SH E LL-SIDE H EAT TRANSFER COEFFICIENTS IN HELICAL COIL HEAT EXCHANGERS C H A R L E S
J .
M E S S A ,
A L A N
S. FOUST, A N D
G A R Y
Department of Chemical Engineering, Lehigh University, Bethlehem, Pa. Geometric construction parameters-winding
P O E H L E I N
W .
18015
angle, radial pitch, and axial pitch-
significantly influence the shell-side heat transfer of helical coil heat exchangers. Shell-side heat transfer coefficients have been calculated from experimental measurements of over-all coefficients for both water and air as the shell-side fluid. Water w a s used in the tubes for all measurements. Tube-side transfer coefficients were determined from the Dittus-Boelter equation, which was checked for several cases by the Wilson plot technique. The shell-side heat transfer coefficients of six heat exchangers have been correlated by the relationship: N," = cN,:N,P. Equations relate parameters a, b, and c to the construction variables.
THEproduction
of refrigeration is difficult and costly in the operation of cryogenic processes and therefore conservation is essential. A vital component of commercial cryogenic systems for conserving refrigeration is the countercurrent heat exchanger, the prime requisites of which are low pressure drop, and small over-all dimensions per unit of heat transfer area to minimize the heat leak to the exchanger. Helical coil heat exchangers possess both these requisites and therefore are used in large scale cryogenic processes. Heat transfer and friction factor relations, derived from studies of flow normal to banks of straight tubes, are utilized by Barrows (1966) and Scott (1960) as design relationships for helical coil heat exchangers. Response of field exchangers and an elementary qualitative analysis of the possible flow patterns indicate that these relationships are only approximate and that substantial deviations, both favorable and unfavorable, can occur. The purpose of this experimental program was to study shell-side heat transfer coefficients of helical coil heat exchangers as a function of the geometric construction parameters. The experimental program was designed to measure over-all coefficients for six uniquely chosen winding configurations and to correlate shell-side heat transfer coefficients in terms of standard dimensionless variables (NRe,N h u ,and NPr)for each exchanger.
level factorial. Later, two additional exchangers within the range of the complete factorial were constructed. Specifications are shown in Table I for the six exchangers tested. The sizes of industrial exchangers vary widely. Some are smaller than those employed in our tests, while others are large enough to require flatcar railroad transportation. Experimental Apparatus
A flow diagram of the experimental apparatus is shown in Figure 2. Temperature differences were measured with 25-junction thermopiles. Temperatures of individual streams were measured with individual thermocouples. The temperatures in the surge tanks were automatically controlled by steam and cold water injection.
n
n
ir
I
V/////////A
c7
SEALED TO FLOW
[MANDREL AXIAL
SEPARATION
W,NDING
?,< \,L //!
n
,'
-
ANGLE D
WOUND ~ U B I N G
Heat Exchanger Construction
Helical coil heat exchangers are a compact shell and tube design consisting of several layers of coiled tubes within a closed shell. The basic construction of the type used for this experimental work, with the appropriate nomenclature, is illustrated in Figure 1. The winding direction for each layer was always in a reverse direction from the previous layer. Tube layers were separated by spacer wires which ran parallel t o the axis of the mandrel. The cost of heat exchanger construction dictated an experiment design which would minimize the number of heat exchangers required. The statistical design initially chosen for this study was a one-half fraction of a two-
TUBE LAYER
R A D I A L SEPARATION
SPACER
-
c c c
c
c
MANIFOLD
MANIFOLD
Figure 1. Experimental exchangers VOL. 8 NO. 3 JULY 1 9 6 9
343
Figure 2. Flow diagram
Table 1. Specifications for Experimental Heat Exchangers
HE
WA
RP
AP
NT NSW T L
1 2 3 4 5 6
18.5' 4.5 4.5 18.5 18.5 4.5
1.29 1.03 1.29 1.03 1.29 1.29
1.03 1.03 1.29 1.29 1.29 1.02
159 37 30 115 126 39
Bundle length, inches Tube o.d., inch Tube id., inch Mandrel diameter, inches No. of tube layers
113 107 117 102 115 107
7.8 26.0 27.9 7.8 7.8 32.3
A,
A:
A,
80.8 63.0 54.7 58.4 64.2 68.6
65.2 59.4 51.8 47.2 51.8 65.2
0.106 0.038 0.127 0.078 0.128 0.104
24.0 0.250 0.152 3.5 7
- 0 4
HEAT
EXCHANGER 2 N, = 778
Q
WATER
A
WATER
Nh
= 228
0
AIR
Np,
= 0.70
4
Flow rates and pressure drops were measured on both shell and tube sides of the exchangers, to obtain a check on the over-all heat balance. Complete details of the experimental equipment and operating procedures are given by Messa (1967). Analysis of Data
The equations describing fluid flow and heat transfer, when written in dimensionless form, indicate a relationship of the type
N,, =
m,,
