-, for
1941
..
I N D U S T R I A L A N D E N G I N E E R I N G CBBMISTRY
the exact condition of the uipment based on pm%%%? where codes exint, the code%odd be followed. 6. Thems~-?A~ on shwld produce a detsiled analye of the t con& on of the equipment. This wiu permit the estab ent of the rate of wnvaion or detezioration and Bstimation
K
logs
This outline msy appear to the nninitiated to be eskuwive, complex,and d y , but when auah a p r o d u r e is made s definite part of the entire plant production progmm, it is readii ahsorbed and, according to published information, can be held well within 0.6 pm cent the totsl plant man-horn. It is a ranall price to pay for protecting plant and psnOnna and guarsnteeing the life and safe operation of prooese pres-
sure equipment. There is Little new in this Buggested routine. It b the correlation of many i d w , definitely attached to the operation edted company executives. of chemical proasae d. For t h m who desire to 7. Through the correlation of tests and inspection data, the obtain sdditional information for guidsnoe in handling the of design, comtruction, and meintenance standt&.blishment u d s f o r t u t u r e e q u i ~ t m a y b e ~ ~ , a n d t h r o u g h t h e c o n -problems in their own plant, I would draw attention to t of defective equipment the pmoees may be the d e s d e d o p d within the American Sooiety of Me %%E=uay l new. chanid Engineers, the d e for d e d preeaure veaaels of 8. The iaspeation pmnud should inveatigata all accidents the Mean PetruIe.um Institute and American sooiety of involving perarms or e uipment. while this may ~ e e mpost Mechanical Engineers, the National Wety Counoil’s publicamorttrm treatment, the lata obtsined ma h o m e important in tion, “Prmre Vessels,Fired and Unlired”, Parts I and 11. devainping future construction and c o n t r o ~ p d u r e . 6. Autest‘ and I’epsirs, when remrded on suitable report forma f o r z f e r e n m , &odd be distributed to inter-
GETTING THE MOSTFROM AUTOMATIC CONTROL a. e. P&
Leeds BE Northrup Company, Philadephia, Penoa.
in auionutic conbol it is the combined charutuirtics d
w&oller and pronr that count. They may be suited to
&.
UnuUdadwy m u l b may mean h i eithn a mom “mlined‘ mode d conboi or additional conbol may be q u i d A m i n some rimple change in the p w e a may solve (he problem. ClVrKtrrhticr d conboi equipmrnt are gennrliy dmple and udly w i l e d . Charas(.risUo d pfocesws a n infinitely vaded. They not only depend upon &e parUculrr application but en frequently subied tu wide *u*Uon rl(h Ume, The UICI d autonutic eonbol equipment should be familiar with cehin general pdnciples which a n
uch
control has long heen to pressure, AUTOMATIC flow, level, and temprature, and recently, to an inaxtent, to such variables as electrolytic conductivity applied
ing
and pH. The great majority of applications have been mcd u l , and, besides freeing operators for other duties, have often paid for themnelvea many fold in in& production and i m p r o d product. An the um of instruments and automatic controllers has i n c d , many plank have assigned s p c i d men to c ~ n tfor them. These men have usually become very proficient, often understanding a mechanism just as well as the manufacturer who supplied it. Generdy they have had a leas complete picture of important relations between controller characteriatiw and p r o w characteristics. To get the most out of automatic control they should be provided with this knowledge, in EO far powile, and, what is just as important, should be given m5cient authority to apply it. Further, they ahould be consulted when new pmcesa equipment is being designed to avoid mistakea which may be dif6cuIt to correct later. What special knowledge should the p h t automatic control expert have in addition to knowing the mechanismsemployed? FW he should understand the modes of control whioh these
helpiui in padlcubr sonbol prcblemr The idul time to apply these to a pmcer is during the M o d of design whcn c ~ U midaka y can be avoided. A brief mview daubnrUc control theory is gira, with examples horn the fields d hmperatumand pH conboi. The impohnce of a suitable relation &em (he con. bol-valve setling and the resultant llow is dmud. C u m am given showing the chanckdrtics d wme pmentday valves when b t e d at consbnt pam drop. The dkd 01 line d m p in ddormining durachhUo in tnrice i s shown by families of gmerrlly applicable cunes plotkd on a percentage bsir
meobanims produce. Then he should know how the results obtSinable with those mcdea depend upon process ahmmtw iatics. FinalIy, he should undmtand the particular prows under consideration well enough to anaIye it from the point of view of ohsracteristics favorahIe or unfavorable to control. oenerslly speaking, a d c i e n t knowledge for moat practical purpoees w q u i ~ little ~ or no mathermatics and is well within the grasp of the average instrument department head. A brief review of the general theory will be given, followed by a discussion of what is called “the &ective valve characteristic”, a subject of considerable practid importanm in both existing and projected &omastic control applications. Examples of M
