Setting u the Mathematical Model Review of Heat & Material Balances Toic Summary... Introduction... Conservation Equations... 3 Use of Intrinsic Variables... 4 Well-Mixed Systems... 4 Conservation of Total Mass:... 4 Comonent Balances... 5 Energy balance:... 5 dditional Relationshis... 6 Excetions to Well-Mixed Process ssumtions... 8 Processes with Dead Time... 8 Flow roximated by Combination of Well-Mixed Blocks... 8 Examles... 9 Tank Liquid Level with Flow Through Valve... 9 State Variables:... 9 Total Mass Balance Leads to a Volume Balance:... 9 Other Tank Geometries... Right Circular Cone... Horizontal Cylinder... Tank flow Change in Inlet Concentration... Overall & Comonent Mass Balances:... Comonent Material Balance:... Using Concentration... Using Mass Fraction... Tank Flow Chemical Reaction... 3 Tank Flow Change in Inut Stream s Temerature... 4 State Variables:... 5 Initial state:... 5 Transient solutions... 5 What if the heat inut is described by Newton s law?... 6 Examle Tank Flow Controlled by Valve... 7 Bernoulli s Law:... 7 Total mass balance reduces to volume balance for constant density:... 8 Examle: Chemical Reaction... 9 Toic Summary Definition of well-mixed system. Combining well-mixed sub-rocesses to describe overall rocess that is not well-mixed. John Jechura (jjechura@mines.edu) - - Coyright 7 ril 3, 7
Dynamic equations from material & energy balance equations Tyical simlifications & modifications dditional relationshis We will show that certain dynamic relations tyically come most directly from certain conservation equations. These tyical relationshis are shown in the following table. Conservation Equation Overall Material Balance Comonent Balance Thermal Energy Balance Tyical Dynamic Relation Outlet volumetric flowrates vs. inlet rates Liquid: Liquid level in system vs. time Gas: Gas ressure in system vs. time Outlet concentration vs. time Outlet mole/mass fraction vs. time Outlet temerature vs. time Introduction Transient behavior of a rocess: Start u. Steady state random disturbances. Change of set oints. Shut down. Stes for math modeling: Develo the relationshis/equations. Simlify. Solve: nalytical Lalace transforms Numerial Euler, Runge-Kutte methods Needs of math model: Quantities whose values describe the nature of the rocess. These are the state variables. Equations that use the quantities & describe how the variables change with time. For algebraic equations, where N is the number of variables & E is the number of indeendent equations: E N, deterministic system. E N, under-determined system. E N, over-determined system. John Jechura (jjechura@mines.edu) - - Coyright 7 ril 3, 7
For differential conditions, also need boundary conditions. For transient roblems, these are normally initial conditions. We will mostly be working with lumed systems i.e., there will be no satial variation. Often termed as a well-mixed system. Conservation Equations We will use the basic rinciles of chemical engineering to guide us in our descritions of our dynamic rocesses: conservation of mass, energy, & momentum. So, the tyes of conservation equations will be: The overall mass balance. Comonent/chemical secies balance (including reaction rate terms). Thermal energy/heat balance. Momentum balance (though we won t usually work with this in this class). General form of the stuff balance equation: Rate of Rate Rate Rate of Rate of ccumulation In Out Generation Consumtion Basic rinciles of ChE: F Q F 6 F Q 3 F 3 F 4 F 5 Q Summary of the balance equations: dm Overall Mass Balance mi dm Comonent Balance m, i m, j RkV m k: rxns j John Jechura (jjechura@mines.edu) - 3 - Coyright 7 ril 3, 7
