Note: Descriptions are shown in the official language in which they were submitted.
~s~
NON~LINE CONS I STENCY
MEP~SUREMENT OF A NON-NEWTONIAN FLUID
FIELD ~ N~ION
This invention relates to ~luid consistency
measurement and, more particularly, to a non-invasive,
in-line means and method ~or calculating consistency o~ a
non-Newtonian, two co~ponent ~luid such as paper stock.
1o BACKGROVND OF THE INVENTION
High speed, automated proce3~ing machine~ now
employed in mill~ ~or paper manu~acture require a high
dogree or consist2ncy ¢ontrol. ~hl~ ontrol i5 necessary
because variatlons in ~eedstock consi~tency may result in
15 dramatic changes in the ~ini~hed produ~t. Ir not .
strictly monitore~, ~uch variakion~ will destroy the
unifor~ity and, th~re~orM, the de~irability Or the
~inished papar product.
In order to monitor reed~tock con~ist~ncy,
20 thereby ~inimi~ing ~uch variations, a number o~ devices
have been adopted in the pap~r indu~try. Devices
dedicated to thi3 purpose may be categorized as ~our
primary type~: non inva~ivQ, in-line; inva ivet in line;
non~invasiv~, o~-line; and inva~ive, o~-lina.
~Turning fir~ to the non-inva~ive, i~va~ive
distinc~ion, the non-in~asiv~ d~vices ~re generally more
modern and contemplat~ th~ us~ o~ ul~rasound or light,
which generally detect consistency variation~ by compara-
tive analysi~ with ~nown sta~dards. Such devi~es are
exempli~ied by that illustratefl in U.5. Patent 4,171~916.
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~Z589~5
In-line, non-invasive measuring devices dedicated to
oth~r purposes are also known. For example, Heine, in
U.S. Patent 4,285,239, describes a device for determining
the density o~ flowing slurry materials.
Returning to con~istency measuring devices,
others employ non-invasive pressure transducers to make
comparative analysis lik~ that depic~ed $n Staege, U.S.
Patent 2,627,788. The most co~mon invasive t~pe of
device is characterized by an impeller. Impeller devices
o~ten are based on comparative driving ~ha~t torqu~
measurements to indicate fluctuating consistency of stock
(See Coats, U.S. 3,155,866). Another type of impeller-
based measurin~ device is illustrated in ~adsen, U.S.
4,1~8,~14, having prQssurs tran~ducer~ located in close
proximity to the impeller bladQs to detect pressurs
difference~ and, consequently, conRistency variations.
Impeller-based device3 ar~ al~o employed in of~-line
device3. Cowan, in U.S. Patent 3,528,281, employa an
impeller to draw ~luid fro~ a conduit into a sample tube
where the variable voluma ~low i~ used to det~rmine
consistency. Staege ¢onstituta~ an of~-line devic~ which
employ~ non~invasive apparatus ~or paper stock pressure
measurement.
All of the abov~-d~s~ribed consis~ency measuring
devices detsrmine consist~ncy by empirical comparativ~
analysis. The impeller-based devices ara recognized to
give repeatable measure~ents and, i~ a~sociated with a
control d~vice, ar~ generally capable o~ regulating con-
sistency to ~0.05%. Howe~er, thas~ devices suffer ~rom
two noted ~hortcomings. First, although repea~able, the
measurement~ are o~ten inaccurate~ Secondly, impeller-
typa devices o~ten become snagged with string, other
s~ron~ fibrous materials or fabric pieces. Hence, fre-
quent clsaning and recalibration are the rule. T~e non-
invasive wav~ ~nergy frequency typ~ (ul~rasound, l$gh~,
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etc.) often produce less repeatable measurements (~0.1%)due to fluctuations in fibex length and flow rate.
Excepting Coats, all recognize a relationship between
consistency and pressure and/or velocity but determine
the relationship in comparative empirical analysis.
In view of the noted shortcoming~ of currently
available con~i~t~ncy mea~uring and control device~ and
the considerable efYort~ to per~ect ~uch devices, the
need still exists ~or accurate, repeatable consistency
measurement-
SUMMARY OF THE INVENTION
It is an object o~ thi~ invention to provide a~eans and method ~or datermining the con~istency of a
flowing, non-Newtonian, two component fluid.
