Dynamic Analysis and Modal Control of Plate Heat Exchange Systems

DYNAMIC ANALYSIS AND MODAL CONTROL OF PLA TE HEAT EXCHANGE SYSTEMS A. Ito and M. Masubuchi Department of Mechanical Engineering, Osaka University, Osaka, Japan

Abstract. Basic governing differential equations are derived by using energy and mass balances for various types of flow passages in plate heat exchange systems. The method of weighted residuals is applied to get approximate systems, and a modal control system is formed. Experimental dynamics and results of control are also given for some typical flow types, and good agreement with theoretical responses is shown. Keywords. Dynamic response; modal control; control theory; distributed parameter systems; MWR approximation; plate heat exchanger. INTRODUCTION

1972) .

Dynamic analysis of heat exchangers has been investigated for a long time and their conventional and optimal control problems have been discussed in the literatures(Kanoh,1974; Koppel, 1968; Masubuchi, 1969; Takahashi, 1952). However, for plate heat exchangers, only the researches on heat transfer characteristics (Buonopane, 1963; Konno, 1967; McKillop, 1960; Watson, 1960), on heat transfer coefficient (Buonopane, 1963; Okada,1968) and on steady state temperature profiles ( Okada, 1970) have been published and there have been few on dynamic analysis.

In this paper, dynamic analysis of plate heat exchange systems has been shown and modal control concept has been applied to the ~MR approximate systems of these heat exchangers. Our system does not require effective diffusion (Gordon-Clark, 1966) and superfluous boundary conditions (Murray-Lasso, 1965). Experimental heat exchangers are constructed by using small size, commercial type heat transfer plates and a water to hot water heat exchange system is considered. DYNAMIC ANALYSIS OF PLATE HEAT EXCHANGE SYSTEMS

Plate heat exchange systems can be classified into flow patterns: series flow pattern, parallel flow pattern, and complex flow pattern. Basic governing equations with boundary conditions for any flow type are generally derived based on heat balance and the dynamics is theoretically obtained as frequency responses for the change of either the inlet fluid temperature or the flow rate (Ito,1977; Masubuchi, 1977). From the results of dynamic analysis it can be shown that the flow pattern of plate heat exchange systems will be classified into three categories - pure parallel flow type, pure counterflow type and transit flow type.

Flow Patterns In the plate heat exchangers, there are various flow types according to the combination and the number of heat transfer plates that have holes or blanks suitably placed at each corner. From these flow types a number of plate heat exchange systems are constructed, but these systems can be classified into three patterns depending on the condition whether fluids split or not. They are series, parallel, and complex flow patterns as shown in Fig. 1.

~lthough the frequency response analyses are :omplicated and troublesome, the method of Neighted residuals (MWR)(Finlayson, 1972; ;ordon-Clark, 1966; Parkins, 1971; Prabhu, 1972) is applied using trajectory approximations which satisfy boundary conditions and sood approximate responses are given for these heat exchange systems.

Derivation of Basic Equations The assumptions used and justified by most of the researches (Buonopane, 1963; McKillop, 1960; Takahashi, 1952) are as follows: (1) The heat loss to the surroundings is negligible. (2) There is no heat conduction in the direction of fluid flow in the plates and the fluids themselves. (3) The temperatures and flow rates are uniform across the flow passage. (4) The film coefficient of heat trans-

10dal control has been investigated and dis~ussed on both lumped and distributed systems In several papers (Murray-Lasso, 1965; Porter, 34 3

A. Ito a nd M. Ma subu chi

344 (a) Stria Fklw Pattem

(b) PanUel Flow "ltern

5.,

S.b

S.d

Po'

le l Co mplu Flo w p.. llnn

c.

P,b

Cb

('..:

Cd

1i

lti,.t •t

6

. J

~

•••J ~'

t m+ I Hh platl'

lilt.J

~~

lillr'lmt

.J

' , ~' ~ ! J

Heat transfer plate.

moth plai t'

.ltH liffi ~,

stlm tpp ' t@t tpu'

Fig. 2(a).

r,

•.J 1 'lm!r~ ~r iw~ 1pv.! lWl ttm

Wili1 imill lillqJ , .. .. ~~ 1fT@

7~

,

~

~_J

~

.. ~W.I Wrrn' I

L.'