dn,, N N r V i j k k: rxns de Energy Balance E E Q W Use of Intrinsic Variables, i j k s m It is useful to factor out intrinsic variables roerties that deend uon the state of the system (ressure, temerature, hase condition) but not the magnitude. It is often convenient to use the roduct of the mass density and volumetric flowrate instead of the mass flowrate. Some of the useful relationshis will be: Cumulative Exression Rate Exression Mass mv mf Moles n V C total V n F C total F Energy E me VE E me FE Enthaly H mh VH H mh FH nh C VH nh C FH total total Well-Mixed Systems The intrinsic roerties of the system at any oint will be the same. dded imlication every stream will ossess the same intrinsic roerties as the system itself. The intrinsic roerties of the inlets can still be indeendent. Conservation of Total Mass: dm d d m i m i : inlet : V ifi Fj dv F F i: inlet j: j i i j j j (use the intrinsic variable ) (well-mixed system) where F i is a volumetric flow rate (in or out). Note that all streams will have the same density as the density within the system s volume. Conservation of total moles? NO! When there is a chemical reaction the total number of moles may not necessarily be conserved. John Jechura (jjechura@mines.edu) - 4 - Coyright 7 ril 3, 7
Comonent Balances The comonent balances can be exressed either in terms of mass (and mass fraction): dm m, i m, j RkdV k: rxns d dv, i ifi, j jfj RkdV d V k: rxns F F R dv, i i i j k k: rxns or in terms of moles (and molar concentration, though mole fractions could also be used): dn N, i N, j r kdv k: rxns d cdv c, ifi c, jfj rk dv d c V k: rxns c F c F r V, i i j k k: rxns where R k is the mass reaction rate exression (kg er unit time er unit volume) and r k is the molar reaction rate exression (moles er unit time er unit volume). It is usually easier to work with moles when there is a chemical reaction. Note that the reaction term can be ositive for generation & negative for consumtion. Energy balance: de E E Q W, i j k s m d U K P w H w H Q W i i j j k s, m F H F H Q W i i i j j j k s, m where i H is the secific enthaly (er unit mass). The energy balance can also be ut on a molar basis: John Jechura (jjechura@mines.edu) - 5 - Coyright 7 ril 3, 7
where d U K P N H N H Q W i i j j k s, m H i is the secific enthaly er unit mole basis. There are additional assumtions normally made to the energy balance: d U K P F H F H Q W i i i j j j k s, m du F H F H Q W, i i i j j j k s m For liquid systems we will generally use the assumtion that (internal energy dominant term) U H : dh F H F H Q W, i i i j j j k s m d HdV F H F H Q W d HV, i i i j j j k s m, F H F H Q W i i i j k s m (well-mixed). For gas systems this is not necessarily the case. However, since we ultimately want a dynamic exression for temerature this is not a significant roblem. d UdV F H F H Q W d UV i i i j j j k s, m (well-mixed). F H F H Q W i i i j k s, m dditional Relationshis These are the basic equations, but now we also need relationshis to the measured rocess variables. Relationshi between mass & volume: m V h (for constant cross sectional area) mf John Jechura (jjechura@mines.edu) - 6 - Coyright 7 ril 3, 7
Thermodynamic relationshis: T H C H H C dt C T P Tref T T T H H C dt C T T ref ref ref ref Tref U H PV H (for liquid systems) U H RT (for ideal gas systems) Equations of state relating density to ressure, temerature, & comosition: T, P, x (the equation of state). Some simle equations of state: PM P xm i i RT RT i i xm i i M i i i i (ideal gas) (ideal volume of mixing/additive volumes) Energy relationshis: K mv Vv mg h h Vg h h (gravitational otential energy. May be other forms) P Relationshi between heat transfer rate & temerature driving force: Q U T T (Newton s law of heating/cooling) a Q T 4 T 4 (Kirschoff s law of radiant heat exchange) Relationshi between flow through a valve & ressure driving force: F F C h (non-linear valve flow exression) Chemical reaction rate relationshis: r r T, x, P (the reaction rate function). For examle: v E / RT r k T c k e c ( st order reaction) E / RT E / RT r k T c k e c or r k e c c ( nd order reactions) B E / RT k T ke is the rrenhius rate exression. where John Jechura (jjechura@mines.edu) - 7 - Coyright 7 ril 3, 7