It is another ob~ect of thi~ invention to permit
repeatable, accurato calculation of a n-~erical
consistency value ~or a ~lowing multi-component ~luid.
Another object o~ thi~ invention i5 to generate
an absolute consistency value ~or a ~lowing non-Nawtonian
fluid using non-inva9ive, in-line means and ~ethods.
It i~ anothar ob~ QCt of this inventIon to ~easure
, consistency o~ a non-Newtonian ~luid with a device
: employing a minimum of moving part~.
Still another ob;ect o~ this invention is to pro-
25 vid~ a means and method ~or non-inva~ive, in-line;
accurat~ measurement o~ consistency according to a power-
law model ~or a non-Newtonian multi-compon~nt fluid flow-
ing through a conduit.
An additional ob~ect of thia invention i5 to
prsvide a means and method ~or calculatiny consi~tency
which i~ not substantially influenced by ~reeness,
density, solids content or pH of th~ non-Newtonian fluid.
Yet another object of thi~ invention i~ to
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~589~5
facilitate control of the consistency of a non-N~wtonian
~luid during processing.
A more narrow object of thi~ invention is to
provide a means and method for calculating the
5 consistency of flowing paper stock using a power-law
based algorithm employing measure~ents of pressure loss
and flow rate a~ well as controlling the consistency with
a respon~ivQ means to ths calculation.
Certain or these ob~ects are satis~ied by a
10 method for measuring and monitoring the consistency o~ a
flu$d havlng at least two component~ and flowing through
a conduik, comprising th~ 5tep3 0~ providing a fluid
feedstock into a conduit o~ a sel~c~ed cro~-sactional
configuration ~n a manner where ths fluid flou~ through
15 the conduit in a non-turbulent manner, ~ensing the valoc-
ity o~ the fluid feedstock flowing through the conduit,
sensing the pre~sure o~ the ~luid at two points separated
by a selected dlstance along tha conduit, detQr~ining the
pressure ~i~ferential between the two points, inputting
20 values o~ th~ pressure di~orential, d$stance, dimensions
and velocity into a calculating dovice, calculating tha
consi tsncy o~ the ~luid according to a power-law
algorithm appliaable to the ~luid and the cross-sectional
geomatry o~ the conduit, and re~ponding with a responsive
25 means to the calculatad consistency value.
Still other object~ of this invention are
satis~ied by an apparatus for monitoring the consistency
o~ a l~guid composed of at least two component~
~imulating a non-Newtonian ~luid wh~n flowing in a
30 substantially laminar manner through a condu$t o~ a given
cross- sectional con~iguration, comprising means ~or
flowing said fluid ln a substantlaliy lamlnar manner
through the conduit, mean ~or ~easuring thQ bulk
velocity o~ thQ fluid flowing through th~ conduit, said
35 measuring means producing a signal representative of the
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:~2S1 3~5
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bulk velocity, at least two means at remotely spaced
points for sensing the pressure o~ the fluid in the
conduit where each sensing means produces a signal
representative of the pressure at each point, means for
5 directly calculating the consistency of the liquid
according to a power-law model requiring as independ~nt
value , said representative signals, and responsive means
for responding to the calculated consistency.
This invention provide a wholly novel and unique
solution for determining conslstency of a non-Newtonian
10 fluid ~lowing in a laminar manner through a conduit. The
invention i-~ primarily directed for U8~ with paper stock
but may be applied to mo~k any pseudoplastic or dilatant
non-Newtonian fluid. Both the mean~ and methods
presented herein contemplatQ in-line, non-invasive
15 consistency determination ~or monitoring or control of
consistency. Rather than comparing consistency values
against known standard solutions ox astablishing ~andom
calibration, this invention accurately ascertains an
actual numarical valua o~ consistency. The consistency
20 calculation rcquires the mea3uremen~ o~ only two
independent variable~ from a non-turbulent ~lowing fluid;
pressure head 1098 over a.~peci~ically qeledted distance
.. and bulk velocity. Thesa two variables ara plugged into
a power-law bas~d algorithm which produces a real value
25 o~ consistency. The alyorlthm, being programmQd into a
calculating devic~ such a~ a computer, allows for a
direct determination o~ the con~istency valua, and
produc~s a signal which may ba displayed ~or monitoring
purposes or may bQ associated with responsive control
apparatus for maintaining the consistency o~ the paper
25 stock.
