Application of fluid mechanics ppt

a brief introduction to fluid mechanics pdf and application of fluid dynamics in daily life ppt
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Published Date:23-07-2017
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Fluid Dynamics and Balance Equations for Reacting Flows CEFRC Combustion Summer School 2014 Prof. Dr.-Ing. Heinz Pitsch Copyright ©2014 by Heinz Pitsch. This material is not to be sold, reproduced or distributed without prior written permission of the owner, Heinz Pitsch. Balance Equations • Basics: equations of continuum mechanics • balance equations for mass and momentum • balance equations for the energy and the chemical species • Associated with the release of thermal energy and the increase in temperature is a local decrease in density which in turn affects the momentum balance. • Therefore, all these equations are closely coupled to each other. • Nevertheless, in deriving these equations we will try to point out how they can be simplified and partially uncoupled under certain assumptions. 2 Balance Equations • A time-independent control volume V for a balance quality F(t) • The scalar product between the surface flux φ and the normal vector n f determines the outflow through the surface A, a source s the rate of f production of F(t) • Let us consider a general quantity per unit volume f(x, t). Its integral over the finite volume V, with the time-independent boundary A is given by 3 Balance Equations • The temporal change of F is then due to the following three effects: • 1. by the flux f across the boundary A. This flux may be due to f convection or molecular transport. • By integration over the boundary A we obtain the net contribution which is negative, if the normal vector is assumed to direct outwards. 4 Balance Equations • 2. by a local source s within the volume. f This is an essential production of partial mass by chemical reactions. Integrating the source term over the volume leads to • 3. by an external induced source s. Examples are the gravitational force or thermal radiation. Integration of s over the volume yields f 5 Balance Equations • We therefore have the balance equation • Changing the integral over the boundary A into a volume integral using Gauss' theorem and realizing that the balance must be independent of the volume, we obtain the general balance equation in differential form 6 Mass Balance • Set the partial mass per unit volume ρ = ρ Y = f. i i • The partial mass flux across the boundary is ρ v = φ , where v i i f i is called the diffusion velocity. • Summation over all components yields the mass flow where v is the mass average velocity. • The difference between v defines the diffusion flux i where the sum satisfies 7 Mass Balance • Setting the chemical source term one obtains the equation for the partial density • The summation over i leads to the continuity equation 8 Mass Balance • Introducing the total derivative of a quantity a combination with the continuity equation yields • Then using may also be written 9 Momentum Balance • Set the momentum per unit volume ρ v = f. • The momentum flux is the sum of the convective momentum in flow ρ v v and the stress tensor where I is the unit tensor and τ is the viscous stress tensor. • Therefore ρ v v + P = φ . f • There is no local source of momentum, but the gravitational force from outside where g denotes the constant of gravity. 10 Momentum Balance • The momentum equation then reads or with for we obtain 11 Kinetic Energy Balance • The scalar product of the momentum equation with v provides the balance for the kinetic energy 2 . where v = v v. 12 Potential Energy Balance • The gravitational force may be written as the derivative of the time- independent potential • Then with the continuity equation the balance for the potential energy is 13 Total and Internal Energy and Enthalpy Balance • The first law of thermodynamics states that the total energy must be conserved, such that the local source σ = 0. f • We set ρ e = f , where the total energy per unit mass is • This defines the internal energy introduced in 14 Total and Internal Energy and Enthalpy Balance • The total energy flux is which defines the total heat flux j . q • The externally induced source due to radiation is • Then the total energy balance may be used to derive an equation for the internal energy 15 Total and Internal Energy and Enthalpy Balance • Using this may be written with the total derivative • With the continuity equation we may substitute to find illustrating the equivalence with the first law introduced in a global thermodynamic balance. 16 Total and Internal Energy and Enthalpy Balance • With the enthalpy h = u + p/ρ the energy balance equation can be formulated for the enthalpy 17 Transport Processes • In its most general form Newton's law states that the viscous stress tensor is proportional to the symmetric, trace-free part of the velocity gradient sym • Here the suffix denotes that only the symmetric part is taken and the second term in the brackets subtracts the trace elements from the tensor. • Newton's law thereby defines the dynamic viscosity. 18 Transport Processes • Similarly Fick's law states that the diffusion flux is proportional to the concentration gradient. • Due to thermodiffusion it is also proportional to the temperature gradient. • The most general form for multicomponent diffusion is written as • For most combustion processes thermodiffusion can safely be neglected. • For a binary mixture Fick’s law reduces to where is the binary diffusion coefficient. 19 Transport Processes • For multicomponent mixtures, where one component occurs in large amounts, as for the combustion in air where nitrogen is abundant, all other species may be treated as trace species and with the binary diffusion coefficient with respect to the abundant component may be used as an approximation • A generalization for an effective diffusion coefficient D to be used for the i minor species is 20

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