Untitled - Technische Universiteit Eindhoven

Numerical discretization of model 157and for the outlet:S n+1o,N= ρn o,Nρ n+1o,NS n o,N +[∆t 2 (ρϕρ n+1 o λ o ) n No,NIn the following definition u n+1α,iblock i:and for the boundary points we find thatu n+1α,0 = u n+1α,inj = q α ρ α,scA ρ n+1α(P out−P n+1o,N )− (ρ∆z 2o λ o ) n N−1]P n+1 n+1o,N −Po,N−1− U n ∆z 2N .(E.10)is the Darcy velocity of phase α at the outlet of grid(u n+1α,i = − 1∆z λn+1 α,i Pn+1α,i+1 − P ) n+1α,i , (E.11); u n+1α,N = − 2∆z λn+1 α,N( )Pout − P n+1α,N . (E.12)The mass fraction woO is updated by a discretized form of the mass balance equationfor oil in the oleic phase, yielding[]ρ n+1o,i S n+1o,i + ∆tϕ∆z ρn+1 o,i u n+1o,i w O,n+1o,i −[∆tϕ∆z ρn+1 o,i−1 un+1 o,i−1]w O,n+1o,i−1= ρ n o,iS n o,iw O,no,i ,which is used for the outlet as well as for the inner grid cells. For the inlet we have:[]ρ n+1o,1 So,1 n+1 + ∆tφ∆z ρn+1 o,1 u n+1o,1 w O,n+1o,1 = ρ n o,1So,1w n O,no,1 + ∆tφ∆z ρn+1 o,inj un+1 o,inj w 0.(E.13)(E.14)The fraction of TMOS at time step n follows from the closure condition: w T o = 1 − w O o .Update of water compositionThe compositions in the aqueous phase are updated implicitly using the following discretizedequations (for water, methanol and silicic acid, respectively):( ϕ∆t ρ wS w + 1 ) n+1∆z u wρ wiw W,n+1w,i − 1∆z (u wρ w ) n+1i−1 wW,n+1 w,i−1= ϕ ∆t (ρ wS w w W w ) n+1i − 4ϕM W S n w,iR n i ,( ϕ∆t ρ wS w + 1 ) n+1∆z u wρ wiw M,n+1w,i − 1∆z (u wρ w ) n+1i−1 wM,n+1 w,i−1= ϕ ∆t (ρ wS w w M w ) n+1i + 4ϕM M S n w,iR n i ,(E.15)(E.16)and( ϕ∆t ρ wS w + 1 ) n+1∆z u wρ wiw SA,n+1w,i − 1∆z (u wρ w ) n+1i−1 wSA,n+1 w,i−1= ϕ ∆t (ρ wS w w SAw ) n+1i + ϕM M S n w,iR n i .(E.17)