Excetions to Well-Mixed Process ssumtions Processes with Dead Time One excetion to the well-mixed assumtion is when there is some iece to the rocess that introduces a significant time delay between when something haens and when this might be measured. One can icture the situation as being similar to lug flow through a ie the material does not change in the ie, but there is a time difference between when it enters and when it reaears at the other end. For examle, if we are interested in the temerature at the exit of a tank, T o, but the thermocoule is in a ie a distance L away, then there will be a delay before the temerature can be measured. This time delay, t, can be estimated from: o t o distance L L velocity F / F c where: c is the cross sectional area of the ie. F is the volumetric flow rate. o c If there is no heat loss in the ie then the relationshis between the temerature of the tank at that temerature measured, T m, is: T t T t t. m o o Flow roximated by Combination of Well-Mixed Blocks Sometimes the flow atterns within a rocess do not roduce a well-mixed system. We may still be able to aroximate the overall rocess as a combination of well-mixed subrocesses. One obvious rocess that is not well-mixed is an annular heat exchanger with counter-current flow. Though the fluids might be well-mixed across any face erendicular to the flow there will be a temerature gradient along the direction of the flow. However, the overall rocess could be aroximated by two series of well-mixed sub-rocesses that flow from one to another, transferring heat across the boundaries of sub-rocesses. Hot Out Cold In Cold Out nnular Heat Exchanger Hot In John Jechura (jjechura@mines.edu) - 8 - Coyright 7 ril 3, 7
Cold In Cold Out Hot Out Hot In Examles Tank Liquid Level with Flow Through Valve F, h F, Look at how the level in a tank changes with changes of flows in and out of a tank. State Variables: Densities Flow rates What are we assuming if we say that the density out is the same as the density in the tank? What are we assuming if we say that the density is not changing? Total Mass Balance Leads to a Volume Balance: Total mass balance: dv F F d h i i j i: inlet j: F F ly other considerations: Constant cross-sectional area: constant John Jechura (jjechura@mines.edu) - 9 - Coyright 7 ril 3, 7
Constant density: Total mass balance becomes a volume balance leading to a differential equation for the level in the tank with resect to time: dh F F Other Tank Geometries The above geometry assumes that the tank is an uright cylinder. There are other common geometries: Right Circular Cone h r r tan h V r h h 3 tan 3 3 dv tan h dh a Horizontal Cylinder b s = arc length r sr ar b sr rb cos r a r r b r b r cos r b r r b r Still need the volume of the cylinder with resect to level: John Jechura (jjechura@mines.edu) - - Coyright 7 ril 3, 7
r h Lr cos L r h r r h if h r r V h r r L Lr cos L h r r h r if h r r The amazing art is that the derivative is the same for both halves: dv L hr h dh Shere. Using similar definitions are for the horizontal cylinder: h 3r h if h r 3 V 3 r r h r h if h r 3 3 and, again, the derivative is the same for both halves: dv dh h r h Tank flow Change in Inlet Concentration F,, C, C o, B, V,, C C B, F,, C, C, B, Look at how the concentration in a tank changes with changes of concentration into the tank. Overall & Comonent Mass Balances: Total mass balance: John Jechura (jjechura@mines.edu) - - Coyright 7 ril 3, 7
d V F F d V F F i i j i: inlet j: What have we assumed here? Constant volume overflow (& well-mixed). If we assume constant density: F F F F. Comonent Material Balance: We can deal with multi-comonent mixtures with the concet of concentrations or mole/mass fractions. Concentration gives the amount of the comonent er unit volume. This is multilied by the volumetric flow rate to get the flux of the comonent. Mole/mass fraction gives the fraction of the total amount corresonding to the comonent. This must be multilied by the overall molar/mass density and the volumetric flow rate to get the flux of the comonent. Using Concentration Total mole balance using concentration: dn C, ifi CFj i: inlet j: d C V C F C F dc V C,F CF, (no chemical reaction & well-mixed) (constant volume) What have we assumed here? Well-mixed, no chemical reaction, & constant volume overflow. If we assume constant density: dc dc F V F C C C C V Using Mass Fraction,, Total mass balance using mass fraction ( ): John Jechura (jjechura@mines.edu) - - Coyright 7 ril 3, 7