Unlik~ the previous consistency measuring
system~, the in-line, non-invasive features enhance the
reliability of the consiRtency valus~ calculated
~5~39~5
according to thi~ invention. The apparatus does not rely
on an invasive impeller, the operation o~ which can vary
due to snagging o~ stringy and fibrou~ materials. As
will becomQ apparent, the contemplated apparatus has a
minimum of moving parts, thus eliminating mechanical
breakdown and increased reliability. Furthermore, the
invention provide~ repeatablQ, accurate consistency
values without the need for continuou~ recalibration
against comparative standaxds.
It is avidant that th~ invention provides a means
and method lending itself to automated s~stem~ involving
proc2ssing of non-Newtonian ~luid~. Automatic
calculation o~ a tru~ numerical value o~ consistency is
clearly supQrior to tedious, generally non-repeatablo
hand ~ea~uremant and th~ comparative mQthods ~laborated
upon above. The fa~t, repeatablo, accurata mcans and
method~ o~ this invention aro highly deqirablQ ~or
monitoring guality and automated control of ~luid
c~nsistency. Particularly in the context o~ computer~zed
consistency contxol~ virtual instantan~ous response to
con~istency variation~ i~ as~ured~ hence, greater
adherenc~ to product speci~lcations and enhanced product
guality.
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BRIEF DESCRIPTION OF THE DRAWIN S
Figure l i~ a partial schematic representation of
tha apparatus us~d in the practics o~ the invention.
Figure 2 i9 a schematic representation Or the
data acqui~itlon and control system of thi~ invention.
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DETAILED DESCRIPTION OF THE EMBODIMENTS
Preliminarily, it must ba noted that the below
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described apparatlls represents one means for practice o~
this invention. It can be readily appreciated, however,
that the elementary apparatus disclosed herein may be
incorporated into more complex systems in a variety of
5 embodiments~ A person o~ ordinary skill in the ~luid
property measuring art, without undue experimentation,
can ~enerate equipment~ dedicated for particular ~luids
in particular situations. For the purpos~ of this
application it i~ neith2r necessary nor desirable to
provide an exhaustiva list o~ po$sibilities. Therefore,
10 apparatus ~or determining t~e oonsistency of paper stock,
a non-Newtonian, pseudoplasti~ ~luid, ~lowing through a
non-rotating, circular conduit is described, in detail,
to illustrate tho invantion.
It i~ nacessary to ~irst point out that the
15 below-described apparatu~ i3 u~eful only when associated
with the below-described algorithm. Ik is through the
algorithm that ~luid consi~tency i9 calculatad from the
measurement~ generated by th~ apparatus. Tha consistency
o~ a multi-componant, ~olid/li~uid non-Newtonian ~luid
20 cannot ba accura~ely detarmined without its use, except
by laboratoxy experiment3.
Secondly, it has been ob~erved that dekermination
oS con~i~t~n~y values accordlng to this invention is
seriou~ly ~lawed when paper stock rlows in a turbulent
25 manner in a conduit. Where ~low become~ non-laminar, thP
means and ~thod~ disclosed herein do not work. I~ plot-
ting the log of bulk velocity against ths log o~ pressure
head 105s, the curve rises at a constant slope unkil
turbulance is experiQnced. It is in this range o~ con-
30 stant slope, that the invention is intended to operate.Employing engineering term~, the Reynolds number ~or
heterogeneous fluids, Re', should not exceed 70 in order
to achieve laminar ~low characteristics. A detailed
description o~ these Reynolds numbers is ~ound in TAPPI
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(Technical Association o~ the Pulp and Paper Industry)
Vol. 33, No. 9, A Study of thQ P1PQ Friction Losse~ of
Paper Stock Suspension, an article by Brecht and Heller.
Thirdly, in general, paper manufacture uses paper
5 stock solutions having a consistency between 2-5%. I~
lesser consistency material is employed, it flow3 ~aster
in order to maintain a constant flow on a dry basis and,
therafore, may give rise to turbulence. In the practice
o~ the invention, on~ solution to overco~ this problem
10 is simply to increas~ the conduit radiu~, thereby
reducin~ the v910city 0~ the ~luid.
