Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978 Krantz, W. B., Martln, P. C., Weers, W. A., Anal. Chem., 9, 799a (1974). Lighthiii, M. J., Proc. Roy. SOC. London, Ser. A., 202, 359 (1950). Martin, P. C., Ph.D. Thesis, Universlty of Colorado, Boulder, Colo., 1978. Mass, R. O., Ph.D. Thesis, University of Illinois, Urbana, Ill., 1967. Merson, R. L., Ph.D. Thesis, University of Illinois, Urbana, Ill., 1964. Merson, R. L., Quinn, J. A., AIChE J., 10, 804 (1964). Mockros, L. R., Krone, R. B., Science, 181, 361 (1968).
353
Sakata, E. K., Berg, J. C., Ind. Eng. Chem. Fundam., 8 , 570 (1969). Springer, T. G., Pigford, R. L., Ind. Eng. Chem. Fundam., 9, 458 (1970). Sutherland, K. L., Aust. J . Sci. Res., Ser. A , , 5 , 663 (1952).
Received for review July 20, 1977 Accepted June 7, 1978
Dynamics of a Shell-and-Tube Heat Exchanger with Finite Tube-Wall Heat Capacity and Finite Shell-Side Resistance K. S. Tan and I. H. Spinner” Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario, M5S 1A4 Canada
Analytic time-domain solutions are obtained for the dynamics of a shell-and-tube heat exchanger with finite tube-wall heat capacity and finite shell-side heat transfer resistance. The transient response for step velocity changes is derived for velocity dependent tube-side heat transfer coefficient without the use of perturbation approximations. Solutions are also obtained for the response to disturbances of shell-side and tube-side fluid temperatures. The equations are in terms of tabulated J functions which facilitate the examination of the effects of the parameters and the determination of the range of applicability of the previous “no-wall” solutions. The transient response characteristics are found to be significantly affected by both the ratio of tube-wall and tube-fluid heat capacities and the fractional heat transfer resistance on the shell side. The response for a limiting “no-wall” model including a velocity dependent heat transfer coefficient shows a significant deviation from the exact solution especially for an increase in velocity.
Introduction The dynamics of shell-and-tube heat exchangers have been examined previously by means of time-domain solutions or by means of transfer function analysis. Cohen and Johnson (1961) examined the frequency response for shell-fluid temperature forcing for a model with finite tube-wall heat capacity. For the same model, analytic time-domain solutions in the form of single and double integrals were obtained for the response to step changes in tube-fluid and shell-fluid temperatures, respectively. They also examined the frequency response to velocity forcing with a model in which the velocity and velocity dependent tube-side heat transfer coefficient was “linearized” by a perturbation method. Yang (1964) used the same linearized model to study the frequency response to velocity forcing. He also obtained time-domain solutions in terms of several definite integrals for step, linear, and exponential velocity forcing for this perturbation model. Although the above authors included the tube-wall capacity in their models, these models are only applicable to small perturbations in velocity. For the response to general velocity changes, methods other than Laplace transformation have been used. Koppel (1962) used a model with constant wall temperature, an arbitrary velocity forcing, and a “linearized” velocity dependent overall heat transfer coefficient. Using the method of characteristics, implicit integral solutions in the time domain were obtained for the tube-fluid temperature response. An explicit analytic solution was presented for a step change in velocity. Ray (1966) presented a mode1 for tube-side dynamics which utilized a forcing wall temperature as an arbitrary function of time, an arbitrary velocity forcing, and a velocity dependent tube-side heat transfer coeffi-
