Modelling steam methane reforming in a fixed bed reactor using - PowerPoint PPT Presentation

Modelling steam methane reforming in a fixed bed reactor using orthogonal collocation Kasper Linnestad Department of Chemical Engineering Norwegian University of Science and Technology December 3, 2014 Kasper Linnestad Reactor modelling December 3, 2014 1 / 36

Outline Theory 1 Governing equations 2 Implementation 3 Results 4 Conclusion 5 Kasper Linnestad Reactor modelling December 3, 2014 2 / 36

Outline Theory 1 Weighted residuals method Collocation points Orthogonal collocation method Governing equations 2 Implementation 3 Results 4 Conclusion 5 Kasper Linnestad Reactor modelling December 3, 2014 3 / 36

Theory Weighted residuals method General problem � � L f ( z, r ) = g ( z, r ) � � B f b ( z b , r b ) = g b ( z, r ) Kasper Linnestad Reactor modelling December 3, 2014 4 / 36

Theory Weighted residuals method General problem � � L f ( z, r ) = g ( z, r ) � � B f b ( z b , r b ) = g b ( z, r ) Approximation P z P r � � f ( z, r ) ≈ a j z ,j r l j z ( z ) l j r ( r ) j z =0 j r =0 Kasper Linnestad Reactor modelling December 3, 2014 4 / 36

Theory Weighted residuals method General problem � � L f ( z, r ) = g ( z, r ) � � B f b ( z b , r b ) = g b ( z, r ) Approximation P z P r � � f ( z, r ) ≈ a j z ,j r l j z ( z ) l j r ( r ) j z =0 j r =0 Residual � � R ( z, r ) = L f ( z, r ) − g ( z, r ) Kasper Linnestad Reactor modelling December 3, 2014 4 / 36

Theory Weighted residuals method General problem Minimize residuals by � � � L f ( z, r ) = g ( z, r ) R ( z, r ) w i ( z, r ) d z d r = 0 , ∀ i � � B f b ( z b , r b ) = g b ( z, r ) Approximation P z P r � � f ( z, r ) ≈ a j z ,j r l j z ( z ) l j r ( r ) j z =0 j r =0 Residual � � R ( z, r ) = L f ( z, r ) − g ( z, r ) Kasper Linnestad Reactor modelling December 3, 2014 4 / 36

Theory Weighted residuals method General problem Minimize residuals by � � � L f ( z, r ) = g ( z, r ) R ( z, r ) w i ( z, r ) d z d r = 0 , ∀ i � � B f b ( z b , r b ) = g b ( z, r ) Orthogonal collocation uses Approximation P z P r a j z ,j r = f ( z j z , r j r ) � � f ( z, r ) ≈ a j z ,j r l j z ( z ) l j r ( r ) j z =0 j r =0 Residual � � R ( z, r ) = L f ( z, r ) − g ( z, r ) Kasper Linnestad Reactor modelling December 3, 2014 4 / 36

Theory Weighted residuals method General problem Minimize residuals by � � � L f ( z, r ) = g ( z, r ) R ( z, r ) w i ( z, r ) d z d r = 0 , ∀ i � � B f b ( z b , r b ) = g b ( z, r ) Orthogonal collocation uses Approximation P z P r a j z ,j r = f ( z j z , r j r ) � � f ( z, r ) ≈ a j z ,j r l j z ( z ) l j r ( r ) P x − x i � j z =0 j r =0 l n ( x ) = x n − x i i =0 i � = n Residual � � R ( z, r ) = L f ( z, r ) − g ( z, r ) Kasper Linnestad Reactor modelling December 3, 2014 4 / 36