Now rsferring to Figure 1, paper stock 11,
conventionally having a con~ist~n~y ranging between 3 4%,
i~ drawn from stock che~t 10 by pump unit 12 and into
15 horizontal ingress pipa 14. Ingress pipe 14 has a sub-
stantially constant diamQker and i~ at least eight pipe
diameters in length. Like all remaining conduit
sections, Pipe 1~ should bQ manu~aatured of corroslon
resistant materials. Stainles~ S~eel 304-L pipes
20 produced by Felker Bros. M~g. o~ Mar~h~ield/ Wisconsin
prov~ ~uitablo ~or this pur~osa. Pipe 14 further
incorporat~ exterior ~lange momber 16 which cooperateg
and mates With spool ~lango 18 to attach ~pool 20 to pipe
14. Spool 20, the prlmary apparatu~ of this invention,
25 incorporate~ T-member~ Z2, located at opposite end~
thereof and elongated central section 24 disposed there-
between. ~he length o~ spool 22 may vary to most any
desired length and the diamet~r may range between 1/2 -
24 inohes, depending on the flow rate necQssa~y to insure
30 laminar ~low. For exampl~ a ~our inch diameter, twelve
foot long spool 20 required a ~low rate of approximately
6-7 feet/sec o~ 3-4% consi~tency paper stock in order to
utilize 30 ton~ o~ solids in one day.
Furthermore, it i~ critical to the proper
35 function o~ thi~ invention that turbulanc~ and othsr un-
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desirable flow phenomena be minimlzed. Hence, it i5important that spool 20 havQ a substantially smooth, con-
tinuous, constant diameter interior sur~ace. It i~ also
desirable that ingres~ pipe 14 and egress pipe 26 possess
substantially simllar characteristics~
Turning br~a~ly to egre~s pip~ 26 it also bears
cooperating, mating flanging 28 for attachment to spool
20~ BacausQ pipe 26 ~ located downstream from spool 20,
; it is pre~erred, at a minimum/ that pipa 26 hav~ P len~th
O~ at least 4 pipe diam~ter~ becau~Q thQ $nternal geomet-
ric configuration requirem~nt~ ara l~ssQr than tho~e ~or
ingre~ pipe 14. Flow di~continuitie~ created by
internal geom~t~ic alteration~ will not have a
pronounced an af~ct on stock 11 as it ~low~ through
spool 20.
Re~erring bacX to spool 20, it ~eatures di~feren-
tial pres~ure trans~itter a~3embly 30 lncluding diaphragm
typQ sens~ng intQr~ces 32, pres~ur~ seals 34 and trans-
ducer 38. Inte~Pa~e~ 32 compri~e extandQd head pressure
seals 34, like th~ Modal ~23EP-IMlSA2KD ~rom the
Foxboro Company o~ ~oxboro, Mass. Tho 3eal~ ~ncorporate
diaphragm3 ( inter~ace~ 3 2 ) o~ 316 ~tainle~s ~teel, a low
coe~iclQnt o~ thermal expanslorl pra~s-lra seal ~luid and
an operating te~p~r~ture rang~ o~ -35 to 1~0 F. Each o~
T-~ne~bsrs 22 ~o~vo ~ a housing ~or Qach o~ seals 34 and
inter~ace~ 32. ~t i~ i~nportant, in order ~o Dlinimiz~
~low turbulerlcQ discontinu~tie~ that inter~ace3 3~ sub-
stan~ially match thc radius o3~ l:he lnterl~r wall o~ spool
20 .
In~erraces 32 sense the absoluta pres~ur~ Or
paper stock ~lowing through ~pool 20 at t~o r~ota
point~. It ha~ been obsQnfQd that to incraa . ~ the
accuracy o~ the ul~imately ob~ain~d consistenGy value,
thQ dt stanc~ between thes~ points must be~ inc:reased. The
3 5 pressure 3i~nals correspondin~ to the ~luid prQssure~ at
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each point are transmitted ~rom seal3 34 to differential
pressure transducer 38 via hydraulic capillaries 36.
Transducer 3~ converts s~nsed di~erential pressure value
into a proportional 4-20 ma analog signal. Thi~ signal
5 i5 then inputted into data acquisitlon system 50 ov~r
wire 39.
Spool 20 al~o ~eature~ ~low transducer 40: a
clamp-on~ wa~erproo~, high-~r~quency, ultra sonic trans-
ducer which i~ capa~lo o~ operation at temperature~
between 40 to 180 F and detect~ ~low rat~ o~ a~ tle
as .5 ~t/~ and 91urrieY down to 25 parts ~Qr m~llion.
