Degrees of Freedom for Steady State Flow Systems
-
development
PARK L. MORSE MlCHlOAN STATE COLLEGE, EAST LANSING, MICH.
C
This work was initiated in order to develop laws for flow HEMICAL engineering tinuous distillation column systems comparable to the Gibbs phase rule which has finds extensive use for that may be assumed to have its main application to isolated static masses. the Gibbs phase rule to prerun for an infinite length of Results of the work indicate that flow systems have detime. The temperaturps and dict the degrees of freedom grees of freedom for material balances, energy and matein some cases the pressures for batches of material under rial balances, design, and operation. General equations of the various trays may be equilibrium nonflow condiare presented that predict the degrees of freedom for a appreciably different. The tions. This paper develops system, regardless of the physical design and whether or phases leaving a given tray comparable equations that not chemical or physical change occurs. may or may not have ($omwill predict the degrees of Application of the derived equations furnishes a means positions corresponding to freedom for steady state flow for carrying out material and energy balances with the those of a comparable batch systems. minimum of effort directed toward analysis and general e q u i l i b r i u m . F o r steady A common example of the measurement. The design of simple and complex flow s t l a t e s y s t e m s i n s t e a d of importance of degrees of systems with the prior knowledge of the correct number thinking of a static mas8 of freedom for a flow system of design variables to be specified is made possible. material one thinks of matemay be found in the operarials in flow, with work and tion of a multicomponent heat possibly being transferred with the surroundings. It is distillation column. It may be desirable for the concentrations of two selected components t o be specified in the two here that such variables as mass rate of flow, kinetic energy, outlet streams but whcn it is attempted to operate the unit EO potential energy, and surface energy come into importance. that the specifications are met, failure results. Clearly this These variables are termed “extensive,” in contrast with “intenmethod of specification is in error. sive,” because they are dependent on the quantity of material. Gilliland and Reed (3) published equations that predict the In addition to the extemive variables the chemical engineer finds that the intensive variables are of importance when considering variance in conventional multicomponent absorption and rcctifyflow problems. ing columns. The equations of Gilliland and Reed are not Up to this point the expression “degrees of freedom” has been intended to apply to systems where chemical change occurs nor to embrace all types of equipment. The development herein used in a broad sense. Actually flow systems may have degrecs of freedom for ( a ) matcrial balances, ( b ) material and energy will present equations for thc degrees of freedom in any steady balances, (c) design, and ( d ) operation. A more precise meaning state flow system whether or not a chemical reaction occurs and for these items will be given as thp article progresses. regardless of the type of equipment utilized. The degrees of freedom, or variance, are the number of independent variables that may be assigned values arbitrarily. Once DEFINITIONS numerical values are assigned t o all such independent variables System. The term “system” describes the material in flow the remaining variables are dependent and invariant. If the and may be limited as desired by enclosing any part of the physical equipment with an imaginary boundary. The terminal points number of arbitrarily assigned values exceeds the degrees of of a system are a t the intersection of the various streams with freedom an inconsistency results unless a fortuitous choice in the this arbitrary boundary. The system under consideration does values selected has been made. The general method of computnot normally extend across a n equipment boundary, although ing the degrees of freedom for a problem in any specialized field, energy may be transmitted to the system across such a boundary. and the one used here, is to subtract the number of independent Thus if alcohol is being vaporized by a closed steam coil the alcohol may be said to be the system, the energy being supplied defining equations from the number of unknowns. through the walls of the steam coil. The steam is not part of There are different concepts involved when degrees of freedom the alcohol system. for nonflowing batches of material and for flow systems are conWhen it is stated that a system is “set” or “tied up,” it is meant Bidered. I n using the Gibbs phase rule development the entire that the potential variahles a t the terminal points are fixed and nonvariant-ither arbitrarily or through actual measurementmaterial is considered to be homogeneous in temperature and as would be the energy transmission rates t o the system if under pressure; work and ordinarily heat are not being transferred consideration as variables. It is not implied that variables a t with the surroundings. The components present in any phase points within the system cannot be altered by changes in equipment or operating conditions that do not affect the above set are said to be in equilibrium in that no net mass transfer is taking conditions. place. The variables encompassed by the Gibbs rule are intenComponents. It will be necessary to define three types of comsive in that they are independent of the mass of material present. ponents: the analytical component, the ultimate component, Thus temperature, pressure, concentration, density, and the and the thermodynamic component. The use of these various components is vital to this development because the number of enthalpy per unit mass are all independent of the quantity of defining equations and unknowns may thereby be established material present and are intenRive variables. in part, and, as indicated previously, this is necessary in deterIn contrast, a flow system may or may not have constant mining the degrees of freedom. Analytical components are temperature and pressure conditions throughout, or equilibrium chemical substances selected so that their number is the minimum required in an analysis to fix the chemical distribution and conbetween existing phases. An example of this would be a con-
1863
INDUSTRIAL AND ENGINEERING CHEMISTRY
1864
centration of all independently variable atomic components. Thus, if a gas containing carbon dioxide, carbon monoxide, oxygen, and nitrogen has its analysis reported on the basis of carbon, oxygen, and nitrogen, the chemical distribution of the atoms is not fixed. Hence the three atoms selected are not analytical components. If the analysis is reported on the basis of carbon dioxide, carbon monoxide, oxygen, and nitrogen, all the atomic concentrations and their chemical distribution are fixed. These four molecular species are then analytical components.
I I
9
Vol. 43, No. 8
or phase of this equilibrium can be described or formed chemically by varying the amounts of calcium oxide and carbon dioxide, or in other words the composition of any stream may be expressed in terms of these two components; hence nt is 2. Findlay ( 2 ) gives additional examples pertaining to the selection of thermodynamic components. Later the use of the term nt is supplemented by use of ntpwhich is the total number of thermodynamic components in all the streams of a given heterogeneous equilibrium. In summarizing it may be said that the term f‘analytical component’, is used to describe concentrations and chemical distribution of atoms in a stream; the expression “ultimate component” is utilized in connection with material balances; and the term “thermodynamic component” is employed in equilibrium conwith nu because the former is siderations. One can contrast nLa evaluated for a given stream while the latter applies to the entire system. If no chemical reaction takes place analytical, ultimate, and thermodynamic components may be considered identical. From the discussion on components it is clear that concentrations of mixtures may be expressed using any one of the three systems. Further, concentrations of ultimate or thermodynamic components in a stream may be expressed as functions of concentrations of analytical components.