m i d m d V m,, j i: inlet j: F F, i i i j i: inlet j: d V, F F i i i j i: inlet j: d V, F F i i j i: inlet j: (no chemical reaction & well-mixed) (constant volume) (constant density) Tank Flow Chemical Reaction Let s assume we have an isothermal, constant volume CSTR with a chemical reaction: k B The inlet stream has no B in it. The molar balances will be: dc V FC FC kc V dcb V FC B kc V To determine into the nd equation. CB t we could first solve for C However, if the reaction is actually t from the first equation & then lug it k k B then the molar balances will be: dc V FC FC kc V kc BV dcb V FC B kc V kcbv John Jechura (jjechura@mines.edu) - 3 - Coyright 7 ril 3, 7
Now the equations are couled. Some additional maniulation must be done to searate the C t terms & visa versa. From the nd equation: CB t terms from the C dc F k V B C k k V and (assuming F constant): dc d C F kv dc k k V B B B Substituting these exressions into the st equation gives an exression in just d C B F kv dcb dcb F kv V FC F kv CB k CBV k kv k kv V d C B F kv dcb F kv dc F B kv F kv FC CB k CBV k k k k V V d C B F kv kv dc F kv F B kv k V C q C k k kv B C B, not C : Notice the system of st order ODEs has been relaced by nd order ODE & one of the original st order ODEs (for C t ). Tank Flow Change in Inut Stream s Temerature F,, T, C, Look at a variable inlet temerature, h Q T, V, C F,, T, C, T t. Start from steady state condition. John Jechura (jjechura@mines.edu) - 4 - Coyright 7 ril 3, 7
State Variables: Densities Temeratures Flow rates Initial state: Steady State total mass balance: d V F F i i j i: inlet j: (well-mixed) F F (steady state) Note that this also imlies that F F w for any ( T ). Steady state energy balance: d HV ifi Hi FjH Q Ws (well-mixed) i: inlet j: F H F H Q (steady state) m C T T H m C T T H Q m C T T Q (constant heat caacity & reference enthaly) ref ref ref ref Q T T mc Transient solutions fter change, total mass balance: d V F F m m (well-mixed) What does this assume? Well mixed. Energy balance: d VH F H F H Q (well-mixed) John Jechura (jjechura@mines.edu) - 5 - Coyright 7 ril 3, 7
d V dh H V F H F H Q d V dh dt H V F H F H Q dt d V dt H VC F H F H Q dv dt H VC m H m H Q dt H m m VC m H mh Q dt VC m H H Q (chain rule) (chain rule) (definition of heat caacity) (derivative from mass balance) (mathematical maniulation) Note that this exression does not deend uon assumtions of constant volume or constant density! dt VC m CT T Q (constant C & reference state) Can do some additional math. We could normalize the form of the ODE so that the coefficient on the time derivative is ; in this class, however, we will normally want the coefficient on the variable without the derivative to be : V dt Q T T m mc VC dt Q T T m C m C Notice that the term V/ m has units of time. This can be thought of as a characteristic time constant for the system. Note Even though the derivation of the ODE does not deend uon whether the system has constant volume and/or density, the integration with time will deend uon this! What if the heat inut is described by Newton s law? Let s assume Q UT T. Then the ODE becomes: s dt VC mc T T U Ts T John Jechura (jjechura@mines.edu) - 6 - Coyright 7 ril 3, 7
dt VC mc T UTs mc U T VC dt m C U T T Ts. m C U m C U m C U The form of the solution is the same, but the characteristic time constant is now VC / m C U & is deendent uon heating arameters. Examle Tank Flow Controlled by Valve F, h, F, Look at flow in valve at of tank. Bernoulli s Law: v g h w What do the terms reresent? f Let s aly Bernoulli s law from the surface in the tank to the entrance of the ie. We will neglect entrance effects & other friction: e s ve vs g h ssume the surface is oen to the atmoshere, so s atm. Because the ie has a much smaller cross-sectional area than the tank ve vs and v v v so: e atm ve gh. e s e If the kinetic energy effect is small comared to the otential energy effect then: John Jechura (jjechura@mines.edu) - 7 - Coyright 7 ril 3, 7
e atm e atm ve gh gh e atm gh. It is normally assumed that at the valve the kinetic energy effect is what creates the ressure dro, so: e v v e v e vv ve vv vv If, then: v atm e atm vv gh vv gh The flow rate through a valve is Fv v vv which means that: F gh C h where C g. v v v v v Oening and closing the valve will change v and hence C v. Total mass balance reduces to volume balance for constant density: d V F F dv F F dh F Cv h This is a non-linear ODE. Sometimes we can get an analytical solution for this articular F t F from F : ODE. For examle, for a ste change in flow such that in h t dh F Cv h d y t F Cv y y dy t F C y h y y y y v y F ln F C v y t Cv Cv y y * John Jechura (jjechura@mines.edu) - 8 - Coyright 7 ril 3, 7