Theory Weighted residuals method General problem Minimize residuals by � � � L f ( z, r ) = g ( z, r ) R ( z, r ) w i ( z, r ) d z d r = 0 , ∀ i � � B f b ( z b , r b ) = g b ( z, r ) Orthogonal collocation uses Approximation P z P r a j z ,j r = f ( z j z , r j r ) � � f ( z, r ) ≈ a j z ,j r l j z ( z ) l j r ( r ) P x − x i � j z =0 j r =0 l n ( x ) = x n − x i i =0 i � = n Residual w i ( z, r ) = δ ( z − z i z ) δ ( r − r i r ) � � R ( z, r ) = L f ( z, r ) − g ( z, r ) Kasper Linnestad Reactor modelling December 3, 2014 4 / 36

Theory Collocation points 2 Legendre polynomials � 1 L m ( x ) L n ( x ) L m ( x ) d x = 0 , m � = n 0 − 1 − 2 − 1 − 0 . 5 0 0 . 5 1 x Kasper Linnestad Reactor modelling December 3, 2014 5 / 36

Theory Collocation points 2 Legendre polynomials � 1 L m ( x ) L n ( x ) L m ( x ) d x = 0 , m � = n 0 − 1 Collocation points L m ( x i ) = 0 , ∀ i ∈ { 0 , . . . , m } − 2 − 1 − 0 . 5 0 0 . 5 1 x Kasper Linnestad Reactor modelling December 3, 2014 5 / 36

Theory Orthogonal collocation method Minimize residuals by � R ( z, r ) w i ( z, r ) d z d r = 0 , ∀ i Orthogonal collocation uses a j z ,j r = f ( z j z , r j r ) P x − x i � l n ( x ) = x n − x i i =0 i � = n w i ( z, r ) = δ ( z − z i z ) δ ( r − r i r ) Kasper Linnestad Reactor modelling December 3, 2014 6 / 36

Theory Orthogonal collocation method Minimize residuals by Inserted � ∀ i z ∈ { 0 , . . . , P z } R ( z, r ) w i ( z, r ) d z d r = 0 , ∀ i R ( z i z , r i r ) = 0 , i r ∈ { 0 , . . . , P r } Orthogonal collocation uses a j z ,j r = f ( z j z , r j r ) P x − x i � l n ( x ) = x n − x i i =0 i � = n w i ( z, r ) = δ ( z − z i z ) δ ( r − r i r ) Kasper Linnestad Reactor modelling December 3, 2014 6 / 36

Theory Orthogonal collocation method Minimize residuals by Inserted � ∀ i z ∈ { 0 , . . . , P z } R ( z, r ) w i ( z, r ) d z d r = 0 , ∀ i R ( z i z , r i r ) = 0 , i r ∈ { 0 , . . . , P r } Orthogonal collocation uses System of equations � � a j z ,j r = f ( z j z , r j r ) L f ( z , r ) = g ( z , r ) � � P x − x i B f b ( z b , r b ) = g b ( z b , r b ) � l n ( x ) = x n − x i i =0 i � = n w i ( z, r ) = δ ( z − z i z ) δ ( r − r i r ) Kasper Linnestad Reactor modelling December 3, 2014 6 / 36

Theory Orthogonal collocation method Minimize residuals by Inserted � ∀ i z ∈ { 0 , . . . , P z } R ( z, r ) w i ( z, r ) d z d r = 0 , ∀ i R ( z i z , r i r ) = 0 , i r ∈ { 0 , . . . , P r } Orthogonal collocation uses System of equations � � a j z ,j r = f ( z j z , r j r ) L f ( z , r ) = g ( z , r ) � � P x − x i B f b ( z b , r b ) = g b ( z b , r b ) � l n ( x ) = x n − x i i =0 i � = n Linearisation w i ( z, r ) = δ ( z − z i z ) δ ( r − r i r ) Af = b Kasper Linnestad Reactor modelling December 3, 2014 6 / 36

Theory Linearisation Example function f ( ψ ) = ψ 2 + ψ exp ( − ψ ) Kasper Linnestad Reactor modelling December 3, 2014 7 / 36

Theory Linearisation Example function f ( ψ ) = ψ 2 + ψ exp ( − ψ ) Fixed-point (Picard-iteration) � − ψ ⋆ �� ψ ⋆ + exp � f ( ψ ) ≈ ψ solves f ( ψ ) = 3 in 114 iterations Kasper Linnestad Reactor modelling December 3, 2014 7 / 36