Figure 1. Combustion of Methane Analytioal components: CH4,02, inert material, COz, CO, Hz0. Ultimate components (nu = 4): C, H, 0, inert material; Z n e , = nm naz nad = 1 2 5 =8
+
+
+ +
If any group of chemical substances that potentially are analytical components has the same relative concentration of each of its components within the group a t the terminal points a t which they appear, the group is termed one analytical component. I t is necessary that sufficient physical data for the group are available to carry out any needed computational work, hence under certain conditions the group composition or molecular weight must be known. Such physical data for the group are assumed known and nonvariant and in no way are they connected with the degrees of freedom or variables considered herein. In the equations presented the number of analytical components, na, within a stream at a terminal point will be of importance. The evaluation of ma together with illustrations of the selection of analytical components are shown for various systems in Figures 1, 2, and 3. The streams for the systems are denoted by encircled consecutive numbers. The terminal points are a t the extremities of the streams. Ultimate components are chemical substances that go through a system without being chemically altered. Their number, nu,represents the minimum that is required to completely describe all the analytical componentsof the system. It follows that n,, also represents the number of material balances that can be made. An ultimate component may be an atom, ion, radical, molecule, or group of chemical substances. As an example, when a stream of methane combines with a stream of oxygen in a combustion reaction there are three ultimate components-carbon, hydrogen, and oxygen. The fact that there may be unreacted methane or carbon monoxide in the products does not alter the number of ultimate components and only three material balances may be made, If the methane and oxygen do not react but simply form a mixture there would be two ultimate components-oxygen and methane. Consider the complete flashing of an arbitrarily selected gasoline from a lubricating oil. In spite of the fact that both the oil and gasoline are composed of an unknown number of compounds, there are only the two ultimate components. If the flashing had not been complete and the more volatile molecules of the original gasoline had flashed predominantly, the gasoline could not be considered one ultimate component. Figures 1, 2, and 3 give additional specific examples of the selection of ultimate coniponents and the evaluation of nu. The idea of a thermodynamic component is one that Gibbs ( 1 ) defined the numutilized in deriving the phase rule. ber of thermodynamic components, nt, as . , , the least number of independently variable chemical substances from which the given (material) in all its variations could be produced.” I n this work the use of the thermodynamic component is largely confined to those single phase streams that are in heterogeneous equilibrium with each other. A4n example of the selection of thermodynamic components may be of help. Calcium carbonate decomposes according to the following equation:
Dad!:
CaC03 = CaO
...
+ COZ
Three phases are present for such a heterogeneous equilibrium and for the case of a steady state flow process each such phase may be considered a separate stream if so desired. Each stream
MATERIAL BALANCE DEGREES O F FREEDOM
Material Balance Equations. Consider the steady state system with N single phase streams of Figure 4. The analytical components in each stream are known. The direction of flow of the streams and whether or not a chemical reaction takes place within the system are of no consequence in the development presented. Such steady state flows appear to have two classifications of equations that are applicable to them: the universd equations that apply to all systems and the peculiar equations that apply only t o certain systems under specified conditions. If only those flows are considered that have no deterioration of mass, the universal equations applying t o material balances are based on: ( a ) law of conservation of mass; ( b ) sum of mass fraction concentrations for a given mass is unity. Equations 1, 2, and 3 summarize these universal equation$. The equations are written for nu ultimatr components, a stream rate of S mass units from the system per unit time, and an ultimate componrnt mass fraction concentration of C. The S and C variables that ent,er into these equations are material balance variables as diffcrrntiated from pressure and temperature which are energy balance variablrs. Streams are denoted by numbers 1 through AT and the components by letters a through nu. The “ith” stream and the “jth” component are used to form summations.
5
SX,,
2=
=
0; j = a, b, c,
. . . . . . nu
(1)
1
N
CS,=O
(2)
2=1
3 =a
For practical application these equations are used in thrir expanded form ap indicatrd by Equations l a through 3N. Expanded form of Equation 1: (SC,)I (SC,)Z . . . . . (SCa)N = 0 (la)
+
(8Cb)l
f
+
+ .
(XCb)z
f
. . . . . . + (Scb)W
+
+
(SC,,)I (SC,,)S ...... Expanded form of Equation 2:
SI
0
+ (SCn,,)X = 0
+ sz + . * . . . + S s = 0
(1b)
(In,) [2(111
*
Expanded form of Equation 3: (ca
+ + . . . . . . $. cb
.......
Cn,)l
=
1
N1)1
INDUSTRIAL AND ENGINEERING CHEMISTRY
August 1951
Equations la through 3N cannot be solved for an S value when no other steam rate has been assigned a numerical valve. This is t.rue regardless of the number of concentration variables known. For this reason in the subsequent discussion of material balances it will be understood that a t least one S will be given a numerical value when the equations are applied.
’
-(a Figure 2.
a. b, c
Flashing of Light Ends
a, CZH4; b, CzH6; c, straw oil. Analytical components: a, b, C. Ultimate components (nu = 3): a,b, C. Zna = no, nlrz n.as = 3 3 2 =8
+
I
+
+ +
Assignment of Numerical Values. There is a limitation to the manner in which independent variables may be assigned numerical values in the above equations. While a system may have a set number of degrees of freedom which are observed, inconsistencies arise if certain variables are assigned values. For a given system there may be k’ sets of streams a t the boundary, each set having stream8 of equal composition of ultimate components within the set. In each set f,’ represents the frequency an independently variable ultimate component stream composition occurs more than once, and b is the number of ultimate components, the concentration of each of which is arbitrarily set in all IC‘ sets of streams. In general, if d’ is the number of independently variable sets of the k’ sets that have all t8hemass stream rates within a set fixed or given numerical values, d’
+b S N
1865
ultimate components and not analytical components has been utilized. In many systems analytical components are identical to ultimate components and the problem is simplified. When two streams have identical analytical component compositions the ultimate component compositions must also be the same. Application of Equation 4 to the case of streams having the same ultimate but not analytical component composition is exceptionallj rare. The termP f,’and b should be examined further. The term f,’ must represent a number of independent, (relative t o the boundary) variables. If only one ultimate component is present, all streams have the same ultimate component cornposition but f,’ is zero because the stream concentrations of the system are not independently variable. Thus for all two-stream systems jc‘ must be zero because both streams inherently have the same ultimate component composition. In evaluating b the ultimate component selected must be independently variable in concentration in at least one of the stream sets. For a binary system b could not equal two Rince the concentration of only one component is independently variable. For the same reason it follows that for a two-stream system b must equal zero. Examples 1, 2, and 3 illustrate the use of Equation 4.
