Chemical Reactor Design

Youn-WLWoo Lee School of Chemical and Biological Engineering Seoul National University 155-741, 599 Gwanangro, Gwanak-gu, Seoul, Korea  [email protected]  http://sfpl.snu.ac.kr CHAP. 1 MOLE BALANCE

Chem ical R eact or D esi gn 化學反應裝置設計

Seoul National University Chapter 1. Mole Balance

Objectives After completing Chapter 1, the reader will be able to:

 Define the rate of chemical reaction.

 Apply the mole balance equations to a batch reactor, CSTR, PFR, and PBR.

 Describe two industrial reaction engineering systems.

 Describe photos of real reactors.

Seoul National University Chapter 1. Mole Balance

111.1 The Rate of Reaction , -rA 1.2 The General Mole Balance Equation 1.3 Batch Reactors 1.4 Continuous-Flow Reactors 1.4.1 Continuous-Stirred Tank Reactor 1421.4.2 Tubular Reactor 1.4.3 Packed-Bed Reactor 1.5 Industrial Reactors 1.5.1 Liquid-phase reaction 1.5.2 Gas-phase reaction Seoul National University Industrial Reactors

Seoul National University 回分式反応器 Batch Reactor Stirring Apparatus

CtiljktConventional jacket

HANDHOLES

Seoul National University 攪拌槽型反應裝置 Cutaway View of CSTR pitched blade turbine

Spiral Agitator

Turbine Type Impeller flat blade radial turbine

Gas Entrainment Impeller

Helix Impeller

Marine Type Propeller Anchor Stirrer Hydrofoil

http://www.jeiopi.co.kr/english/prd/impeller.htm Seoul National University 攪拌槽型反應裝置 CSTR/batch Reactor

Seoul National University 攪拌槽型反應裝置 Type of Jacket

Half pipe jacket Conventional jacket

Baffles

Dimpled Jacket

Inner coil type Heat exchange

www.centralfabricators.com/hpipej.htm Seoul Nationalimpeller University 攪拌槽型反應裝置 Stirred Tank Reactor

Seoul National University 攪拌槽型反應裝置 ppyolymerization reactor

Seoul National University 管型反應裝置 High Pressure Tubular Reactor for LDPE (Low Density PolyEthylene) plant

ExxonMobil's tubular process technology for Sasol's new high-pressure low density polyethylene (LDPE) plant in Sasolburg , South Africa. The new 220,000 ton-per- year plant is expected to be completed in 2005.

Seoul National University 管型反應裝置 Tubular Reactor for SCWO

The Shinko Pantec Plant, Capacity: 1100 kg/h Seoul National University 管型反應裝置 Tubular Reactor for production of ethylbenzene

Ethylene 400oC, 20bar Benzene

177oC 510oC

a bank of catalyst packed tubes (Zeolite)

430oC, 20bar Ethylbenzene

The default configuration catalytically reacts ethylene (reactant A) with benzene (reactant B), an exothermic reaction, to produce ethylbenzene (product C), an intermediate chemical used in the manufacture of styrene monomer. (http://www.simtronics.com/catalog/spm/spm2200a.htm)Seoul National University 管型反應裝置 Static Mixer in Tubular Reactor

Seoul National University 固定層型反應裝置 Industrial Reactor Photos

Reactor System Used at Amoco

Seoul National University 固定層型反應裝置 “Ultraformer Reactor”-Reforming Petroleum Naphtha

Spherical Reactor at AMOCO Spherical Reactors Connected in Series Seoul National University 固定層型反應裝置 Hydrotreating Unit

Catalytic hydrotreating is a hydrogenation process used to remove about 90% of contaminants such as nitrogen, sulfur, oxygen, and metals from liquid petroleum fractions. These contaminants, if not removed from the petroleum fractions as they travel through the refinery processing units, can have detrimental effects on the equipment, the catalysts, and the quality of the finished product. Typically, hydrotreating is done prior to processes such as catalytic reforming so that the catalyst is not contaminated by untreated feedstock. Hydrotreatingisalsousedpriortocatalytic cracking to reduce sulfur and improve prodtduct yildields, and to upgrademiddle- distillate petroleum fractions into finished kerosene, diesel fuel, and heating fuel oils. In addition, hydrotreating converts olefins and aromatics to saturated compounds.