An
A of Control
a basis for discuening the combined dwt of controller
characteristics and pmcass characteristics, three of the most important mcdea of control w i l l be considered: two-position control, proportional-position control, and p i - ~ p ~ i ~ ~ l position plus proportional-aped-floatingcontrol. For brevity the latter will be referred to 88 proportionalplusfloating control in further discussion. In explanation of these mcdea, consider a furnace heated by gas to be controlld at about 600’ C. The temperature is
Vd. 33, No. 0
INDUSTRIAL A N D BNQINBBRING CHEMISTRY
10%
valve position is here shown to be linear within the working range (690' to 610' C.). Thje applies to many a d d controllern when the valve position is &&en aa the paition of the valve stem. An will be pointed out later, the tion between temperattaw and rate of gas flow, or heating rate, is not umdly linear and usylly should not be. Figure 1 shows that, if the range of load change r e q h that
automatidy measured by suitable means, and from the d t a of the measurement a valve in the gas line is automatically positioned in emordance with the control employed. In two-position control the valve automatidy takas one of two positions. In the 6rst the rate of heating is always lees and in the second always grater than is needed to mSintaia the deaired temperature. (On-andd control is the special form of two-psition control in which, for one position, t h e OWID heating etieot in
in one case thevalvemust be nearly closed and in another nearly full-open, the controlled temperature will range from approximately 690' to 610' C. This temperature interval iu d y referred to as the proportional-band or throttling range. It would be possible to so adjust the contmller that the valve would be full-open at 696' C. and a l o d at 806' C., but this might introduce undesirable OJrOling of the temperature, as will be sxplsined later. When this is the 0888, the temperature c ~ l lbe held more o l d y to the desired value only by manually ohsnginer the relation between valve poSition and temperabm, while the width of the band for automatiowntml is held constant. For example, the entira muve of Figure 1 could be moved to the left en that the valve would be full-open at 580' C. and c l o d at 800" C. Proportional-plun-floating control combines the smooth action and stability of pmportionalnal-positionoontrol with the ability to control to a speaified temperature under all load conditions. This is accomplished by adding to the propartiod-pwition control d o n a floating mode of control whiah move8 the valve oontinually, if the temperature deviatea from the eat pint, in a diraotion depending upon the direation of the deviation and at a rate proportional to the deviation. Further explanation of the resultant control action will be given in the next section. 0888
reduoed to m by closure of the valve.) F l O u a B 1. FIBLATION OF R M P I B n As t h e temlwBm AND VaLVl PmON IN F%cperature is to be POBTlONAL-PWITION CO-L held- at ut ~. - ah -o .-. 800" C., it may that when the temperature rises tQ 8020 c., be a& the valve will be suddenly moved to the more nearly dosed position. The temperature will then fall until at a predetermined temperat-y, 688' C.-the valve will be suddenly moved to provide the h@er rate of heating, which rate will continue until 6 0 2 O C. is again reschad; then a new oycle of valve d o n and temperature chsnge will begin. With twc-position control the time int8rv& dnring whiah the valve is in each position are automatidy en related to eaoh other that the average nb of heat supply oorreapndn with the average mqhment for heat. With this type of eontml, continuous cyoling of the temperature is inevitable. With proprtiod-position control the gas valve may be autamstioally set anywhere between itu limita of travel. There is dwam a delinib wsitional relation between the
Contr0ll.n Applied to a P-I
Figure 2 repreants two simple temperature prooesses; a in e b in relatidy d i 5 d t . Eech consists of a d i d water bath heated by gas. The load or demand, subjeot to
eaag to contml M
variatioq is a now of wate; whiah enters the tank through valve V I
and ovdowa at B. In the control instntment the temprature of thermometer T is messured, and from it the position of valve V. is determined in accordance
b
a
valve stem and the temperature index as indicated in Figure 1. This o w e &om the valve to be full-open when the temperature in 690" C. or lower and o l d when the temperature is 610' C. or higher. Forintermedhbtempem turea the valve tgkescorresponding intermedhtepositim. Therelation between temperature and
with the parti& mode of control used. For present purpoaes it is convenient to dissegsrd any time lag in the thermometer or the messuringmeahaninm and to wnme that the control &om are at all timea made in accordanoe with the true temperature of the hath.