EXAMPLE 1. APPLICATION OF EQUATION 4. Discuss the manner in which variables may be assigned numerical values for the system shown in Figure 5 . The number of sets of streams having independent but unique compositions, k‘, is three; the sets being made up of streams 1, (2and 3 ) , and (4,5, and 6). N is 6. Starting with stream 1 and proceding clockwise around the system the value off: for. set I is 0, for set I1 is 2, and for set I11 is 1. Therefore, by Equation 5 E&’ = 0 2 1 = 3. Substitution in Equation 4 gives d‘ b S 6 - 1 - 3 = 2. If each of the stream rates in sets I and I1 is fixed then all the stream rates of set I11 may not be fixed a t the same time, although all but one of the rates in this set may be fixed. Or if the concentration of a specific ultimate component is fixed in each of the sets simultaneously ( b = 1) together with St, then the stream rates of all the streams in either set I1 or I11 may not be fixed. However, in this example j’: stream rates in sets I1 and I11 could be set.
+
+ +
lo‘
-1 -
fl,; d’ 2 1
(4)
i= 1
where (5) i= 1
Equation 4 describes the manner in which independent variables- may be assigned values and does not limit the number of variables that may be given numerical values. I n a sperial sense it, is a degrees of freedom equation. The equation may be derived from inspection of Equations la through 2(1) together with the above discussion. From Equation 2(1) it is apparent that there are N - 1 independent stream rates. Allowing for the fact that some streams have equal compositions and that the stream rates of certain of the k’ sets are assigned numerical values it follows that N - 1 - Zfc’ - d’ is the number of independmtly variable sets of stream rates of the k’ sets. This number of unknown sets of stream rates will equal the number of solvable indepcndent equations [(la through 2( 1) J when this sum is equal to b, but in no case must b be greater if an inconsistency is to be avoided. Equation 4 expresses this mathematically. Previously it was pointed out that a t least one stream rate must be assigned a value in order for a material balance to be made using Equations la through 3 N . For Equation 4 then d‘ 2 1. In the development of Equation 4 the mass concentration of
Figure 3.
Contact Process
Components in stream 2 in equilibrium. Analytical e o m ponents: SOz, 0 2 , Nz, SOs. Ultimate c o m p o n e n t s (nu = 3): S,0 , Nz. Zna = nai naz = 3 4 =7
+
+
EXAMPLE2. APPLICATION OF EQUATIONS 4 AND 10. Consider the combustion process shown in Figure 1 and discuss the independent material balance variables. There is no chemical homogeneous equilibrium, and no two streams have the same composition or stream rate; R’ of Equation 9 is 0. Utilizing the values of Zna and nuobtained from the figure and substituting in Equation 10 gives F‘ = -nu Ena - R‘ = 4 8 0 = 4. Then SI,S2, per cent oxygen in stream 2, and per cent carbon monoxide may be set (measured) as independent variables and all other material balance variables may be determined from a combination or equivalent of Equations l, 2, and 3. Another interpretation is that if a material balance on an operating unit is carried out a minimum of three variables in addition to one of the stream rates must be measured. For practical ap lication the oxygen and inerts would ordinarily be from air a n t hence the composition of stream 2 would be fixed. F’ then would equal 3 not 4. Equation 4 gives d‘ b 3 - 1 - 0 = 2. The following may not be given as examples of independent variables to be set: 1. Per cent methane in stream 1. 2. Ss if SI and Sa have been set in mass units (by Equation 4).
+
- + -
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
1866
3. The concentration of inert material in stream 2 if the oxygen content in stream 2 has been set. 4. The weight per cent inert material in streams 2 and 3 at the same time if SIand SZhave been set in mass units (by Equation 4). EXAMPLE 3. APPLICATION OF EQUATIONS 4 AND 10. Discuss the selection of variables that are necessary to measure for a material balance on the unit illustrated in Figure 2.
Figure 4.
Schematic Drawing of Steady State Flow System
As in the preceding example R' is arbitrarily said to be zero. Obtaining values from the figure and substituting in Equation 10 8 - 0 = 5 . Care should be used in selecting gives F' = -3 the five independent variables to measure. Equation 4 gives d' b 5 2. If SIand 8 2 are set (measured), b = 0 and the concentration of any given component may not be specified in all three streams simultaneously. Conversely, it SIand the concentration of ethane in three streams are set neither S2 nor S a may be fixed and the concentration of either ethylene or the straw oil may not be fixed in streams 1 and 2 at the same time. If heterogeneous equilibrium exists between the vapor and liquid leaving the system of Figure 2, R' is not zero and the degrees of freedom are lessened by 6 which equals 5 - 3 = 2 . In essence, there are two equilibrium equations that tie the material balances of the system together. Equation 10 then gives F' = j - 2 = 3. For this latter case, if a material balance is to be made on a system in operation, a minimum of three variables must be physically determined. In such a case it is understood that the temperature and pressure of the equilibrium are known and are not considered variables. If the equilibrium is questionable or the user wishes to ignore it in the material balance computations, it would be necessary to measure a minimum of five variables. /
+ +
Degrees of Freedom Equations. At this point it would be well to reiterate the over-all methodology of the development to come. The number of independent equations that apply to the material balances of Figure 4 will be subtracted from the number of unknowns thus giving the material balance degrees of freedom. The development is itemized in Table I.
c m
F R E E D O X FOR RfATERIAL BAL.4NCES
Xumber of Equations or Variables IJnknowns Analytical components Stream rates Equations Universal Peculiar Homogeneous equilibria Heterogeneoua equilibria Equal stream composition Eoual stream rates Degree's of freedom, F ' : unknowns equations
Zna
N
-
--nu
+ Zna
-
nu
+N
(?LO
+ (26 + + zA)nt > I f
ZfO(E(nl7
- 1)
V s J
N
nat. Tf there are A7 unknown stream i= 1
+
61
-t 6 2
+ . . . . + 6m I
.