•

d m· th passqe

Fig. 2(b). Fig. 1.

Flow patterns of plate heat exchange systems.

fer is greatly dependent on the fluid velocity and is proportional tO,an exponential function of fluid velocity. The other physical properties of the materials are constant. (5) Except in the case of series flow pattern, the fluids will split equally between passages. Figure 2(a) shows an example of heat transfer plates which is used here. Under the above assumptions, energy balance in the elements shown in Fig.2(b) yields (1) and (2). For m-th fluid temperature:

o~",+ "e", -M - V",?)["

rirnW(e e) ti"'W(e PIM\'VI,. ~",- '" + W",

e) In

(1)

For m-th plate temperature:

3 er'"

= CX,.-IWce

at

-8

)+O("'W($ -~ )(2)

rw~", "'-I P'" Wprn III (>lit "+" in the double sign means the flow direction in Fig.2(b), while "_" does the other direction.

Boundary conditions can be expressed as linear combinations of fluid temperatures and can be described as the following equations using vectors 8, 8 ao , 8Bl , and matrices D,E with suitable sizes. (3)

It is assumed that the hot-side and the coldside fluid flow through odd- and even-numbered passages and split equally into M and N streams, respectively. To generalize these expressions, the nondimensional terms l=t/Th (Tn=V/Fh), ~= ~ /L should be substituted. Equations (1) and (2) will be as follows. For fluid temperature in odd-numbered passage: ",:!: ~h = IX .. ~ C6 ~-c F~ o~ ~YhFh/M l'1li (4)

Mae

-ell»

ce..

+ c~rh ~~ /M WplI+"\-e,.)

For fluid temperature in even-numbered passage: NI"3e.. + "&'06",_ d.",_A -$ ) I"

bT - F~ ~

-

y.t::F./N

(e p,.

,.

+ C,~;. ;'/N (eplM1-ello)

(5)

For plate temperature:

~!r

"

., fI.~!! P.. (B",-,-e~ .. )+ IV,rI,,,,!,v, (q",-Q,,,,,)

(6)

Elements used to derive the energy equation.

For boundary conditions:

De(O,T)= eBo(T),

Ee(1,T)= eBtI-r)

(7)

Dynamic Analysis The basic equations (4),(5), and (6) can be generalized as

C ~~ + V ~~ + He = 0

(8)

where C, V, and H are coefficient matrices. Now a dynamic analysis approach is shown to give frequency responses for the change of inlet fluid temperatures and flow rates. Small deviations from the steady state values are considered to linearize the equations. The variables may be considered to have time dependent and time independent terms as follows.

e(~. T) = e (~)+Lle(~,T). $60{<:)

=6Bo +A(780{T),

eBt(1:)=eBI+LlftB~(T),

rn:)=F +L1FlL\ V(T)=V+6VCt),

(9)

H(T) =H+.1 rj('Tl

By substituting the expressions (9) into (7) and (8), neglecting the second-order terms, and subtracting the equations describing the steady state behavior, the basic linear equations describing the dynamic behavior are obtained. Laplace transforming these equations with respect to 1 and subtracting all plate temperatures yield

~L1e(~,S) = y(~),1e(~S)+Z(e(mL1f(5) DtlecO,S")=.16so(S),

E:-,1eC1,S)=,dGB\(S)

(10)

(11)

vlhere l\f=col(lIFh(,)/F'h, lIFdl)/Fc ), lI8(~,s) is a vector whose components are deviations from steady state fluid temperatures, Y is a square matrix with a size corresponding to the number of passages, Z is a matrix whose components are functions of a steady state,

345

Dynamic Analysis and Modal Control and D, E are matrices representing boundary conditions and have different sizes from those in (7). Putting s=jw and solving (la) with (11) yield frequency responses of fluid temperatures at any position along the flow, which leads in general to a two-point boundary-value problem. The boundary conditions (11) are reduced to the following equation.

Pu 1$(O,S) +p~t1e<1,S)=Lleit5)

(12)

where 68~ is a vector whose components are inlet fluid temperatures and PI, Plo are square matrices whose components are determined by the following rules. (1) First, a boundary condition with an inlet fluid temperature is arranged as a diagonal component which has a positive sign. (2) When a row of one matrix is determined by a boundary condition, the corresponding row of the other matrix is put as a zero vector.

t

-2 ~J=af(ef'!-e,)+ al(~-e,)

r*++2~t=~1(~-e4)+al(~-~) (16)

For plate temperatures:

~I =

bf(e,-~)

~= b,(ef-er,) +b,.($,.-6f'!)