Y
k!
c
D
e TlYC
TIYE
a
b
ROWBE a. Tno-Pasmon C-
I~COB OBTM
(W)m 'a 0
WITH
Pnocuss on R a m ~ 2~
INDUSTRIAL AND B N Q I N E E ~ ~ N ~ , C H ~ M I S T R Y The olrly difieranoe between the two processes is that in Figure 20 the flame playa d i i t l y upon the hottom of the tank,whereas in b the heat must lirat pea through a dab of iron, N,and a sheet of asbe~ba,M. The iron is introduced to IEplW& thllld Oa&tY and the SsbeStoS thermal reaistsnce. In a we have thermal caprmity only, while in a WB have two the& o~pacitie~ ~eparatedby a
thermal re%istanca.
Two-position control applied t o Figure 2a givea the temperature record of Figure 3a. Each t i e the temperature reaohea the value H , the gas in turned off; egoh time it drops to the value L,the gas is turned on. The diraction of t n the temperature d trend changes e imm&tdy in we. By reducing the temperature interval between H and L, the range of temperature wriat i o n could be reduced. The valve would then moye from one position to the other more often, and the frequency of temperature OaailIation would be correspondingly increased. Applied to Figure Zb, twc-position contml gim the temperature m r d of Figure .%. The valve is o p e d at temperature L and c l d at tamperature H as More,but the direotion of temperature tmnd does not change until wme time later. h e n though the temperature interval between H and L were redud to m,a eimilar m r d would be obtained. The reason in 88 follows: The iron plate ha9 considerable heat Ospaaity and, because of the prsspmce of the
E
~sbastossheet,muatbersisedtoatemperaturewellabo~that of the water More the lmenmry rata of haattraorder can tak0 pkoe. At m (Figure Sa) the heat is turnd ofi but theiron plate is coneiderahly hotter thanthe water,and the heat &nv to the water exceedn the heat requirament for a time until point n is resohed. S i l y , when the heat is turned on at p, the full rate of heating is immediataly applied to the bottom of the plate, but nome time elspaen Mom the water is being heate3 at a rate d m e n t to reverae the temperature trend. The &e& of hest eapacitr olwdy asaooietedwith the w a h in the tank may be designsted 88 demanddde eapscity Isg and that due to the &eat of thwnal capscity Nand thermal rwktanw db may be tfmned as tranefer h. At any given value of deansnd,a pmoess @mdy hewmea more favorable to control 88 the d m a n d d e capaaity in increased and tnmefer lag between the automatidly adjusted heat supply and the primary element (e. g., thermomtm) in de-
tiod-po&tion control tskes care of a sudden ebge in demand. Curves b, c, and dare for su&vdy narmm proportional band or thmttling ranges. Am the temperah ebge neto full&ke the valve is reduced, the drop in temperature rssultingfrom the inmeseed load is reduced, but the O S d h t i O I l E followhg the sudden w increased. T h m set a limit 88 to how narmw the thmttling rang^ can be made. The more favorable the p rom time lags are, the na~owsrthe throttling renge that oan be used without introducing undwirable cedations. Figure 6 is an a d & of the respom of proportional-plw flostins control to a sudden, &d, load disturbance. Curve a represante the disturbance. Curve b is the resulting, experimenWy obeerved tempmatme chsnge. Curve e shows change in rate of heating or valve position. In e the d v e overshoots ita final equilibrium paeition quia!dy, then gradually basks ofi. In doing BO it i n t d u w a Mock of energy, represented roughly by m’d, whiah two purposes: It the wat4r bamk to temperature by making up for energy loat hefore the eon- can completels do ita work, and it pmvidea the energy to the kon plate to the new tempmature at &oh the new required rate of heatiug can be continndy suppkl to the water. curvee c and dshow the mparate && ofthe two aomo i the ponent modas aating eimul~ously. The &e& y to that of proportionalcomponent correaponding d position control is &own in 6. It is the came in form ~d the temperature m. Tbia is the component whiah supplia the extra energy necessary for Able return to the contml point. The floating component is represented by aurved,andthia is the component responsible for the ahift to the new steadyvalve position. c
-
rb
Rcretion and
.
lag h a v e been mand-side
d t v lan and
dum two m
.
