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

 

 

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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.

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Nomenclature

A_o= outside surface area of bare tube

A_t= total surface area of finned tube

a= constant

b= constant

C_N = function of number of tube rows

c_p= specific heat of fluid

D_e= volumetric equivalent diameter

D_f= outside diameter of fins

D_o= 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

S_t= transversal tube pitch

s= distance between adjacent fins

t= fin thickness

V= actual average velocity

v = superficial velocity

ϵ = void fraction

η= fin efficiency

μ_a= 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 A_t/A_o, 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 D_e, 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 D_e/L and the finning factor of Schmidt [3] A_t/A_o, 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 S_t/D_e, the ratio of the tube pitch to the equivalent diameter. The consistency ratio K/K_f or K/K_w, 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 ∆T_lm = [(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

D_i 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

h_w = (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 = h_m = 16,11 W/m².°C

From equation (6)

t = fin thickness = 0,003 m

k_t = thermal conductivity of tube = 53 W/m.°C

m = [(2 x 16,11)/(53 × 0,003)]^1/2

= 14,24

D_of = diameter at base of fins = 0,0566 m

D_f 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

D_e = 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\K_f = 2625,5/2414,4

= 1,087

L = Length of flow channell = 1,42 m

D_e/L = 0,04833/1,42

= 0,03404

Finning factor

A_t/A_o = 204,4/1476,52

= 0,1384