Experimental identification of transfer functions for diffusive and/or advective heat transfer for linear time invariant dynamical systems Denis Maillet U niversity of L orraine & CNRS, Nancy, France U niversity of L orraine & CNRS, Nancy, France L aboratoire d' E nergétique et de M écanique T héorique et A ppliquée (LEMTA) New Trends in Parameter Identification for Mathematical Models IMPA, Rio de Janeiro, Brazil, October 30 – November 3, 2017 Contribution: Waseem Al Hadad
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17 Experimental inverse problems in heat transfer and engineering METTI Group, SFT (French Heat Transfer Society) Recently: interest in convolutive models and associated inverse problems * Pollutant source identification in a ventilated domain (turbulence, transient concentration measurements) * Transient thermal behaviour of heat exchanger (PhD W. Hadad, Fives Cryo postDoc) 2 * Virtual sensor construction in a furnace under vacuum conditions (PhD T. Loussouar, Safran Group)
Scope 1. Forced thermal response of Linear advective/diffusive systems with Time Independent (LTI) coefficients 2. The calibration problem 2.1 Case of a heat exchanger 2.2 Experimental Impedance/transmittance estimation for a half heat exchanger 3. Analysis of deconvolution deadlocks 3.1 Reference case: 1D transient conduction 3.1 Reference case: 1D transient conduction 3.2 Noisy matrix and Total Least Squares 3.3 Comparison of calibration methods 4. Rectangular deconvolution 4.1. Point versus averaged values for input and unknown 4.2 Rectangular deconvolution (using « stairs » parameterizing) 4.3 Rectangular estimation with n < m non uniform (NU) time steps 5. Conclusions/perspectives 3 Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17 1. Forced thermal response of an advective/diffusive system with time constant coefficients Material multicomponent system = K solid or fluid domains solid fluid solid - Ω 3 Flowing solid - Ω 1 fluid fluid Ω K - 1 solid - Ω 4 Flowing solid - Ω K fluid Ω 2 solid - Ω K -2 Assumptions: time constant thermophysical properties and velocity field 4
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17 Initial uniform state or steady state temperature field + one single separable thermal excitation ≠ ≠ T ∞ ) ( t T = & in Q ( t ) m c T ( t ) Q ( t ) or T ( t ) T init in in in b s s init h (P) P P s s P s solid - Ω 3 Flowing Q v ( t ) fluid solid - Ω 1 Ω K - 1 solid - Ω 4 Flowing Flowing solid - Ω K fluid Ω 2 solid - Ω K -2 Time part of thermal excitation u ( t ) (starts at time t = 0) : Q v ( t ) • volumetretric heat source Fixed geometrical support : • surface heat or temperature source Q ( t ) or T ( t ) s s • point ≠ • line T ∞ ) ( t T • change of external fluid temperature init • surface • volume 5 T in • change of temperature at one fluid inlet ( t ) b
Change of perspective: one single heterogeneous fluid in one single domain (if solid part : zero velocity) ≠ ≠ T ∞ ) ( t T = & & in Q ( t ) m c T ( t ) Q ( t ) or T ( t ) T init in in in b s s init h (P) P P s s λ thermal conductivi ty : (P) r ρ Q v ( t ) volumetric heat : c (P) velocity field : v (P) P P heat heat transfer transfer coefficien coefficien t t ≡ y ( t ) T (P, t ) (external boundaries ) : h (P) Transient separable thermal excitation : Point response at any point P : ⇒ ≡ y ( t ) T (P, t ) u ( t ) 6 6 Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17 Recap: Physical system: Set of solids AND fluid(s): 3D forced convection with constant velocities (in time but not in space) P = ANY point in the system One single thermal excitation defined by its support One single thermal excitation defined by its support Assumptions : Transient heat equation + boundary conditions with time-invariant coefficients + uniform initial temperature or steady state (the system is Linear and also Time-Invariant LITI) ( ) ∂ r r r r ( ) T ( ) Q ( t ) . . ρ + ρ ∇ = ∇ λ ∇ + v c P (P , t ) c P u (P) T (P , t ) (P) T (P , t ) f (P) ∂ t V source Conduction Transient Advection Internal source 7