.
P VI
which may be dpeignatea as reeotion bg and didmaevdoaity lag. The team “mmtionlag‘ is uaed to denote delay between the time of intduation of a oontrol agent into a treated solution aod the time when &emid equilibrinm is reached. It m d y d. E v a I d t h e a B ~ a h e e t m c m l i t t e d , a c r m e o ~ entern into the contml problem hdbedy. In Figum 6 the shooting would amur sinoe iron ia not a PerMOularly good pH e k i t d w may be plaoea very d w t o the tsnt,but if the thermd mduotm and every h e n t involvm both thermal perticular reaction in one that requires applecisble time,and Cawmty and thermal lenhnce. although the pH is o k d y oontrolled at the point of memm. h p r t k ~ t i o control n and pmpc1~nal-plw5o&ng ment, a memmment made mme distsnat down the line w i l l oontrol will be oondmd in cormeation with Figure ab (the &ow that an additional change ha9 ooolvred en mute. It is m 0 m ~ t p m o a a a ) o n I y . Rg3ue.4lhoarsboapropnst~then,~wsnttooontmlatspointfsr~~awsyso
loge
INDUSTRIAL
VoL 33, No. 9
A N D ENGINEERING CHEMISTRY
-that the reaction is essentially wmplete. As a reault, d i e t9nce-velocity lag is intmdud, representi the time required for tlu dution to flow from the tank to the point of measurement. This type of lag is particnlarly unfavorable in that channes occurrine in the tank durine the delav interval are inevide, since tie contro~erm n o i act untii it knows about them. With a tank of d c i e n t capacity, good wntml may be ohtsined in spita of wnsidemhle distance-velocity lag. IncreaainK the storm capacity incthe time available for react& and also&ts down-the rate at which vaiationn in the in9owing material can change the pH. Both of these are &e& favorableto wntml. In increasing stow cap&ty to make a pmcem more favorable to wntml, it must be kept inmindthatthe deeired Sdvantagpa will not be gained unless provision is made forthomughmixing. s t i i g of the type which FIG- 6. Smpm ImuTBnlmon OF OHCO-L merelymovee the material about without q u i d y distributing inconung material throughout the tank is usually insu5cient. One &eat of i n d capacity which should be noted, although it seldom proves to be didvantageow, is that the m t e r the capacity, the dower h the mpw to change in control-pint setting. When lseg in the measuring and controlling equipment are of the eame order of magoitude 88 thoee of the p r o w , they enter into the control pmhlem and may require s p e d wnsideration.
1
I
whereh
-
a a t o f PmPdioditY AF hA5 01 F whbereb =, k~/loo
-
-
The wmpnding ditrerentd dues.of hF and AS is dF/F W
(2)
for i&My
-
log F
Asanming S
-
(3)
+
h5‘ a oonstaot (g)
- -
= 0 for minimum flow Fa,then
K
in Equation 4.
log FO
log FIFO
and
(4)
W
Or
-
-- -
To obtain the aquation of curve p 1 of F&ure 10, wnd e r maximum flow F . = 100 when S 100. Further, wmider ratio of maximumto minimum flow to be 50, or FJFo 60; Fa 2
-
Then Equation 6 becomes F 2X
l(FS1ma
(6)
Equations 5 and 6 plot 88 ntraight linen on semilog paper and thus provide an easy way to obtain intermediate d u e s , wrreaponding to any particular end dues of Fa and F,.