cient. Implicit integral expressions were obtained which provided a framework for numerical integration when the arbitrary time functions were specified. An explicit analytic solution was given for a step change in velocity and constant wall temperature. In this work, a model with finite tube-wall heat capacity, finite shell-side heat transfer resistance, and a velocity dependent tube-side heat transfer coefficient will be used. Explicit time-domain solutions will be presented for step changes in tube-fluid temperature and shell-side fluid temperature. Time domain solutions will also be presented for step changes in tube-fluid velocity for a model without any “linearization”. The derived equations are compact because of the use of normalized parameters and the use of known mathematical functions. For these reasons, the solutions are easy to use for examining the parametric effects in the study of the dynamics of shell-and-tube heat exchangers. The significance and physical interpretation of the derived equations will be examined and discussed. Appropriate numerical values of the important parameters will then be used to illustrate parametric effects on the system. Some conclusions will be drawn with respect to the range of validity of the no-wall capacity solution as previously used in the study of the dynamics of shelland-tube heat exchangers. Model Equations Constant physical properties of the tube fluid with plug flow are assumed. The shell-side fluid temperature is assumed t o be uniform. The dependence of physical properties of the tube wall on temperature is assumed to be negligible and longitudinal conduction is neglected. The heat transfer coefficient on the shell side is assumed to be constant but the variation of tube-side heat transfer
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coefficient with velocity is taken into account. The wall resistance to heat transfer is assumed to be negligible so that the overall heat transfer coefficient is solely determined from the resistances of the tube side and the shell side. With the above assumptions the model equations used in this study are
aT aT C ~ ~ U A+- C,pA- = h(u)A(T, - T ) az at aT, M C - = hgAg(T,- T,) - h(u)A(T, - )'5 pw
at
020
-
(1) (2)
Letting (y
=
h(u)Az
8=--
-'
h(u)At
MWCPV c=-*
C,pAu ' C,PA ' CpPA ' l/(hgAg) -Uo(U)Ao UO(U)AO - . 1-f=h(u)A = l/(UOAO) h A ' then eq 1 and 2 become aT -aT + - = T,- T (3) aa a8
aT, 1 (Tg- T,) + ( T - T,) (4) a8 Cf Equations 3 and 4 can be rewritten in terms of deviations from an initial steady state by letting P[8,a] = T [ ~ , c Y - ]T*(a*); Pg(0)= Tg(B)- Tg*; T,[~,cu]= T,[~,cY] - T,*(cu*); Po = T[8,O] - To* -=
I
+
(Vn - l)f*]. It is noted that Uo(u)= Uo*Vnonly i f f * = (Vn- l)f* a. The corresponding equations for the response of tube-wall temperature can also be obtained but will not be discussed here. where The analytic forms of eq 7,8, and 9 are not difficult to f*vn evaluate since only exponential functions and two other * F1 = V / V " - 1; a = 1/ccn; f = mathematical functions are required. The J function is (v" - l ) f * > available in tabulated form (Sherwood et al., 1975; F2 = (1/V" - 1 ) / C Helfferich, 1962) or graphs (Hougen and Watson, 1947). The values of the J, function can be obtained by series R3 = expansion and tabulated or presented in graphical form y2{[1+ a - (1- f * ) V / V " ]+ {[I+ a - (1 - f*)V/Vn124a[(l - f ) - (1 - f * ) V / V n ] } 1 / 2 } as shown in Figure 1. It was found that over a wide range of C, f , and a that dropping terms containing the jb function in eq 8 and 9 resulted in response temperatures R4 = y2{[1+ a - (1 - f * ) V / V n ]- ([l+ a which were usually less than 1% lower than the values (1 - f * ) V / V " ] 2- 4a[(l - f ) - (1 - f * ) V / V " ] ) 1 /=2 ) obtained from the exact solutions. Neglecting terms aC(1 - f , - (1 - f * ) V / V " ] / R , containing the J, function is equivalent to neglecting, in R:3and R4 are real and R3 > 0 for all values of n, V ,f * , and part, the smaller "effective" time constant ( l / R l or l / R J C. In particular, for Newtonian fluids and regions of of the interacting wall and tube fluid system in the time practical interest 0 < f < 1, 0 5 n I 1, and C finite: R3 domain of 8 > a only. > a > R4 > 0 for V < 1 and R3 > a > 0 > R4for V > 1. Limiting Analytic Solutions In eq 9, F,(W and F 2 ( V ) are the velocity forcing The exact solutions of eq 7, 8, and 9 reduce to the functions. The function F 2 ( V ) is due solely to the delimiting ("no-wall") solutions when C approaches zero with pendence of h on velocity and is the direct velocity forcing f finite function for the tube wall (eq 