Diffusion equation January 28, 2009ME 501B – Engineering Analysis1The Diffusion EquationThe Diffusion EquationLarry CarettoMechanical Engineering 501BSeminar in Engineering Seminar in Engineering AnalysisAnalysisJanuary 28, 20092Overview• Review last week• Diffusion equation– Physical meaning and derivation– Relation to Laplace equation– Solution by separation of variables– Sturm-Liouville orthogonal eigenfunction expansion for initial conditions• Only possible for homogenous boundary conditions• Treatment of nonhomogenous boundary conditions3Review Sturm-Liouville• Homogenous equations for a ≤ x ≤ b• Solutions, ymare complete set of orthogonal eignenfunctions that can be used to expand any function[]0)()()(=++⎟⎠⎞⎜⎝⎛yxpxqdxdyxrdxdλ0)(0)(2121=+=+==bxaxdxdybydxdykaykll∑∞==0)()(mmmxyaxf4Review Orthogonal Functions• Defined as inner product integral with p(x) from Sturm-Liouville equation• Get coefficients in eigenfunction expansions()∫∫==bammbammmmmdxxyxyxpdxxfxyxpyyfya)()()()()()(,),(()ijibajijiydxxpxyxyyyδ2*)()()(, ==∫5Review Fourier Series• Based on periodic functions defined over –L < x < L∑∞=⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛+=10sincos)(nnnLxnbLxnaaxfππ∫−=LLdxxfLa )(210∫−⎟⎠⎞⎜⎝⎛=LLndxLxnxfLbπsin)(1∫−⎟⎠⎞⎜⎝⎛=LLndxLxnxfLaπcos)(16Review Even/Odd Functions• Odd function: f(-x) = -f(x) (like sine)• Even function: g(-x) = g(x) (like cosine)• For odd f(x)0)( =∫−LLdxxf• For even g(x)∫∫=−LLLdxxgdxxg0)(2)(• Product of odd times even is oddsine times cosinecosinesineDiffusion equation January 28, 2009ME 501B – Engineering Analysis27Review Half-Interval Series• For even functions, there are no sine terms in the Fourier series• For odd functions there are no cosine terms in the Fourier series• In these cases can use integrals from 0 to L for the coefficients• Can also use equations from 0 to L for all functions, but get periodic behavior outside this region8Review Half-Interval Series II• Look at function below for L = 2• Full series gives periodic resultsFull Series9Review Half-Interval Series III• Expanding function for 0 ≤ x ≤ L = 2 with a sine series gives odd periodic repetitionSine Series10Review Half-Interval Series IV• Expanding function for –L ≤ x ≤ 0 with a sine series gives odd periodic repetitionSine Series11Review Half-Interval Series V• Cosine series for 0 ≤ x ≤ L = 2 with gives even periodic repetitionCosine Series12Review Half-Interval Series VI• Cosine series for –L ≤ x ≤ 0 gives even periodic repetition of different regionCosine SeriesDiffusion equation January 28, 2009ME 501B – Engineering Analysis313Diffusion and Laplace Equations• Partial differential equations related to conservation principles of fluxes governed by potentials– Heat transfer from temperature gradient– Mass diffusion from concentration gradient– Current from electrostatic potential– Magnetic fluxes– Ideal fluid flow from velocity potential• Laplace is steady state case of diffusion14Derivation Basics• Consider a flux, f, that is a gradient of a potential, u, f = –ρα∂u/∂x• The net outflow is f|x+dxΔA–f|xΔA• Dividing by ΔV = ΔAΔxgives the net outflow per unit volume = (f|x+dx–f|x)/Δxwhich is ∂f/∂x as Δx approaches zero• Substituting the potential definition gives the net outflow (per unit volume) as ∂[–ρα∂u/∂x]/∂x15Derivation Basics II• A conserved quantity has zero net outflow for steady processes with no source terms– This gives ∂[-ρα∂u/∂x]/∂x = 0 or ∂[-ρα∂u/∂x]/∂x + ∂[-ρα∂u/∂y]/∂y + ∂[-ρα∂u/∂z]/∂z = 0; constant properties ρα gives Laplace’s equation02222222=∇=∂∂+∂∂+∂∂uzuyuxu• Can be two-dimensional• Changes with coordinate systems16Derivation Basics III• For transient processes, a net outflow causes a decrease in ρu– This gives ∂ρu/∂t = ∂[ρα∂u/∂x]/∂x + ∂[ρα∂u/∂y]/∂y + ∂[ρα∂u/∂z]/∂z – For constant properties ρ and α we have the diffusion equationuzuyuxutu2222222∇=⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂=∂∂αα• Can have one or two space dimensions• Changes with coordinate systems17Derivation Basics IV• Diffusion equation gives transient process for systems where steady state is Laplace’s equation022222222=∇∇=⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂=∂∂uuzuyuxutuααtransient steady-state• Can have different space dimensions• α, called the diffusion coefficient or diffu-sivity, has dimensions of (length)2/time18Multidimensional Equations• General diffusion equation for three dimensionsutu2∇=∂∂αφφφθφθ∂∂+∂∂+∂∂+⎟⎠⎞⎜⎝⎛∂∂∂∂=∇∂∂+∂∂+⎟⎠⎞⎜⎝⎛∂∂∂∂=∇∂∂+∂∂+∂∂=∇urururrurrruSpherezuurrurrrulCylindricazuyuxuuCartesian222222222222222222222222cot1sin1111Diffusion equation January 28, 2009ME 501B – Engineering Analysis419Variable Properties• Included in outer space derivativeφ∂∂φραφ∂∂φ+θ∂∂ραθ∂∂φ+∂∂ρα∂∂=∂ρ∂∂∂ρα∂∂+θ∂∂ραθ∂∂+∂∂ρα∂∂=∂ρ∂∂∂ρα∂∂+∂∂ρα∂∂+∂∂ρα∂∂=∂ρ∂ururrurrrtuSpherezuzurrurrrtulCylindricazuzyuyxuxtuCartesiansinsin1sin1111222222())( ugraddivutuρα=∇ρα⋅∇=∂ρ∂20Diffusion Equation Solutions• Governs heat conduction and species diffusion for t ≥ 0 and 0 ≤ x ≤ xmax– u(x,t) is temperature, species concentration– Initial condition u(x,0) = u0(x)– Boundaries u(0,t) = uL(t); u(xmax,t) = uR(t)22xutu∂∂=∂∂α• Diffusivity, α, is material property (length)2/(time)21Separation of Variables• Assume u(x,t) = X(x)T(t)[][]222222)()()()()()()()(xxXtTxtTxXxuttTxXttTxXtu∂∂=∂∂=∂∂=∂∂=∂∂=∂∂ααα• Divide by αX(x)T(t)22)()(1)()(11xxXxXttTtT ∂∂=∂∂αResult is function of t equal to function of x22Separation of Variables Works• Assumed solution, u(x,t) = X(x)T(t), gives a function of x equal a function of t• Since x and t are independent, both sides must equal a constant for this to be true• Choose negative real constant, –λ2, to simplify later solution of resulting ordinary differential equations (ODE)222)()(1)()(11λα−=∂∂=∂∂xxXxXttTtT23Solve ODEs to Get u(x,t)• Have simple ODEs with known general