Effective Valve Characterictic
By efiective valve-c
’ ‘ 0 is meant the actual relation between valve-stem position and flow of a control agent when the valve is applied on a job In twc-position control, the only flow values of inter& are those for the two Positjom at which the valve is to be used. For controls in which the valve may he automatidy positioned anywhere within its range, the wmplete relation between stem position and flow musthewnsided. Thiaisbecausethebestthmttling-lrmge setting when operating with a particular flow IW4UhDIent is directly proportional to the rata of change of flow with stem position in the vicinity of the o p t i n g @t. The term “inherent d v e obmack& . d ’ w i l l b d t o denote the relation hetween atam p i t h and Bow 88 found by teat under wnditiom of wnstgnt pressure drop. ”vw B to F of Figure 7 are experimmkdly determined inherent characteristirs of particular valves of the typa indicatd. AU of them sppmximate more or less closely the form of m e A, which is a theoretidy &eat equal-percentage characteristic. With this charaderistic,e q d movementa of the valve stem result in equal percentage changes in flow, r e m of the actual flow. “Eqd-pereenhge 9ow oharsctenst ‘ ic” meam a relation such that andl equal changea in valve-stem wition result in equal percentage ahanges in a w , wardem of the actual flow. If AS denotes a small change in stem position and AF d e n o h a corresponding lrmall change in flow F, then the above statement 18 erpreased mathemstically by AF x loo = k,A5 (l)
.
While, thearetically, valves with perfect equal-pecentage ’ ’e8 can never reduce the flow helow Fa, provision is made for tight-segting of practical valves. Another theoretical charactens ‘ tic which is approximated by certain d w is the linear obsracteristic. This would lm reprasented hy a straight line in Figure 7. (In this and ‘foU0wingGgures, alliflows are plotted in per cant ofmaximum flow9nd all atem pcmitions in per cent of total movement.) -C
#-
N D U S T R I A L AND E N G I N E E R I N G C H E M I S T R Y
1
Ehctr of Pressure Variation Many factom, in addition to the inherent &m*tic, may influencethe effeative charactmistic. The most common in the effectof variations in pressure drop 8 0 ~ the 8 ~valve with Bow through it. The influence of this factor waa calculated for the special conditions of inatallation shown in Figure 8 and for inherent chsraoteristies of both the equal-percentage and linear form. The derived cnrvea apply to turbulent flow ofa liquid. The method of derivation is explained later.
operating poeition of the valve stem is noted togetbr with the drop acMB% the valve and the total dmp. Ttta ratio of the former to the latter preaum dmp is p’. As an example, wnme a linear valve at astern positionof S 40. Thedrop 8 0 ~ the 8 ~valve is noted to be 30 pnuads, and thetotaldrop is 76 pounds. Therefore., p’ SOP5 = 0.4. Point S 40, p’ 0.4 of Figure 11nearly fallson the curvemarked p 0.1. Curve p = 0.1 of Figure 9 may therefore be considered SI the effective characteristic.
-
-
-
--
n
,.
m-Plm.. m--h
h i . M Fm VIOL-OPEN VALVL
A B FIQW 8. M~DT~ODS OF APPLYINQCONTROL VALVES(A), m f h m m s ros U8ma V A L m IN A BY-PMB OB % ’ LINE I ”(B)
Figure 8 . 4 shows a armmoo arrangement (b) for repulatine, he1 flow to burners. The preeanre upstream from the valve is constant. The presnva downstmam is determined by drop through the burner 8yntem and thedore d e s with the rate of flow. When the flow ie verg emall, 8% with a nearly closed control valve, p r s a i d y the full available pressurrr drop appears aemn the valve. With the valve fd-open the p-dropaerossitmsybeonlyafractionofthetotaldmp. Figure SA indicates the g e n d arrangement (a) which WBS considered in making calculations; (b) is a special CBBB. Curves of Figures 9,10, and 11 apply to the arrangements of Figure 88. In Figure 9 t h curvemarked p 1 isa linear flow charw teristic. The letter p designates the ratio of the drop a m the valve M the totsl drop when the valve is delivering 100 per cent Bow. When p = 1, the entire preseure drop is taken by the valve and the effective characteristic is identical with the inlierent characteristic. Curve8 marked 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05 are for caees in which the drop acroea the vdve at 100 per cent opening is the designated fraction of the total drop. As the proportion of the drop available for the control valve is reduced, the efienive characteristic flattens out more and more at the higher flows. As will be brought out in more detail later, this is frequently undesirable with proportionalposition or proportional-plus-Eoatingcontrol. Figure 10 shows that with a valve of inherent equal-percentage characteristic the dective charaoteristic becomes straighter for d e r values of the ratio p. The fact that this type of inherent characteristic “resists” flattening due to line drops is probably the principal resson that it has proved to be satisfactory for a large percentage of automatio control applications. In sizing up an automatic control installation applied to an operating prooess, it is sometimes convenient to be able to determine the dfective valve characteristic without qperimentally determining the ratio p. If the inherent valve sharacteristic in known, this is readily done. The curvw . . of Figure 11 are lor nea when the inberent cbamctmhc @ either linear or equal percentage. To nea these c k s ,th.
