I
G.
T. WESTBROOK'
and R. ARlS
Department of Chemical Engineering, University of Minnesota, Minneapolis 14, Minn.
Chemical Reactor Design For homogeneous flow reactions, a digital computer can determine the optimum combination of reactor type and operating conditions
TIE
ENGINEER, faced with the problem of a suitable reactor selection, has a broad spectrum of types from which to choose. Even when the decision has been made to use a homogeneous reactor system, much freedom still exists for the final choice. For a continuous process, the two principal alternates are the familiar continuous stirred tank reactor (CSTR) and the continuous tubular reactor (CTR). T h e conventional approach to this problem has been to find the residence time required, in a C S T R and in a C T R . to achieve some desired conversion. I t is then a simple matter to compute the reactor volume and cooling area for each type. This would be done by means of a steady state material and energy balances, This could then be followed by a n economic comparison. T h e reader is referred to a series of articles by Corrigan ( 3 ) Lvhich considers this problem in some detail. This writer dramatically illustrates the difference in volume requirements through a n analysis of a simple, first order reaction system. I t can be readily shown that:
Here x is the fractional conversion of the reactant (Figure 1). For this system, as a higher conversion is demanded, the C S T R would become less and less artractive. I t is quite probable that a t some conversion level, it ivould become more economical to sivitch from the relatively cheap tank volume, to the more expensive coil volume. However, several additional alternatives become apparent. Thus it is quite possible that a series of stirred tank reactors could achieve the required conversion more economically. Again, the initial conversion might be accomplished in one or more tanks, follow.ed by a coil to complete the conversion. 'IVith these complications, the selection of the most suitable homogeneous reaction system is no longer a simple choice between a single tank, or a length of tube. Rather, it is a very complicated process and economic problem. T h e design of such a reactor train \.\-auld require additional specifications such as the per cent of total conversion to be allo-
cated to each stage, or the reaction temperature to be assigned to each of the reaction stages (2, 72). T h e nvo objectives of this study can noiv be stated. T h e first is to illustrate the over-all magnitude of the problem of homogeneous reactor selection and design. A computer method is then suggested by which this might be accornplished without recourse to bench scale models or pilot plants, when all the necessary physical data are available. T h e second objective is to develop a small portion of this over-all design scheme with a new design method for a single CSTR. This method, for optimum stable design, has been set u p on the Univac 1103 Scientific Computer. \Vhy is a new approach to the design of a C S T R Ivarranted? There is one principal reason for this. Initially all previous analysis has either started from assumptions of fixed reactor temperature, throughput, and conversion, with the problem of finding the required volume; or from the assumptions of fixed tem-
RECORD I REACTOR TYPE 2 OPTIMUM T 8 q
SET-INITIAL VALVES FOR
REPLACE COMPARISON OF A L L PROCESS ALTERNATIVES
71
All of these steps are required to find the true optimum reactor design
-e-
The dotted line shows the scope of this article
OPTIMIZE REACTOR
FRACTIONAL
Figure
1.
CONVERSION
X
-@
TUBULAR REACTOR
COMBINATION
A t high conversions, the
CSTR becomes less attractive than a CTR Present address, The Dow Chemical Co., Midland, Mich.
COMBIN-
DECISION
OPTIMIZE 8 STABILIZE FOR COMBINATION
VOL. 53,
NO. 3
MARCH 1961
18
perature, throughput, a n d volume, with the object of predicting the resulting conversion. These fixed conditions would probably be the result of pilot plant experience, plus management specifications. I n general there would be little guarantee that some other set
of fixed conditions might not be more attractive. Indeed, the only inherent factors which a r e fixed at the start of a plant design would be the level of production of the desired product, and the temperature range within which the reaction would be feasible. However,
once these have been specified, the reactor temperature, T , and throughput, q, may be varied to any set of values within the allowable ranges. Once a throughput has been chosen, the outlet concentration, A , for the desired product is also specified since:
Example of Design Method Defining the System
D-l REACTOR VESSEL
Assume a reaction system, with the properties of the react-