Temperature rise at any point P: θ = (P , t ) T (P , t ) - T (P) init Its Laplace transform : ∞ ∫ θ = θ (P , p ) exp (- p t ) (P , t ) d t 0 Laplace parameter Assumptions : Transient heat equation + boundary conditions with time-invariant coefficient + uniform initial temperature (the system is Linear and also Time-Invariant LITI) ( ) r r r ∂ r ( ) T ( ) Q ( t ) . . ρ + ρ ∇ = ∇ λ ∇ + v c P (P , t ) c P u (P) T (P , t ) (P) T (P , t ) f (P) ∂ t V source Conduction Transient Advection Internal source Consequences : Laplace transformed heat equation (no time derivative) ( ) r r r r ( ) ( ) Q ( p ) . . ρ θ + ρ ∇ θ = ∇ λ ∇ θ + v c P p (P , p ) c P u (P) (P , p ) (P) (P , p ) f (P) V source Transient Advection Conduction Internal source 8 Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
Linear system with a single excitation Temperature or flux ⇒ response at any point P in the system = simple product (Laplace domain) = y (P, p ) H (P , p ) u ( p ) or convolution product (time domain) excitation ∫ ∫ t = = − y (P , t ) H (P , t ) * u ( t ) H (P, t t ' ) u ( t ' ) d t' response response 0 0 Excitation : u ( t ) Transfer function Response in any specific point P : y ( t ) = − − init init u ( t ) Q ( t ) Q or Q ( t ) Q v v s s = θ = y ( t ) (P, t ) T (P, t ) - T (P) H (P , t ) init or T ( t ) - T (P ) s init s or local heat flux in init or T ( t ) - T ϕ ∞ ∞ « init »= initial steady state any direction (P, t ) x in in , init or uniform temperature field or T ( t ) - T b b 9
t ∫ = = − y (P , t ) H (P , t ) * u ( t ) H (P, t t ' ) u ( t ' ) d t' 0 excitation response y ( t ) u ( t ) ss y ss u H ( t ) t t 0 0 H t 0 0 ∞ ss y ( ) dt ∫ Steady state (ss) version = = ss H H t ss of a transfer function u 0 Time distribution asymptotic values − = Φ T T R case : in out Thermal resistance, flux pipe ≡ − SS u Q or Q Q (thermal power) between 2 isothermal surfaces ≡ θ = − y T T (P) (temperatu re variation) init − = ss ss ⇒ ≡ T T Z Q H Z (thermal impedance) ∞ Generalized resistance, no flux pipe 10 Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
2. The calibration problem 2.1 Case of a heat exchanger Assumptions : � Constant thermo-physical properties (fluid and walls) and velocities (LTI heat equation) : ∂ β ∂ = β ≡ λ ρ / t 0 u , u , , , .... hot cold � Uniform initial conditions/initial steady state) = ) = ) = = = = = = ss ss T T (P (P , , t t 0 0 T T T T (P (P , , t t 0 0 ) ) T T (P) (P) init init � Heat losses through convection/(linearized) radiation with environment through a uniform = { heat transfer coefficient h at temperature T T ∞ init � One single heat source � Transient/unsteady thermal regime (inlet temperature increase) with observed responses at any point q : ⇒ that starts at t = 0 + : ( ) ( ) θ = − t T t T θ = ≠ q q init ( t ) T ( t ) - T 0 ( ) ( ) 1 1 init θ ≤ = θ > ≠ t 0 0 and t 0 0 q q Cause Consequences θ = = 11 ( t ) T ( t ) - T 0 4 4 init
Calculation of convolution products (transmittance case) Parameterizing with piecewise constant functions, square case response transmittance unique pseudo source θ = θ (P , t ) W (P , t ) * ( t ) 1 ∫ t = − θ W (P, t t ' ) ( t ' ) d t' 1 0 t ∫ = θ − (P, t t ' ) W (P, t ' ) d t ' 1 0 m ∑ θ ≈ θ ∆ = = = = (P , t ) t W (P) ∆ ∆ t 0 ; t i t for i 1 to m ; t t / m i 1 , j − + 0 i final i j 1 j = = j 1 1 sampled averaged over 1 time interval ( ) 1 1 t ∫ = i ≈ + = θ z z ( t ) d t z ( t ) z ( t ) for z ( t ) or W (P) − i i 1 i 1 ∆ t 2 t − i 1 q = 2 , 3 or P θ θ 1 θ θ ( t ) , ( t ) q 1 2 3 q W 1 θ 1 t ( ) Heat exchanger θ P1 t ( ) t t K K t 0 0 t t t t t 1 2 m 2 m 1 12 Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
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