Microcombustor modeling using the RBF-FD method andez-Tarrazo 2 and M. S´ M. Kindelan 1 , V. Bayona 1 , E. Fern´ anchez-Sanz 2 1 Mathematics Department Universidad Carlos III de Madrid 2 Fluid Mechanics Department Universidad Carlos III de Madrid ICERM 2017, Providence August 6-11 2017 Kindelan, Bayona (UC3M) Microcombustor May 16 1 / 38
Introduction 1 Mathematical model 2 Numerical implementation 3 Validation and convergence 4 Numerical experiments 5 Conclusions 6 Kindelan, Bayona (UC3M) Microcombustor May 16 2 / 38
Introduction Introduction 1 Mathematical model 2 Numerical implementation 3 Validation and convergence 4 Numerical experiments 5 Conclusions 6 Kindelan, Bayona (UC3M) Microcombustor May 16 3 / 38
Introduction Micro rotary engine Figure : Sketch of a micro-rotary engine from the Micro-Rotary Combustion Lab, University of California, Berkeley. Kindelan, Bayona (UC3M) Microcombustor May 16 4 / 38
Introduction Microscale combustion nano and micro technology devices require compact and rechargeable power supplies at present these devices rely on batteries but energy density of batteries is very low: 0.7 MJ/kg for lithium-ion batteries several hours to recharge possible alternative: micro-engines hydrocarbon fuels have 45 MJ/Kg of stored chemical energy an efficiency of 5% in converting this energy to electricity will outperform batteries small-scale rotary engine (Fern´ andez-Pello, 2002) high specific power low cost due to: minimum number of moving parts and no valves required for operation mechanical shaft output can be directly coupled to electric motor Kindelan, Bayona (UC3M) Microcombustor May 16 5 / 38
Mathematical model Introduction 1 Mathematical model 2 Numerical implementation 3 Validation and convergence 4 Numerical experiments 5 Conclusions 6 Kindelan, Bayona (UC3M) Microcombustor May 16 6 / 38
Mathematical model Model Combustion chamber is approximated by a 2D channel. Bottom wall moves with velocity ± V relative to the other. Upper wall has a notch which modifies the combustible flow and facilitates the attachment of the flame. The velocity profile at the inlet is the sum of a Poiseuille flow and a Couette flow. When the mixture flows through the channel, a recirculation zone appears due to the notch. If the mixture is ignited, a steady flame might be established in the channel. Its structure and location depends on the flow rate which determines the attachment position, among other parameters. Kindelan, Bayona (UC3M) Microcombustor May 16 7 / 38
Mathematical model Micro rotary engine: flow results ( x c ,y c ) R c h V h V Figure : Channel configurations for an inner notch (up) and an outer notch (down). The flow field is illustrated by selected streamlines. Kindelan, Bayona (UC3M) Microcombustor May 16 8 / 38
Mathematical model m = 2, wall velocity V = − 0 . 5. pressure 1 20 15 10 5 0 −6 −4 −2 0 2 4 6 u velocity 1 3 2 1 0 0 −6 −4 −2 0 2 4 6 v velocity 1 0.5 0 −0.5 0 −6 −4 −2 0 2 4 6 Kindelan, Bayona (UC3M) Microcombustor May 16 9 / 38
Mathematical model m = 2, wall velocity V = − 0 . 5. vorticity 1 20 0 −20 0 −6 −4 −2 0 2 4 6 streamlines and reaction rate 1 0 −6 −4 −2 0 2 4 6 Kindelan, Bayona (UC3M) Microcombustor May 16 10 / 38
Mathematical model m = 2, wall velocity V = − 0 . 5. pressure 12 10 1 8 6 4 2 0 −6 −4 −2 0 2 4 6 u velocity 1.5 1 1 0.5 0 0 −6 −4 −2 0 2 4 6 v velocity 0.2 1 0 −0.2 0 −6 −4 −2 0 2 4 6 Kindelan, Bayona (UC3M) Microcombustor May 16 11 / 38
Mathematical model m = 2, wall velocity V = − 0 . 5. vorticity 15 10 1 5 0 −5 0 −6 −4 −2 0 2 4 6 streamlines and reaction rate 1 0 −6 −4 −2 0 2 4 6 Kindelan, Bayona (UC3M) Microcombustor May 16 12 / 38
Mathematical model Parameters m mass flow Pr = 0 . 7 Prandlt number Peclet number Pe Re = Pe / Pr Reynolds number Ze = 10 Zeldovich number u p = u p ( Le ) S L / U L γ = 0 . 7 heat release parameter wall velocity V κ heat loss coefficient θ m temperature in combustion chamber Kindelan, Bayona (UC3M) Microcombustor May 16 13 / 38