6). When n = 0, F 2 ( V ) = 0 and the only effect of velocity on the wall is that due to P - = H[8-a] exp[-(1 - f ) a ] the coupling to the tube fluid and its change in residence (7L) .iih time. The forcing function F,(V) defines the direct effect of velocity changes on the tube-side dynamics (eq 5 ) and T -- 1 - exp[-(1 - fib'] - H [ 8 - a] exp[-(1 - f ) a ]+ _ arises from both changes in h and changes in the fluid residence time. When n = 1, Fl(V)= 0 and the direct H[B-a] exp[-(1 - f ) B ] (8L) effect of velocity on the tube-side dynamics disappears. Koppel (1962) also noted this condition for his no-wall F = 1 - exp[-R40] H[8 - a ] exp[-R48] model. Thus for n = 1the only effect of changes in velocity Tg*- T* on the tube-fluid dynamics is that due to the coupling to H[B - a ] exp[-R,a] (9L) the wall transients, which in turn are affected only by the change of fractional resistance on the shell side. Limiting solutions of identical form are obtained from The final steady state, obtained from equation 9 is eq 7,8, and 9 with C finite when f approaches zero. In this Pa case, the wall attains the shell-side temperature very = 1 - exp[-(1 - f , a + (1 - f*)a*] (9s) rapidly since an infinite heat transfer rate is approached. T,* - T* Discussion of Parametric Effects Solutions are presented here only for step changes in tube-side inlet temperature and the shell-side fluid Numerical results are calculated from eq 7 , 8, and 9 to temperature (eq 7 and 8). However, time-domain solutions show the quantitative effects of values of C, f , n, and V for general time-dependent inputs are readily obtainable on the transient response to the several disturbances. For with the use of linear superposition and the Duhamel compactness in presenting the figures, the tube-side formula. temperature is normalized in terms of fractional attain+
c!
+
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0.6-
f 04 L
/’
i”
// / /
0.4
-
/Y,
n ~0.0 = 1.0
c
f ’0.2
’I
OZC
////A
a-.
1.0
0 2 .-
-
I
e/a
Figure 3. Response to step change in shell-side fluid temperature for different shell-side resistances.
O i
0
0.2
0.4
0.6
0.8
1.0
1.2
e/a
Figure 5. Response to step changes in tube-fluid velocity with constant h, and C = 1.0. Solid lines, exact solution; dotted lines, limiting “no-wall” solution. 1.0
08
0.6
i .-
.
.// / /
;”
n 0.8
c.
0.4
1.0
f“. 0.2
..
v = 0.8
/// 0.2
e/a
Figure 4. Response to step change in shell-side fluid temperature for different heat capacity ratios.
ment of the final steady state and is plotted against the throughput ratio (d/a). The response to a step change in inlet tube fluid temperature for various values of C is shown in Figure 2. The effects off and a on the response can be determined by examining the tabulated J function or by the use of its known properties. The response to a step change in shell-side fluid temperature is shown in Figures 3 and 4. From Figure 3 it is seen that for C = 1 and f smaller than 0.1 that the “no-wall” solution provides a good approximation. For values of Cf less than 0.1 the approximation to the ‘‘tiowall” solution is seen to be better in Figure 3 as compared to Figure 4. Thus the applicability of the limiting solution depends on the individual values of both C and f . Given a finite tube-wall heat capacity, the fractional heat transfer resistance on the shell side is also important in determining the behavior of the transient response. Figure 5 shows the response to a step change in velocity assuming a constant tube-side heat transfer coefficient ( n = 0). For C = 1 and for a wide range of velocity changes there is little transient after the passage of one throughput volume and the “no-wall’’ solution is an excellent approximation. Calculations show that a large increase in tube-wall capacity (C > 5) is required before significant departure from the limiting solution is obtained. The results for the response to step change in velocity using a heat transfer coefficient correlation of h = h* V‘.a are shown in Figures 6 to 9.
Figure 6. Response to step change in tube-fluid velocity for different space parameters.
i ?-
@/a
Figure 7. Response to step changes in tube-fluid velocity with h = h* V0.8. Solid lines, exact solution; dotted lines, limiting “no-wall” solution.