For the special case when nt is 1no equilibrium relationship exists between components in the various phases, hence the restriction nt > 1 for each equilibrium must be placed on Equation 6. When nt = 1, Equation 6 is not evaluated because the equilihrium relationship exists between temperature and pressure. The restriction nt > 1 applies only for material balance degrees of frtedom and not for energy and mat(eria1 balance degrew of freedom which are considered later. Examples 3 and 7 illustrate the evaluation of 56.
SET I
0
(a' SYSTEM BOUNDARY
System Having Streams of Equal Composition
No chemical reaction. Ultimate and analytical com onents identical. Streams 2 and 3 have same composition. &ream8 4, 5, and 6 have same composition. Components present8 a, b , c, d. a, b , c , din stream 1. a, b , c, in streams 4, 5, and 6. b, c, d, in streams 2 and 3
+
Because each stream has noanalytical components, the total unknown concentrations are
6i =
2=1
Figure 5.
TABLEI. DEGREESO F
+
and 3. Of the nu N 1 equations only nu N are independent since Equation 2 may be implied from Equations 1 and 3. Certain systems have additional equations that help define their material balances, and a8 stated before such special relatiomhips may be spoken of as peculiar. For each chemical homogeneous equilibrium that is made up of products and reactants within phase a t a terminal point bhere is one indepcndent equabion relating material balance variables, and for ho such equilibria there are an equivalent number of independent rela,tionships. The fact that these cquations are dependent on temperature and pressure is not' taken into account at, this time since only material balance degrees of freedom are under consideration. Heterogeneous equilibria also furnish peculiar equations. In order to arrive at. the number of additional equations t.hat are introduced, consider one such equilibrium between single-phase streams at the terminal points of the system and select one thermodynamic component for attention. There is one less equilibrium equation than the sum of the streams (phases) in which the component, appears. The total number of equilibrium equations for d l component,s and all streams of the heterogeneous equilibrium is (nt, - nt) which is defined as 6. Then for m seta of heterogeneous equilibria a t the terminal points the additional indepmdent equations are given by
+
+
Voi. 43, No. 8
A'. The number of univerrates, the total unknowns are E n , sal equation8 may be obtained by inspection of Equations 1, 2,
The Gibbs phase rule may be written as Fa = nt 2 - P, where F G is the Gibbs degrers of freedom and P is the number of phases (streams) in equilibrium. Equation 6 may be used t o derive the phase rule by subtracting the number of equations from the number of unknowns. If temperature and pressure are not ronsidered variable., the phase rule reduces to FG= nt - P . Equation 6 was written to allow in part for heterogeneous equilibrium. The Gibbfi phase rule states that in effect each added phase reduces the degrees of freedom by one. Hypothetical phases that could exist in equilibrium with saturated phases present at the terminal points without changing the conditions of these phases must be considered when computing degrees of freedom.
'
August 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
1867
TABLE11. DEQREES OF FREDDOM CHARTFOR STEADY STATE FLOW SYSTBMS
.
De rees of Freedom and kelated Equations
Application
+
-
-
b N 1 Z f J ; d’ 2 1 (Eq. 4) Material balances of systems in operationa d’ F’ is minimum number of material F’ = rn Zn. R’; . 2 . 1 (Eq. 10) balance variables that may be deter- F a = nt P (for each equilibrium) mined. F a is not to be exceeded if equilibrium relationships are used in R’
-
+
+
-
R‘
+
Conditions for Usin ho, 8 , and A with R’ and R fofEquilibrk
I. nt > 1. Delete 8 and A from R‘ if nc = 1 11. Necessary temperature-pressure-composition equilibrium relationships available. Necessary temperatures and pressures known 111. If conditions I and I1 are met use of ha, 8, and A is arbitrary. If deleted do not use equilibnum relationships in material balance computations I. nr 2 1 11. Necessary temperature-pressure-composition equilibrium relationships available 111. If conditions I and I1 are met use of ha, 6, and A is arbitrary. If deleted do not use equilibrium relationships in material and energy balance computations I. nt > 1. Delete 6 , and A from R’ if nt 1 11. If nr > 1 use of ha, 8 , and A is mandatory
b (see Eq. 4) R Material and energy balances of systems d‘ nu R (Eq. 15b) F = 5N q 10 e Zna 1 in operationb F is minimum number of material If potential, kinetic, and surface energies are negligible, and energy balance variables that change 5N to 2 N (Eq. 1 6 ) 2 P (for each equilibrium) may be determined. F a is not to be F a = nt exceeded if equilibrium relationships are used in R R‘ b (see Eq. 4) Design of systems by use of material bal- d’ ance vaTmablesa ZF’ ZM’; d 2 1 (Eq. 18) = nt P (for each equilibrium) FI’ is number of independent material balance design variables. For F’ (see Eq. 10) any combination of unit systems F a and F’ are not to be exceeded R I. If nt 2 1 use of ha, 8 , and A is mandatory b (see Eq. 4) Design of systems by use of material and d’ Fx = Z F Z M (E 17) energy balance variablesb 2 P 8or each equilibrium) Fr is number of independent material F a = It: and energy balance design variables. F (see Eq. 15b and Eq. 16) For any combination of unit systema F a and F are not to be exceeded a Material balance variables: stream rate, concentration, ratio of two stream rates. b Energy and material balance variables: material balsnce variables, temperature, pressure, energy transfer rates, stream height, stream velocity, stream surface.
+
7 2
+ + + + + -
- -
-
-
-
+
+- -
If A represents the difference between the number of saturated strcams that theoretically could exist in a heterogeneous equilibrium and the number of saturated streams: that actually do exist in the equilibrium at terminal points, the added peculiar equation m
A$. As in connection with
for m heterogeneous equilibria are P-1
Equation 6, when material balance degrees of freedom are considered, nt > 1 and A is not evaluated when nt is 1. When evaluating A only saturated streams that potentially could be in equilibrium with additional streams at the terminal points need be considered since any other stream will have a A value of zero. Another source of peculiar equations is found in those system that inherently have the same composition of analytical components in two or more streams a t the terminal points. If there are i% sets of streams each set having streams within it of equal analytical component composition, and if fc is the frequency an independently variable (relative to boundary) analytical component stream composition occurs more than once, then the number of addrd peculiar relationships are
2=1
. ..
