CE 329, Fall 2015 First Mid-Term Exam

Transcription

1 CE 39, Fall 15 First Mid-erm Exam You may only use pencils, pens and erasers while taking this exam. You may NO use a calculator. You may not leave the room for any reason if you do, you must first turn in your exam, and you will not be permitted to resume taking the exam. his exam is printed on two-sided pages there are 5 short answer questions, each worth 5 points and three problems worth 5 points each for a total of 1 points. Answer All questions, including the short answer questions, in the blue book that has been provided to you. When the exam ends, put these sheets inside your blue book and turn both in. For the problems you only need to fully show what equations to use and to fully explain how to use them to obtain the requested answer. You do not need to perform any calculations, and you do not need to perform any algebra beyond that needed to obtain the equations to be solved. If you need to use numerical methods to answer a question, provide the information listed below Numerical Methods If you need to fit a linear model to data, you must state that it is necessary to fit a model to the data numerically and you must explicitly identify (a) the specific linear model being fit to the data, (b) the response and set variables in the model and (c) the parameters in the model. hen you must (d) show how to calculate the value of each response and set variable for an arbitrary data point. Once you have provided that information, you may assume that the correlation coefficient, the best value of each model parameter and its 95% confidence interval and either a model plot or a parity plot and residuals plots have been found using appropriate numerical software, and you may use those results as you need to complete the problem. If the solution to a problem involves solving a set of non-linear algebraic equations, you must state that it is necessary solve a set of non-linear algebraic equations numerically and you must (a) explicitly identify the equations to be solved and an equal number of unknowns to be solved for by writing the equations in the form, fi(unknowns list) expression. You then must (b) show how to calculate every quantity that appears in those functions, assuming you are given values for the unknowns. Once you have provided (a) and (b), you may assume that the values of the unknowns have been found numerically, and you may use those values as needed to complete the problem. If the solution to a problem involves solving a set of initial value ordinary differential equations, you must state that it is necessary to solve a set of initial value ODEs numerically and you must (a) explicitly identify the equations to be solved, the independent variable and the dependent variables by writing the equations in the form, (derivative i ) fi(independent variable, dependent variable list) expression. hen you must list values or show how to calculate (b) initial values of the independent and dependent variables, (c) the final value of either the independent variable or one of the dependent variables and (d) every quantity that appears in those functions, assuming you are given values for the independent and dependent variables. Once you have provided (a), (b) and (c), you may assume that the

2 final values of the remaining independent and dependent variables have been found numerically, and you may use those values as needed to complete the problem. If the solution to a problem involves solving a set of boundary value ordinary differential equations, you must state that it is necessary to solve a set of boundary value ODEs numerically and you must (a) explicitly identify the equations being solved, the independent and dependent variables in those equations and the boundaries of the range of the independent variable over which the equations are to be solved, (b) list values or show how to calculate boundary conditions for each dependent variable the number of boundary conditions for a particular dependent variable must equal the highest order of derivative of that dependent variable appearing in the equations being solved and (c) list values or show how to calculate every quantity in the equations being solved other than the derivatives, assuming you are given values for the independent and dependent variables. Once you have provided (a), (b) and (c), you may assume that the value of each dependent variable and its first derivative with respect to the independent variable is known at any position between the boundaries, and you may use those values as needed to complete the problem. a ax x n xn+1 n +1 (n 1) a + bx ln x a + bx x 1 ax + b a Indefinite Integrals x ln x x a + bx 1 1 ( b 3 a + bx ) a( a + bx) + a ln( a + bx) x a a + bx a + bx 1 b 3 a + bx aln a + bx ΔH ΔH K ( 98 K) ν i, ΔH f,i i all ( ) ΔH 98 K ( a + bx ) 1 ln( a + bx) b x( a + bx) 1 a + bx ln a x a + bx b( a + bx) 1 x a + a + bx a + bx 1 b ln a + bx x a + bx x b a ln a + bx b hermodynamic Relationships 98 K ( ( 98 K) ) K ν i, ΔH c,i i all ΔG 98 K + ν i, d i all 98K ( 98 K) exp ΔG 98 K R( 98K ) K a i y i ϕ i P 1 atm a i γ i x i a i x i a i h i x i K 98 K ν i, ΔG f,i exp a i ν i, ( 98 K) 98 K ( ) d R ΔH a i y i P 1 atm