Tran ducer 40 generates ultrasonic wave energy, directing
it into ~lowing 5tock 11 and detQct~ th~ quantum o~
refl~cted energy. A signal corresponding to tha
15 re~lectQd energy i~ f~d to tran~mittar 42 which
dQtermlnes the doppler ~hi~t cau~d ~y tho ~low rate o~
~tock 11. TransmittQr 42 than gQnQrates a 4-20 ma analog
signal proportional to the 8tocX~ bulk velocity that is
transmitted to data acquisition ~ystem 50 over wirQ~ 44.
20 Th0 ~odsl ~FT-30013-TS00 }~F ~lowm~ter aasembly
manu~acturscl by Dyn~13oni¢, Inc., Naporville, ~llinois,
inco~por~te~ both ~ tran~ducar and t~ansmitt0r mseting
the above-de~cribad prR~err~d reguiremant~.
The si~nal~ ~ed into data a~ ition syst~m 50
25 reprasent th~ pra~sure drop ~pressur~ head lo~) over the
length o~ spool 20 and the bulk v~locity o~ ~tock 11.
Data acqu~ition 5ye~C8~11 50 employ~ the balow-described
algorlthm to calculatu the con~i tency of stock ll u~ing
the data g~nerat~d ~rom thQ tran~ducers. ~lthough a
30 measurement o~ ~toc~ con~i. tsncy may h~ o~ valu~ rOr
~uali~y cont~ol ~onitorlng purpo~, it i~ pr~ferred to
a~sociat~ da~a acqui~ition ~yE~l:Q~a S~ with a con~istency
control systQm. ~uch a control syst~m is now described.
Ideally, th~ control system i~ provided with a
35 measls to establish a partlcularly d~ired con~istency.
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C 11
The consistency is then set at that value which is in-
putted into data acqui~ition system 50. Once inputted,
the system is able automat~cally to calculate and compare
the consistency of stock 11 with the preset value and
5 make any required adjustment3.
A 4-20 ma ~ignal representing th~ di~ferential
between tha actual and set consistency value~ is tran~-
mitt~d over wlre~ 52 to Standard I/P converter 54 such
aq the Dy~a~onia ' g Model 512000. Converter 54 receives
10 tha signal and conv~rt~ it to a pne~matlc signal o~
bet~ean.3-15 p.~.i. The pressurized air trav~ ro~
conv~rter 5~ to valve positioner 58 through copper tube
56 which po~itions the valvQ porportlonally in responsa
to the a~oresa~d pnQumatic ~ignal. Th~ valvQ i~ actually
15 moved by ~upply air 57 at B0 p~ n this particular
arrangement, a Foxboro Power Po3itioner meeting th2
following specirications: air ~upply: max. 150 po~
air delivery: 7.4 ~c~m at 60 p~8~ relay bleed: .75
~c~m at 50 p.s.i~, accuracy: 1~ o~ ~troke, sensitivity: 1
20 inch water ~ign~1 pra0su~e and a temperatura range: 20
to 160 F~ was e~ployad. Th~ positloner i~ attached to
the ste~ oP and ontrol~ the movement o~ valve 60.
Stainl~ teel val~ 60 i~ characterized by a V-
seat in~rt ~d an air cylinder ~ount ~uch as the
25 Kni~Q G~t~ #37R-316-V-HOl ~v~ilablo rrom Fabri-
Valve o~ Portland, OragonO Valve 60 controls thequantity o~ w3ter ~lowing into pipa 62. Pipo 62 i~
conn~ctad to pump unlt 1~ so ~he ~mount o~ water flowing
into pump 12 gov~rn~ th~ con~istency Or stock 11 p~mped
3Q into p~pe 14. Conseguently, ~he consisten~y o~ s~ock 11
i5 automatically and continuou.ly stabiliz~d by thi~
~eedback control ~ystem.
To summarlz~ the "hardwarel' e~ployed in th~ prac-
tics o~ thQ inv~ntion, pump 12 moves ~ock 11 to spool
35 20 wher~ in-lina, ~on-inva~iva mea~urement o~ pre~ure
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head loss and flow rate is made. The generated signals
are fed into data acquisitio~ system 50 where the
measurements are calculated by the below-described
algorithm to determine consistency. Where the
consistency value 50 obtained differs from a pre-set
value, a proportional signal generated by system 50 is
~ed into converter 54 which governs control valve 60
thereby controlliny water flow quantity into pump 12 and,
accordingly, the cons$stency o~ stock 11.