ing liquid taken as those of benzene. Assume there are two parallel reactions, at least one of which is exothermic:
7
MI X E R 7
EFFLUENT
Assume both reactions are of first order, and the rate
equations are r i A l = -kiAl, rZAl = -k?Az moles/hr. gal. (3) where r l A , is the rate of production of A1 in Reaction 1, and r2A2 is the rate of production of A2 in Reaction 2. Assume the rate constants are of the Arrhenius form, then k l = file-Ei/RT', kz = f i g - W R T '
Material Balance Equations
-
A2 = A4 - A i A : - Ai = ( A t - A : ) f ( A 4 A:) (4) For A I , q(Ay A i ) 4- r i A I V 7 2 ~ ~=V V ( d A i / d t ) For A t , q(Ai A z ) f r2A-V = V ( d A z / d t ) (5) For this problem, A i and A: are zero, A ; and Ai are known constants. Further, for any set of reactor conditions ( T , q ) A4, k l , and k z will be constant. A;
+
-
-
Energy Balance Equation q C p p (TO- T )
+ 71.4,V (-AH,") f 72a2v(-AH,") U a ( L M T D ) = VCpp
dT
(6)
Sizing the Reactor
Chemical reactor system
b Cooling water rise is from 75" to 100" F. b A constant cooling water velocity of 3 ft./sec. in the tubes b A constant mass flow rate of 50 lb./ft.2 sec. for the
reaction medium in the shell A water-fouling factor of 0.002 (B.t.u./lb. F. ft.2)-' b A reaction-medium fouling factor of 0.005 (B.t.u./lb. O F. ft.2)-1. b Standard 1-in., 16 BWG, 16-ft. exchanger tubes O
At steady state, Equations 4,5, and 6 reduce to A: - kiOAi - k2AzO = 0 (7) A ; - A2 - kiI3A2 = 0 (8) and k l A I V ( - ~ H , " ) k 2 A z V ( - ~ H , "= ) qCp(T - T o ) U u ( L M T D ) (9) For the initial conditions assumed above, kzOAn = A: - A2 = An (10) or 0 = A4/kzAz = Ad/[kz(A: - A 4 ) l (11) The reactor volume can be readily found from this equation. Equations 7 and 10 can be combined to form,
+
FEED PUMP P-l q GPH
+
Sizing the Exchanger b Heat to be removed by the pump-around circuit b Evaluating over-all heat transfer coefficient 5
1 / U = CRi i=l
Assumptions for Exchanger Design b Vapor pressure of benzene is given by P = 3.3 X 10-6T3 p.s.i.a. (13) b Design pressure should be the greater of 50 and (25 + P )
p.s.1.a.
T < 200'F. T > 200°F. b Temperature drop through the cooler is 10" F. and for C, = 0.50 B.t.u./lb. ' F. A heat balance across the cooler then gives Q P = W'0Cp4T W, = 0 . 2 0 4 ~ b An amount of bypass fluid of 25y0 of that through the cooler is adequate for temperature control. W , = 0.25qp
Pn PD
1 82
= 50 = 25
+ 3.3 x 10-673
INDUSTRIAL AND ENGINEERINGCHEMISTRY
where R1 = tube-side film resistance = 0.00103 R2 = tube-side fouling resistance = 0.00229 Rs = tube-wall resistance = 0.00022 Ra = shell-side fouling resistance = 0.00500 R5 = shell-side film resistance = l/ho. Here ho is temperature dependent, and may be evaluated by standard methods. Evaluating R 5 in terms of temperature, and summing, U = 43 0.0452T b Calculating L M T D ( T - 85) - ( T - 100) LMTD = In [ ( T - 8 5 ) / ( T - l 0 0 ) l or L M T D = T 94. b Calculating cooler area For any set of conditions ( T , q )
+
-
a =
QP
(43
+ 0.0452T)(T - 94)
A = -P L- 1b.-moles 4 gal.
(2)
where P , is the hourly production level of the desired product to meet the expected annual demand (1b.-moles/hr.). This then is the method to be used for
the CSTR design, At each set of conditions, the computer will calculate the complete reactor design. check stability, estimate costs. and compute an economic parameter, E. the capitalized earning rate, commonly referred to as the per cent return. T h e optimum design will
then be found a t those conditions which maximize E, if design is stable. How this method could fit into the over-all design scheme for the prediction of the correct processing alternatives, operating conditions, a n d equipment size is shown on page 181, lower right.
Chemical Reactor Costs. Basis: 1959 Costs Equipment Costs Vessel Cost. The vessel length was arbitrarily fixed by an l / d ratio of 13/4. A vessel skirt, 10 feet long, of the same thickness and diameter as the reactor cylinder, and a '/*-inch corrosion allowance was specified. Estimating metal thickness: t c = ( P ~ d / 3 2 , 0 0 0 ) 0.0104 (walls) fL = ( P ~ d / ' 6 4 , 0 0 0 f ) 0.0104 (ends)
+
A value of 16,000 p.s.i. was used for the allowable working stress. PD, t c , and t, increase with reactor temperature. Reactor volume is u = qB, and allowing 5% of this volume to the pump-around circuit, the reactor diameter is d = q m The total steel weight in the vessel is thus U' = I0.0909 d 3 0.482 d 2 ] P o 36.6 d 2 160.5 d Data for steel costs (7) give
+ +
+
ce =
3.50
WJ.782
(T/150)- 1
T
> 200'
+ 0.0133T)d2
T
.An allo\vance is needed for piping.. instrumentation, installation, bui1ding.s. and contingencies. Schweyer ( 70) recommends increasing delivered equipment cost by a factor of five for fluid processes:
F.