W'= b1ce,.-e/'l) + Dl

($1-8",)

~= Dd e~-~r-) + l?).cB..-ept)

~~= D,.cBt-°rt ) Double sign means upper for S4a, lower for S4d. And al=CihA/C"kYhF\, a2=a,A/C rc.y,Fc., bl=CihA/w,Vh, bl=a
Splitting 68 into two variables x, y satisfying the two relations,

a\-X(~,jW)= yx +Z..1f, ~~(f,ciW)= Y'd.

X(O,jW)=O

R

(13)

(17)

R

~ (O,jw)+ R~(/,iW)=.1ei.
+ {-P(~( Pl -I (I (~ . (15) + R~(()) R).p(f-rIZ«()at; + )o~(~-()Z(~',)d( ~ L1{cjw)

where ~(s) is a fundamental solution matrix of (14). The first and the second term in the righthand side of (15) represent frequency responses for inlet fluid temperature and flow rate change, respectively.

B,(-i,1) = /1,.(T) e4(-f.l) = e,.n:) el( /:() -

81 ( (, -n =

\

(S4d)

fi)(/,"C)

~+C 1,1)

Let {~4(n)} be the orthonormal Legendre polynomials and approximate trajectories of fluid and wall temperatures of the following forms be assumed (n is the order of approximation):

e,("l)~

e.(,,-tl - l1+,) 1>(1)XJCl)+O,.[T) '" j( f+, 1~(?) ~(l) + ell fT)

(S4a)

el.t"t)~ e,(1,-q =

MODAL CONTROL OF PALTE HEAT EXCHANGERS

(1-1) !f(1)lJ:r) +11'(0 :4(T)+8d(-n (S4d)

~t".n~ ~Cp:)= (1-~)

MWR Approximate Systems of Plate Heat Exchangers

$'l-li,l)~

MWR approximate systems are derived by using trajectory approximations which satisfy boundary conditions and by requiring that the residuals of basic equations be orthogonal to the spatially dependent coordinate functions (Finlayson, 1972). From physical considerations such as steady state temperature profiles of heat exchangers orthonormal Legendre polynomials defined on [-1,1) are used as a complete set of coordinate functions. Introducing a new variable n=2s-1 for the convenience of using the above coordinate functions, S4a and S4d types will be considered as examples of plate heat exchangers. For fluid temperatures:

~~' +2 ~' = o.l(~ -e,)+a1(~-~) r~~2~ =o.l(~-el)+ tA,.(e,,-&1)

-

e+(t,)=

fr,n ("t)+2. fC()l,(tl+&",(r) 1

j fc,l, ~-n+2f{flllt)+$dtt) ((-1)

(S4a)

(f+1)t(1)l~+ec(T)

(S4d)

eM
k= f,'-', ,.

(18)

where

1r?)=CO{ (cf~'"

f'1H)

(Pt (1) : i-tn

cf.(?)=J2~f

L~ndrtt.

P.(1)

pofynomit/ ()

Xitr)=~((Aci "-XlHi) <-1,"',4, XtF(Tj-Col (~""-4
i=AX-I-C,e..+c.,.e",+4,e4 +d,.eol

(19)

346

A. Ito and M. Masubuchi (20)

where, A, ~(n), cl' c2' d l , d 2 , e~,and e 4 are matrices and vectors. Now defining new variables yeT) by ~(L) = X(L) -Cl em - d t 4 to avoid the differentiation of the inputs such as the inlet fluid temperatures, (19) and (20) yield the following expressions.

Thus, the modal control law m is given as follows. 1\

tn=k'v:n

d

(25)

where

~' 1

e

Ac, +C2., cl = Ad, +011-

v-~ Now, controlled systems with reduced-order control law are derived eliminating some selected time-dependent state variables by using linear equations related to fast modes (Marshall, 1966; Wilson, 1973). Using (25) yields the following reduced-order control law.

CJ-ifd,+eJ.