Seoul National University 固定層型反應裝置 Packed Bed Reactor Fisher-Tropsch synthesis reaction at Sasol Limited Chemical

Gas inlet (50% conversion)

Catalyst Steam injector

K2O/SiO2 on Fe BET=200m2/g

Tube bundle (2050 tubes) Product 5cm ID X 12 m H = Light hydrocarbon + wax (candle & printing inks)

Seoul National University 流動層型反應裝置 Straight Though Transport Reactor

Fisher-Tropsch synthesis reaction at Sasol Limited Chemical

Settling hopper

Riser (Straight-Through Transport Reactor) (Circulating Fluidized Bed)

Standpipe Reactor 3.5 m ID x 38 m H

Seoul National University Straight Though Transport Reactor流動層型反應裝置 waxes and distillate fuels Tail Gas (T) catalyst

38% H2 Synoil 35% CH4 12% CO Volume=365 m3 2 11% C -C 353.5m ID IDX38 X 38m H 2 5 7% CO

150 ton Recycle (R) CtlCatalys t R/T=2 P=25atm T=350oC

Catalyst Feed 6-9.5 ton/sec 300,000 m3/hr @STP

58% H2 32% CO

9% CH4 1% CO2 http://www.hcasia.safan.com/mag/hnov02/it60.pdf Seoul National University 流動層型反應裝置 Sasol Advanced Syy()nthol (SAS) Reactor light olefins and gasoline fractions

Seoul National University 流動層型反應裝置 Fluidized Catalytic Cracking Unit in the petroleum refining industry

Seoul National University Fluidized Catalytic Cracking Reactor 流動層型反應裝置

Stone & Webster’s plant Seoul National University 氣泡塔型反應裝置 Slurry Phase Distillate Reactor

Seoul National University 氣泡塔型反應裝置 Bubble Column Reactor FFihFor Fischer-ThRtiTropsch Reaction

www.fe.doe.gov/programs/.../tl_liqphase_schematic .html Seoul National University 流動層型反應裝置 Fluidized Bed Gasification Reactor

GTL reactor for Sasol www.fao.org/DOCREP/T0512E/ coal-gasified gas into synthetic oil T0512e0a.htm Seoul National University 流動層型反應裝置 Residual Oil Fluidized-Bed Catalyygtic Cracking reactor

Seoul National University 管型反應裝置 Dimersol G unit (Two –CSTR and one PFR in series) Institute Français du Petrόle Process

http://www.ifp.fr/ Dimerization propylene into isohexanes Seoul National University 管型反應裝置 Plug-flow reactor for Dimersol ™ process

The finishing reactor (“the snake”) to comply with LPG specification in the USA (less than 5% olefins) Seoul National University 固定層型反應裝置 Automotive Catalytic Converter

2NO → N2 + O2

2NO2 → N2 + 2O2

2CO + O2 → 2CO2

Seoul National University 管型反應裝置 Microreactor and Lab-on-Chip

Microreactor made of silicon Lab-on-Chip made of glass and polymer anodically bonded with glass for DNA amplification and detection

Rutherford Appleton Laboratory (RAL) in the UK Seoul National University 管型反應裝置 Microreactor for DNA analysis 10 nano g / 

 0.1C

50 nano liter

(50h X 500w m)

Seoul National University Oxidation Reactor in Semiconductor Processing

Seoul National University Diamond film is synthesized through CVD (Chem ica l Vapor Depos it ion )

CVD Diamond coated tools

1 to 2 micron/hour

SEM of Diamond Films on Si-wafer substrate Seoul National University Metallization

II Integrated Circuit Wafer bis-hexafluoroacetyl-acetonate-Cu and Packaged Device (CF3COCHCF3CO)2Cu Seoul National University Chemical Reactions in Microelectronics Processing

Chemical Mechanical Polishing Chemical Vapor Deposition

Plasma Etching Electrochemical Deposition

Seoul National University Metal Deposition in Microelectronics Processing

Oxidation reactor silicon-diffusion furnace

Metal Deposition Reactor Seoul National University Ozone Depletion Reaction in Stratosphere

Paul Crutzen Mario Molina (Seoul National University) (MIT)

