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A computer based iterative solution for accurate estimation of heat transfer coefficients in a helical tube heat exchanger

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A computer based iterative solution for accurate estimation of heat transfer coeﬃcients in a helical tube heat exchanger
P.K. Sahoo
a
, Md.I.A. Ansari
b
, A.K. Datta
b,*
a
Department of Food Engineering, Faculty of Agricultural Engineering, Bidhan Chandra Krishi Viswavidyalaya, Mohanpur 741 252, Nadia,West Bengal, India
b
Department of Agricultural and Food Engineering, Indian Institute of Technology, Kharagpur, West Bengal 721 302, India
Received 19 February 2002; accepted 09 September 2002
Abstract
An iterative technique is developed and reported for accurate estimation of heat transfer coeﬃcients in a helical triple tube heatexchanger. Based on the experimental temperature rise of whole milk in a helical triple tube heat exchanger, accurate values of ﬁlmheat transfer coeﬃcients and overall heat transfer coeﬃcients based on the outside area of the innermost tube and the inside area of the middle annulus are obtained from ﬁrst principles. Three diﬀerent ﬂow rates of milk were used giving rise to three Reynoldsnumbers. The procedure described can be applied to any heat exchanger with minor modiﬁcations, if necessary.
2003 Elsevier Science Ltd. All rights reserved.
Keywords:
Heat transfer coeﬃcient; Heat exchanger; Iterative solution; Helical tube
1. Introduction
Accurate estimation of heat transfer coeﬃcients inheat exchangers is diﬃcult, particularly in tubular heatexchangers where an annulus is used to heat or cool theworking ﬂuid. Since both the inside and outside surfacesare responsible for heat transfer, it is important that theﬁlm coeﬃcient in the working ﬂuid is accurately esti-mated. In turn the overall heat transfer coeﬃcientsbased on the inside area of the middle tube and theoutside area of the innermost tube should be obtained.In this article an iterative computer based solution isreported for accurate estimation of such heat transfercoeﬃcients. Experimental data on a concentric helicaltriple tube heat exchanger (Fig. 1) is used to validate theprocedure. The middle tube of the heat exchanger wasused to convey the working ﬂuid (i.e. milk). The inner-most and outermost tubes were used to carry the heatingmedium (i.e. steam). The temperature rises of milk atthree diﬀerent ﬂow rates were obtained. The iterationwas based on equations obtained from ﬁrst principlesand is described in Section 2.
2. Theory
The condensate ﬁlm heat transfer coeﬃcients wereobtained from the following equations (Kern, 1984):
h
c1
¼
0
:
5754
k
3c
q
2c
g
kl
c
D
i1
ð
T
s
T
i1wall
Þ
0
:
25
ð
1
Þ
h
c2
¼
0
:
725
k
3c
q
2c
g
kl
c
D
o2
ð
T
s
T
o2wall
Þ
0
:
25
ð
2
Þ
The condensate ﬁlm temperature (
T
f
) was obtained fromthe following equations (Cuevas & Cheryan, 1982):
T
f1
¼
T
s
0
:
75
ð
T
s
T
i1wall
Þ ð
3
Þ
T
f2
¼
T
s
0
:
75
ð
T
s
T
o2wall
Þ ð
4
Þ
Initial estimation of the ﬁlm heat transfer coeﬃcient onthe milk side was carried out using the following equa-tion (Chopey & Hicks, 1984; Geankoplis, 1997; Zuritz,1990):
h
m
¼
k
m
D
eq
0
:
027
N
0
:
8
Re
N
0
:
33
Pr
D
i2
D
o1
0
:
53
1
þ
3
:
5
D
eq
D
c
ð
5
Þ
T
i1wall
and
T
o2wall
were obtained from the followingequations based on the heat ﬂux equivalence concept(Cuevas & Cheryan, 1982):
Journal of Food Engineering 58 (2003) 211–214www.elsevier.com/locate/jfoodeng
*
Corresponding author.
