The use of graphical transfer-function methods for residence-time distribution studies

Shorter Commumcations Vol 32, pp 786-788

The use of graphwal

Pergamon Press

Pnntcd1x1 Great Bntan

transfer-function

methods for residence-time

dlstributlon

studies

(Received‘J June 1976, accepted 24 January 1977) One may analyse tracer-response data for the study of resldencetime dtstrrbution (RTD) of flmds wtthm process vessels III a number of ways These methods mclude the analysts of moments , curve-fitting m the time or frequency domam, and methods usmg Laplace transforms Of the last kmd are graphlcal transfer functions methods mtroduced by Mtchelsen and Ostergaard [ l-31 to study the axmlly dispersed plug-flow (ADPF) model Besldes the general advantages of transform methods, which Include the mmmusmg of tadlng effects through the wetghtmg factor, exp (- st), m the Laplace transform and of noise by the mtegraUon procedure, the use of a graphtcal techmque enables the vahdlty of the model to be tested through a condltlon that the plot must be linear We extend this procedure to analyse other more complex models wluch may be appropnate m some cases

T,

(al

--E--6‘

Pe

ADPF

+-L&=(b) Tanks I” sene~ 8 plug-flow

PREVIOUS WORK The non-normahsed

transfer function 9(s)

s(s)

LSdefined by

= KG(s)/~,.(s)

(1)

where K IS an arbitrary coefficlent and G,(s) and E.(s) are the Laplace transforms of the outlet and mlet concentration curves respectively The coefficlent K is Introduced because, m many practical sltuatlons, only the relative change m the tracer concentration and not Its absolute value can be measured The followmg character&lc functions were Introduced by Mlchelsen and Ibstergaard U,(s) = In 9(s) U,(s) = - U;(s) G(s)

(e)

= - U:(s) = F(s)/%(s)

Fig

(3) - {g’(s)/9(s#

(4)

where the primes denote dlfferentlatlon with respect to s If the U-functions are to be evaluated from experlmental data, numerlcal dfierentiatlon can be avolded by the use of “we&ted moments”[2], which mvolve numerlcal mtegration of the data From the propeties of the transfer functions It can be shown that the U-function of a system of closed vessels m series are the sum of the mdlvldual U-functions For example, the addition of a plug-flow element (S(s) = exp { - Tps)) to a system m series wdl contrlbute a further term ( - Tps) to U.,, a term T, to U,, but wdl not affect U, Mlchelsen and 0stergaard[3] show that for the ADPF model (FE la)

ANAL.YSIS

We now derive the correspondmg sable models of flmd motion

expresslons

1 Tanks-an-senes with plug flow (TSPF) (Fig The transfer-function for this model 1s s(s)

=

K ew (- Zs) (1 + T&n)”

for other pos-

motion

from which UO= -nIn(l+T,s/n)-T,s+lnK

(7)

U,=--+T, n/T, + s

(8) (9)

u2 = (n,Tr+ s)2

Smce this 1s a three parameter model, at least two equations must be plotted to find T,, T, and n It can easily be shown that

&J*)

_-V(n) T,+

_.A_ d(n)

= V(n) (I+ T,ITz) + VA’(n))s

(10) (11)

wluch serve as the equations to be tested 2 hmrnrshmg backmuc (DBM) model The previous model assumes that the flow changes suddenly from a backmix regrme to a plug-flow regme It IS desuable to have an alternative model whereby the degree of backmlxmg gradually decreases We first consider a series of n tanks, the size of wluch decreases m a geometnc series with ratlo k The holdmg times of the tanks are given by T,, kT,, k*T k”-‘To The transfer- and U-functions of this system are

lb) 9(s)

(6)

system

1 Mlxmg models of fled

& Thus, d UXm21s a plotted agamst s and the data resolve on to a straight lme, then the model may be assumed to be vahd The intercept of the lme yields l/? and the slope 4/rPe, from which the RTD parameters may be estlm_ated Some care must be exercised m choosmg the range of st for thts procedure If the_ errors m the response curves are randomly dlstrrbuted, then st should he m the vlcmrty of 1

cells

(f) Second-order

(2) = - F(s)lz%Q)

Backflow

n--l = K II ~

1

1-O 1f k’T,s

(12)

n-1 U, = In K - x In (1 + k’T,s)

(13)

1--o

787

Shorter Commumcatlons (14)

u1=v/(1

(25)

+ T,‘s’)

nT,‘(l + 2b + T,s) u2 = (I + 2(1 +Zb)T,s + T~zs2)3’2

(15) In this model the degree of backmlxmg changes by discrete steps as one progresses along the system However, a contmuously varymg degree of backnuxmg can be consldered by defimng a model with the followmg characteristic equations, by analogy to eqns (13)-(15)

+ 2(1+ ;;,s

(26)