3 ξ g k ( n i n i ) ξ! ν i, f k f k equil Rate, Composition and Reaction Progress Relationships ( ) N ind n i n i + ν i, ξ n k n k n k n k ν i, equil C i n i 1 f k n n k k n k V C i V! V!!n R total P V! r r i, ν i, 1 V dξ ( constant ρ ) P n R total V P!n total R P i n i R V P!n R i i V! P y P i i µ r g C cells r AB forward N Av σ AB C A C B r ABC forward 8N Av σ AB σ BC lc A C B C C r forward r k, f r RI, q Elementary Reaction Relationships 8k B πµ exp E R r N σ C k B AA forward Av AA A πµ exp E R k B Nq AB q C h exp ΔE k B AB C k B π µ AB µ BC exp E R i ν i, k,r i ν i, k, f i reactants s all steps ν RI,s r s C cat products reactants ν i, 1 i ν i, K,eq r i, ν i,s r s r r srd s all steps C cat, free + ν cat,i C cat,i C isurf C sites θ i θ vacant + θ i 1 i all catalyst complexing i all adsorbed F ( λ) w w t F ( λ) 1 exp λ w f w t Age Function Relationships F λ t!m w out t t w m tot for t < t 1 for t t F λ F λ x total xn ( x) x total x dn ( x) y total y( x) N ( x) y total y( x)dn ( x) x x N total N x N total dn(x) y average x x x x y( x) N ( x) x N ( x) x y average x y( x)dn ( x) x dn ( x) 3

4 τ V SV 1 dn τ i V!Q! W n i Ĉ p,i d +V ν r i, Reactor Relationships r ΔH dp V P dv!q W! Ĉ p i d +V ( r ΔH ( )) +V z π D 4 P z 1 ε G ε 3 ρφ s D p g c π DU ( e ) z dn i!n i +V ν i, r + d Ĉ p i ν i, r t P z G 4 g c π D z fg ρd 15( 1 ε )µ Φ s D p G V ν i, r!q W! ĥ i ĥi,stream D ax d C i dz λ er r + d dz u s C i + 1 r r Ĉ p i + π D 4 + d ν i, r r ΔH + π D 4 ( n i Ĉ pi ) + V r ΔH C D i er + 1 r r u sρ fluid C! p, fluid z r ΔH C i r t z u C s i d P dv Ĉ p i dp V P dv ν i, r V dp P t V 4

5 hot,out Other Relationships,hot d +,cold d,hot d +UAΔ hot,in Δ AM cold,out + hot,out Δ cold hot,out cold,in R R reactor in cold,out cold,in cold,in + hot,in, feed d +,r d feed reactor out ( recycle ) hot,out hot,in hot,out cold,in Δ LM hot,out cold,in ln hot,in cold,out ( hot,in cold,out ) recycle flow process exit flow, feed + R R,reactor out 1+ R R,reactor in reactor in Short Answer Questions (5 points each) 1. rue or false: If you want to use a differential method of data analysis on batch reactor data, then the reactor must be operated differentially when the data are being collected.. rue or false: In an ideal plug flow reactor, perfect radial mixing is assumed. 3. rue or false: External transport limitations that affect the apparent rate of a heterogeneous catalytic reaction are the result of concentration and/or temperature gradients along the length of the catalyst pores. 4. he quantity, 1 exp λ t, represents a. he response of an ideal CSR to a step change input. b. he age function for an ideal CSR. c. he concentration gradient between the bulk fluid and the external surface of a heterogeneous catalyst. d. he Weisz-Prater criterion for the absence of internal mass transport limitations. e. An impulse stimulus for measuring the residence time distribution. 5. Write the Arrhenius expression. Problems (5 Points Each) 6. he macroscopically observed reaction given in equation (1) takes place at the molecular level via the mechanism given in equations () and (3). Without making any additional assumptions beyond the Bodenstein steady state approximation, derive an expression for the rate of the overall reaction. Your answer should not include the concentration or partial pressure of any reactive intermediates. ICl + H I + HCl (1) 5

6 ICl + H HI + HCl () HI + ICl HCl + I (3) 7. Gas phase reaction (1) below was studied in an isothermal, batch reactor. Initially the 1 L reactor contained pure reactant A at a temperature of 35 C and a pressure of 1 atm. he reaction proceeded at 35 C, with measurements of the total pressure being taken versus time. he resulting data are given in the table below. Use the data to test the validity of the rate expression given in equation () below using an integral analysis. A(g) B(g) (1) r 1 k 1 P A () t (min) P (atm) he liquid phase reaction between A and B, equation (1), was studied in a 5 ml CSR at 4 atm total initial pressure and 139 C. he reactor was fed a solution containing A at a concentration of.8 M and B at a concentration of 1. M. he fractional conversion of A was measured as a function of space time time the data are given below. Determine whether the rate expression given in equation () adequately describes the data. A + B R (1) r A kc A C B () space time (h) fa