6. Energy balance on chemical reactors Most of reactions are not carried out isothermally BATCH or CSTR heated (cooled) reactors Inlet tubing Coolant outlet Reaction Mixer mixture Double shell Coolant Outlet Internal inlet tubing heat exchanger The balance of total energy involves: Internal energy mechanical energy (kinetic energy) potential energy R.B.Bird, W.E.Stewart, E.N.Lightfoot : .... Transport Phenomena, 2nd Edition, J.Wiley&Sons, N.Y. 2007 Transformation of various kinds of energy Balance of total energy Input x Output Input x Output of total energy of total energy by convective by molecular flux flux Rate of change of total energy EUEE Work done kin p Work done by molecular by external interactions forces Main reason to study energy balances : assesment of temperature of reacting system (reactor) Application of the 1st law of thermodynamics on the open homogeneous reacting system Rate of work done on surroundings [W] Heat flux [W] E e dV V VR Single phase reacting system eo,e1 – specific total energy of inlet (outlet) streams [J/mol] 3 VR – volume of reaction mixture [m ] dE F e Fe Q W dt oo 11 eeo ,1 molar energies of inlet and outlet streams [J/mol] Rate of work W done by the reacting system on the surroundings consists of: • Flow work of inlet stream(s) VPFVPo o o mo o • Flow work of outlet stream(s) V1 P 1 FV 1m 1 P 1 • Work provided by stirrer Ws dVR • Work done by volume change P dt • Work done by electric, magnetic fields Wf dE dV FeVP FeVPQP R WW dto o mo o1 1 m 1 1 dt s f Neglecting potential and kinetic energies (EU ), we have dU dV F h F h Q PR W W dto mo11 m dt s f hhmo, m1 molar enthalpies of inlet and outlet streams [J/mol] If WWsf0, 0 dU dV F h F h Q P R dto mo11 m dt From enthalpy definition dH dU dP dV dH dP VP R V F h F h Q dt dtR dt dt dtR dt o mo11 m Introducing partial molar enthalpies of species N oo Fo h mo F i H i i1 N F11 hm F i H i i1 We have finally NN dH dP oo VFHFHQR i i i i dt dt ii11 BATCH reactor dH dP VQ dtR dt Enthalpy is a function of temperature, pressure and composition Total heat capacity [J/K] Partial molar enthalpy [J/mol] HHH N dH dT dP dni T P,, n P T n i1 ni jjT,, P n ji HH NN CP dT dP H i dn i m V R c P dT dP H i dn i PP T,, njjii11 T n Molar density [mol/m3 ] Molar heat capacity [J/mol/K] We know that (from thermodynamics) H VR VTVTR R1 p PT Tn, j Pn, j where the coefficient of isobaric expansion is defined as 1 VR p VT R Pn, j and we obtain N dH dT dP dni mc P V R V R1 p T H i dt dt dti1 dt dn Finally by substitution of i in the energy balance of the batch reactor dt NR dT dP N dn dni i VrR ki V, k mc P V R p V R T H i Q dt k1 dt dti1 dt Using definition of the enthalpy of k-th reaction N rHH k ki i i1 we have dT dP NR mc P V R p V R T V R r H k r V, k Q dt dt k1 dP Isobaric reactor ( 0 ) dt dT NR mc P V R V R r H k r V, k Q dt k1 VR f() t we need state equation ! Homework 8: Energy balance of ideal gas isobaric batch reactor dV Isochoric reactor ( R 0 ) dt NR dT p mV R c V V R r H k T V kr V, k Q dt k1 T Pf ()t the coefficient of isobaric expansion the coefficient of isothermal compressibility 11VVRR pT VTVP RR P, nj T the volume variation due to k-th chemical reaction N V VVVk ki i, where i is the partial molar volume of species i i1 ni T,, p n ji the specific heat capacity at constant volume 2 PP VR p CVPPR C T C TV pC P TV R TTT Vn, j Pn, j Vn, j T Variation of the pressure can be derived from total differential of volume N VR dT dni Vi