After rearrangement, eqns (25) and (26) give (27)

(28) ln(l+k’T,s)dz+InK

(16)

(17)

T,(k” - 1) =slnk(l+T,s)(l+T,k”s)

+‘In s21nk

(w)

(18)

For this model the parameters T,, k and n have lost their orlgmal meanmg, but are still measures of the ongmal degree of backnuxmg, the rate of decrease of backnuxmg and the total vessel size respectively The holding time 7 1s given by 7 = U,(O) = T,(l - k”)/( - In k)

(19)

wbch can be plotted manner

for

parameter

findmg m the foregoing

4 Second-order model (Fig If) This model, conslstmg of two stured tanks m series (with holdmg times T, and Tz respectively), IS frequently mentioned m the literature, especially m the field of process control (5) It can be shown that 9(s)

=

K (T,s + 1) (T~s + 1)

(29)

U,=ln(T,s+l)+ln(Tzs+l)+lnK

(30)

T2

U,-T-+-

l+T,s

(31)

l+Tzs

T1” U’=(l++f(l+T&Z

T,’

(32)

and the varmnce of the response curve by For parameter fiudmg, eqns (31) and (32) are rearranged to gve u* = U2(0)/?

= -1n k(1 + k”)/Z(l

- k”)

WV

For parameter findmg, two cases wdl be consldered m the first case, the end of the vessel IS reached while there IS stdl some backnuxmg, m the second case, backmlxmg decreases to zero with the last part of the vessel occupied by plug flow (I) No plug-flow zone (Fig lc) From eqns (17) and (18) [(s + l/T,)(U,

-s&)1-’

= -Ink(1 _ k”)

[Wls

(21)

The value of T, IS adjusted untd equation 21 gwes the best strmght lme, from wluch k and n can be estimated from the slope and Intercept of the p1ot (11)Decreasrng backmlx zone followed by plug-jlow region (Fig Id) The correspondmg expression for tlus sltuatlon 1s found from eqn (21) by replacmg U, by U, - T, and settmg n to mfmlty There results the relatlonshlp 1

(U,-Tp-su2)

=(-lnk)s-Ink/T,

(22)

In this case, the value of T,, LSadjusted to give the best straight hne fit, k and T, ate found from the slope and mtercept of the lme so found

___- 2 u,z- u,

s*

T, f T, =?iV+TlTz

1

which can be plotted as a strmght line If a plug flow element T, 1s present, UI must be replaced by (U, - T,), and T, can be adjusted to give the best straight hne m the manner previously described No doubt other models could be mvestlgated with this approach, but the models considered m this commumcatlon were chosen because they appeared relevant to au motion m slender vessels, which 1s of particular concern to us The apphcatlon of these models to descnbe the mow mslde a particular tall-form spray chamber will be treated m a subsequent paper Acknowledgements-This work was undertaken as part of a research contract from the New Zealand Dairy Research Institute One of us (Q T P ) acknowledges the award of a Colombo Plan Scholarshp to enable him to carry out tbs work Department of Chemical Unrverslty of Canterbury New Zealand

Engmeenng

Q T PHAM R B KEEY

NOTATION

3 Backflow-ceil model (Fig le) ns model has been used to represent the behavlour of systems wherem mlxmg occurs as a result of eddymg, as m packed beds[4] The transfer function, negiectmg recychng effects at the mlet, IS aven by s(s)=

.(&1+2b

+ T,s -d/(1

+2T,s(l+2b)+

T:s’))r

(23)

from which U.=F;;(;nl;(l+2b+T,s-~(1+2T,s(1+2b)+T,Zs2)) (24)

recycle ratlo m backflow cell model concentration m arbitrary umts unnormahsed transfer function, in arbitrary umts parameter m dlmmlshmg backnux model proportionahty constant of transfer function number of tanks m multi-tank models, or parameter dlmlmshmg backmlx model Peclet number Laplace transform parameter holdmg time of n stirred tanks holdmg time of a plug-flow component parameter m dlmlmshmg backmlx model function defined by eqn (2)

m

Shorter

788 U, U2

fun&on function

defined defined

Commumcatlons [2] Mlchelsen M L and Ostergaard K , Chem Engng Scl 1970 25 583 [3] Ostergaard K and Mchelsen M L, Can J Chem Engng 1969 47 107 [4] Semfeld J H and Lapldus L , Mathematrcal Methods m Chemtcal Engmeenng, Vol 3 Prentice-Hall. Englewood Cbffs, New York 1974 [S] Perry R H and Chllton C H (Eds ), Chemwzal Engineers’ Handbook, 5 Edn, pp 22-12 McGraw-H& New York 1973

by eqn (3) by eqn (4)

Subscripts 1 at vessel mlet * at vessel outlet REFERENCES [l]

