Chemical reactors - chemical transformation of reactants into products Classification: a) according to the type of equipment o batch stirred tanks small-scale production, mostly liquids o continuous stirred tank reactors (CSTR) medium to large-scale production, arrangement into cascades (liquids) o tubular continuous reactors gases or liquids flow through a pipe, which may be containing a fixed bed of granular material catalyst, large-scale production b) according to number of phases o homogeneous just one phase (gas or liquid) o heterogeneous fixed bed of catalyst particles General mass and enthalpy balance - mass of component k: m k,i + dm kr inflow + source/sink = outflow + accumulation = m k,e + dm ks - mass of all components: m i + = m e + dm S - enthalpy: H i + = H e + dn S H has thermal contribution, pressure contribution (often negligible) and reaction contribution ( source - like) Stoichiometry - chemical reaction is written as: M k = stoichiometric coefficient, negative for reactants, positive for products M k molar mass of component k Conservation of stoichiometric ratios n kr n lr = ν l ; l and k are two components taking part in a reaction (reactants or products) it follow that n kr = n lr ν l = ξ extent Extent is independent of the choice of component, it is a measure of the reaction. If n k is amount of k at the beginning and n k is amount of k at the end of the reaction then
unit of ξ: [moles of reaction turnover] For constant volume (liquids): ξ = V s c k c k Conversion of a reactant k - for constant volume: ζ k = c k c k c k - relation between ξ and ζ k : ξ = n kζ k - another relation: - for V S = const also ξ = n kr = n k n k = m k m k M k ζ k = n kr = n k n k = m k m k n k n k m k n l n l ν l = n k n k c l = c l + ν l (c k c k ) n l = n l + ν l (n k n k ) Thermodynamics Enthalpy of the reaction mixture is: n l = n l + ν l n k ζ k ; for V S = const: c l = c l + ν l c k ζ k dh = nc p dt + ( H p ) dp + h r dξ T,ξ n c p h r number of moles molar thermal capacity molar reaction enthalpy For liquids and ideal gases ( H ) = p T,ξ Standard molar reaction enthalpy is: h r = h fk = h ck where h fk standard molar enthalpy of formation of component k () h ck standard molar combustion enthalpy of k
can be found in tables h r varies with temperature: h r = h r + c pk dt T T The Gibbs free energy is dg = SdT + V S dp + g r dξ S g r a k K entropy molar reaction Gibbs energy activity of k equilibrium constant ν g r = g r + RT ln ( a k k ) K k=1 K At equilibrium g r and g r is found in tables Calculation of K: K true equilibrium constant ln K = g r RT K C concentration equilibrium constant K C = K ν c k k=1 k K depends on temperature: ln K ( T ) p,ξ = h r RT 2 source/sink: dn kr Kinetics = R = rdv V S R extensive reaction rate [moles of reaction turnover/s] reaction rate of component k r intensive reaction rate [moles of reaction turnover/s/m 3 ]
R K = R r K = r From the definition: R = 1 dn kr = d (n kr ) = dξ For ideally mixed systems r does not depend on position dv R = rv S Reaction rate r depends on molar concentration of reactants For example aa + bb r = kc A a c B b where k = k frequency exp factor ( activation energy E RT ) Arrhenius relation For reversible reactions: r = r + r For example: aa + bb pp + qq r = k + c A a c B b k c P p c Q q at equilibrium r = k + c A a c B b = k c P p c Q q and K = k + = c p q P c Q k c a b A c B Batch reactor Assumptions: - one reaction only - ideal mixing R = rv S - constant density V S = const Balance on moles of k (no inflow, no outflow) source = accumulation dn kr = dn ks