A value of a* = 1,which corresponds to a reasonable size of exchanger and operating conditions for liquid-vapor heat exchangers, was used in most of the calculations. The effect of variation of a* on the response to velocity change is illustrated in Figure 6. The larger the value of cy* the lesser the transient after the passage of one throughput
Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978
n.08 1.0
c
d= 1.0 v = 1.2
E.-LL
‘0
02
04
06
08
II O
12
14
I8
20
Wa
.’ /
0.2
04
0.6
changes is derived without the use of perturbation approximations. Solutions obtained for the response to tube-side fluid and shell-side fluid temperature disturbances are more compact and easier to evaluate than those of previous work. The use of known tabulated functions in the derived equations facilitates the examination of a wide range of parametric effects and enables a quantitative determination of the applicability of the approximate limiting “no-wall’’ solutions. Numerical results obtained from the derived equations indicate that, for step changes in tube-side fluid temperature or shell-side fluid temperature, the product of C and f must be less than about 0.2 to justify the use of a “no-wall” solution for approximation. Comparison of the transient response to velocity changes for a model with constant h with that for a velocity-dependent h indicates significant differences. Good approximation by the “no-wall’’ solution may be obtained if tube-side heat transfer coefficient can be assumed to be constant. However, the use of the approximate “no-wall’’ solution for a velocity-dependent h model can lead to significant error in predicting the transient behavior.
! A
16
Figure 8. Response to step change in tube-fluid velocity with h = h*VO and V = 1.2, for different shell-side resistances.
‘0
357
OE
e/a
Figure 9. Response to step change in tube-fluid velocity with h = h*P8and V = 1.5 for different heat capacity ratios.
volume and hence the better the approximation by the ‘‘no wall” solution. Similar effects of CY*are found for the response to shell-fluid temperature changes. Figure 7 shows the response for a wide range of increased or decreased velocity. It can be seen that except for a large decrease in velocity, there is a substantial deviation between the exact and the limiting solution. For a decrease in velocity the resistance to heat transfer on the tube side increases; hence the fractional resistance on the shell side decreases. If the velocity is decreased sufficiently, f will approach zero and the “no-wall’’ solution may become a reasonable approximation. Figures 8 and 9 show the important role off* and C on the dynamic behavior for velocity increase. Similar effects are found for decreases in velocity. Comparing Figures 5 and 7 shows that there is a significant difference in the response for n = 0 and n = 0.8 even for small velocity changes. It is therefore reasonable to conclude that the overall response to velocity change depends to a larger degree upon the variation of h with V rather than upon the magnitude of the residence time change. Thus the use of a model with constant h could lead to appreciable error in predicting the exchanger dynamics even for moderate changes in velocity. Summary and Conclusions Analytic time-domain solutions are obtained for the transient behavior of a shell-and-tube heat exchanger with finite tube-wall heat capacity and finite shell-side heat transfer resistance. The transient response to step velocity
Nomenclature a = reciprocal of the product of C and f 4 = cross-sectional area of the tube 8 = tube-side heat transfer area per unit length of exchanger Ag = shell-side heat transfer area per unit length of exchanger A. = overall heat transfer area per unit length of exchanger C = heat capacity ratio of tube wall to tube fluid C, = heat capacity of tube fluid C,, = heat capacity of tube wall f* = initial fractional heat transfer resistance on the shell side f = final fractional heat transfer resistance on the shell side h* = initial heat transfer coefficient on the tube side h = final heat transfer coefficient on the tube side h = heat transfer coefficient on the shell side f! = Heaviside function Io = modified Bessel function of the first kind of order zero J = mathematical function, defined in text k = counter number in the series defining $ M , = mass of tube wall per unit length of exchanger n = exponent in heat transfer correlation equation R = roots in quadratic transform equation t = absolute time T = tube-fluid temperature T = shell-side fluid temperature T“, = tube-wall temperature Uo* = initial overall heat transfer coefficient Uo = final overall heat transfer coefficient u* = initial linear velocity of tube fluid u = final linear velocity of tube fluid V = normalized linear velocity of tube fluid x = argument of mathematical function X = terms containing x in \c function y = argument of mathematical function z = axial distance from inlet of exchanger Greek Letters a = normalized space parameter X = dummy variable
0 = normalized time parameter & = mathematical function p
= mathematical function = 1 = density of tube fluid
-
J
$ = mathematical function
Subscripts w = tube wall g = shell side 0 = tube fluid inlet Superscripts ^ = deviation from initial steady state
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Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978
* = initial steady-state value w
= infinity
L i t e r a t u r e Cited Cohen, W. C., Johnson, E. F., Chem. Eng. Prog. Symp. Ser., No. 36, 57, 86 (1961). Goldstein, S., Proc. Roy. SOC.London, Ser. A , 218, 151 (1953). Helfferich,F., "Ion Exchange", McGraw-Hill, New York, N.Y., 1962. Hiester, N. K . , Vermeulen, T., Chem. Eng. Prog., 48, 505 (1952).