E-mail addresses:
pks_1973@lycos.com (P.K. Sahoo), irfan262@yahoo.com (Md.I.A. Ansari), akd@agfe.iitkgp.ernet.in (A.K. Datta).0260-8774/03/$ - see front matter
2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0260-8774(02)00370-9
T
s
T
i1wall1
h
c1
D
i1
¼
T
i1wall
T
bmln
D
o1
D
i1
2
k
ss
þ
1
h
m
D
o1
ð
6
Þ
T
s
T
o2wall1
h
c2
D
o2
¼
T
o2wall
T
bmln
D
o2
D
i2
2
k
ss
þ
1
h
m
D
i2
ð
7
Þ
Overall heat transfer coeﬃcients were obtained from thefollowing equations based on conduction and convec-tion in cylindrical coordinates:
U
o1
¼
1
h
m
0@
þ
D
o1
ln
D
o1
D
i1
2
k
ss
þ
D
o1
h
c1
D
i1
1A
1
ð
8
Þ
U
i2
¼
1
h
m
0@
þ
D
i2
ln
D
o2
D
i2
2
k
ss
þ
D
i2
h
c2
D
o2
1A
1
ð
9
Þ
Introducing
q
1
¼
U
o1
A
o1
D
T
lm
ð
10
Þ
and
q
2
¼
U
i2
A
i2
D
T
lm
ð
11
Þ
_
mm
m
C
m
D
T
m
¼
q
1
þ
q
2
ð
12
Þ
3. Methodology
The dimensions of the tubes in the concentric helicaltriple tube heat exchanger are shown in Table 1. Wholemilk was preheated to 93
C in steam jacketed pan andsupplied to the inlet of heat exchanger by means of acentrifugal pump. Milk was raised to various tempera-tures inside the heat exchanger by steam supplied from a‘‘VASPA’’ electrode steam boiler. The various outlettemperatures for diﬀerent ﬂow rates are given in Table 2.The developed iterative method consists of the followingsteps:1. Tube diameters, coil diameter, ﬂow rate of milk,length of helical triple tube heat exchanger, tempera-ture of steam, inlet and outlet temperatures of milkare the inputs.2. Milk side ﬁlm heat transfer coeﬃcient is estimated us-ing Eq. (5).3. Wall temperatures (
T
i1wall
and
T
o2wall
) are initializedby assuming
T
i1wall
<
T
s
and
T
o2wall
<
T
s
.4. Condensate ﬁlm temperatures are estimated usingEqs. (3) and (4).5. Condensate ﬁlm heat transfer coeﬃcients are ob-tained using Eqs. (1) and (2).
Nomenclature
A
heat transfer area, m
2
C
speciﬁc heat, kJ/kgK
D
tube diameter, m
D
c
coil diameter, m
g
acceleration due to gravity, 9.81 m/s
2
h
ﬁlm heat transfer coeﬃcient, W/m
2
K
k
thermal conductivity, W/mK
L
length of tube, m
_
mm
mass ﬂow rate, kg/s
N
dimensionless number
T
absolute temperature, K
U
overall heat transfer coeﬃcient, W/m
2
K
Greek symbols
D
diﬀerence
k
latent heat of condensation, kJ/kg
l
dynamic viscosity, kg/ms
q
density, kg/m
3
Subscripts
b bulk or average between two heat transfersurfacec condensate ﬁlmh heatingeq equivalenti insidem milko outsides steamss stainless steellm log mean
Re
Reynolds number
Pr
Prandtl number1 tube of smallest diameter2 tube of intermediate diameter
Fig. 1. Isometric view of helical tube UHT milk sterilizer.212
P.K. Sahoo et al. / Journal of Food Engineering 58 (2003) 211–214
6.
T
i1wall
and
T
o2wall
are obtained using Eqs. (6) and (7).7. Steps (3)–(6) are repeated until
T
i1wall
and
T
o2wall
assumed in step (3) matches with those obtained instep (6).8. Overall heat transfer coeﬃcients (
U
o1
and
U
i2
) arecalculated using Eqs. (8) and (9).9.
q
1
and
q
2
are obtained using Eqs. (10) and (11).10. Obtain ratio
r
¼
q
2
q
1
.11. Substitute experimental data and
q
2
¼
rq
1
in Eq. (12).12. Maintaining ‘‘
r
’’ obtain new
q
2
.13. New
U
o1
and
U
i2
are calculated from Eqs. (10) and(11).14. New
h
m
is calculated by using new values of
U
o1
and
U
i2
in Eqs. (8) and (9) and obtaining an average of thetwo values.15. Steps (8)–(13) are repeated until the two sets of over-all heat transfer coeﬃcients match.16. Replace the value of milk side ﬁlm heat transfer coef-ﬁcient (
h
m
) in step (2) with the value obtained in step(14).17. Repeat step (7).18. Repeat step (15).19. Stop when
h
m
obtained in step (14) in the last itera-tion is within
0.01% of the previous iteration.