9
Np, 1
(1)
for single-phase heat transfer. Experimental results of many investigators have been correlated by equations of the type
N," = CN,; Np,b
(2)
for single-phase flow in ducts or over objects, where c, a, and b are constants dependent only on geometry. This form of correlation was used in our work. The Nusselt, Reynolds, and Prandtl numbers are defined in a manner similar to those for flow normal to banks of straight tubes (McAdams, 1954). The mass flux is equal to the mass flow rate divided by the free flow area. The free flow area was determined by subtracting the volume of the tubing, mandrel, and spacer wires from the bundle volume of the finished exchanger and then dividing this volume by the bundle length. The outside heat transfer area was calculated from the total length of tubing used in an exchanger. 344
l & E C PROCESS D E S I G N A N D DEVELOPMENT
6
8 IO4
NRe
Figure 3. Experimental data
The values of a, b, and c of Equation 2 were determined by multilinear regression analysis of the heat transfer data. Shell-side pressure drop was also measured for all exchangers. Various forms of the friction factor were compared to the friction factor predicted for flow normal to banks of straight tubes. Results a n d Discussion
Heat Transfer. Shell-side coefficients were measured with both water and air with Npr = 2.28 and N,, = 0.70, respectively. Several runs were made with low temperature water a t NPr Y 7.8. The Reynolds number range for most exchangers tested was from 400 to about 10,000, with some runs as high as 45,000. Figure 3 shows the data for heat exchanger 2 a t three Prandtl number levels. These data were fitted to Equation 2 by a regression program to yield
N,, = 0.0085 NR:.'' Np;"j
(3)
This correlation provides a good fit of the data, as shown in Figure 4. The data for all exchangers were treated in a similar manner. The "best-fit'' constants for each exchanger and the correlation statistics are shown in Table 11.
Table II. Constants Derived from least Squares Fit
N,, Heat Exchanger 1 2 3 4 5
6 a.
=
cNR. N& CL.Q,, CL,hp,
a
a0
b
bb
InC
C
0.71 0.84 0.85 0.83 0.71 0.86
k0.01
0.93 0.57 0.70 1.14 1.02 0.83
10.02 Ito.O1 10.04 +0.03 It0.02 k0.02
-2.8698 -4.7682 -3.4893 -3.4410 -2.4642 -3.8449
0.0567 0.0085 0.0305 0.0320 0.0851 0.0214
10.01 1t0.03 1t0.02 *0.01 10.02
3 3 3 3 3 3
3 3 3 3 3 3
R'
Residual Mean Square
Data Points
0.99774 0.99557 0.98732 0.99778 0.99895 0.99596
0.003148 0.000796 0.012976 0.003190 0.001826 0.003717
45 24 24 30 30 26
Standard deviation.
R2. Square of multilinear regression coefficient. CL. Confidence limit.
HEAT EXCHANGER 2
0
WATER AIR
N,,
N,,
= 778
Table 111. Predicted Values of a, b , and c for Exchangers 5 and 6 Using a Y J - Factorial ~ ~ Analysis
Heat Exchanger
(1
Calcd
= 0
Exptl
Difference
I. Prediction of a 1 2 3 4 5 6
0.78 0.78
0.71 0.84 0.85 0.83 0.71 0.86
9.6 9.6
11. Prediction of 6
Figure 4. Least squares correlation
1 2 3 4 5 6
1.10 0.53
0.93 0.57 0.70 1.14 1.11 0.83
1.2 36.4
111. Prediction of c
Examination of the statistical quantities in Table I1 indicates the excellent consistency of the fit of experimental data to Equation 2. The R 2 values represent the goodness of fit of the correlating equation to the measured data. R 2 = 1.0 is a perfect fit. The exponent of the Reynolds number was approximately 0.8 for all exchangers tested. This average value is equivalent to the rule-of-thumb value suggested for many heat transfer processes. I t differs from the 0.6 recommended by Colburn (1933), McAdams (1954), and Kreith (1961) for heat transfer to fluids flowing normal to banks of straight tubes. The Prandtl number exponent varied from 0.57 to 1.14, differing significantly from the usual 0.33 value. In view of the widespread use of the 0.33 exponent, a discussion of possible reasons for the significant difference is appropriate. Several steps in the calculation of the exponent are potential aggravators of this disagreement. The most basic fault is that fluids with only two Prandtl number values (0.7 and 2.3) were used for the majority of the heat exchangers tested. The potential for error in the evaluation of the Prandtl number exponent is great, because these values are near 1.0, a very insensitive range for a power function correlation. One purpose of this work was to relate constants a, b , and c (Table 11) to the construction parameters: winding angle, radial pitch, and axial pitch. This was done by use of a fractional factorial analysis on the results from exchangers 1 through 4. The values of a , b , and c (generated as In c in the regression program) for exchangers
1 0.0575 2 0.0086 3 0.031 4 0.0326 5 0.082 0.086 5.0 6 0.0218 0.0218 0.0 Coefiients a,, Calculated from Results of Exchangers 1 to 4 aOl= -0.036, ao2= -0.028, a,%= -0.0322 ad = 0.807, am = 0.835, U b i = 0.20, as? = -0.0228, ab3 = -0.0837 aCl= 0.487, a,, = 0.462, a,?= 0.177 ad = -3.642,
1 to 4 were used to determine the constants in the linear models shown below.