(26)

~ =A O+b~1I'I +deq e(l,"t)=(fo+c.",&m+c...ed where

D=

C.",-iIcl+e""

(21)

(22)

To obtain frequency responses for e~, Laplacetransforming (21) with 8 4 =0, solving it with respect to Y4'S, putting s=jwand using (22) yield approximate frequency responses of fluid temperatures at any position along the flow. Numerical results corresponding to the order of approximation n show that the MWR approximate systems can show good agreement with the exact systems corresponding to an increase n from the low frequency range in the frequency characteristics, and besides that the approximate systems can also show excellent agreement to the fairly high frequency range with only the first few terms of the coordinate functions. Therefore, the MWR approximate systems presented in this paper can give good approximations to plate heat exchangers. Modal Control The dynamics of the final control element producing 8rn is assumed ·to have the following expression. Therefore, the controlled system involving the control element is given. ;..

"'''''

where

A

'a+bm+c\6J. ~

(T)

= CAJ (

A=

A

a= Z dR +~m

~~ = (,C[ ( 0 ~/ 0 ~,.' 0 f; 0 y,/ () &", ) Z CA (l~ n+1 )x Cgn+l) mr.triX ().

Z

q n+l

Then the feedback gains are derived which may give a desired response by modal analysis, and a modal control system and controlled results will be shown using the gains in this section. Since in (26) consists of Yl-Y4, 8~, the time-dependent state variables should be calculated by measuring each fluid temperature at some positions along the flow and by using (22). Thus, the following equations are obtained:

YR

-I

(

~:Jt) =j[( l+'j\.) ~(1~)1~{ $"C7Jl,"t)- C4 d(?j).) ~cI('t)1 [(I-?j~) CP(?j~]

if r 8hi

0 b= -1("';-

cl= d 0

A

I

: Tv

1

I

f.

(t=l, ... , m)

(24)

j,pl

(S4d) (27)

rp'('lllJ {~(7)J,\) -Ch.!1i')£f,,('"()

G4-tT) =

-I

f r1j+)J {O~(~i+IT.)-c,4(1i+}&d(T) - 2. rt I) I.Jr)} -\ (\+1j+l f(1jotl J ( &+('jt,'t)- Ci~(1Jt )&..(1) I

l((-1pt) lJ

tI'\

(S4a)

-21"(1) tl<<:) \

Thus, the feedback gains k, (i=l"",m) that will modify the distinct eigenvalues A,(A) to some desired values are given as follows (Porter, 1972).

F{ TT (AP~<) 3,,1

-I

1\

Tv

p",'" (fj-A,)

{81.C1p.;t> -44( 1N ~tl{t) - 2 fUl t4-(T) \

)

11l"'C) = [(I-til)

A :b --O--;-r

vector

Modal Control System and Controlled Results

(23)

I I

1\

....

~,(1) = [(1+1jt)fr1j,)1{ e,(1j,;'r)-C.l"(7)1)&"'('t)~

Tv 6",+9",= kv m ~=A

where

(S4a) (S4d)

where np, n;2,'" are measuring positions of 8 1 , 8 2 "", respectively. c2"" C 44 ,.··are the components of the vectors c~ and Cd, respectively. The measuring points n;;.' s will have only to be chosen so that the inversion of the matrices in (27) exists. A modal control system is given in Fig.3.

where The output is

1\

ALAi=Ai()':

v, [\j~ 1 =

1

ct=1'" n+f) I

(

I

,

U=[U\'l

I = U- ) I f i = Vi' b 1\

(#- 0 )

110= where

qPS (SI h

+ GDS (0;) C1.1

(28)

Dynamic Analysis and Modal Control

o [el+?}!)

o

~

r'(1J1) Y\:qu (Jjl, s)

W+/iS.) r'cll>-) Tt {&r+A (1}1,)) - ~('j'.l ~ _ { [( I-~j>-) (n[( I+Jj4o) 1>'(J;,4o)] Y494 (1i4. 5)

-

9~ -

0 [(I-1i')

347

f(1jl) r 1 [ 61,,, (lj,,5)

- '2

cf'(1) U1+?jo f(1i1i

-

L9t/(7i"JJ

r'q>d (1)1,S»)

o JHt-'j4\1:~i+)1) qp./{Ji~~) -cs
l[ (1+1j+) Ch?i"') J { G,.U?i+,S) - C6A o

(1jf-)