F. Sherwood Rowland (U. C. Irvine)

Seoul National University 工業反應裝置分類

1. 固定層型反應裝置 (Fixed bed) 2. 移動層型反應裝置 (Moving bed) 3. 流動層型反應裝置 (lididbd)(Fluidized bed) 4. 攪拌槽型反應裝置 (Stirred Tank) 5. 氣泡塔型反應裝置 (Bubble cap tower) 6. 管型反應裝置 (Tubular) 7. 火炎型反應裝置 (Flammed) 8. 氣流型反應裝置 (Pneumatic conveying) 9. 段塔型反應裝置 (Multi-staged) 10. 回轉圓板型反應裝置 (Rotary) )

Seoul National University 固定層型反應裝置 (Fixed bed)

Gas (Liq) Gas inlet Liq Gas Catalyst bed

Gas flow

Gas Gas outlet

Gas (Liq) Liq (a) Fixed bed (1) (b) Fixed bed (2: Countercurrent) (c) Radial flow type

Outlet gas

Ammonia + Air Pt/Rh gauze Catalyst Pd/Au screen 2 cm bed Metal Porous bed mesh StSupport screen

Nitric Oxide

(d) Parallel flow type (e) Thin bed catalysis reactor (Ammonia Oxidation)

Seoul National University 移動層型反應裝置 (Moving bed)

Gas Catalyst solid solid Gas

Gas Gas

Gas Gas Gas solid solid (a) Countercurrent (b) Countercurrent (c) Cross flow (gas-solid cat. rxn) (gas-solid rxn)

solid Gas solid Gas

solid Gas (d) Moving grid

Gas solid Gas Solid

Gas solid (e) Rotary kiln (rotated) (f) Multistaged

Seoul National University 流動層型反應裝置 (Fluidized bed) Gas Gas

Solid

Gas Gas (a) Fluidized bed (b) Fluidized bed (gas-solid cat.) (gas-solid rxn) Gas Gas

Liquid Solid Gas Flow Solid Particle Solid Solid Particle

Liquid

Gas Gas Gas (c) 3 Fluidized bed (d) High flow fluidized bed (e) Spray flow bed

Seoul National University 攪拌槽型反應裝置 (Stirred Tank)

Reactant inlet Gas

Baffles Steam Liquid

Steam Heat Transfer Jacket Coil

Impeller Liquid Gas Products outlet

(a) CSTR (Jacket) (a) CSTR (Coiled) (c) CSTR (G-L)

Products

Impeller

Reactant inlet Products

Reactant inlet

(d) Series CSTRs (e) Multi-Staged CSTRs

Seoul National University 氣泡塔型反應裝置 (Bubble cap tower)

Gas Gas Gas

Liquid Liquid Liquid

Heat Transfer Multi hole plate Ring Tubes Sparger Single hole nozzle Gas Liquid Multi hole Gas distributor Gas Liquid Gas Liquid ()Ri(a) Ring sparger t ype (b) Sing le nozzl e ()Mltihl(c) Multi hole gas distribution gas distribution

Seoul National University 管型反應裝置 (Tubular)

HT media

Reactant inlet Products outlet

Jacket (a) Single tube type Gas

Reactant inlet Liquid Gas Liquid Boiler for Steam generation Liquid HT Media naphtha naphtha Jacket preheating tubular tubular reactor reactor burner Cooling Gas burner water HT Media Gas quenching quenching

Liquid Products outlet Liquid (b) Multi tube type (c) Burner heated type (d) Wetted wall type (e) Spray tower

Seoul National University 段塔型反應裝置 (Multi-staged)

Liquid Gas

Rotating Disc Axis

Liquid Level Water

Vessel

回轉圓板型反應裝置 (Rotary)

Gas Liquid

Seoul National University Heat Transfer Mode in Fixed bed catalytic reactor

Gas Gas

Catalyst Bed Catalyst ic Bed tt Catalyst

Bed adiaba Cooling Coils HT Media

Gas

Gas Gas Gas (a) Self-heat exchange (b) Multi-tube heat exchange (c) Internal cooler

Seoul National University Heat Transfer Mode in Stirred Tank Reactor

Reflux Condenser

HT Media

HT Media Heat Exchanger

Outer Heat Exchanger

Seoul National University Selection of Reactor Type

Phase Reactor GGLLSCSC GS GL GLS LL LG SS

Fixed bed 1 2 3 MiMoving b bded 4 Fluidized bed 5 6 Stirred tank 7 8 9 10 11 Bubble cap 12 Tubular 13 Pneumatic 14