4. Results and discussion
Two sets of values of milk side ﬁlm heat transfercoeﬃcient (
h
m
) and overall heat transfer coeﬃcients (
U
o1
and
U
i2
) are presented in Table 3. The ‘‘calculated val-ues’’ refer to those obtained after completing all thesteps involved in the iteration whereas the
predictedvalues
refer to the values obtained on the basis of Eq. (5)and iteration completed up to step (13) only. The reasonfor such comparison is to obtain the eﬀect of the semi-empirical Eq. (5) on the ﬂuid values of overall heattransfer coeﬃcients. Fig. 2 compares the two sets of values of milk side ﬁlm heat transfer coeﬃcients forvarious Reynolds numbers. Fig. 3 plots the ‘‘predicted’’and ‘‘calculated’’ values of overall heat transfer coeﬃ-cients based on outside area of the innermost tube asaﬀected by Reynolds numbers. Fig. 4 is similar except
Fig. 2. Variation of milk side heat transfer coeﬃcient with Reynoldsnumber.Fig. 3. Variation of
U
o1
with Reynolds number.Table 1Dimensions of tubes in helical triple tube heat exchangerInnermost tube diameters (mm) Middle tube diameters (mm) Outermost tube diameters (mm) Length (m)Inside Outside Inside Outside Inside Outside6.00 8.50 12.7 15.0 22.23 25.4 2.28Table 2Milk ﬂow rates, Reynolds numbers and outlet temperaturesFlow rate (l/h) Reynolds number Outlet temperature (
C)135 5159 142153 5791 140176 6587 138Table 3Calculated and predicted values of heat transfer coeﬃcientsReynoldsnumberCalculated ﬁlm heattransfer coeﬃcient(
h
m
) (W/m
2
K)Predicted ﬁlm heattransfer coeﬃcient(
h
m
) (W/m
2
K)Calculated
U
o1
(W/m
2
K)Predicted
U
o1
(W/m
2
K)Calculated
U
i2
(W/m
2
K)Predicted
U
i2
(W/m
2
K)5159 1508 1516 1226 1102 1318 14005791 1587 1596 1275 1140 1376 14666587 1701 1713 1345 1196 1459 1558
P.K. Sahoo et al. / Journal of Food Engineering 58 (2003) 211–214
213
that it plots the overall heat transfer coeﬃcient valuesbased on the inside area of the middle tube. The mag-nitudes of diﬀerence between the two sets vary from0.55% to 0.70% for milk side heat transfer coeﬃcient.The diﬀerences between
U
o1
values vary from 10% to11%. The same diﬀerence between
U
i2
may vary from6.00% to 7.00%. The source of error in the experimentaldata is in the measurement of volume ﬂow rate of liquidin the heat exchanger. Since every liter measurement hasa
10 ml error we can say that there is a
1% error inthe reported Reynolds
s numbers. This shows up as a
0.8% error in the values of ﬁlm heat transfer coeﬃ-cients
h
m
. The corresponding error values for
U
o1
and
U
i2
are
0.72% and
0.77% respectively. So for ﬁlmheat transfer coeﬃcients the experimental error is com-parable to the errors between predicted and calculatedvalues. On the other hand the experimental error isnegligible compared to the spread between calculatedand predicted values of
U
o1
and
U
i2
. The method dem-onstrates how accurate estimation of heat transfer co-eﬃcients is possible when two heat transfer surfaces areinvolved. This method is easily adaptable to any con-centric heat exchanger involving two heat transfer sur-faces. The input to the programme must be altered tosuit other tubular heat exchangers. Appropriate heattransfer equation also has to be substituted for Eq. (5).
References
Chopey, N. P., & Hicks, T. G. (1984).
Handbook of chemical engineering calculation
(p. 33). New York: McGraw-Hill Bookcompany (Chapter 7).Cuevas, R., & Cheryan, M. (1982). Heat transfer in a vertical liquidfull scraped surface heat exchanger: application of the penetrationtheory and Wilson plots method.
Journal of Food ProcessEngineering, 5
, 1–21.Geankoplis, C. J. (1997).
Transport processes and unit operations
(3rded., p. 239). New Delhi: Prentice Hall of India Pvt. Ltd.Kern, D. Q. (1984).
Process heat transfer
(pp. 263–269). New York:McGraw-Hill International.Zuritz, C. A. (1990). On the design of triple concentric tube heatexchangers.
Journal of Food Process Engineering, 12
, 113–130.Fig. 4. Variation of
U
i2
with Reynolds number.214
P.K. Sahoo et al. / Journal of Food Engineering 58 (2003) 211–214

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