+
[fc(%
-
l ) ] k (7)
It should be noted that fc and f:, and k and k‘ are defined differentiy. When no chemical reaction occurs the !ifferences between kinds of components disappear and fo = f, and k = k’. Example 4 illustrates how Equation 7 is applied. EXAMPLE 4. APPLICATION OF EQUATION 7. How many restrictions are placed on the degrees of freedom of the system shown in Figure 5 due to the equivalent analytical component compositions of some of the streams? There are three sets of streams of unique composition hence k is 3. Starting with stream 1 and proceding clockwise the restrictions will be given by E uation 7, Zfc(ma - 1) = 0 X 3 (2 X 2) (1 X 2) = 6. For %e sake of simplicity any stream that does not have its composition repeated may be ignored in the calculation.
+
-
+
Certain systems have stream rates at the terminal points that are equivalent. If fa is the frequency with which an independently variable stream rate (in the units of 8)occurs numerically more than once a t the system boundary and there are r sets of strcams having equal numerical rates within the set, then the added peculiar cquations are
2
fai
= fa1
+ + .. .... 2.f
+fer
i1
I n view of the wayf, is defined its value must be zero for a twostream system. This is illustrated in Example 6. The total number of peculiar relationships ordinarily encountered, R’, that apply to material balances of steady state flow system is given by Equation 9.
+ (Z6 + Z A h > + V4n a - 1) + 2fa
R’ = ho
1
(9)
The number of unknowm, universal equations, and peculiar equations have now been ascertained and the degrees of freedom, F’, are obtained by subtracting the number of equations from the number of unknowns as indicated in Table I. The concentration terms appearing in Equations 1 and 3 can be replaced by functions of analytical component concentrations; equilibrium relationships may also be expressed in terms of analytical component concentrations. It follows that the degrees of freedom as given by Equation 10 are valid even though three different component systems are used in its derivation,
+
F’ = -n. Zno - R‘ = (ho (z6 Z A ) n t > 1
+
+
+
--71,
+ En, - 1 ) + Zfa);
zfc(m.(n.
d
5 1 ) (10)
Discussion of Development. I n connection with Equation 4 it is necessary to consider the unknown concentrations in terms of ultimate components, hence d’ 2 1 IS a necessary restriction. The components embraced by F’ of Equation 1 0 are analytical components and from inspection of Equation 1 it follom that unless one of the k sets of streams has all its stream rates fixed no stream rate may be computed. If d is the number of independently variable sets of k sets that have all stream rates fixed within the set, d 2 1 is a neceseary restriction for Equation 10. In addition to thk restriction each of the other sets of the k sets of streams must, at least have fc stream rates fixed or assigned numerical values from the F’ degrees of freedom. Equation 4, of course, should not be violated when utilizing the degrees of freedom predicted by Equation 10. The preceding equations have been derived on the basis of mass units-that is, the stream rates and concentrations have been defined by the use of mass. The question arises as to whethcr or not Equations 4 and 10 apply when other Rets of units are utilized. I n the following discussion the expression “molecular” is used loosely to denote number (moles, atoms, etc.) or
*
INDUSTRIAL AND ENGINEERING CHEMISTRY
1868
relative number (mole fraction, atomic fraction, ctc.) of analytical romponents. For the special case where no chemical reaction occurs stream rates and compositions may be expremed in molecular units and substituted directly in Equations 1, 2, and 3. Therefore, Equations 4 and 10 are valid when molecular units are utilized and no chc 1r.ica1reaction occurs. For systems that have chemical react L t T ~ and S mhcie niolccnlar units are used to express stream rates and component comatrations the analysis is somewhat different. The component balances represented by Equation 1 may still be made using molecular rates of flow and molecular concentrations, and molecular concentrations may, of course, he
MATERIAL FLOW WS
f
Vol. 43, No. 8
the temperatures and pressures of the equilibria are fixed Table I1 summarizes the uses and limitations of Equation 10 and other pertinent relationships that are discussed in this papcr. The main peculiar relationships encountered with material balances are included in Equat,ion 10, but if other restrictions exist for a particular system they must be accounted for. Each additional relationship has the effect of reducing the degrees of freedom by one provided additional unknowns are not employed. Subsequently it will be shown how F’ may be utilized in design work, but the reader is cautioned against using F‘ by itself a8 representing the design degrees of freedom because the potential variables within the system boundary have not a6 yet been taken into account. Figures 1 and 2 are schematic drawings of steady state flow systems that are utilized in conjunction with Examples 2 and 3 to show the application of Equation 10. Unless otherwise specified molecular concentrations and molecular rates of flow arc used in the examplee.
M45 MATERIAL A N D ENERGY BALAXCE DEGREES OF FREEDOM
t
MAT~RIAL FLOW Figure 6 .
0
Combination of T Unit Systems
substituted directly into Kquation 3. Equation 2 is not valid u h e n using molecular rates of flow, for clearly the number of molecular particles per unit time into and out of a process are not always equal. In effect this means that for systems in which a chemical change occurs Equation 4 is invalid if molecular units rather than mass units are used. Equation 10 remains valid regardless of Thich system of units is utilized. This follows since Equation 2 wae used in the derivation of Equation 4 but not in the dcvelopment of Equation 10. In summary it can be concluded that the equations PITwnted are valid whether concentrations and stream rates are given in mass or molecular units-the one exception being Equation 4 which is not applicable when molecular units are used for a process in which chemical change occurs. For such cases Equation 4 has limited application, hence this restriction has little practical significance. Significance of F’. The physical significance of the 8” degrees of freedom given by Equation 10 should be thoroughly understood. When a complete material balance for a system in actual operation is desired, F‘ represents the minimum number of variables that must be physically determined. When F’ is so used it may be necessary t o measure more variables than predicted by Equation 10. If a heterogeneous equilibrium exists but equilibrium relationships of satisfactory accuracy are not available, or the user wishes t o ignore them in the subsequent material balance computation, 6 and A are deleted from R’. The number of variables to be measured as predicted by Equation 10 are then increased accordingly. The same treatment may be used for the ho homogeneous equilibria. I n the application of Equation 10 if equilibrium relationships are to be used it is understood that
Development. The degrees of freedom for steady state flow systems M hen energy and material balance variables are considcrcd will be developcd in a manner similar to that utilized for the material balance degrees of freedom. The universal equations that apply to a system under steady state flow include thosc discussed in connection with material balances and the law of ronservation of encrgy. For a given system an energy balance is applied by summing all the energies associated with outlet streams and subtracting from this the energies associated with inlet Ftreams, and equating this to the energy supplied the system from its surroundings. Each stream is considered to possesa that energy associated with enthalpy, position, motion, and surface. Magnetic and electrical effects of the material in flow are ronsidcred constant. I n this development the types of energy that may be transferred with the surroundings are heat, mechanical shaft work, and electrical. The law of conservation of energy is summarized in the form of Equation 11.