In c = ad + aClW A + ac2RP + a,:,AP a = aao+aa1W A + aaeRP + a,.3AP b = abo + abl W A + ab2RP + ab?AP
(4) (5) (6)
Equations 4, 5 , and 6 were then used to predict the performance of heat exchangers 5 and 6 (Table 111). All predictions are accurate except for the Prandtl number exponent for exchanger 6 (a 36% difference). We do not recommend use of Equations 4, 5, and 6 outside the ranges of geometry of the test exchangers. VOL. 8 N O . 3 JULY 1969
345
Additional experimental data were obtained a t higher Prandtl number (approximately 7.8) for exchangers 2 and 5, since the Prandtl number exponents were widely scattered and always above the expected 0.33. These data did not change the 0.57 exponent for exchanger 2. The exponents computed for each pair of Prandtl number values in exchanger 5 varied from 0.55 to 1.0, suggesting considerable uncertainty in this constant. Since the Reynolds number exponent was near 0.8 for all exchangers, the experimental data were also fitted to the correlation given by Equation 7.
N N u= c N ~ : ~ N ~ : ' ~ ~
The following forms of friction factor relationships for flow normal to banks of parallel tubes are given by McAdams (1954) and Kreith (1961).
(7)
The correlation suggested by Colburn (1933) for banks of in-line straight tubes is:
Nsu = 0.26 NRFNp:.33 The shell-side geometry of a helical coil heat exchanger is considerably different from bank of in-line straight tubes, as illustrated in Figure 1. However, a comparison of our results with Colburn's equation still seems justified. To make this comparison Equation 7 is rewritten as follows: Table IV gives the Reynolds number range of the experimental data, the constant e, and the range of ( C N ~ ' ) values for each exchanger. The results of most of the exchangers bracket Colburn's predictions over the range of Reynolds numbers tested. Exchangers 3 and 6, which are most nearly like the in-line tube geometry, are closest to predictions for in-line tubes. The geometry of exchanger 2 differs considerably from the in-line tube geometry and the heat transfer results are also different. Frictional Losses. Shell-side pressure drop was measured over a Reynolds number range from 3000 to about 50,000 with water as the shell-side fluid. Friction factors can be defined in several different manners. A standard definition can be written as:
fi
=
force wetted area
. .
(
1
kinetic energy unit volume
I
Conclusions
Table IV. Comparison of NNu= with Colburn's Equation
Exchanger
NRe Range
c
1
500-45,000 900- 7,500 400- 7,500 800-12,000 400-37,000 500- 7,100
0.037 0.015 0.060 0.084 0.059 0.052
2 3 4 5 6
Range of c NR:
'
0.128-0.26 0.057-0.088 0.199-0.354 0.316-0.55 0.096-0.42 0.180-0.307
Table V. Friction Factors from Water Flow Measurements
Heat Exchanger 1
2 3 4 5 6
346
Friction Factors at Re
=
f3 can be computed without experimental data. All three friction factor forms were evaluated for each exchanger. The following conclusions can be drawn: fi and f2 describe the data very well because of the forced fit. f3, except in two fortuitous cases, does not accurately predict either fl or f2. The values of fi and f2, in most cases, vary less with changes in the Reynolds number than the -0.15 power in f3 would suggest. Table V lists all three friction factors for each exchanger a t NRe= 10,000. These values actually apply a t other values of the Reynolds numbers because of the observed flatness mentioned in the third result. A comparison of the heat transfer and friction factor data shows that some of the heat exchangers represent impractical construction. In particular, exchanger 2, which has small values of both radial pitch and axial pitch, requires large pressure losses for relatively small heat transfer coefficients. The only other exchanger (exchanger 4) with a small radial pitch also has a large friction factor. In this case, however, heat transfer was also higher.