G=Gas; phase, L= Liquid phase, SC=Solid catalyst, GS=Gas-Solid phase, GL=Gas-Liquid phase, GLS=Gas-Liquid-Solid phase, LL=Liquid-Liquid phase , LG=Liquid-Gas phase, LS=Liquid-Solid phase, SS=Solid-Solid phase

Seoul National University Selection of Reactor Type 1. Partial Oxidation of Propylene Ammonia Synthesis Naphtha Reforming Reaction 2. Hydrodesulphurization 3. Immobilized Enzyme Reaction 4. Production of Steel in Furnace 5. Sohio Process for Production of Acrylonitrile Fluidized Catalytic Cracking 6. Gas phase Polymerization of propylene Fluidi zed Coa l Com bus tion 7. Bulk Polymerization of Styrene 8. Production of Antibiotics 9. Production of Terephthalic Acid Hydrogenation of Edible Oil 10. Emu lsion Polymeri Pol merization of SBR 11. Production of HDPE 12. Liqqpuid phase Oxidation of Olefin 13. Production of Ethylene by Cracking of Naphtha 14. Production of Syngas Seoul National University Batch Reactor

Characteristics No charge or discharge during reaction

Phases Gas, Liquid, Liquid/Solid

Application Small scale production Intermediate or one shot production Pharmaceutical Fermentation agricultural chemistry

Advan tages High conversion per unit volume for one pass Flexibility of operation (same reactor can produce one product one time and a different product the next) Easy to clean

Disadvantages High operation cost Product quality can be changed batch to batch Seoul National University SemiSemi--batchbatch Reactor

Characteristics Either one reactant is charged and the other is led continuously (at small concentrations) or else one of the product can be removed continuously to avoid side reaction.

Phases Gas/Liquid, Liquid/Solid

Application Small scale production CtiCompeting reactions

Advantages High conversion per unit volume for one run Good selectivity Flexibility of operation (can beused with a reflux condenser for solvent recovery or in bubble type runs)

Disadvantages High operation cost Product quality more variable than with continuous operation

Seoul National University ContinuousContinuous--StirredStirred Tank Reactor (CSTR)

Characteristics Run at steady state with continuous flow of reactants and products: the feed assumes a uniform composition through the reactor, exit stream has the same composition as in the tank

Phases Liquid, Gas/Liquid, Liquid/Solid

Application When agitation is reqq,uired, Series configurations for different concentration streams

Advantages Continuous operation Good temperature control Easily adapts to two phase runs Low operating (labor) cost Easy to clean

Disad van tages LtLowest conversion per unit volume By-passing and channeling possible with poor agitation Seoul National University Plug Flow Reactor (PFR)

Characteristics One long reactor or many short reactors in a tube bank No radial variation in reaction rate (concentration) Changes with length down the reactor

Phases Gas

Application Large scale production/Continuous Production Fast reaction High TtTemperature

Advantages High conversion per unit volume Low operating (lab or ) cost Continuous operation Good heat transfer

Disadvantages Undesired thermal gradients Poor temperature control (hot spot) Shutdown and cleaning may be expensive

Seoul National University PackedPacked--BedBed Reactor (PBR)

Characteristics Tubular reactor that is packed with solid catalyst

Phases Gas/Solid catalyst, Gas/Solid

Application Heterogeneous gas phase reaction with a catalyst

Advantages High conversion per unit mass of catalyst Low operating (labor) cost Continuous operation

Disadvantages Undesired thermal gradients Poor temperature control (hot spp)ot) Channeling Shutdown and cleaning may be expensive

Seoul National University Fluidized-Bed Reactor (PBR)

Characteristics Heterogeneous reaction Like a CSTR in that the reactants are well mixed

Phases Gas/Solid catalyst, Gas/Solid

Application Heterogeneous gas phase reaction with a catalyst

Advantages Good mixing Good uniformity of temperature CtlCatalyst can becontinuousl y regeneratdted with the use of an auxiliary loop

Disadvantages Bed-fluid mechanics are not well known Severe agitation can result in catalyst destruction and dstdust formation Uncertain scale-up Seoul National University General Mole Balance on control volume