2=1
of a stream; 2, the height above a datum plane; V , the velocity; s, the specific surface energy, gL, the local acceleration of gravity; g., the gravitational constant; &, the rate of heat transfer to the system; Wa,the rate of mechanical shaft work received by the surroundings; and E , the I ate of electrical energy transierred from the surroundings. Surface tension is an intensive variable encompassed by the Gibbs n here H i s the specific enthalpy
TABLE
111. DEGREES O F FREEDOM FOR EXEROY AND BALANCES
Unknowns in addition t o material balance unknowns Temperature a n d pressure of all streams Potcntial, kinetic, and surface energy of all streams Heat transfers Shaft works Electrical energy transfers Equations in addition t o material balance equations Univereal
h~ATERlhL
&-umbel of Eauation, or \‘ariai;le 2
‘ \ A
3 !\’ 4
IC
e
1
zft ZfP
5 N + q + w + e - l Total energy a n d material balance degrees of freedom, F
[W
F’+5A1+q+w [2jt
+
+ ZfPl
+e-
1
V p l ; 721
2
1
August 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
phase rule but surface energy is dependent also on the shape or surface of the mass. The specific surface energy (energy per unit mass) is thus not defined by the Gibbs phase rule. The development of the degrees of freedom for energy and material balances which follows is itemized in Table 111. In order to define H for each stream (phase) the temperature and pressure, in addition to the concentration variables already accounted for as material balance variables, must be considered unknowns. For N streams then, 2 N additional unknowns exist. In like manner it follows that there are N 2,N V , and N s variables or a total of 5N required to define the left-hand member of Equation 11. The S unknowns have been included as material balance variables. If q is the number of separate heat transfers; w , the number of mechanical shaft works; and e, the number of individual elecw e are the number of variables retrical transfers, then q quired to define the right-hand member of Equation 11. Thereq w e unknowns in addition to those fore, there are 5 N itemized for material balances. Added peculiar relationships come into evidence. If there are 21 sets of streams a t the terminal points of the system, each set having streams of equal temperature within the set, and if ft is the frequency any given stream temperature occurs more than once within a set, the added peculiar relations are expressed by
+ + + + +
fii
= ftl
i- 1
+ ftr + . . . . . . + ft.
(12)
The same treatment is used t o allow for streams of equal pressure a t the terminal points. If there are y sets of streams a t the terminal points of the system each set having streams of equal pressure within the set, and f p is the frequency any stream pressure occurs more than once within a set, then the added peculiar relations are
i- 1
In summary there is one additional universal equation (EquaZ f p peculiar relationships supplementary to tion 11) and Zft the material balance equations discussed previously. Therefore, the added degrees of freedom are the difference in the extra unknowns and the additional equations, or 5 N q w e - 1 - Zft - Z f p , The total peculiar relationships, R, that exist for material and energy balances are given by Equation 14.
F
5N + q
+ W
+ C
+Zne
1869
-1
R
=
R'
+ 25 + Zfp = h~ + (26 + 2 A ) n t k l +
V4n. - 1) + .V8+ Zft
F = 2N + q + w + e
(14)
The main peculiar equations are accounted for by Equation 14; however, for added restriction R must be adjusted accordingly. For example, if the flow of gas through a nozzle is reversible R is increased by one. Equations 9 and 10 allowed for heterogeneous equilibrium by utilizing the expressions 26 and 2 A with the stipulation that nt > 1. When nt = 1 the equilibrium equation that results involves temperature and pressure, which are not material balance variables, rather than component concentrations. When pressure and temperature are also variables there is no restriction on nt, hence the notation nt 2 1 is used for emphasis with the expressions Z6 and Z A when material and energy balance variables combined are considered. The total degrees of freedom, F , that exist for material and energy balances may be obtained by combining the additional degrees of freedom with F' as indicated byEquation 15a.
F = F'
+ 5 N + q + w + e - 1 - Z ~ I- Z f p
Substituting Equations 10 and 14 in Equation 1% givee
(15a)
(15b)
+ Zn. - 1 - n * - R
(16)
I n Equation 11 S is defined basis mass units. This may be changed to molecular units provided stream molecular weights are introduced. Because these molecular weights are functions of analytical component concentrations which have been accounted for as unknowns, it follows that Equations 15a through 16 are valid whether mass or molecular units are used. Significance of F. The physical significance of F should be considered. For a steady state flow system in operation F represents the minimum independent variables that are necessary to measure in order to compute a complete energy and material balance. When such a balance on an operating unit is made ho, 6, and A should be deleted from R if the necessary equilibrium relationships are not to be utilized in the computation. In general, F does not represent the number of variables that must be set prior to design. A further development in this paper will indicate how F may be used to determine design variables. Figures 1 and 3 are flow systems used in conjunction with Examples 5 and 6 t o show the application of Equations l5b and 16.