I

o The controlled results will be examined in frequency characteristics. The frequency responses of Gp$ with and without modal control loop are given in Fig .4. It is shown that the dyn~mics of the controlled system can be improved. EXPERIMENTAL INVESTIGATION

capacity per unit length O.209[kJ/n"C). And water is used as both hot and cold fluids, cross-sectional area of passage O.OOOl19[J), passage volume O.0000546[m 3 ), heat capacity per unit length O.498[kJ/m"C). The flow rate F~ in the cold-side fluid is fixed at 1.6[1/ min) while Fh is fixed at or varied about 2.4 [l/min). The correlation between the film coefficient of heat transfer a and the fluid velocity V"

Experimental Apparatus The test conditions for a heat transfer plate are A=O.00258[m 2 ), W=O.0565[m), L(=A/W)=O.457 [m), cross-sectional area O.000058[m~), heat

!ll-10

.,..0 . 4. 1 1'-1.0

c: k'V..' 'Z l-It'V..'1

•

"-20 •

- 30

\

:~:~ ,-1.': !' Itv-, 0 T.,"4 . 48 b.-I.

I

-240

i - I,', 4

j.'

-300

10 C"cul.,. fr<1uon ey dl

Fig. 4. Fig. 3.

Hodal control system.

Frequency responses with and without modal control loop.

348

A. Ito and M. Masubuchi TABLE 1

is obtained using the plate heat exchanger with three transfer plates. a is assumed to be proportional to an exponential function of v and calculating experimental data and averaging yield the following expression a= 1146v 0.4"1 [kJ/m2 hroC]. The temperatures are measured with thermocouples of copperconstantan wires.

Pure parallel flow type pattern

Frequency Responses for the Fluid Temperature Input In Fig.5 frequency responses for the tempe·rature input are obtained by using sinusoidal temperature changes of inlet hot-side fluid, by measuring temperatures at some positions along the flow and by calculating amplitude ratios and phase angles. The sinusoidal chan.ge can be given by mixing hot and cold water using a feedback temperature signal at the inlet of the heat exchanger. Forcing sinusoidal signals at A are applied from a function genera·tor. FCl and FC2 are flow controllers used to fix flow rates at B equal to 2.4[1/min] and at C equal to 1.6[1/min], respectively. Frequency responses obtained through these experiments show good coincidence with theoretical curves from the steady state to the fairly high frequency ranges. Figure 6 shows temperature effectiveness vs. NTU for C4b and also shows both pure parallel flow type pattern and pure counter flow type pattern. Points 1 and 3 correspond to the counterflow-pattern region where the phase angle curve does not lag beyond -180 degree line. Points 2 and 4 correspond to the para"" llel flow pattern region where the phase angle curve lags beyond -180 degree line. There may be a separate line between these regions. Table 1 shows the classification obtained

Classification of the flow types from the dynamic characteristics Pure counterflow type pattern

Ser i es (loo pattern

S. • S. b S. c

S2d

Parallel (loo pattern

P ••

P.b

C'*P lu (l ... pattern

C. • CCc COb Ck Cif C" eSi C'j COk

Transit flow type pattern

eu

S .d c ept S2d )

C.d «b

cOe

C.b C,. C•• COl

from the dynamic characteristics discussed as above . And also the experimental frequency responses for the change of the flow rate show good coincidence with theoretical curves (Ito, 1977). Experimental Results of Hodal Control Experimental control system will be apparent from Fig.3. PI controller with~200[%], Tt = O.S[min] is used as the main controller. SW is a switch which cuts off the modal control loop. Outlet temperature responses of the cold-side fluid for the step temperature disturbance of the same fluid are shown in Fig.7 on the S4d type plate heat exchanger. u ~I

.

"

~

r-

riot woter--oOIO)-~--!--j~

20

Col d woter - Three woy val ve

Fig. 7. Outlet terr.perature responses of the cold-side fluid to the upset of the inlet temperature of the same fluid on the flow type S4d.