Mathematical model Thermo-diffusive model of flame propagation The flow is assumed to be independent of the combustion, and is described by the continuity and momentum equations ∇ · u = 0 (1) 1 Pe Pr ∇ 2 u ( u · ∇ ) u = −∇ p + The propagation of premixed flames subject to the previous flow is described by ∂θ ∇ 2 θ + Pe 2 ω ( θ, Y ) ∂ t + Pe ( u · ∇ ) θ = (2) ∂ Y 1 Le ∇ 2 Y − Pe 2 ω ( θ, Y ) + Pe ( u · ∇ ) Y = ∂ t where θ is the temperature, Y is the fuel mass fraction and ω ( θ, Y ) is the reaction rate, � Ze ( θ − 1) Ze 2 � ω ( θ, Y ) = Y exp . (3) 2 Le u 2 1 + γ ( θ − 1) p Kindelan, Bayona (UC3M) Microcombustor May 16 14 / 38
Mathematical model Boundary conditions ∂θ ∂ Y y = 0 : u = V , v = 0 , n = κ Pe ( θ − θ m ) , n = 0 , ∂� ∂� ∂θ ∂ Y y = y s ( x ) : u = v = 0 , n = 0 , n = 0 . ∂� ∂� u ( y ) = − 6 m y 2 + (6 m − V ) y + V , v = 0 x → −∞ , Y = 1 , θ = θ m ∂ u ∂ x = ∂ v ∂ Y ∂ x = ∂θ x → + ∞ , ∂ x = 0 , ∂ x = 0 . Kindelan, Bayona (UC3M) Microcombustor May 16 15 / 38
Mathematical model Domain and initial conditions Channel length: [ L 0 , L f ]. Width = 1. Upper boundary: y s ( x ) = 1 + ae − bx 2 , (4) where a and b control the depth and width of the notch. Initial conditions: Hot spot: θ ig e − r 2 /δ 2 , r 2 = ( x − x ig ) 2 + ( y − y ig ) 2 θ (0) = (5) Y (0) = 1 where x ig , y ig , θ ig and δ are parameters that define the location, intensity and decay rate of the initial hot spot Planar flame speed in channel (for a = 0): � 1 + e c ( x +1) � Y (0) = 1 / (6) θ (0) = θ m + (1 − θ m )(1 − Y ( x )) Kindelan, Bayona (UC3M) Microcombustor May 16 16 / 38
Numerical implementation Introduction 1 Mathematical model 2 Numerical implementation 3 Validation and convergence 4 Numerical experiments 5 Conclusions 6 Kindelan, Bayona (UC3M) Microcombustor May 16 17 / 38
Numerical implementation Navier-Stokes equations Stream function formulation, u = ∂ψ v = − ∂ψ ∂ y , ∂ x ; the Navier-Stokes equations take the form � ∂ψ ∂ ∆ ψ − ∂ψ ∂ ∆ ψ � ∆ 2 ψ + PePr = 0 (7) ∂ x ∂ y ∂ y ∂ x with boundary conditions ∂ψ y = 0 : ψ = 0 , n = V . ∂� ∂ψ y = y s ( x ) : ψ = m + V / 2 , n = 0 . ∂� As x → −∞ , ∂ψ ψ ( y ) = − 2 m y 3 + (3 m − V / 2) y 2 + Vy , ∂ x = 0 . (8) As x → + ∞ , ∂ 2 ψ ∂ 2 ψ ∂ x ∂ y = 0; ∂ x 2 = 0 . (9) Kindelan, Bayona (UC3M) Microcombustor May 16 18 / 38
Numerical implementation Navier-Stokes equations Equation (7) is solved with Newton’s method. Initial approximation ψ (0) at each iteration compute ψ ( i ) = ψ ( i − 1) + ξ , where the correction ξ is the solution of � ∂ψ ( i − 1) − ∂ψ ( i − 1) ∂ ∆ ψ ( i − 1) ∂ ∆ ψ ( i − 1) � ∂ ∆ ξ ∂ ∆ ξ + ∂ξ − ∂ξ � ψ ( i − 1) � ∆ 2 ξ + PePr = R , ∂ x ∂ y ∂ y ∂ x ∂ x ∂ y ∂ y ∂ x (10) with boundary conditions B ξ = g ( x , y ) − B ψ ( i − 1) . (11) � ψ ( i − 1) � is the residual at iteration i R � ∂ψ ( i − 1) ∂ ∆ ψ ( i − 1) − ∂ψ ( i − 1) ∂ ∆ ψ ( i − 1) � = ∆ 2 ψ ( i − 1) + PePr � ψ ( i − 1) � R , (12) ∂ x ∂ y ∂ y ∂ x � � ψ ( i − 1) �� iterations continue until � R � ≤ ǫ � � RBF-FD with polynomial augmentation is used to discretize differential operators. at each iteration equations (10) are solved using a direct solver. Kindelan, Bayona (UC3M) Microcombustor May 16 19 / 38
Numerical implementation Combustion equations ∂θ � ∇ 2 − Pe ( u · ∇ ) � θ + Pe 2 · ω ( θ, Y ) = ∂ t (13) � 1 ∂ Y � Le ∇ 2 − Pe ( u · ∇ ) Y − Pe 2 · ω ( θ, Y ) = ∂ t Spatial differential operators: are discretized (in a preprocessing step) using RBF-FD augmented with polynomials. ⇒ sparse differential matrices ∇ 2 − Pe ( u · ∇ ) , D θ = 1 Le ∇ 2 − Pe ( u · ∇ ) . D Y = Kindelan, Bayona (UC3M) Microcombustor May 16 20 / 38
Numerical implementation Combustion equations Time integration: semi-implicit CN-AB2 (implicit for the linear terms and explicit for the non-linear terms). � I − ∆ t � � I + ∆ t � θ k + ∆ t 3 G k − G k − 1 � � θ k +1 2 D θ = 2 D θ 2 · (14) � I − ∆ t � � I + ∆ t � Y k − ∆ t 3 G k − G k − 1 � � Y k +1 2 D Y = 2 D Y 2 · (14) together with boundary conditions, are solved at each time step using iterative solver BiCGSTAB with iLU as preconditoner. G k representes the non-linear term Pe 2 · ω ( θ k , Y k ) . G k = � θ k − θ k − 1 � � Y k − Y k − 1 � � ≤ tol and � ≤ tol (tol = 10 − 8 ). � � Iterations continue until Kindelan, Bayona (UC3M) Microcombustor May 16 21 / 38
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