At this point it i~ desirabl~ to brie~ly describe
th~ operation o~ d~ta acqui~ition system 50. A schematic
diagram o~ thiA operation i~ ~ound in Flgure 2. Analog
signal~ ~ro~ th~ valocity transmitter 42 and differential
pressure transmitter 38, are recei~ed by tho signal
conditioning module 70 and converted to digital signals
by digital conversion module 71. The constant parameters
a, ~, n, R & L (identi~ied below) as well a~ a desired
consi3tancy ~etpoint ar~ Xey~d in by conventional.
thumbweels 72 to digital input module 74. Processor
module 76 which ha3 b~en programmed with the program
~ound in Appondix A submitted herewith vla RS-~32 T~rial
cc~unications cable 78 and computer 80 ~an IBM-PC ,)
receive~ the values stor~d in module 71 and module 74 via
,~ standard bu~ 73 ~a commercially avallabl~ d~vice
compri~ing a ~et o~ parallel lines ~or trans~ission of
digital information between computer components) and
calculates the consistency according to the below
described algorith~0 Thi~ consistency value, a~ well as
the other s~gnals~ are th~n sent to digital outpu~ ~odule
82 which in turn displays the ~ignals and consistency on
conventional LED's 84. I~ ~ control system i~ associated
with the consistenGy dQtermi~lng as~embly, processor 76
al50 co~pares the value o~ the consisten~y with th~
consistency setpoint and calculate~ the prop2r control
action u~ing standard PID (Proportional, Integral~
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Derivative) software. The control action signal is
transmitted via BUS 73 to digital/analog converter module
86 which outputs analog signal 88 (52 in Fig. 1) to I/P
converter 54 and finally valve 60.
Referring now to the heart of the invention, it
is an algorithm which pe~mits consistency o~ a fluid to
be calculated as an absolute value employing only
pre-~sure head los~ and flow rate a~ independent
variables. A summary o~ the derivation o~ the algorithm
is now provided.
The equation o~ motion which describe~ (in
cylindrical coordinates) a fluid in laminar flow at
steady state through a non rotaklng circular pipe i~
described cn pa~e B5 o~ Bird, R.B., Stewart, W.E. and
Lightfoot, ~.W., "Transport Phenomena," John Wiley &
Sons, Inc., NQW YOrk 1960.
r ~ ~rTrz) + pgz
or
~P = 1 ~ J (2j
where P - P - Pg i~ th~ absoluk~ pre~sure
r Y radial dimen~ion
Z 3 leng~h di~en~ion
2S and
. Trz i~ ~e shear fo~c~.
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1~
For water, the shear force i a linear function
of ~he velocity gradient in the pipe:
dVz (3)
Tr 2 ~dr
Vz = velocity
~ = viscosity (constant~
A ~luid which obeys Equation 3 i3 called a Newtonian
~luid. The shear force of paper stock, however, i~ not a
linear funckion o~ the veolocity gradient. The "apparent
viscosity" i5 a~fected by the consi~tency (% solids) of
the paper stock. The Du~y corrQlation~ below, described
and discus~ed in Waller, M.H., "~easurement and Con~rol
of Paper Stock Consistency", Instrument Society o~
America-Monograph 5, ~1983), r~lates tha head loss of
paper stock in a pipe to the consistency and v~locity,
using an ela~tic de~ormation o~ the ~iber nQtwork model.
. . .
~ dll ~ RC~V~DY ~4)
wher~, dll/L ~ head 10~5 / lengt~ of pipe
C ~ consistency
V ~ bulk velocity
D = pipe ~iameter
K,,~, and y - suitable coe~icients
This eguation ha~ been compared with data, and values of
K, a, ~ & y have been detaxmined to giv~ ~h~ best
correiation. This correlation is used quite o~ten in
calculating pre sure dro~s ln the design o~ paper stocX
flow systems. See pg. 247-252, Du*~y, G~G., "~ow to
Determine Pipe Friction Los~ ~or the Design of S~ock
Piping System~ PPI Engineering ~onference
Proceedings Book 2, 1979.
