volves a very large operation error. For common isothermal data, consistency means that the same Q function fits both Q and the log(yI/y*) profile and for isobaric data consistency means an additional agreement between estimated and experimental H M values. Of course, the algebraic functions used imply less flexibility than the graphic means to compensate for the cited advantages.
Z y
= =
a designated term in Gibbs-Duhem equation liquid phase activity coefficient
SUBSCRIPTS 1 , 2 = component identity = azeotrope c, b = arbitrary liquid composition a
literature Cited Ac knowledgment
Sidney G. Martin and Adrian C. Snyder assisted in the computer work. Nomenclature
GE = excess free energy HJf = heat of mixing of liquid solution
P
total vapor pressure of a liquid solution dimensionless excess free energy gas constant = absolute temperature V+f= volume of mixing of liquid solution = liquid solution composition, mole fraction x
Q R T
= = =
(1) Black, C., Ind. Eng. Chem. 50, 391 (1958). (2) Brown, C. P., Mathieson, A. R., Thynne, J. C. J., J. Chem. SOC.1959, p. 4141. (3) Chao, K. C., Ind. Eng. Chem. 51, 93 (1959). (4) Hougen, 0. A., Watson, K. M., Ragatz, R. A., “Chemical Process Principles,” Part 11, 2nd ed., pp. 904-7, Wiley, New York, 1947. (5) Ibl, N. V., Dodge, B. F., Chem. Eng. Sci. 2,120 (1953). (6) Myer, H. S., Znd. Eng. Chem. 47, 2215 (1955). ( 7 ) Naphtali, L. M., Ibid., 51, 1318 (1959). (8) Redlich, O., Kister, A. T., Zbid., 40, 345 (1948). ( 9 ) Smith, L. C., Tao, L. C.,Chem. Eng. 70,193 (1963). (10) Tao, L. C., Znd. Eng. Chem. 56, 36 (1964). RECEIVED for review June 13, 1966 ACCEPTED September 16, 1966
DEGREES OF FREEDOM FOR UNSTEADYSTATE DISTILLATION PROCESSES G . M I C H A E L HOWARD
University of Connecticut, Storrs, Conn. The usual method of determining the number of degrees of freedom for a distillation column is shown to b e incorrect when applied to unsteady-state processes. The error is in neglecting the various holdups in the system, which provide extra degrees of freedom. The additional degrees of freedom exactly equal the number of holdups which may b e specified. The design of columns for steady-state operation is not affected by these additional degrees of freedom, but they should b e considered when developing mathematical models or analyzing possible control systems in which the number of independent controllers is equal to the degrees of freedom.
THE problem
of deciding how many variables must be specified to ensure a unique solution of steady-state distillation tower design problems is a familiar one. T h e basic concept is that of the phase rule. I n a system with a large number of physical variables and a set of relationships among them, how many of the variables may be freely chosen? T h e rest will be uniquely determined by the various relationships. If too few variables are specified, the problem will have an infinite number of solutions. If too many variables are specified, the problem will have no solution unless the extra variables happen to fit the exact solution. The problem has been treated by several authors (7-6) with Smith’s summary of Kwauk’s (3) work the most lucid explanation of the general theory. Most recently, Murrill (4)applied the concepts to the selection of instrumentation schemes for a distillation tower. The author recently attempted to use this approach in determining the conditions which could be specified in solving the equations describing the unsteady-state behavior of distillation columns and found that the previously published tables of degrees of freedom do not apply. T h e holdup in the systems has not been considered. In making steady-state calculations, the amount of holdup is immaterial; but in 86
I&EC FUNDAMENTALS
transient operation, the response time is a direct function of holdup. Thus, some provision must be made for including holdup in the analysis. Determination of Degrees of Freedom with Holdup
Expressions for the degrees of freedom of the various parts of a distillation column may be derived by noting that the holdup is analogous to a stream. I t has the same variables, except that a quantity is involved instead of a flow rate. T h e development here follows the notation and procedure of Smith. A single phase stream has C 2 variables which completely specify the stream. These are C - 1 concentration variables, a rate (or quantity) and two other intensive variables (temperature and pressure). There are no restrictions among these variables other than the condition that all values be positive. An ideal plate is shown in Figure 1. I t involves five material streams (vapor holdup is neglected) and one heat stream to give a total number of variables
+
N , = 5(C
+ 2) + 1 = 5 C + 11
The relationships among the variables depend on the way in which the plate is defined, but the total number of relation-
Figure 1,. Ideal plate with four mass streams, holdup, and heat leak
N Plates
ships will be a constant, If the plate is considered a single mixed pool with the li’quid leaving having the same properties as the liquid on the plate, the number of relationships is
NE= 3 c + 4 These are: a total material balance, C - 1 component balances, C distribution relationships between V, and Hn, C - 1 concentration identities between L, and H,, four temperature and pressure identities between V , and L, and H,, and one energy balance, The degrees of freedom for the plate is the difference between these two quantities:
N c = N,
- N,
= 5 C + 11
-
3C-
Figure 2. Batch distillation column with N plates, total condenser, reflux splitter, and total reboiler
4 = 2 C f 7
A cascade of N ideal plates would have free choice of the number of plates plus the degrees of freedom for each plate less the degrees of freedom of the interstreams between the plates. There are 2 Irl - 2 interstreams, so that the number of degrees of freedom is given by NfY= N ( 2 C + 7 )
+ 1. - 2 ( N - l ) ( C +
2)
=
3N + 2C
5
The number of degrees of freedom for several types of process units are given in TabYe I. The combination of process units into complete columns is shown in Figure 2 and Table I for a batch distillation column and in Figure 3 and Table I for a conventional column. As shown in the table, the number of degrees of freedom available to the designer for a batch column is 3 N 9 and for the conventional column the number is 3N 3M C 16.