Insulation costs per unit area depend on this thickness according to Nelson (6) and the total cost for this item is C, = (17.1
+
S
Insulation Cost. Reactor temperature and vessel surface area determine the amount of insulation needed. Its thickness is ti = 0 T 5 200" F. ti =
Assuming same pump efficiency and using Equation 13 for P: HP3 = [6.95 X lo-' f 4.59 X 10-"T3]q Since the feed would always be at 100' F., the first unit cost would apply: Cj = 580 (HPs)"46i $ Pump-around Cooler. The heat exchange area is defined in Equation 1 5 . The area per shell was restricted to 50 < a < 4000 sq. ft., with a minimum desiyn pressure of 150 p.s.i.a. Above 330" F. this is not sufficient and a pressure-correction factor is required: Pcp = 0.962 1.68 X 10-9T3 .4pproximate exchanger costs (5)are: C6 193 n?a@ 646 T 5 330" F. Cg = 193
[email protected]$ T > 330' F. Total Plant Investment. The total delicered cost for this process will then be:
> 200'
F.
Both vessel and insulation costs may be combined :
Annual Operoting Costs Four types of operating costs were incorporated into this model: LABORCOST. This process was arbi-
For T 5 200" F. 3.5[4.55d3 f 60.7d2 160.5d]oi8* For T > 200' F. 0.482dz)Pn 36.6d2 160.5d10'82 [17.2 f 0 . 0 1 3 3 T ] d 2
+
Ci CI = 3.5[(0.0909d3
+
+
Miscellaneous Reactor Costs. Platforms, foundations, rtc., were estimated as: C?
=
1000
+ lOOd
$
Reactor Mixer. The basis for this cost was a constant power input (0.05 hp. per unit volume). Hence. HP1 = 0.05V. From cost data for mixers ( 9 ): C3 625(HPi)O.30 = 2 5 5 ( V ) @ 3o $ Pump-around Pump. The pump-around flow rate was set at W p = 0.25 Q p , and a reasonable pressure increase for this pump is 40 p.s.i. Assuming a constant efficiency of 70 yo: HP? = 1.88 x 10-5 Q~ Two-unit pump costs were used to reflect the higher costs for packing glands above 2.50' F. ( 8 )with these: Ca 580 (HP2)@ 467 $ T 5 250' F. C, = 922 (HP2)0.4'' $ T > 250' F. Reactor Feed Pump. This pump boosts P p.s.i. the reactor throughput to 50
+
+
+
trarily charged with the use of 4 manhours per shift at a wage of $2.52 per hour. On this basis the labor cost, L , ~ o u l dbe $1 1,000 per year. P L ~ NBURDEN. T .Allowances were made as per cent of total plant investment : depreciation 10% maintenance 470 taxes 2Y0 supplies 1% insurance 1 70 The resultant plant burden can be expressed : B = 0.18 Z UTILITY COSTS. The basis used to evaluate the four utility costs has 1 cent per kw.-hr. of power, 2 cents per 1,000,000 gallons of water, and an annual operating time of 8200 hours. Utility costs are then: Mixer power. C, = 3.10 V; $/yr./reactor Feed pump power, CS = 61.10 HP3; f /yr. ,'reactor Pump-around pump power, Cg = 1.15 X Q p ; $/yr./reactor Cooling water, Cla = 0.79 X 10-3 Q p ; $ /hr./reactor
Hence total utility charges are 10
IJ,
=
ci
n, i=7
RAW M A ~ E R I ACOSTS. L Although the amount of product A , produced will be constant, only the amount for one reactant A2 will also be constant. The other raw material A I will be consumed in varying amounts. The cost equation for this factor is R = 8200 q Pal(A7 - A ) 164,000 PAZ These prices ($/mole) were used : PA1 = 0.05 PA3 = 0 PA? 0.50 Pad 3.00
+
Process Profitability-Capitalized Earning Rates O n the basis, the total value, S, of the product would be $492,000 per year. h'et profit is taken as 50% of the gross product. II = 0.50 ( S - 0 ) . The capitalized rarning rate, or per cent return, is thus E = 5o(s - oj/r. Nomenclature inlet concns. moles/gal. outlet concns., moles/gal. cooler area, sq. ft. process burden, $/yr. equipment costs, $ utility costs, $/yr./reactor total delivered equipment cost: 6/yr. vessel cost, $ reactor diameter, ft. capitalized earning rate, 70 mixer horsepower pump-around pump hp. feed pump hp. fixed plant investment, 3 reactor length, ft. labor costs, $/yr. log mean temp. difference, F. no. of reaction processes number of cooler shells/process total annual operating costs reactor operating press. p.s.i.a. pressure correction factor reactor design pressure, p.s.i.a. (price of reactant Ak) $/mole annual profit, $ reactor feed rate, g.p.h. pump-around flow rate, g.p.h. pump-around heat-removal duty, b.t.u./hr. raw material cost, $/yr. sales revenue, $/yr. reactor temperature, O F. reactor cylinder wall thickness, ft. reactor end-wall thickness, ft. insulation thickness, in. reactor residence time, hr. over-all heat transfer coefficient, B.t.u./hr./' F./sq. ft. reactor vol., cu. ft. reactor vol., gal. wt. of reactor steel, lb. Pump-around mass flow rate, lb./hr.