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The shear force of non-Newtonian ~luids, such as
paper stock, is subjected to a power law relationship
like that presenked by the Ostwald-de Waele model:
~rz ~ ~m¦d Zl ~ ~5)
m = .418 lbf - sn/ft2
. n = .575
IdV I n-l .
10 -m I ~ - 1 8 appar~nt viscosity
For rluids with a valuR o~ n~ 1, the behavior i~ said to
be pseudoplastic. Metzner A~vance~ in Chemi~al
Engineering, Vol. I, Academic Pres~, NQ~ YOrk (1956), pg
163, ~ound that a 4~ paper stocX solution had the
15 ~oregoing value~ oP ~ and n. Thus, papex stock ¢an be
considered a pseudoplastia rluid.
E~tations 2 and 5 complotoly desaribQ the ~low o~
any psQudoplastic (inoluding pap~r ~tock) in a non-
xotating, ci~cular pipe. It i~ evident that fluids other
20 than paper stock requirQ di~erent m and n values. Such
values may be dHtermined ~rom appropriatQ experimentation
or may be available ~rom the literature. ~o deter~$ne
relationship between the "apparent visco3ity" and consis-
tency, Eqyatio~ 5, ~ound on page 11 o~ the 8ird publica-
25 tion, i~ substltuted into Eguation 2 and solved ~or thevelocity pro~ile V (r~:
Vz(r~ n~)(2~P ) n [-(R) n ~ Rn+l (6)
;' ,~ ' ; ,`: . .~
,,
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16
Vz(r) - v~locit~ profile
aP - pressure change
m = constant
n = constant
R = radi~3 of pipe
L = length of pipe
The next step i~ to determine the bulk velocity o~ the
fluid by integrating the velocity profil~ over the cross~
sectional area, and dividing by the cross~sectional area:
1 n~l . .
~Vz> = (2~mL) n ~- n ~3n+lj (?)
cVz~ = bulk velocity
: R - radius of pipe
.
Finally/ tha head 1088 i~ detQrmined by integratlng the
local rate o~ dissipation of mechanical enerqy over the
volume o~ a pipa o~ length L as described on pag~ 215 of
the Bird publlcaton:
3 -~V ( T: V V) dV (8)
. V ~ volume of pipe length L
where, Ev 1~ the friction 1033 and T~V ls the rat~ o~
'~ irreversible conver~ion to intsrnal ensrgy. Wh~n solving
Equation 8, with the power law model described in
Equation 5, ~h~ resulting equation is (see page 23~ of
Bird):
~v = ntl ~3~-~ (93
p ~ d~nsity
Examining paper stock ~riction lo~ data correlated by
Brecht and Heller (TAPPI, Yol. 33t9), Pg. 144, (1950)~
supra. and correla~ing it wi~h Equa~ion 9, the ~ollowing
relationship resulted:
s
17
m - ccB ~10)
Pressure los~ data is compared to m using Equations 9
and 10 for various pipe sizes, velociti~s and
consistencies.
5 Table 1 compares data obtained from Brecht and Heller
with t:hat obtained from the power law model.
Table 1
C~6 cV~ :et/~ D inches Hf ~t/100 ft E ft/100 ft.
(Article) (from Equation 9)
. . , _~_ .
2.0 2.0 ~.0 5.16 5.15
2.0 4.0 6.0 6.63 6.7s
3.0 2.0 6.0 11c02 10.93
4.0 3.4 6.0 23.10 23.16
3.0 4.0 ~.0 24.5~ 25.~3
lS ~.0 5~1 4.0 4S.9~ 47.6~
4.5 3.8 4.0 51.80 53.16
2.0 ~.0 8.~ . 4.. 47 ~ 4,49
; 3.0 ~0 8.0'3.65 9.~1
.0 ~.0 8.0~1.39 21.6
where a - 0.024~, ~ =1.88,n 8 0.39 and L = 100 .t.
It is readily apparent that the power-law model
reproduces actual friction loss data accurately. To
illustrate the relationship o~ consistency to bulk
. velocity and thQ pressur~ drop in t~e pipe, Equation 10
is su~stituted into Equation ~ and consistency isolated.
.:
- ,
..