+
+
Plates
F
d
+ +
Comparison with Resirlts of No-Holdup Analysis
When Smith’s rules are used to determine degrees of freedom, a much smaller number is found. T h e batch column has a net 2 N 8 degrees (offreedom, which is N 1 less than the number found using the holdup analysis. The conventional 2 AI C 13, which is N M 3 less. column has 2 N These additional degrees of freedom correspond exactly to the number of holdups which could be specified. Thus, the holdup on each plate and in the condenser may be specified for the batch column and the plate holdups and condenser and reboiler holdups for the conventional column. The possibility of using the additional degrees of freedom to specify variables other than the holdups is intriguing. Unfortunately, the extra degree of freedom is illusory. The nature of the distillation column equations does not permit an unrestricted choice of specified variables. Normally, the variables in the over-all material and energy balance equations are used as completely as possible and an additional degree of freedom for the whole system does not provide any additional freedom in these equations.
+
+
+ +
Plates
+ + +
9, Figure 3. Conventional distillation column with total condenser, reflux splitter, and partial reboiler Applications
Knowledge of the true number of degrees of freedom is of greatest benefit to the control engineer, who can use it to determine the points of control of the column. I t has been frequently pointed out that the number of independent controllers is the same as the number of degrees of freedom not VOL 6
NO. 1 F E B R U A R Y 1 9 6 7
87
Table I. Inventory of Degrees of Freedom for Various Process Units with Holdup
Unit
Degrees of Freedom
Single-phase stream Ideal plate Cascade of A' ideal plates Feed plate Stream splitter (no holdup) Total condenser and accumulater (perfectly mixed) Total reboiler (no liquid taken off as product) Partial reboiler For a Batch Column Total condenser Reflux splitter hrPlates Total reboiler Total
C+
5
c+
5
c+
4 5
c+ c+ c+
3 N f
5 5 2c+ 5 c+ 4
3 N +
5C+19
Restrictions caused by 5 interstreams 5C+10 Net system degrees of freedom 3 N + 9 For a Conventional Column Total condenser c+ 5 Reflux splitter c+ 5 A' plates above feed 3 N + 2c+ 5 Feed plate 3c+ 9 M plates below feed 3 M + 2c+ 5 Partial reboiler c- + , -5 Total 3N 3M 10 C 34 Restrictions caused by 9 interstreams 9Cf18 Net system degrees of freedom 3 N + 3 M + C+lG
+
Table II.
+
+
Assignment of Degrees of Freedom for Typical Control Scheme Shown in Figure 4 Variable
Determined by inherent relationships or design specifications within system Pressures on all plates, reboiler, and reflux splitter Heat leaks on all plates and reflux splitter Holdups on all plates Reflux temperature Number of plates above and below feed ( N and M ) Uncontrolled input variables Feed composition Feed pressure Controlled variables Feed rate, by FRC 1 Feed temperature, by TRC 1 Reboiler duty, by FRC 2 Condenser pressure, by PRC and FRC 4 Reflux flow rate, by T R C 2 and FRC 3 Condenser holdup (determines distillate rate) Reboiler holdup (determines bottoms rate) Total degrees of freedom assigned
Degrees of Freedom
N +
M +
3
N + N +
M + M +
2 1 2
c-
1
3N
+
ILEC FUNDAMENTALS
STEAM
Figure 4. Typical distillation column control scheme, as shown by Murrill
out to control any of the uncontrolled variables. If the noholdup rules are used in assigning the variables, the column seems to have two extra controllers. Any operating distillation column is completely determined, since all the column variables will have consistent values, whether controlled or not. A mathematical model of this same column must generate the same values from a set of equations and specifications chosen by the designer. Knowledge of the true number of degrees of freedom will be of great help in making these choices. Suppose one wished to simulate the uncontrolled response of the column of Figure 4 and Table I1 to a change in feed composition. An equation for each relationship among the variables at each location in the column would have to be written. Thus, each single mixed pool plate will require 3 C 4 equations. These equations may be made as complex as desired. For example, the distribution relationships between V,, and H, could be equilibrium equations, equilibrium equations modified by a constant efficiency term, or equilibrium equations modified by an efficiency term which is a function of vapor and liquid flow rates and accounts for varying mass transfer characteristics in a dynamic situation. The unique solution to this set of equations would then be assured by the specification of the 3 iV 3 M C 16 variables listed in Table 11. Some of these variables would probably be chosen as constants, but others could be specified by writing additional equations. The holdups could be specified by making the constant molal or constant mass assumptions, or by writing the fluid dynamics equations relating the holdups to the liquid flow rates through a weir equation. The pressures could be specified as constant or by writing force balances over each plate. If a more complex model of the plates, such as dividing the liquid into several mixed pools, or some other part of the column were used, the number of degrees of freedom would be different from those listed in the tables. The same principles
+
+3M +C+
1
16
used by inherent relationships within the column. Strict application of the steady-state rules, however, usually indicates that the control system is overinstrumented. This is illustrated by the typical control system used by Murrill (4) in Figure 4. The degrees of freedom associated with column operation and control are shown in Table 11. The column is completely determined, since all 3 N 3M C IG degrees of freedom have been assigned. The column is not completely controlled, since some scheme could be worked 88
-+ B +
+ +
+
+ +
of combining streams into elements and elements into process units would have to be applied to the new model. The use of the degrees of freedom would be the same.