Hougen, O., Watson, K . M., "ChemicalProcess Principles", Part 111, Wiley, New York, N.Y., 1947. Koppel, L. B., Ind. Eng. Chem. Fundam., I,131 (1962). Ray, W. H., Ind. Eng. Chem. Fundam., 5, 139 (1966). Sherwood, T. K., Pigford, R. L., Wilke, C. R., "Mass Transfer",McGraw-Hill, New York, N.Y., 1975. Yang, WJ., J . Heat Transfer, 86, 133 (1964).
Received for reuiew July 6, 1977 Accepted July 6, 1978
EXPERIMENTAL TECHNIQUES An Inexpensive and Simple Correlator for Velocity Measurement' Katsuya Okl, Walter P. Walawender, and Llang-tseng Fan* Department of Chemical Engineering, Kansas State University, Manhattan, Kansas 66506
An inexpensive and simple correlator was constructed for use in the velocity measurements based on the correlation technique. I t consists of a small number of digital integrated circuits, an operational amplifier, and a low-cost time delay device. The correlator was successfully used in conjunction with a fiber optic probe to measure the local velocity of solid particles.
Introduction Since the correlation technique was first employed for measuring the velocity of steel sheets (Butterfield et al., 1961), it has been widely used as an effective tool for determining velocities in various flow fields including multiphase flow where traditional measuring techniques failed. For example, the local velocity of fine solid particles within equipment can be determined by optically detecting individual particles at two fixed points. The transit time, T,, of particles between the two points separated by a distance of 1 corresponds to the peak position T , of the cross-correlation function of two random signals detected a t the two points; the velocity of particles up is obtained as 1 1 ~ ~ . One drawback in applying this correlation technique is that the computation of the cross-correlation function is difficult. Although real-time correlators especially designed for rapid correlation analysis are now available commercially, they are rather expensive. The present work describes a simplified correlator that can be used to estimate the maximum of the cross-correlation function of the amplitudes of two random signals that are in a strict statistical sense "stationary" functions of time. In other words, the signals must be such that the individual signal means and variances as well as the cross-correlation function of the two signal amplitudes do not change over the time necessary to conduct the signal analyses. In this study specific attention was given to Department of Chemical Engineering Contribution No. 63j, the Kansas Agricultural Experiment Station (Project 0940), Kansas State University. 0019-7874/78/1017-0358$01.00/0
testing the applicability of the correlator by measuring the local velocity of spherical particles under steady-state conditions. A fiber optic probe was used to detect reflected light from the moving particles which served as the signals to be correlated. Although the scope of testing in the present work is quite limited, it is believed that the basic methodology can be of far more general applicability. Examples of potential applications include the measurement of local particle velocity of free flowing granular materials in gravity flow from bins and in standpipes, measurement of solids velocity in pneumatic conveying, and perhaps the measurement of local particle velocity in slurry and emulsion flow. The nature of the probe employed (Oki et al., 1973) is such that solids velocity profiles in pipes and equipment can in principle be obtained with minimum disturbance to the flow. The simple correlator is orders of magnitude lower in cost than available real-time correlators; however, this reduction in cost is not without penalty. The simple correlator cannot be used under conditions where velocities change with time, whereas real-time correlators can be used under such conditions. Once processed, data are not available for further treatment and consequently we are limited to signals that can be considered stationary over the courses of the signal analysis. The correlator was constructed from a small number of digital integrated circuits (I.C.), an operational amplifier, and a charge-coupled device for signal delay. The total cost of the correlator, including a regulated power supply, has been estimated to be approximately $300. An X-Y recorder is also required as an output device. Some experimental results obtained by using it are given for local 0 1978 American Chemical Society