Seoul National University Balance on control volume

N  moles j control volume

Nj Fj0 Fj Gj = rj · V moles moles G   volume j time timevolume

A mole balance on species j, at any time, t, yields

Rate of flow Rate of flow Rate of generation Rate of of j into the of j out of of j by chem. rxn accumulation of j system -+the system within the system =within the system (mole/time) (mole/time) (mole/time) (mole/time) in - out + generation = accumulation

dN F  F  G  j j0 j j dt Seoul National University Rate of formation of species j by chem. rxn

Suppose that the rate of formation of species j for the reaction varies with the position in the control volume. The rate of generation, Gj1,intermsof rj1 and sub-volume V1 is

V V2 G j1  rj1 V1 1 r If the total control volume is divided into j1 r M sub-volume, the total rate of generation is j2 M M V G j   G ji   rjiVi i1 i1 By taking the limits (i.e., let M →  and V → 0) and making use of the definition of integral, we can rewrite the foregoing equation in the form

V G  r dV j  j r j can have different values at difference locations in the reactor since the properties of the reacting materials (e.g., conc., temp.) Seoul National University 1.2 The General Mole Balance Equation (()GMBE)

V dN j F  F  r dV  (1-4) j0 j  j dt

With this GMBE, we can develop the design equations for the various types of industrial reactors: batch, semi-batch, and continuous-flow. Upon evaluation of these equations we can determine the time (batch) or reactor volume (continuous-flow) necessary to convert a specified amount of reactants to products.

Seoul National University The most common industrial reactors

batch reactor CSTR (bac km ix react or)

Reactants

Products

PFR PBR (tubular reactor) (packed-bed reactor) Reactants Products Reactants Products

Seoul National University Ideal Reactor Type

Batch Reactor  uniform composition everywhere in the reactor  the composition changes with time Continuous-Stirred Tank Reactor (CSTR)  uniform composition everywhere in the reactor (well mixed)  same composition at the reactor exit Tubular Reactor (PFR)

 fluid passes through the reactor with no mixing of earlier and later entering fluid, and with no overtaking.  IiIt is as if ifh the fl fliduid moved di in si ngl e fil filhe through hh the reactor  There is no radial variation in concentration (plug-flow reactor) Seoul National University 1.3 Batch Reactors

If the reaction mixture is 0 0 V perfectly mixed so that there V dN j GMBE r dV  Vr is no variation in the rate of F j0  Fj  rj dV   j j  dt reaction throughout the reactor volume, we can take rj out of the integral and write the GMBE in the form

Design Equation dN A for Batch reactor  r V (1-5) dt A

dN In tegrati ng with limit s th at dt  A at t = 0, N = N r V A A0 A at t = t1, NA = NA1

What time is necessary to N reduce the iiilinitial number of A 0 dN A moles from NA0 to a final t1  (1-6) desired number N ? N A1 A1  rA V Seoul National University 1.3 Batch Reactors

A B

dN A  r V dt A

N A0 N A 0 dN A N t1  B1 N r V A1 A N NA B

NA1

t t1 1 t t

Moles of A change with time Moles of B increase with time Seoul National University 1.4.1 ContinuousContinuous--StirredStirred Tank Reactor (CSTR)

0 Fj0 V dN j V GMBE F  F  r dV  r dV  Vr j0 j  j dt  j j Fj

The CSTR is normally run at steady state and Design Equation Fj0  Fj for CSTR V  (1-7) is assumed to be perfect mixed.  rj - No tltemporal,spatial varitiiations in conc., temp., or rxn rate throughout the vessel - Conc. and temp at exit are the same as they are elsewhere in the tank F  C v j j (1-8) - Non-ideal mixing, residence-time distribution moles moles volume F  C v   j j model is needed time volume time

The reactor volume, V, necessary to reduce the v0C A0  vCA entering flow rate from F V  (1-9) j0  r to the exit flow rate Fj at A reaction rate of rj. Seoul National University 1.4.2 Tubular Reactor (PFR)

- The reactants are continually consumed as they flow down the length of the reactor - The concentration varies continuously in the axial direction through the reactor. - CtlConsequently, the reac tion ra te w ill a lso vary ax ia lllly. - To develop the PFR design equation, we shall divide (conceptually) the reactor into a number of sub-volumes so that within each sub-volume V, the reaction rate may be considered spatially uniform.

y

Fj0 Fj ,exit y y  y

Fj (y) V Fj (y  y)