Mlchelsen 37

Chemrcol

Engmeenng

M L

Scmce

and 0stergaard

1977 Vol

K

32 PP 78X-789

Chem

Engng J 1970 1

Perwmon

Press

hted

I” Great

On generalized (Received

Bntam

dispersion 13 October

Marietta and Swan[l] m analyzmg the unsteady convective ddTuslon process which occurs m an electrostatic precipitator, mdlcate that the methodoIogy developed by Gdl and Sankarasubramaman[2-71 IS mapphcabIe for the solution of their system equations This conclusion IS the result of a mlsmterpretatlon of our approach, and one of the purposes of this note 1s to clarify when and how our procedure should be employed Also, the general form of the superposltlon pnnclple applicable to Inlet dlstrlbution problems ~111 be denved here m a simpler manner than in earher works [4,8] Generahzed dlsperslon theory, as developed m Refs [2-7] refers to a systematic solution procedure which 1s apphcable to transient convective dtiuslon problems m bounded flows In the mam flow duectlon, the system 1s assumed to be unbounded, and the coordmate measured m this dlrectlon will be referred to as the “axial coordmate” m subsequent discussIons In applying the theory, one has to dlstmgmsh clearly between two basic classes of problems (a) Inrtral dwtnbutron problems This class refers to problems m which solute mtroductlon occurs only at time zero At this mstant m time, a fhnte amount of solute is introduced in the form of a prescribed uutlal dlstrlbutlon m the spatial coordmates, and 1s subsequently subJected to convective dlffuslon No further mtroduction of solute mto the system may take place However, solute removal at the boundary IS permlsslble as dlustrated m[6] It 1s a characteristic of this class of problems that no boundary condltlons are prescribed at any value of the axial coordinate except the requuement that the concentration dlstnbutlon fall off to zero as one approaches axial stations at infinity The steady state solution m these problems IS the trivial one of zero concentration everywhere Our series solution[3,6] which actually corresponds to a formal power series in Fourier space[9], can be applied directly to this class of problems (b) Inlet drstnbutron problems This class refers to problems which are characterized by the contmuous mtroductlon of solute at the system Inlet m a prescribed manner starting from time zero This IS usually represented mathematically by prescrlbmg the local concentration at the mlet as a function of time and transverse posltlon Such a boundary condrtron on the axral coordmate cannot be satisfied by the senes solution used for problems m class (a) above However, for sufficiently large values of the Peclet number, the contribution of axial diffusion m such systems 1s relatively unimportant and may be dropped This permits the reformulation of these problems to formally equivalent ones m which the solute mtroductlon at the system inlet 1s represented by an mhomogenelty (a contmuous source term) in the governing part& dlfferentlal equation as shown by Sankarasubramaman and Gill [5] The resultmg equations may be solved by usmg the superposltlon integral given by Courant and Hllbert [lo] Physically, the superposition prmclple reduces the

theory

1976)

contmuous source problem to a series of uutlal dlstibutlon problems which may be solved by usmg the series presented m[3] For actual detads, the reader may consult the articles referred to above Here, we shall bnefly sketch the outline of the procedure m broad terms m a more general form than has been done so far Turning our attention to Ref [l], the problem under consideration 1s the unsteady convective dlffuslon of particulate matter in turbulent flow between two wide parallel plates in the presence of an electric field between the plates Initially, the concentration C IS zero everywhere m the system At time zero, a step change m the inlet concentration 1s uutlated and mamtamed for all future tune Marietta and Swan give the proper form of the convective dlffuslon equation satisfied by C(t, x, y, z) where t IS time, x IS the axial coordmate and y and z represent the transverse coordmates For our purposes, we shall note that this problem IS a special case of the general class represented by

..%I’+ u(t, y. z) C(O, 4

at, Here

8 1s a hear

operator

ST=D(y,

Y,

z) $

(24

2) = 0

0, Y.2) =

ccxt,Y, 2)

(2b)

of the form

and ZL may contain derivatives with respect to y and/or I and coefficient functions which may depend on t, y and z Speclfically, 2, contains no derlvatlves m x and no coefficients which depend on x u(t, y, z) IS the axial velocity and D(y, z) 1s the dlffuslvlty A complete statement of the problem will involve writing one more boundary condltlon on x and boundary conditions m the y and I coordmates For our purposes, these will not be necessary We shall, however, assume that these conditions are homogeneous m what follows It 1s clear from eqns (1) and (2) that we have posed an inlet dlstnbutlon problem Therefore, the senes solution of [3] cannot be applied directly for Its sol&on The use of generahzed dispersion theory in this problem has to be preceded by the apphcatlon of the superposition principle In order to do this, the right hand side of eqn (l), which represents axial dtiuslve transport, should be dropped to gve ~c+u(t,r,z)~~=o

From

a practical

point

of view,

(4) the relative

contmbutron

of the