VOL. 53, NO. 3
MARCH 1961
183
Reactor Design-A Specific Reaction System T h e equations involved in the specific example are shown together on page 182 and need little more than a running commentary. T h e two parallel reactions were chosen for study, and, for simplicity, they were taken to be of the first order (Equations 3). This involves no essential restriction of generality in the method, but allows a convenient reduction in the number of parameters. A full material and heat balance gives the Equations 4, 5, and 6 which are needed for later stability analysis. For the calculation of the size and cost of the reactor only the steady state Equations 7. 8, and 9 are needed. If the production level PL of the desired product A4 is to be attained, A4 = P J q . T h e n Equations 7 and 8 can be rearranged to express A1 and A , in terms of ,44 (Equations 12 and lo), and so all concentrations are determined as functions of T and q and known constants. T h e amount of heat to be removed in the pump-around circuit is the difference between the net heat generated by reaction and that required to bring the feed stream u p to reaction temperature. I t is most desirable to construct a model of the reaction process which will accurately account for change of temperature and throughput and reflect this in the cost analysis. Both the heat transfer coefficient and the L M T D will vary with temperature and flow rate and depend on the physical properties of the reaction mixture and physical layout of the exchanger. I t was assumed that the physical properties of the reactants were approximately those of benzene. Accordingly, the design pressure had to be sufficiently greater than the vapor pressure of the reactants to keep them in the liquid state and an excess pressure of 25 p.s.i.a. and a minimum pressure of 50 p.s.i.a. were chosen. .This introduces a breakpoint a t a temperature of 200" F. above which the design pressure PD increases rapidly with temperature. This is most important in determining the cost of the vessel as the thickness of the metal varies with the pressure. T h e external pump-around interchanger was chosen as providing a more flexible system than an internal cooling coil. I t was intended to do many calculations with a wide variation of parameters, and, with no a priori knowledge of how these would affect the relative volumes of reactor and interchanger, it was felt probable that unrealistic situations would be encountered if a n internal cooling coil were used. A temperature drop of 10" F. through the cooler was chosen. This factor could itself be optimized but such a n additional study would add little. With the assumptions listed, the L M T D and so the required interchanger area (Equation 15) can be
184
calculated, and this is the basis on which its cost is evaluated. T h e area of interchanger was held between the limits of 50 and 4000 sq. ft. If the calculated a was smaller than 50, it was automatically set up to that limit; if it was found to be greater than 4000, the number of shells, n2, was increased until the area per shell was satisfactory. Sufficient design data can thus be calculated to allow the cost to be evaluated. This is outlined in the table of chemical reactor costs. There are three important factors in this cost analysis: 0 Total process investment I , including all equipment, piping, instrumentation, overheads, and contingencies. 0 .4nnual operating charges including utility costs, labor. raw materials and process burden. This burden will consist of such factors as depreciation, taxes, insurance, and maintenance. annual expected net profit n. Since production of the desired product A? is constant and no value is placed on Aa, the potential sales value will be constant. This again is a simplification that could be removed without difficulty. The net profit will be the difference between these sales. S, and the total operating charge, 0, corrected for taxation. Using a tax level of 507, this gives r~ = 0.5(S- 0 ) .
T h e single economic parameter chosen to evaluate the profitability of the process is 100 II/Z = 50(S - O)/Z
E
Since the design and all costs have been evaluated as functions of T and q, the surface E ( T , q ) can be constructed. I t is this that is later shown in Figures 3. Additional Criteria for Selection of Reactor Conditions-The Reactor Stability I t Lvould be highly unreasonable to expect that purely mercenary considerations would alone be used to establish the process conditions for the reactor. Indeed. each specific chemical system would have its own unique factors to be considered. If, for example. the reaction system were not irreversible as used here, then the effect of the reaction equilibria xvould be most important. For an exothermic reaction, the equilibrium constant, k,, will fall with a rise in temperature, since the following equation will apply :
''
a- =In bT
AH' and AH'
RTZ
I
/
