ChE 471 Lecture 10 Fall 2005 SAFE OPERATION OF TUBULAR (PFR) ADIABATIC REACTORS

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Philomena Taylor
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1 SAFE OPERATION OF TUBULAR (PFR) ADIABATIC REACTORS I a exothermic reactio the temperature will cotiue to rise as oe moves alog a plug flow reactor util all of the limitig reactat is exhausted. Schematically the adiabatic temperature rise as a fuctio of space time, measured from the reactor etrace, takes the form show i Figure 1. FIGURE 1: Temperature as a fuctio of space time i a adiabatic PFR with exothermic reactio. The fial adiabatic temperature : ( ) C Ao T ad 1+ Δ H r A ρ C p [ 1 + β] (1) is ofte excessive for highly exothermic systems (β large) ad eeds to be avoided. I such situatios the old rule of thumb suggests that we should operate i such a way that the iflectio dt poit, i.e. the poit of maximum temperature rise, is ever reached. This implies that dτ max we operate with τ < τ * where τ * is the value of space time at which the iflectio poit d T dτ occurs. 1

2 Let us ow develop a coveiet ad simple formula for use by practicig egieers which would guaratee safe operatio. Cosider a -th order irreversible reactio: R A k o e E RT C Ao ( 1 x A ) () At adiabatic coditios temperature ad coversio are related by the adiabatic lie equatio T ( 1 + β x A ) (3) If we defie, as i the case of wall cooled reactors, the dimesioless temperature by θ γ T (4) the coversio i Eq. (3) ca be expressed i terms of dimesioless temperature as: x A θ δ (5) where with δ β γ (6) γ E R (7) The rate of reactio evaluated at adiabatic coditios i terms of dimesioless temperature is obtaied by usig the adiabatic lie eq (5) to replace coversio. The result is: ( R A ) ad ( R A ) o e γθ θ + γ ( ) (8) where the rate evaluated at the feed coditio is: ( R A ) o k o e E R C Ao (9) The eergy balace for adiabatic PFR operatio is: ρ C p dt dτ ( Δ H r A ) ( R A ) ad (1) Writte i terms of dimesioless temperature it becomes: dθ dτ δ e γθ τ R ( θ + γ ) (11)

3 where the characteristic reactio time is: C Ao τ R ( R A ) o (1) The iitial coditio is τ θ (13) We ca rewrite Eq. (11) i the most compact way by defiig the Damkohler umber as the ratio of characteristic process time ad reactio time, i.e. ( ) o τ Da τ R A τ R C Ao (14) Equatio (11) ad iitial coditio (13) become: dθ d ( Daδ) eγθ ( θ + γ ) (15) At Daδ θ (16) Now we wat to have the reactor short eough (limit the coversio achievable) so that we ca keep the iflectio poit d θ out of the reactor i.e. we do ot let it occur i the reactor. dτ The critical temperature at iflectio poit,, is obtaied by settig d θ ( ) (17) d Daδ which results i a equatio for θ θ if yieldig: θ if γ [ γ + 4 ( γ + δ) γ] ; (18a) θ if δ, (18b) The result for the zeroth order reactio, eq (18b), ca be obtaied by applyig the L Hospital rule to eq (18a). A better approach is to otice that for equatio 15 idicates that dθ/d(daδ) > always, so that there is o iflectio poit as the rate of temperature rise keeps risig util all reactat is depleted. The at x A 1, from eq (5) it follows that θ max δ ad therefore T max T ad. 3