EXAMPLE 5. APPLICATION OF EQUATION 15b. Figure 1 represents a combustion process that is transferring heat to the surroundings and was dismssed under material balance degrees of freedom, Example 2. Discuss the variables to be measured for an energy and material balance. There are no homogeneous or heterogeneous equilibria, no two streams of e ual composition, equal rate of flow, equal temperature or e q u a pressure assumed at the terminal points; hence R by hquation 14 is 0. Substituting in Equation 15b gives F = 5 X 3 1 +.O 0 +,8 - .l - 4 0 = 19. F' for this example was 4, the difference in this particular case representing energy balance variables. When assigning values for the F degrees of freedom no more than four material balance variables may be included; however, if equilibria were present such reasoning would not hold. Variables that possibly could be assigned values are:
+
+
Height of each stream above datum plane Velocity a n d surfaoe of each stream ?&O!in inlet stream Ratio of rate of stream 2 to stream 1 R a t e of stream 1 %CO in flue gas Temperature an+ pressure of 3 streams
Total
+ ZfP
-R
For a great share of systems the kinetic, potential, and surface energy effects may be considered negligible and Equation 15b may be simplified to
+
+ + +
-nu
Number of Variables 3 6 1 1 1 1 6
19
The first nine variables accounted for above are of negligible importance for such a process, hence Equation 16 would have been suitable to use. EXAMPLE6. APPLICATIONOF EQUATION 16. The contact process for the conversion of sulfur dioxide to sulfur trioxide is shown in Figure 3. How many numerical values should be furnished a student in order that he may qompute all the energy and material balance variables? The exit stream is in homogeneous chemical equilibrium hence ho is 1. Substituting in Equation 14 gives R = 1 0 0 0 0 0 0 = 1. In spite of the fact that streams 1 and 2 have the same numerical rates when mass units are used Xf, = 0 since the rate of stream 2 is not independently variable in relation to stream 1. Potential, kinetic, and surface effects are negligible hence Equation 16 may be used to compute the degrees of freeIf dom. F = 2 X 2 + 1 + 0 + 0 + 7 - 1 - 3 - 1 = 7 . the sulfur dioxide in the inlet stream is produced by burning sulfur with air, as is usually the case then the ratio of nitrogen content to the sum of the sulfur dioxide and oxygen contents in stream 1 is a constant for all possible feed conditions and the degrees of freedom are 6 not 7. I n either case F represents the number of variables that may be given arbitrary values. Temperatures, pressures, concentrations, stream rates, and heat transfer rates may be furnished as data. However, such data as heat capacities and the standard enthalpy change accompanying a reaction are not classified as degrees of freedom, for in no sense are they potential variables.
+ +
+ + + +
INDUSTRIAL AND ENGINEERING CHEMISTRY
1870
DESIGN DEGREES OF FREEDOM
Development. The equations derived above apply to systema n herein the streams and energy transmissions concerned cut an imaginary boundary. Within such a boundary a multitude of other independent variables may exist. In some instances it is important to have an equation that will express the sum of the energy and material balance degrees of freedom throughout the system; this may be defined as the internal or design degrees of freedom, F I . Figure 6 is a schematic representation of a specific steady state flow system; however, the equation derived from it will have general application. The circles represent each of the T unit systems in which physical or chemical change takes place. A unit system is one in which hypothetical knowledge of the numerical values of all independent variables at its boundary would be sufficient to fix the physical design of the unit system equipment. Examples of unit systems are: pump, combustion chamber, pipe line, heat exchanger, and evaporator. The lines connecting the circles in the figure denote linkages that provide for flow of material or energy from one unit system to another. M is the number of mutually independent energy and material balance variables in common a t a linkage with any two of the T systems. The notation A!fl,z means the M value bctween systems 1 and 2. Starting with system 1 and proceeding through the unit systems and summing the degrees of freedom give8 the following analysis for 5” unit systems and L linkages: System
Degrees of Freedom Fi Fz Mi,z Fs Mz,i F1
Fa
T
-Mz,~
- M2,6 - M4,a . -MT-I,~~
............................... FT
m
Adding, Fr =
CFi - CAfi 1
1,
i=l
i=l
(IT)
Equations 4, 15b (or 16), and the Gibhs phapc rule must not be violated when assigning the FI numerical values for design purposes. This means that for this particular situation F and F G should be treated as maximum values which perhaps may not be met but must not be exceeded for any combination of unit systems. For the many systems that are designed solely from the standpoint of material balance and without the aid of an energy balance a similar equation may be derived and written in the form of
1=l
2=1
where F; is the internal or design degrees of freedom for systems where only material balance variables are considered and M’ is the number of mutually independent material balance variables at a stream linkage. In using Equation 18 it is understood that temperatures and pressures of the equilibria will be fixed and that necessary equilibrium data are available. If this is not true and equilibria exist the design may not be carried out by use of material balance variables. FGand the F’ of Equation 10 are maximums not to be exceeded for any combination of unit systems when assigning F; numerical values. In addition Equation 4 should not be violated. Discussion of Development. Equations 17 and 18 simply mean that each unit system of a combination of connected unit systems is considered it?olated and its F (or F’) value computed. The M (or M’) value for each linkage is then calculated. The sum of all of the F (or F ‘ ) values minus the sum of all the M (or M ’ ) values gives the total design degrees of freedom. Frequently there is only one unit system (2’ = 1)and F and F‘ then become equivalent to FI and FI, respectively. It is not implied that relationships in addition to those given in
Vol. 43, No. 8
this paper will not be used to compute the physical dimensions of the equipment. Since the design of the equipmcnt brings into play additional unknowns and equations it may be worth while in some instances to add in the resulting increase in degrecs of freedom to the equations already derived in order to obtain an overall design degrees of freedom equation that includes the dimcnsions of the equipment. Previously it was shown for units in operation that it is satisfactory if desired to delete ho, 6, and A from F‘ and F but for design work this is not permissible when evaluating 8’; and PI. It will be of value to consider the variables embraced by the terms 1Tf and M‘. Such variables must be mutually independent for both of the unit systems being considered. For evaluation of AT‘ when the linkage is a stream, the components the same throughout, and no special restrictions exist (such as equal stream rates, etc.) the variables are 8, and na - 1 concentrations or a total of n. variables for each stream. For M under the same restrictive conditions the variables are S , na - 1 concentrations trmperature, pressure, 2,V , and s or a total of no 5. Under the same circumstances if potential, velocity, and surface effects are considered negligible, M = na 2 for each stream. If the linkage represents energy transfer (Q,W,, E ) from one system to another, iM is increased by one for each type of transfer. In the derivation of Equations 17 and 18 a given combination of T unit systems was stipulated and the effect of increasing the number of unit systems is not in general reflected in the results. For certain operations this may be of importance and a degrec of freedom may be added. Example 7 illustrates the application of Equation 18.