N TPn, dti1 dt dPj p dT 1 dni Vi dtVR dt Vi1 dt TRT P Tn, j Homework 9: Energy balance of ideal gas isochoric batch reactor dV R 0 Summary of energydt balance of BATCH reactor NR dT p mV R c V V R r H k T V kr V, k Q dt k1 T Pf ()t dP 0 dt dT NR mc P V R V R r H k r V, k Q 0,cc dt NR If p p V dT k1 liquid (condensed) systems mc P V R V R r H k r V, k Q Rate of change Rate of heat generation Rate of heat of reactiondt mixture k 1by chemical reactions loss (input) enthalpy Heat fluxV : f() t RQSTT H ( e ) the overall (global) heat transfer coefficient [W.m21 .K ] 2 SH the heat exchange area [m ] Te temperature of external cooling (heating) fluid Limiting cases dT NR Isothermal mc P V R0 V R r H k r V, k Q dt reactor k1 Adiabatic dT NR Q0 c V V H r reactor m p R R r k V, k dt k1 Example Adiabatic reactor with 1 reaction, constant heat capacities Energy balance on adiabatic BATCH reactor dT c V V H r m p Rdt R r V Molar balance of key component dc dX jj cro dtj dt j V N mV R n, c p y i c pi i1 NNN Vcnyc nc noo i nXc m R p i pi i pi i j j pi i1 i 1 i 1 j NNNoo oonj X j n j X j ni c pi i c pi n i c pi c p i1 j i 1 i 1 j Assuming that cpi const c p const yo H() T X TT j r o j TX o N o j j oo vj y i c pi y j c p X j i1 yo H() T j r o the adiabatice rise of temperature j N oo vj y i c pi y j c p X j i1 Trajectories of T(t) and Xj(t) Xj = 1 Exothermal reaction T(t) Xj(t) r H 0 Xj(t) o yj r H() T o j N oo vj y i c pi y j c p X j T(t) i1 To t Homework 10 The reversible reaction A1 + A2 A3 is carried out adiabatically in a constant-volume BATCH reactor. The kinetic equation is r k c1/2 c 1/2 k c fb1 2 3 31 k f (373 K)=2x10 s E1 100 kJ/mol 51 kb (373 K)=3x10 s E2 150 kJ/mol Initial conditions and thermodynamic data cco 0.1 mol/dm3 25 J/mol/K 11p o 3 cc220.125 mol/dm p 25 J/mol/K oo rpH298 40 kJ/mol c 3 40 J/mol/K T = 373K Calculate X1(t),T(t). Example Acetic anhydride reacts with water (CH3CO)2O + H2O 2CH3COOH in a BATCH reactor of constant volume of 100 l. Kinetics of reaction is given by 46500 r2.14 107 eRT c mol.m 3 .min 1 V 1 Data o -1 -1 -1 1 3 c1 0.3 mol.l , cpM 3.8 kJ.kg .K , H r 209kJ.mol , =1070 kg.m 1 oocpi .SH =200W.K T e T 300 K, c p 0, c pM w i c pMi , c pMi i M i wco initial mass fractions, - mass heat capacities (kJ.kg-1 .K -1 ),M - molar weight (kg.mol 1 ) i pMi i RK 8.31446 J.mol11 . In neglecting variation of heat capacities with temperature, calculate T(t) and X1(t) for an non-adiabatic and adiabatic case. By numerical integration we get dT 1 46500 0.1 209E3 2.14E7xexp - 300 (1 X1 ) 200 (300-T) dt(1070 3.8E3 0.1) (8.31446 T ) dX1 46500 2.14E7 exp - (1 X1 ) dt(8.31446 T ) 316 1.20 T [K] 314 1.00 X1 312 0.80 310 0.60 308 0.40 306 0.20 304 0.00 0 10 20 30 40 50 60 t [min] Continuous (perfectly) stirred reactor (CSTR) Energy balance on CSTR NN dH dP oo VFHFHQR i i i i dt dt ii11 Total differential of enthalpy N dH dT dP dni mc P V R V R1 p T H i dt dt dti1 dt NN dP oo VFHFHQR i i i i dt ii11 Molar balance on CSTR NR dni o Fi F i V R ki r V, k , i 1, N dt k1 We get N dT dP N NR rHH kc V ki i TV H Fo F V r Enthalpy mPRdt pR dt ii iR kiVk, i1 ik11change for NN oo k-th reaction FHFHQi i i i ii11 dT dP NR mc P V R V R p T V R r H k r V, k dt dt k1 N oo FHHQi i i i1 dP 0 dt The CSTR usually works at constant pressure (no pressure drop) NR N dT oo QSTT Tm() mPRc V V R rkVk H r, + F ii H H i Q dt ki11 Steady state dT dn i 0 dt dt NR N oo VR r H k r V, k + F i H i H i Q 0 ki11 Assuming ideal mixture, i.e.