Y m Y m i i ADV i ρ = ′ ′ = ′ ′ , (3.4) For the purposes of this course, the diffusive mass flux is given by Fick’s Law : x Y D m i DIFF i ∂ ∂- = ′ ′ ρ , (3.5) Fick’s Law states that the mass flux is proportional to the gradient of the mass fraction of the...
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This note was uploaded on 09/16/2011 for the course ME 563 taught by Professor Staff during the Spring '11 term at Auburn University.
- Spring '11