10,000
fl
12
fi
0.12 2.5 0.39 1.5 0.33 0.14
0.24 11.0 0.58 4.1 0.60 0.24
0.24 5.1 0.24 2.8 0.24 0.24
I & E C PROCESS D E S I G N A N D DEVELOPMENT
The results of this research can be considered accurate only for helical coil heat exchangers constructed of % inch tubing, but indicate trends that might be expected for similar geometry and different absolute sizes. The results indicate that the normal construction parameters can have a significant influence on both heat transfer and pressure drop in helical coil exchangers. Acknowledgment
The financial support of the M. W. Kellogg Co. and the Gardner Cryogenic Corp. is acknowledged. Nomenclature
A P = axial pitch - (axial separation + tube diameter) / tube diameter a = correlation Reynolds number exponent (Equation 2) a,, = regression constants (Equations 4, 5 , and 6) Ai = shell-side free flow area, sq. ft. Ao = total shell-side heat transfer area, sq. ft. A $ = bundle-only heat transfer area, sq. ft. b = correlation Prandtl number exponent (Equation 2)
e = correlation constant (Equation 2)
W A = winding angle, degrees p = fluid density, lb./cu. ft.
CI, = statistical confidence limit fa
= friction factors
F = shell-side flow rate, cu. ft./sec.
0
gc = conversion factor, (1b.-ft./ sec.’-lb.,) H E = heat exchanger m = shell-side mass flow rates, Ib./sec.
NT = NSW = n = AP = R‘ =
RP
=
TL
=
= standard deviation
Literature Cited
Nusselt number Prandtl number Reynolds number number of tubes transverse to flow (defined for flow normal to straight tubes) number of tubes in the exchanger number of spacer wires exponent in friction factor relationship Equation 12 (dimensionless) = 0.43 + 1.13/AP shell-side pressure drop, lb., /sq. ft. square of multiple linear regression coefficient (dimensionless) radial pitch - (radial separation + tube diameter) / tube diameter average tube length, ft.
Barrows, Randall, “Cryogenic Systems,” McGraw-Hill, New York, 1966. Colburn, A. P., Trans. Am. Inst. Chem. Engrs. 29, 174210 (1933). Kreith, F., “Principles of Heat Transfer,” International Textbook Co., Scranton, Pa., 1961. McAdams, W. H., “Heat Transmission,” McGraw-Hill? New York, 1954. Messa, M. S. thesis, Lehigh University, Bethlehem, Pa., June 1967. Scott, R . B., “Cryogenic Engineering,” Van Nostrand, Princeton, N. J., 1960. RECEIVEDfor review July 5, 1968 ACCEPTED April 2, 1969
EXPERIMENTAL STUDY OF THE ICE-MAKING OPERATION IN THE INVERSION DESALINATION FREEZING PROCESS S H E N - Y A N N CHIU,
LIANG-TSENG
FAN,
A N D
RICHARD
G.
A K l N S
Department of Chemical Engineering, Kansas State University, Manhattan, Kan. 66502
Experimental equipment was designed and built to test the ice making of the inversion desalination freezing process a t atmospheric pressure. A backmixed type contactor was used to produce ice from salt water by direct contact freezing, using a mixture of n-tridecane and n-tetradecane as a working medium. The thermal driving force, degree of mixing, and nominal residence time were studied as controlling variables for the ice production rate, using 47 experimental runs and four working media. The agitation speed ranged from 300 to 900 r.p.m., and the nominal residence time from 86 to 238 seconds. Pictures obtained under different operating conditions showed that ice crystals were fairly large, well-shaped, and smooth. The results of the production rate study can be used to design a larger scale plant. Very reasonably sized equipment may be used. The quality of ice indicates that the ice-washing operation should present no problems.
THEso-called
inversion desalination process utilizes a unique way of upgrading heat (Cheng and Cheng, 1967). Because of the difference in the effect of applied pressure on melting points, a substance which melts a t a temperature lower than the freezing point of an aqueous solution may melt at a temperature higher than the melting point of the solution a t a sufficiently high pressure. Therefore, a working medium (or refrigerant) can be selected to form a cyclic auxiliary system to remove the heat of crystallization in a partial freezing operation and supply the heat required t,o melt the ice so formed. The process is distinct from the conventional freezing processes in that only condensed (liquid and solid) phases are present. This has a considerable effect on the energy require-
ments of the process and favors the control of ice crystallization. The purpose of this work was to study the ice-making operation, which occurs at low pressure. The rate of heat transfer between aqueous and organic phases and the ice crystal size and shape were quantities of primary interest. The major independent variables studied were the temperature driving force, the amount of mixing, and the nominal residence time. Theoretical
The shape and direction of a univariant pressure us. temperature curve for the melting point of a pure substance and the eutectic temperature of a binary mixture VOL. 8 N O . 3 JULY 1 9 6 9
347