y y F F Cv y ln t Cv Cv F Cv y h h F F Cv h ln t C v Cv F Cv h If the inlet flow is more comlicated or if this equation is art of a larger set of equations, there is no guarantee that an analytical solution exists. Examle: Chemical Reaction F F T coil In the main section we started to analyze a CSTR with a first order reaction start setting u the equations lets only make a coule assumtions: B. To Well-mixed system within the reactor. There is a constant liquid volume within the reactor. heat transfer fluid is used within the coils to control the temerature. hase change occurs to rovide the heating or cooling (e.g., steam condensation for heating or refrigerant boiling for cooling). This will kee the temerature uniform throughout the coil. Pure is fed to the reactor. The reaction has elementary fist order kinetics: r kc k e C E / RT Total mass balance: d V F F John Jechura (jjechura@mines.edu) - 9 - Coyright 7 ril 3, 7
d V F F m m (Constant volume) Mole balance on : dn d CV CF CF rv dc E / RT V CF CF ke CV (Kinetic exression) Mole balance on B: dn B d CBV CBF rv dcb E / RT V CBF ke CV (Kinetic exression) Energy balance: d VH F H F H U T T coil d H V F H F H UTcoil T dh d V H V F H F H UTcoil T dh V H F F F H F H U Tcoil T dh V F H F H UTcoil T dh V F H H U T T coil (Constant volume) (Chain rule) (Mass balance) (Cancel like terms) ( little algebra) John Jechura (jjechura@mines.edu) - - Coyright 7 ril 3, 7
coil dt VC F H H U T T (Chain rule & C definition) This energy balance equation is general and has the same form whether there is a heat of reaction or not. So, where is the heat of reaction? (Or heats of mixing, or temerature deendent heat caacities, or comosition deendent heat caacities for that matter.) It was noted before that the heat of reaction is embedded in the difference in the enthalies, H H ; this is all well and good to say, but it doesn t really hel in the ractical matter of setting u the energy balance equation to relate all of the relevant temeratures. To simlify the math, let s make two other assumtions: The heat caacity is constant with resect to temerature (though not necessarily with resect to comosition). The enthalies mix ideally (i.e., no heat of mixing effects). With these assumtions the enthalies can be exressed as: H C T T H ref ref H C T T H ref ref and the energy balance is: dt VC F C T Tref Href C T Tref Href U T coil T dt VC F C T Tref C T Tref F H ref Href U Tcoil T The heat of reaction is still embedded in the term relating the secific enthaly values at the reference conditions of temerature ( T ref ) and comosition. Notice that assuming the heat caacity is not comosition deendent does not affect this reference state term, it only simlifies the first term relating the net flow of enthaly to the system; if we assume no comosition deendency then C C C and: dt VC F C T Tref C T Tref F H ref Href U Tcoil T dt VC F C T T F Href Href U Tcoil T.. John Jechura (jjechura@mines.edu) - - Coyright 7 ril 3, 7
Coule things to note about the reference state term that has the heat of reaction embedded in it: The units on the term are energy/time, such as cal/min. In this articular formulation the bracketed term is energy/mass & the leading term is mass/time; the two terms could just as easily be slit in molar units. The heat of a reaction is usually calculated by determining the differences between the heats of formation of the reactants and the roducts. Heat of formation is simly a reference enthaly there is little difference between what we ve done in thermo class & what we want to do here. The change in the reference state enthaly term only comes from that ortion of the stream that reacts. We can exress the term reference state enthaly term as: F H ref Href rv H rxn. Using this exression in the energy balance equation: dt VC F C T Tref C T Tref rv H rxn U Tcoil T For the secific rate exression considered here: dt E RT VC F C T Tref C T Tref k e C V H rxn U Tcoil T / or if the heat caacity is not comosition deendent: dt E RT VC F C T T k e C V Hrxn U Tcoil T /. John Jechura (jjechura@mines.edu) - - Coyright 7 ril 3, 7