(-~,

-<_~~~

___

d i p ce ll

J

Contro l ... a lv e Hea t Insu lot lon (rubber ) f o ur- possoQe

CONCLUSIONS

plo le heat elchonoer

( S40 1

Fig. S. Schematic drawing of experimental apparatus. !.D' r - - - - - r - - - - - - - , - - - - - - - , - - - - , I .'

)1:0. "91 )I ~ 1. 0

L - - - -- - j · ,

,..,.,-z .n

)'

0. 6

O. ~

~;;:;----..." . ,

I

It has been shown that the MWR approximate systems with only the first few terms of the coordinate functions may give good approxi·· mations to plate heat exchangers with no axial diffusion, and that dynamic responses can be improved by applying modal control to the approximate systems. Numerical results have indicated that the bandwidth of the controlled systems can be widened to some high frequency range and experimental step responses have been improved.

- HI '

0 '

I I

ACKNOI-ILEDGEMENTS .0

Fig. 6. NTU vs. temperature effectiveness (C4b).

10 . 0

7he authors thank Associate Professor H.Kanoh for his invaluable suggestions and also wish

Dynamic Analysis and Modal Control to express their gratitude to graduate students Mr. K.Matsunami, Mr. M.Kumura, Mr. M. Terasaki and Mr. R.Narimatsu for their help in performing this work. REFERENCES Buonopane, R.A., R.A.Troupe, and J.C.Morgan (1963). Heat transfer design method for plate heat exchangers. Chem. Eng. Prog., 59, 57. Finlayson, B.A. (1972). The Method of Weighted Residuals and Variational Principles. Academic Press, New York. Gordon-Clark, M.R. (1966). Dynamic Models for Convective Systems. Ph.D. Thesis, MIT. Ito, A., and M. Masubuchi (1977). Experimental frequency analysis of plate heat exchange systems. Technology Reports of the Osaka University., lI, 261. Kanoh, H. (1974). Control of stirred tank heat exchanger and parallel-counter heat exchanger by manipulation of flow rate. Technology Reports of the Osaka University., 24, 257. Konno, H., and co-workers (1967). Fundamental studies on the heat transfer characteristics of a plate type heat exchanger. Kagaku Kogaku (in Japanese), 31, 872. Koppel, L.B., Y.-P. Shih, and D.R. Coughanowr (1968). Optimal feedback control of a class of distributed-parameter systems with space-independent control. Ind. & Eng. Chem. Fund., L, 286. Marshall, S.A. (1966). An approximate method for reducing the order of a linear system. Control, 10, 642. 11asubuchi , M. (1969). -Dynamic response of crossflow heat exchangers. Proc. 4th IFAC Congress (Warszawa), Sess. 47, 84.

349

Masubuchi, M., and A. Ito (1977). Dynamic analysis of a plate heat exchanger system. Bull. JSME, lQ, 434. McKillop, A.A., and W.L. Dunkley (1960). (Plate heat exchangers) Heat transfer. Ind. & Eng. Chem., 52, 740 Murray-Lasso, M.A. (1965). The Modal Analysis and Synthesis of Linear Distributed Control Systems. Sc. D. Thesis, MIT. Okada, K., and co-workers (1968). Studies on the heat transfer coefficient and pressure drop of several kinds of plate heat exchanger. Kagaku Kogaku (in Japanese), 32, 1127. Okadi; K., and co-workers (1970). Temperature distribution of fluid in plate type heat exchanger. Kagaku Kogaku (in Japanese), 34, 93. Parkins, E.S., and R.L. Zahradnik (1971). Computation of near-optimal control policies by trajectory approximation: Hyperbolic-distributed parameter systems with space-independent controls. AIChE ~, 17, 409. Porter, B., and R. Crossley (1972). Modal Control Theory and applications. Taylor & Francis, London. Prabhu, S.S., and I. McCausland (1972). Optimal control of linear diffusion process with quadratic error criteria. Automatica, 8, 299. Takahashi, Y. (1952). Transfer function analysis of heat exchangers. In A. Tustin (Ed.), Automatic and Manual Control, Butterworths, London. pp.235-248 Watson, E.L., A.A. McKillop, and R.L. Perry (1960). (Plate heat exchangers) Flow characteristics. Ind. & Eng. Chem., 52, 733. Wilson, R.G., D.E. Seborg, and D.G. Fisher (1973). Modal approach to control law reduction. JACC, 554.