' ~' '
39~
The resulting equation is:
1 n+l n
C = (~P~L) R /~ (3n 1) ~ll)
<~ >
Thus, an equation i~ obtained for determining the
consistency Q~ a pseudoplastic in a non-rotating,
circular pipe having a~ the only independent ~ariables,
pressure head loss over a 5p8Ci~iC length and the fluid
~low ra~e. Employing the above-described apparatus,
the~e va.riable~ ar~ determined and, there~or~, the ~luid
consistency i~ calculable. The con~istency calculation,
~n th~ above-described embodi~ent o~ tha invention may be
performed on the above described data acquisition system.
u~n*~B s~
h~Es~rit~-- .
The deter~inat~on o~ consistency o~ a non-
Newtonian ~luid ~lowing through a conduit 1~ not limited
to tha speci~ic yeom~try o~ tho conduit. The detailed
derivation o~ consi~tency ~lowing through a pipe having a
circular aro~s-section i~ applicabl~ to other geometries
albeit sub~ect to some mo~i~ication. For example, the.
alternativu expression ~or calculating the con~ t~ncy of
a non-Newtonlan ~luid through a re.ctangular, slotted
spool 20 is ds~cribed below. First it should be noted,
however, that the apparatus in ~hls embodimen~ should
posses~ the ~amQ charac~eristic~ as those describe~ above
for a circ~lar conduit, i.e. smooth, continuous lnterior
surfac'e, etc.
Moving now to ~he establishm~n~ o~ th~ algorithm
necessary ror consis~ency ~alculatIon ~or the slo~ted
geometry, the equation o~ motion describing ~n
rectan~ular coordinates the laminar flow of a fluid at
~teady-state t~rough a non-rotatinq, rectangular slotted
.
'
~S~3~3~5
19
conduit is:
~p _ l~cz
~Z l)X PgZ (12 )
5 ,~jp ~i T or
liZ = ~ (13)
where P = P~Pgz.isthe absolute pres~ure and TXZ iS the
shear force.
The equivalent of equation 5, but r and R being
replaced bv rectangular coordinates x and B is:
¦ d ¦o-i
~xz = -m ¦-d Zl ~ ~14)
Equation 14 iq ~hen sub~titu~ed into Equation 13 and ~he
resultlng axpre~#lon i~ solv~d for the velocity profil~
V (X):
Vz n+l [ mL ~ (B~ ) P (15)
x is the width dimensi:on and B = half the width of slotted
rectangle.
~ext, the bulk velocity i~ deter~ined by integrating the
velocity profile over th~ cro~s-~ec~ional area, and then
dividing by the oro~-sectional axea.
., .
., :, .
' ~.~..,'' . ~ ,. .
~2~8~
l n+l
cV ~ = (mL) (n+~ n+l) B (16)
Substituting Equation 10 fro~ above, into Equation 16,
consis~ency ~or a fluid flowing through a rectangular,
slotted condu$t is then expressed as:
1 n~l -
C = (~P) ~ n In~ 2n+l)~ ~17)
cvx >
It should no~ be apparont to one o~ ordinary
skill in th~ art that consistQncy determinations for
other conduit goomQtrie~ ar~ easily solvable by employing
the equation o~ motion ~or a power-law type ~luid, in
laminar ~low, and at ~teady-~tate. Hence, the
consi~tency o~ a ~luid ~lowing through any non-rotating
conduit i3 detorminable by variation o~ the pxoper
coordlnat~ ~ystQm as appli~d to the above-identi~iQd
equations. A~ two such ~y~te~s hava been described, $t
i8 not beliaved to b2 necessary to elaborate further on
such variations for the purpos~ o~ thi~ application.
This invention also contemplates d~velopment o~
derivatlve in~ormation employing the calculated consis-
tency value~. For example, having the actual value of
consistency, the tons per day o~ paper stock 11 flowing
through spool 22 can al50 be calculated u ing th~ follow-
ing equation.
, ' `i' ' '
., ~ .
:~5~L5
21
S = 84. 60 <Vz~ R~C (18 )
where:
S Y tons/day
R = radius of pipe ( ft )
C = consistency (~6)
~Vz> = bulk velocity (ft/s)
once giverl the above disclosur~, various other
modlfications and improvement~ wl 11 beco~e appaxent to
ths skilled artisanO As ~uch, thay ar~ considered to be
10 part OI ~he invention, ~he scope of which is to be
deterTQined by the following claims:
.