Let Fj(y) represent the molar flow rate of species j into volume V at y Fj(y+y) represent the molar flow rate of species j out of volume VatV at (y+y)

In a spatially uniform sub-volume V, V r dV  r V  j j Seoul National University 1.4.2 Tubular Reactor (PFR)

0 GMBE V dN j V in V F  F  r dV  r dV  r V j V j V V  j dt  j j

 F  F  j V V j V lim    rj (1-10) V 0 V  

Integrating with limits that Design Equation dF j (1-11) at V = 0, F = F for PFR  rj A A0 dV at V = V1, FA = FA1

The reactor volume, V , 1 F necessary to reduce the A1 dFA entering molar flow rate FA0 V1  (1-13) to some specified value F FA0 A1 rA at reaction rate of rA. Seoul National University 1.4.2 Tubular Reactor (PFR)

A B dF A  r dV A

F F A0 A 0 dF A V1  FB1 FA1  rA FA FB

FA1

V V 1 V 1 V

Figure 1-12 profiles of molar flow rates in a PFR Seoul National University 1.4.3 PackedPacked--BedBed Reactor (PBR)

For a fluid-solid heterogeneous system, the rate of reaction of a substance A is defined as gmol A reacted  r '  A sec g catalyst

The mass of solid is used because the amount of the catalyst is what is important to the –r’A W

FA0 FA W W  W

FA (W ) F A(W  W ) ' rAW ' FA (W )  FA (W  W )  rAW  0 In - out + generation = accumulation

FA Design Equation dFA W  No pressure drop for PBR F ' A0 rA No catalyst decay

Seoul National University Example 11--11 How large is it? (PFR)

rA=-kCA The first -order reaction (liquid phase rxn) CA0 v0 V CA A  B is carried out in a tubular reactor in which the volumetric flow rate, v0, is constant.

(1) Derive an equation relating the reactor volume (V) to the entering concentration of A (CA0), the rate constant k, and the volumetric flow rate v0.

(2) De term ine the reac tor vol ume necessary tdto reduce th thitie exiting concentration (CA) to 10% of the entering concentration (CA0) when the volumetric flow rate (v0) is 10 ℓ/min and the spp,,ecific reaction rate, k, is 0.23 min-1.

Seoul National University Example 11--11 How large is it? (PFR)

C V dF v0 A dCA GMBE A r   dV for PFR  A C 0 dV k A0 CA

dFA  rA v C dV 0 A0 Tubular, V  ln 1st order rxn k CA  r  kC (1st-order A A reaction) 10l / min C dF d(C v ) dC V  ln A0 A  A 0  v A 0.23min 1 0.1C dV dV 0 dV A0 10l dC  ln10 Combine v A  kC 0.23 both side 0 A dV 100l

v0  dCA  A reactor volume of 100L is necessary to     dV convert 90% of species A entering into k  CA  product B for the parameter given. Seoul National University P1P1--66B How large is it? (CSTR)

Fj0 The first -order reaction (liquid phase rxn) V F A  B j rA=-kCA is carried out in a CSTR in which the volumetric flow rate, v0, is constant.

(1) Derive an equation relating the reactor volume (V) to the entering concentration of A (CA0), the rate constant k, and the volumetric flow rate v0.

(2) Determine the reactor volume necessary to reduce the exiting concentti(Ctration (CA) to 10% of th e ent eri ng concent rati on (CA0) whthhen the volumetric flow rate (v0) is 10 ℓ/min and the specific reaction rate, k, is 0.23 min-1.

Seoul National University P1P1--66B How large is it? (CSTR)

For CSTR, Fj0 the mole balance on species A was shown to be V

F  F C v  C v Fj V  A0 A  A0 0 A 0 r =-kC  rA kCA A A

1 CA  0.1CA0 , v0 10 / min, and k  0.23min

C v  C v 0.9v V  A0 0 A 0  0 0.1kCA0 0.1k 1  (9)(10 / min) /(0.23min )  391.3 The CSTR is almost 4 times larger than the PFR for getting 90% conversion

Seoul National University Mole Balance on Different Reactor

Reactor Differential Algebraic Integral

dN N A dN A  r V t  A A N BthBatch ddt A0 rAV

FA0  FA CSTR V   rA

F dFA A dFA  rA V  PFR dV F A0 rA

dF FA dF PBR A  r W  A A F dW A0 rA

Seoul National University