+
+
EXAMPLE7. APPLICATION OF EQUATION 18. Compute the number of independent material balance variables necessary to design an extraction battery made up of T equilibrium stages with two streams in and two streams out of each stage. All equilibrium temperatures and pressures are fixed. The analytical, ultimate, and thermodynamic components are identical since there is no chemical reaction. The number of components in all streams is the same and designated by na. For each isolated unit system (stage) the degrees of freedom are given by Equation 10; F‘ = --no 4n, - (2n, - n,) = 2n, For T stages ZF‘ = 2Tn,. M ’ , the number of mutually independent variables that exist in the two streams between any two stages, ie 2(n, - 1) 2 = 2na since no - 1 concentrations and 1 stream rate are the independent variables for each of the two streams that form onc linkage. Since there are T - 1 linkages 2‘14’ = ( T - 1)2n, Substitution in Equation 18 gives F; = 2Tn, ( T - 1)2n, = 2n,. In other words 2na variables must be set in order that this extraction battery of a given number of stages may have all its material balance degrees of freedom fixed. If it is desired to consider the number of stages as a variable, the degrees of freedom are 2n. 1. In order to utilize effectively these degrees of freedom in design work one might set n, - 1 concentrations and 1 stream rate for each incoming stream at the terminal points and also one concentration in the outlet extract stream, for a total of 2n, 1 independent variables. The number of equilibrium stages and the other unknowns can then be computed with the aid of the necessary equilibrium relationships. In contrast, if design work is being done on an extraction battery having a fixed number of stages, then only 2n, variables may be set. If the extraction battery stages had not been in equilibrium Equations 10 and 18would have taken this into account when they wew applied.
+
+
+
+
COMMENTS AKD CONCLUSIONS
The equations derived for design in no way give assurance that the physical characteristics of the equipment will allow the design values to be obtained in actual operation. W x t h e r or not this occurs depends on a multitude of equipment variables that exist within the boundaries of the system. ,4 unit that is in actual operation with all the physical equipment variables (such as exchanger area, tower diameter etc.) fixed has a different degrees of freedom than allowed for herein, This is the operating degrees of freedom mentioned at the beginning of the article. The special
INDUSTRIAL AND ENGINEERING CHEMISTRY
August 1951
restrictions on the degrees of freedom encompassed in the terms R’ and R must be adjusted for any work in this direction. The equations derived should have application to design, instrumentation, and general energy and material balance work. Table I1 presents a summary of the pertinent equations together with their uses and limitations. The cost of material and energy balances can be placed a t a minimum by observation of the equations. In addition, the chemical engineering teacher who is forever confronted with devising problems that can be solved by the student may find the equations and conceptions of help.
,
k
= sets of streams, each set having a common analytical
L
=
m
=
M’ = M
nt
=
ntp = N = P =
Q
Xomenclature applies at the boundary of the system = number of ultimate component concentrations set in all b k’ sets C‘ = ultimate component mass fraction concentration I d’ = sets of k’ sets having all stream rates fixed d = sets of k sets having all stream rates fixed 6 = ntp - nt A = maximum minus actual number of saturated streams e = number of electrical transfers E = rate of electrical energy transfer fl = frequency an ultimate component stream composition occurs more than once fo = frequency an analytical component stream composition occurs more than once fp = frequency a pressure occurs more than once f8 = frequency a stream rate occurs more than once ft = frequency a temperature occurs more than once F‘ = material balance degrees of freedom F = material and energy balance degrees of freedom FG = Gibbs degrees of freedom F; = material balance design degrees of freedom F I = material and energy balance design degrees of freedom go = gravitational constant gi = local acceleration of gravity ho = number of chemical homogeneous equilibria H = specific enthalpy k‘ = sets of streams, each set having a common ultimate component composition
=
na = nu =
(I
NOMENCLATURE
1811
= =
r
=
s
=
S T v B
= =
R‘ = R =
w
W,
y
2
= =
= =
= =
component composition number of linkages between unit systems sets of heterogeneous equilibria number of independent material balance variables a t a linkage number of independent material and energy balance variables at a linkage number of analytical components in a stream number of ultimate components number of defining thermodynamic components total thermodynamic components in an equilibrium number of single phase streams number of phases number of heat transfers rate of heat transfer sets of streams having common stream rates number of material balance restrictions number of material and energy balance restrictions specific surface energy mass rate of flow of a stream from system number of unit systems sets of streams having common temperature velocity of stream number of shaft works rate of mechanical shaft work sets of streams having common pressure height of stream above datum plane
Subscripts i = “ith” stream, system, ete.
j = “th” ultimate component a, b, c , etc. = ultimate components 1 , 2 , 3 , etc. = stream numbers, system numbers etc. LITERATURE CITED
(1) Dodge, B. F., “Chemical Engineering Thermodynamics,” p. 11,
New York, McGraw-Hill Book Co., 1944. (2) Findlay, A,, “The Phase Rule and Its Application,” London. Longmans, Green and Co., 1931. (3) Gilliland, E. R., and Reed, C. E., IND. ENG.CHEM.,34, 551 (1942). RECEIVED May 11, 1950.
locating Fluidized Solids Bed level in a Reactor
Enggtring Process
HOT WIRE METHOD
I
G. L. OSBERG NATIONAL RESEARCH COUNCIL, OTTAWA, ONTARIO, CANADA
A n alternative to the usual pressure drop method of locating the fluidized solids bed level in a reactor is often desirable, because plugging of pressure taps which are in contact with the bed is encountered unless special preventive measures are taken. The hot wire method described here exploits the excellent heat transfer properties of a fluidized solids bed. The thermally sensitive elements are a part of a thermal conductivity bridge and are mounted in a probe. Details of construction and performance in a laboratory unit are given. This type of level probe should be particularly useful in high pressure units. The probe could readily be incorporated in automatic bed level control equipment.
A
HOT wire method for locating the fluidized solids bed level in a reactor offers a practical alternative to the pressure drop method commonly employed for determining and controlling bed depth. The hot wire method described here is based on the well-established observation that the heat transfer rate from a hot surface is much higher in a fluidized solids bed than in a fixed bed or in gas alone (1). Thus, when a hot wire, which has a large positive resistivity temperature coefficient, is submerged in a fluidized solids bed, its temperature will be lowered, and hence its resistance will be less, compared with its temperature and resistance in air alone. The hot wire is made a branch of a Wheatstone bridge circuit, so that changes in its resistance alter the balance of the bridge. A somewhat similar application of a thermal conductivity
