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R&D Journal
versão On-line ISSN 2309-8988
versão impressa ISSN 0257-9669
R&D j. (Matieland, Online) vol.9 Stellenbosch, Cape Town 1993
Heat transfer and friction loss in extended surface heat exchangers for non-Newtonian fluids in laminar flow
E. E. A. Rouillard
Senior Lecturer, Department of Chemical Engineering, University of Durban-Westville, Private Bag X54001, Durban 4000, South Africa
ABSTRACT
Empirical correlations for predicting the heat transfer coefficient and pressure drop in cross-flow annular finned tube heat exchangers with pseudoplastic non-Newtonian fluids in laminar flow are presented. The correlations were derived from measurements conducted on industrial heat exchangers with both staggered and in line configurations, and with different combinations of fin spacings and fin sizes.
Nomenclature
Ao= outside surface area of bare tube
At= total surface area of finned tube
a= constant
b= constant
CN = function of number of tube rows
cp= specific heat of fluid
De= volumetric equivalent diameter
Df= outside diameter of fins
Do= outside diameter of tube
dv/dy= shear rate
f = friction factor
h= heat transfer coefficient
K= consistency index
k= thermal conductivity
L= length of flow channel
l = fin height
Nu= Nusselt number
n= flow behaviour index
∆P= pressure drop
Pr= Prandtl number
Re= Reynolds number
St= transversal tube pitch
s= distance between adjacent fins
t= fin thickness
V= actual average velocity
v = superficial velocity
ϵ = void fraction
η= fin efficiency
μa= apparent viscosity
ρ= density
ψ= function of tube layouts
φ= function of fin geometries
Subscripts
f = measured at average film temperature
w= measured at wall conditions
Introduction
During the process of sugar manufacture, the low grade product, consisting of a mixture of fine crystals and molasses, is treated in cooling crystalizers from which it comes out in a supersaturated state. This fluid is pseudo-plastic with an apparent viscosity of up to 6000 Pa.s. The apparent viscosity is given by the equatio
Where K is the consistency index, and n is the flow behaviour index. The greater the departure of n from unity, the greater the non-Newtonian behaviour of the product. The next step is the separation of the crystals in centrifuges, and to facilitate this operation it is necessary to reduce the viscosity. This is done by warming the product using finned tube heat exchangers while the product is brought back to saturation. As a result of the high viscosity of the fluid and of the low velocities, laminar flow prevails in these heat exchangers.
Empirical correlations, derived experimentally, have been proposed for estimating the heat transfer coefficient in cross-flow extended surface heat exchangers in turbulent flow [1]. These equations are of the type
Different equations have been proposed for square and triangular pitch arrangements, and the effect of fin geometry is taken care of by a factor which differs in each equation. Briggs and young [2] used the finning factor s/l, fin spacing/fin height, for high finned tubes, and Schmidt [3] used At/Ao, the total area/outside bare tube area.
Similar empirical correlations derived for the Fanning friction factor in turbulent flow include the parameters shown below
Since the correlations that have been proposed do not apply to laminar conditions it was necessary to derive new equations experimentally.
Equipment and procedure
Measurements were conducted on six industrial cross-flow extended surface heat exchangers having the characteristics shown in Table 1, with square annular fins as shown on figure 1. Both square and equilateral triangular pitches were used, and fins were either in line or staggered.
The range of the physical properties and operating conditions that prevailed during the tests are shown in Table 2.
All the heat exchangers had a single shell pass, and from 6 to 60 tube passes. For this combination of tube and shell passes with both fluids unmixed the corrections to the temperature difference were small. Corrections were also applied for actual fin efficiencies using the method of Weierman [4] which is applicable to both staggered and in-line layouts.
Since the heat exchangers had square fins, an average fin diameter based on the area of the fin was used.
The length of the path of the cold fluid on the shell side was assumed to be equal to the height of the tube bundle for the square pitch exchangers and one and a half time the height of the tube bundle for the equilateral triangular pitch arrangement. Fouling resistances on the inside and outside of the tubes were neglected.
Results
Because the fluids were non-Newtonian with pseudoplastic properties, it was necessary to use the generalized Reynolds and Prandtl number
Where De, the volumetric equivalent diameter, was calculated from
The average velocity in the equations above was equal to the superficial velocity divided by the void fraction, that is v/e.
In addition to the Reynolds and Prandtl number, the parameters included in the heat transfer correlation were the ratio De/L and the finning factor of Schmidt [3] At/Ao, which gave a better correlation than the factor of Briggs and Young [2] s/l. The correlation was also improved when the Prandtl number was based on the film temperature rather than the bulk temperature. The equation obtained was
with a correlation coefficient of 0,95.
