and and ECMI week 2008
Outline and
Problem Description find model for processes consideration of effects caused by presence of salt point and numerical solution
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and heat equations liquid phase: T L t = k L ρ L c L 2 T L x 2, 0 x h(t) solid phase: T L (0, t) = T A T S t T S (l, t) = T G = k S ρ S c S 2 T S x 2, h(t) x l thermophysical constants: k S, k L thermal conductivities c S, c L specific heat ρ S = ρ L densities
interface movement T S (h(t), t) = T L (h(t), t) = T M ( point) domain Ω = [0, l] separated by interface into [0, h(t)) and (h(t), l] one-dimensional Stefan condition: T S k S x (h(t), t) k T L L (h(t), t) }{{ x } heat flux difference h(0) = 0 slap initially frozen = ρl dh dt }{{} velocity and L - latent heat (e.g. water: 334 J/g)
Two-phase Stefan problem Heat-Equations: Interface conditions for t > 0 : Initial conditions: Boundary conditions for t > 0 : { Tt = α L T xx : 0 < x < h(t), α L := k L ρc L, T t = α S T xx : h(t) < x < l, α S := k S { T (h(t), t) = T M, ρlh (t) { h(0) ρc S. = k S T x (h(t), t) k L T x (h(t), t). = 0 (material initially solid), T (x, 0) { = T init < T M : 0 x l. T (0, t) = T A > T M, T (l, t) = T G < T M. and
Simplification: solid phase at uniform temperature T init = T M, only liquid phase is active, Ω = [0, ) = one phase Stefan problem Find T (x, t) and h(t) such that Heat equation: T t = α L T xx, 0 < x < h(t), t > 0, { Interface conditions T (h(t), t) = T M, for t > 0 : h (t) = βt x (h(t), t), β := k L ρl Initial condition: h(0) = 0. Boundary condition: T (0, t) = T A (t) = T A > T M : t > 0. and similar processes: solidification, flows through porous media, combustion, reactions
and (a) without salt: dynamic equilibrium (a) at point with salt (i.e. NaCl): disruption of equilibrium (b) foreign molecules (i.e. Na +, Cl ) dissolve in the water the chemical potential of the solvent is decreased by dilution equilibrium between solid and liquid phase is established at another depressed point (b)
Raoul s law: T M = K M m B point : T M = T M(pure solvent) T M(solution) K M - the cryoscopic constant m B - the molality of the solution (no. of moles of solute per kg solvent) molality m B = m solute i, i- van t Hoff factor (number of individual particles in solution) colligative property sugar: i = 1 NaCl: i = 2 (Na + and Cl ions) CaCl2: i = 3 (Ca 2+ and 2 Cl ions) and
Analytical If the initial temperature distribution is ( ) x erf T (x, 0)= T 2 αt A (T A T M ) 0 erf(λ), 0 x h(0) T M, x > h(0) with: λ - the solution of equation λe λ2 erf(λ) = C L(T A T M ) πl = solution of the Stefan problem: h(t) = 2λ α(t + t 0 ) erf T (x, t) = T A (T A T M ) ( x 2 α(t+t 0) erf(λ) and t 0 = h(0)2 4λ 2 α ) and
solution of Stefan problem with FDM: For each time step: 1. Update the temperature distribution by T t = αt xx using implicit FDM: T (x x, t + t) T (x, t + t) = T (x, t)+α x 2 t 2T (x, t + t) T (x + x, t + t) α x 2 t 2. Update the boundary state by h t = βt x (h(t), t) using explicit FDM: h(t + t) = β 3. new point T (h(t), t) T (h(t) x, t) t x and
1. Temperature distribution Using matrix notation T A T T 2 (t) 0... 0 1 (t + t) 1 2 1 0 0.. T (t + t)= T n(t) + t... T n 1 (t + t) T h F 2 T 1 2 1 n(t + t) T n+1 (t + t).. 0.. 0. 0... 0 T F }{{} T N+1 (t + t) =A leads to update step algorithm: T (t + t)=(i N+1 A) 1 [T A,T 2 (t),,t n(t),t F,,T F ] T and
2. Interface movement Update step for interface movement h(t + t) = h(t) + β T n+1(t) T n (t) t x 3. Assumption: constant amount of substance of solute per unit area n A, V = h(t) A n A A solution T M (t) = T M(pure) K M m b =... = T M(pure) K M i ρv solution (t) and n A = T M(pure) K M i ρh(t)
Example and
Further Issues Possible improvements extend model to more dimensions and other geometries consider more than one phase supercooling effects ρ L ρ S (ρ L < ρ S void formation) variable thermophysical properties, i.e. c(t), k(t), ρ(t) diffusion model for salt concentration include other strategies for solution FEM pertubation methods and
Further Issues Possible improvements extend model to more dimensions and other geometries consider more than one phase supercooling effects ρ L ρ S (ρ L < ρ S void formation) variable thermophysical properties, i.e. c(t), k(t), ρ(t) diffusion model for salt concentration include other strategies for solution FEM pertubation methods Thanks for your Attention! and