Reaction time: The use of conversion: ζ k = c k c k c k rv S = V S dc k τ τ = c k = dc k r c k c k = c k (1 ζ k ) dc k = c k dζ k ζ k τ = c k dζ k r r must be expressed in terms of either c k or ζ k Example 1: isothermal reaction of first order τ = c A A B; r = kc A ; ν A = 1 τ = c A c A dc A kc A = 1 k ln c A c A ζ A dζ A kc A (1 ζ A ) = 1 k ln 1 1 ζ A Example 2: isothermal reaction of second order c B is expressed via c A : and then: τ = c A aa + bb r = kc A a c B b ; ν A = 1, ν B = 1 dc A kc A (c B + 1 1 (c A c A )) c B = c B + ν B ν A (c A c A ) c A = 1 (c A c B )k ln (c B c A ) c A c B c A + c A Enthalpy balance (neglecting pressure term) Q i = Q e + dh Q i Q e = d (nc pdt) + d ( h rdξ)
Q i Q e + ( h r ) dξ dt = nc p Mass and enthalpy balances must be solve together since R = rv S depends on molar concentrations. R Continuous stirred tank reactor CSTR - assumptions: o one reaction o ideal mixing (R = rv S ) o constant density: V i = V e = V = const (liquids) o steady state Balance on moles of k: Using conversion: Again r is expressed in terms of c k or ζ k. Enthalpy balance n ih i = n eh e + Q e Q i n ki + rv S = n ke V c ki + rv S = V c ke τ = V S V ζ k = c ki c ke c ki = c ke c ki r τ mean residence time τ = c kiζ k r h i and h e include thermal and reaction terms! Q e Q i = K(T e T c )A heat transfer equation After some algebra it can be shown that the enthalpy balance is: n ic pi (T i T ) + ( h r )rv S = n ec pe (T e T ) + K(T e T c )A By choosing standard temperature T = T i : ( h ri )rv S for exothermic reaction this is sigmoidal function of T e H R = n ec pe (T e T i ) + K(T e T c )A linear function of T e H P
Intersection of the line steady states three A unstable state B, C stable states with low and high temperature respectively (high temperatures operational limits, explosions!) Cascade of CSTRs - assumptions as for one CSTR - conversion in n-th reactor: ζ kn = c k c kn c k - molar flow: n kn = V c kn - reactors may have different volumes: V n ; n = 1,, N Balance on moles of k in the n-th reactor n k,n 1 + r n V n = n k,n c k,n 1 c k,n + r n τ n = c k (ζ k,n ζ k,n 1 ) + r n τ n = Convenient calculation is from reactor N to reactor 1 because r n depends on c k,n and c k,n 1 is easy to evaluate Graphical solution
It does not matter if from reactor 1 to reactor N or vice versa. Rewrite mass balance: ν k r n > for reactant, generally a curve = 1 τ n (c k,n c k,n 1 ) straight line with slope 1 τ n passing through point [c k,n 1 ;] The straight lines are parallel if the reactors have constant volume ( τ = const) Analytical solution for isothermal 1 st order reaction A ; r n = kc A,n ; ν A = 1; τ = const. Mass balance: c A,n 1 c A,n + ( 1)kc A,n τ = c A,n 1 c A,n = 1 + kτ Then c A, c A,1 c A,N 1 = c A, = (1 + kτ ) N c A,1 c A,2 c A,N c A,N Enthalpy balance: (reference state at T = T ) Q in Q en + n n 1c p,n 1 (T n 1 T ) + ( h r )r n V n = n nc pn (T n T ) K(T n T c )A n Can be used to calculate heat exchange area A n. - assuming: o one reaction Tubular reactor (continuous, for gases or liquids)
o o o turbulent flow approximation by plug flow density may not be constant for gases steady state Conversion is defined as: ζ k = n ki n k n ki dn k = n kidζ k Balance on moles of k in dv Reaction volume is: n k + rdv = n k + dn k d (dζ k ) V V = dv n ke = dn k = n ki dζ k r r a) liquids: density = const V = const τ = V V is well defined mean residence time: n ki ζ ke τ = V V c ke = dc k r c ki Formally the same formula that is for calculation of reaction time in a batch reactor b) gases: instead of τ we define a spacetime at input τ i = V V i then Now we need to express r in terms of the conversion. We know that: 1) n l = n li ν l n kiζ k 2) gas follows the equation pv = n RT = l n l r depends on c l ; l = 1, and c l is expressed as ζ ke τ i = V = c ki dζ k V i r RT
n c l = n l = n l = p l n l l n l = c V n RT RT n n p l n l Thus r is expressed in terms of y li, y ki and ζ k. Enthalpy balance where n li ν l n kiζ k l n l = c (n li ν = l l ν n k kiζ k ) l n l dq i + H = H + dh + dq e c molar density y li ν l y ki ζ k 1 y kiζ k ν l ν l k Thus dξ dh = n c p dt + h r d ( ) m c pm dr=rdv h r rdv = m c pm dt + dq e dq i