4 We ca itegrate the differetial equatio (15) by separatig the variables to obtai the critical value of Daδ * ad, hece, of τ * Daδ * Daδ * d ( Daδ) θ if e γθ ( θ + γ ) 1 θ dθ (19) Clearly for give values of parameters γ, β, δ βγ, ad reactio order, we ca evaluate θ if from eq (18a) ad the calculate the value of Daδ * by umerically evaluatig the itegral i eq (19). Usig the the defiitios of Da ad δ we get the criterio for safe operatio Daδ < Daδ * which ca be expressed as: k o e E R C Ao ( Δ H ra ) E τ * R ρ C p Daδ * () To get a coveiet, easy to remember value of Daδ *, the followig approximatios are ofte made. First, the Arrheius depedece of the rate costat o temperature is replaced by a expoetial depedece, i effect e γθ ( θ + γ ) e θ (1) Substitutig this approximatio i eq (15) yields via eq (17) to a ew approximate value of the temperature at iflectio poit ( θ if ) app δ () Substitutig eq (1) ad eq () ito eq (19) yields ( Daδ * ) app δ e θ 1 θ dθ δ e δ δ e u u du (3) The fial approximatio (which is coservative i ature as it assumes the worst possible case of zeroth order reactio) igores the slowdow of the temperature rise due to the cosumptio of the reactat, which is the same as takig i eq (3). 4

5 This yields ( Daδ * ) e δ e u du e δ e u app, δ e δ e δ 1 ( ) δ [ ] 1 e δ 1 ( for large eough δ) (4) For highly exothermic reactios ad clearly ( Daδ * ) 1. app, o Substitutig this ito eq () gives the coservative criterio for safe operatio. It is costructive to ote that this same equatio (a) with time t replacig τ k o e E R C A o ( Δ H ra ) E t * R ρ C p <1 (a) is used to determie the so called time of o retur or time to explosio i batch systems. This time to iflectio poit may be very log for low but becomes quite short if the system of high activatio eergy is exposed to higher. Hece, chemicals that may be safe to store at 5 C may be explosio proe if exposed to 4 5 C! SAFE OPERATION OF ADIABATIC CSTR The mass balace for a irreversible -th order reactio is: C Ao x A ( R A ) τ (5) The adiabatic equatio relates coversio ad temperature T T x A o (6) ( Δ H ra ) C Ao ρ C p Upo substitutio of dimesioless temperature we get from eq (5) let 1 Da δ θ eγθ G ( θ) e γθ θ + γ L ( θ) 1 Da δ θ ( θ + γ ) ( ) (7) (8a) (8b) 5

6 We kow from before that G is a sigmoidal curve i θ ad represets heat geerated by reactio. L is the heat removal rate (i.e., heat removed by sesible heat of the fluid that flows through the CSTR). We kow that up to three itersectios are possible betwee G ad L lies. To avoid the itersectio leadig to excessively high temperatures we must assure that itersectios at low temperatures are available. The last permissible operatig coditio is the oe whe lie L is also taget to curve G as schematically show i Figure. FIGURE : Schematic of the G, L vs. θ Clearly as Daδ icreases the slope of the L lie decreases so that ( Daδ) 1 < ( Daδ) < ( Daδ) 3 (9) While operatig adiabatic temperatures at θ 1 ad θ are acceptable, θ 3 represets too large a temperature jump. Hece, we must assure that the L lie always itersects the G lie at its lower temperature brach. The critical poit is reached whe L is also taget to the G lie. For safe adiabatic operatio we therefore require L G (3) dl dθ dg dθ (31) Applyig the above to eqs (8a) ad (8b), ad usig the equality sig i eq (31), we get the equatio for the maximum permissible temperature θ max perm θ *. The critical value of the space time τ * ca be obtaied from the critical value of Daδ *, which i tur results from substitutig the expressio for θ * ito eq (3). 6

7 To get a simple, easy to remember expressio, usually we agai replace e γθ ( θ + γ ) with e θ. This yields θ * 1 δ + 1 δ + 1 ( ) 4 δ (3) The egative sig i frot of the square root eeds to be take as we are iterested i the lower of the two temperatures at which the L lie could be taget to the curve G. The equatio (3) yields: * ( Daδ) app θ * 1 θ* A coservative estimate, with, yields θ * 1 ad e θ * (33) Daδ * e 1 (34) For safe operatio the k e E R C Ao R ρ C p ( Δ H ra ) Eτ e 1 (35) 7

The axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c.

The axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c. 5.4 Applicatio of Perturbatio Methods to the Dispersio Model for Tubular Reactors The axial dispersio model for tubular reactors at steady state ca be described by the followig equatios: d c Pe dz z =