-
A COWrmL
..-
Another method of applying a contr01 vdve is to throttle a return Bow 8% indicated by (a) and (b) in Figure 88. The arrangement of (a) may be considered to repwent control of a beat exchsnger by b y - w a portion of the heating agtmt. In this C B B ~the resistsnce to flow is ofiered by the heat exchanger and any ess0ciat.d oriiim or valm. It is wnmed that the total flow is constant. Effeotive olmmter& ’ ‘caabave been dculated for various valuea of ratio r, which is the ratio of the returned flow to the useful flow when the valve in wide Open.
The cnrves of Fignre 12 are for a valve w i t h a k inhsrant. characteristic. These aurvea are plotted with the closscl position of the valve stem to the right and the fnllapen posC tion to the left, h u s e in this camthe useful flow incBS the valve opening is decreased. For the cnrve r 16 of Figure 12 the ratio of maximum to minimum flow is slso 16. Larger ration could be obtained either by deareasing the reaistgnceto flow offered by the valve or by incragsing the resistance of the p a d e l path, but it i~ evident that an effectivecharaoterietic of incrmsing curvature would result. The curvature of the r = 16 charsoteristiais already too steep for even moderakly di5icnlt control jobs unless the valve is to be operated over a narrow range only. Figure 13shows the results to be expected when the valve has an inherent equal-percentage c b W c . The curve for r = 16 m not far from a linear effective characteristic. In general, better wults can be expaoted with an equal-percent age valve than with a linear valve when the valve is plaeed as in Figure SB. Cornparkon of F i 12 and 13 shows that, by using a valve with an inherent characteristic with emne what less c ~ ~ a t uthan m the equal-parcentsgeform, effective chmctenstica ranging between those of the two finures can be obtained. In determining pinta for the of Figurea B, 10, 12, and 13, it w88 neceBsBly to make c a l d t i o for ~ those based on a linear inherent charaoteristio only. As will be shown later, the effective characteri%tia for an equal-prcentage valve or, in fact, a valve of any other inhppent characteristic ia readily obtained from the res* fer-the linear valve.
-
INDUSTRIAL AND ENQINEERING CHEMISTRY
1100
VoI.33, No. 9
Curves of Figure 10 were obtained from thw of J?igum 9 as follows: Consider, for exampk, 8 flow of 60. For a hear valve this would wour at stem position 60. For the eqdpercentage valve (curvep 1, Figure 10)it occtlls at stem position 82.4. Therefore, d that is neceaary is to plot for stem position 82.4 in Figure 10 all poi& for ntem pceition 60 of Figure 9. Other pinta for Figure 10 are similarly deter-
-
Y
mined.
p
-
-
PI
By following this wane procedure, seta of ourve8 of eftective oharaatariatics may be quiokly plotted from an inherent cbaraoterbtio of any form, on a pcaamtsge besis. Cwms OF WQWB 11. The ordinate p' is defined by the equation,
-Pa.,
P, - Pa
(11)
Equation 8 may be written in the fom,
P
i:I(P,
- P.) - - Pd1
From Equations 9 and 11,
(Pn - Pd
-
Fmm Equations 11 and IS,
(PI
(la)
Ilk3
(la)
Fmm Equation 7,
Fmrn Equation 7, Fmm Equation 9, , Substituting in Jtquation 12 froin 13 and 14,
(PI
- Pa-.)
-
-e1
Subtitlltingin 19 from 20 and 21, hanging Equation 10 similar to 12, dividing the Equation 9, and mMtutingfrom 11,
p'
result by
(W
In Figure 11 scales are given for both linear and equalvalws. A special scale can be providd, or a oomplete mries of ourvea can be drawn, for any valve by merely noting that the cmespondh&Btem p o s i t i ~for ~ the linear and any othet valve are those for which the flowngiven peacantsge
0uWtutingin 16 fmm 16 and mlving for F,
F'p/S'
[email protected], 1941
INDUSTRIAL AND ENGINEERING CHEMISTRY
by the inherent charadens ' tios are the m e on a percentage
baeis. AUUANQEVS~ OF F I Q 8B. ~ Derivation for curves of
-
Figurea 12 and 13.
Fr P
kI,h
-
constants
Fa
-
(W
4