Theory Linearisation Example function f ( ψ ) = ψ 2 + ψ exp ( − ψ ) Fixed-point (Picard-iteration) � − ψ ⋆ �� ψ ⋆ + exp � f ( ψ ) ≈ ψ solves f ( ψ ) = 3 in 114 iterations Taylor (Newton-Raphson-iteration) � f ( ψ ) ≈ f ( ψ ⋆ ) + ∂f � ψ − ψ ⋆ � � � ∂ψ ψ ⋆ solves f ( ψ ) = 3 in 8 iterations Kasper Linnestad Reactor modelling December 3, 2014 7 / 36

Outline Theory 1 Governing equations 2 Continuity equation Energy equation Species mass balance Ergun’s equation Initial and boundary conditions Implementation 3 Results 4 Conclusion 5 Kasper Linnestad Reactor modelling December 3, 2014 8 / 36

Governing equations Fixed bed reactor r z Kasper Linnestad Reactor modelling December 3, 2014 9 / 36

Governing equations Steam methane reforming Assumptions Pseudo-homogeneous Efficiency factor = 10 − 3 Reactions CH 4 + H 2 O = CO + 3 H 2 CO + H 2 O = CO 2 + H 2 CH 4 + 2 H 2 O = CO 2 + 4 H 2 Kasper Linnestad Reactor modelling December 3, 2014 10 / 36

Governing equations Continuity equation General form ∂ρ ∂t + ∇ · ( ρ u ) = 0 Kasper Linnestad Reactor modelling December 3, 2014 11 / 36

Governing equations Continuity equation General form ∂ρ ∂t + ∇ · ( ρ u ) = 0 Simplified ∂ρu z = 0 ∂z Kasper Linnestad Reactor modelling December 3, 2014 11 / 36

Governing equations Continuity equation General form ∂ρ ∂t + ∇ · ( ρ u ) = 0 Simplified ∂ρu z = 0 ∂z Linearised ∂ρ ⋆ ρ ⋆ ∂u z ∂z + u z = 0 ∂z ���� � �� � ⇒ b uz ⇒ A uz Kasper Linnestad Reactor modelling December 3, 2014 11 / 36

Governing equations Energy equation General form � ∂T � ρc p ∂t + u · ∇ T = −∇ · q − ∆ rx H Kasper Linnestad Reactor modelling December 3, 2014 12 / 36

Governing equations Energy equation General form � ∂T � ρc p ∂t + u · ∇ T = −∇ · q − ∆ rx H Simplified � � ∂z = λ eff ∂T ∂ r ∂T − ∆ rx H ρc p u z r ∂r ∂r Kasper Linnestad Reactor modelling December 3, 2014 12 / 36

Governing equations Energy equation General form � ∂T � ρc p ∂t + u · ∇ T = −∇ · q − ∆ rx H Simplified � � ∂z = λ eff ∂T ∂ r ∂T − ∆ rx H ρc p u z r ∂r ∂r Linearised � � ∂r + ∂ 2 T ∂T 1 ∂T ρ ⋆ c ⋆ p u ⋆ = λ eff ,⋆ − ∆ rx H ⋆ z ∂r 2 ∂z r � �� � � �� � ⇒ b T � �� � ⇒ A T ⇒ A T Kasper Linnestad Reactor modelling December 3, 2014 12 / 36

Governing equations Species mass balance General form ∂ρω i + ∇ · ( ρ u ω i ) = −∇ · j i + R i ∂t Kasper Linnestad Reactor modelling December 3, 2014 13 / 36

Governing equations Species mass balance General form ∂ρω i + ∇ · ( ρ u ω i ) = −∇ · j i + R i ∂t Simplified � � = D eff ∂ρu z ω i ∂ rρ ∂ω i − R i ∂z r ∂r ∂r Kasper Linnestad Reactor modelling December 3, 2014 13 / 36