The Fanning friction factors were calculated from
and correlated with the generalized Reynolds number and the parameter St/De, the ratio of the tube pitch to the equivalent diameter. The consistency ratio K/Kf or K/Kw, which is included in the pressure drop equation of Gunter-Shaw [4], was not found to be a significant variable. The equation obtained is
with a correlation coefficient of 0,97.
Acknowledgements
The author thanks the Sugar Milling Research Institute for permission to publish the results of this work which was carried out at that Institution.
References
1. Cheremisinoff, N. P., Ed., "Handbook of Heat and Mass Transfer, Vol. 1, Heat Transfer Operations", Gulf Publishing Company, Houston, 1986.
2. Briggs, D. E., and Young, E. H., "Convection heat transfer and pressure drop of air flowing across triangular pitch banks of finned tubes", Chem. Eng. Prog. Symp. Ser., 59, 1963, p. 1. [ Links ]
3. Weierman, C, Oil Gas J., Sept. 1976, p. 94.
4. Gunter, A. Y., and Shaw, W. A., Trans. ASME, 1945, p. 643.
First received August 1992
Final version January 1993
Appendix
Sample Calculations
Mass flow rate of water = 19,30 kg/s
Inlet temperature of water = 59,20 °C
Outlet temperature of water = 58,08 °C
Inlet temperature of massecuite = 46,6 °C
Outlet temperature of massecuite = 58,0 °C
Overall Heat Transfer Coefficient
Enthalpy of water at 59,2 °C = 247 786 J/kg
Enthalpy of water at 58,08 °C = 243 104 J/kg
Heat transferred = ö= 19,30 (247 786-243 104) = 90 363 W ∆Tlm = [(59,2 - 58,0) -
(58,8 - 46,6)]/ln[(59,2 - 58,0)/(58,8 - 46,6)]
= 4,55 °C
Correction factor
Where
R = (58,0 - 46,6)/(59,2 - 46,6) = 0,9048
S= (59,2 - 58,08)/(58,0 - 46,6) = 0,09825
Then
CF = 0,9982
Heat transfer area (including tube plates) = 1497 m2
Then
k = 0,653 W/m.°C
μ = 4,81 x 10"4 Pa.s
Pr = 3,08
Di tubes = 0,0447 m
Number of tubes per pass = 12
Mass flow rate of water =
19,30/(12 × π/4 × 0,04472) = 1024,9 kg/m2.s
Re = (0,0447 × 1024,9)/4,81 x 10-4
= 95 243
Using Dittus Boelter equation
Nu = 0,023 (95 243)0,8 (3,08)0,4
= 346,9
hw = (346,9 × 0,653)/0,0447
= 5068 W/m2.°C
Massecuite side heat transfer coefficient corrected for fin efficiency
Assume corrected massecuite side heat transfer coefficient = hm = 16,11 W/m2.°C
From equation (6)
t = fin thickness = 0,003 m
kt = thermal conductivity of tube = 53 W/m.°C
m = [(2 x 16,11)/(53 × 0,003)]1/2
= 14,24
Dof = diameter at base of fins = 0,0566 m
Df outside diameter of fins = (0,122 × π/4)1/2 = 0,1354 m
l = (0,1353-0,0566)/2 = 0,0394 m
From equation (4)
l' = 0,0394 + 0,003/2
= 0,0409
From equation (5)
X = 0,9005
From equation (7)
Y = 0,9005(0,7 + 0,3 × 0,9005)
= 0,8736
From equation (8)
This result is close enough to the assumed value. Tube surface temperature
ρ= 1505,4 kg/m3
De = 0,04833
Nusselt number
Nu = (16,13 ×0,04833)/0,3201
= 2,435
Volumetric flow rate of massecuite
= 90 363/(1443,6)(58,0 - 46,6)( 1505,4)
= 3,647 ×10"3 m3/s
Sectional area of heat exchanger = 18,1 m2
Void fraction = 0,7862
Massecuite velocity
= 3,647 ×10"3/(18,1 x 0,7862)
- 2,563 x 10-4 m/s
n = flow behaviour index = 0,8003
From equation (10) the Prandtl number is
Average bulk temperature of massecuite
= (58,0 + 46,6)/2 = 52,3°C
K = 2625,5 Pa.s
From equation (9) the Reynolds number is
Viscosity ratio
K\Kf = 2625,5/2414,4
= 1,087
L = Length of flow channell = 1,42 m
De/L = 0,04833/1,42
= 0,03404
Finning factor
At/Ao = 204,4/1476,52
= 0,1384