N , = independent restricting relationships
Conclusions
V = vapor rate
T h e number of degrees of freedom associated with the design of a distillation co1um.n should include the holdups on the plates and in the csndenser and reboiler systems. These additional degrees of freedom will have no effect on the specification of variables for steady-state calculations but should be considered when developing mathematical models or analyzing possible control systems.
SUPERSCRIPTS
Nomenclature
B C
D H,, F L M N
Ni = degrees of freedom; number of variables which must be specified
N , = number of variables which must be considered q = general designation for a heat stream
e u
= element where element is some part of a process unit =
unit which is a combination of elements
literature Cited
(1) Gilliland, E. R., Reed, C. E,, Znd. Eng. Chem. 34, 551 (1942). (2) Hanson, D. N., Duffin, J. H., Somerville, G. F., “Computation of Multistage Separation Process,” Chap. 1, Reinhold, New York. 1962. -----> --
bottom product rate number of components = overhead produc:t rate = liquid holdup of any plate. Subscripts c and s refer to condenser and stillpot, respectively = feed rate = liquid flow rate = number of plates below feed plate = number of plate:; above the feed plate, or total number of plates = =
(3) Kwauk, M., A.Z.Ch.E. J . 2,240 (1956). Process. Petrol. Refiner 44, No. 6, (4) Murrill, P. W.,. Hydrocarbon 143 (1965). (5) Robinson, C. S., Gilliland, E. R., “Elements of Fractional Distillation,” 4th ed., p. 215, McGraw-Hill, New York, 1950. (6) Smith, B. D., “Design of Equilibrium Stage Processes,” Chap. 3, McGraw-Hill, New York, 1963. RECEIVED for review August 16, 1965 RESUBMITTED April 8, 1966 ACCEPTED September 19, 1966
TURBULENT HEAT TRANSFER T O A NONEQU ILIBRIU M CHEMICALLY REACTING GAS P. L . T . B R I A N A N D S . W . B O D M A N Department of Chemical Engineering, Massachusetts Institute of Technologv, Cambridge, Mass.
Numerical solutions have been obtained for heat transfer to a reversibly reacting gas in the turbulent boundary on the surface of a rotating cylinder. By restricting the study to cases of a small temperature driving force, very gteneral results are obtained in terms of an arbitrary reaction rate expression. By presenting these results in a dimensionless form, they are found to be completely insensitive to wide variations in the Reynolds and Prandtl numbers when the Lewis number is fixed. As the Lewis number is changed, significant variations in the results are noted; these variations are correlated in an efficient manner that permits accurate interpolation of the effect of the Lewis number. Finally, the general theoretical results obtained in the present study are compared with previously published data for heat transfer to decomposing nitrogen dioxide gas iln the turbulent boundary layer on the surface of a rotating cylinder. HE rate of heat transfer from a solid surface to a turbulent T g a s may be greatly affected by the presence of a reversible chemical reaction in the gas phase. As a result of the space program, many works have appeared recently which seek to predict quantitatively the effect of chemical reaction kinetics on the rate of heat transfer in a gas. Such information is particularly important in the design of re-entry surfaces ( 7 , 74, 75, 20, 27). Several experimental investigations with the chemical sys2 NOz substantiated early theoretical studies in tem N204 which the gas being studied was assumed to be in chemical equilibrium; the cases of heat transfer through a stagnant film of gas as well as heat transfer in a turbulent gas system were investigated (7, 75, 77, 23, 24, 27, 29). A logical extension of
e
this work followed when several workers examined heat transfer rates in reacting gases having finite kinetics (2-6, 8, 73, 74). Early investigators linearized the reaction rate expression, thus allowing solutions which are very general with regard to chemical system but are limited to small values of the temperature driving force. T h e results of these studies are conveniently presented in terms of the dimensionless parameter, Cp, which is the factor by which the chemical reaction enhances the heat transfer rate. The value of Cp is found to be a function of two dimensionless parameters : m represents the chemical kinetic rate relative to the diffusion rate in the gas, and 7 is the ratio of the equilibrium thermal conductivity to the frozen thermal conductivity a t bulk temperature and pressure. I n general, Cp is equal to unity a t low values of m VOL. 6
NO. 1
FEBRUARY 1967
89
