Laminar flow and heat transfer of non-newtonian falling liquid film on a horizontal tube with variable surface heat flux

In;. Comm. Heat Mars Transfeer.Vol. 28, No. 8, pp. 1125-l 135. 2001 Copyright 0 2001 Elsevier Science Ltd F&ted in the USA. All rights reserved 073%1933/01/$-see front matter

PII: So7351933(01)00315-3

LAMINAR FLOW ANB HEAT TRANSFER OF NON-NEWTONIAN FALLING LIQUIB FILM ON A HORIZONTAL TUBE WITH VARIABLE SURFACE HEAT FLUX

D. Ouldhadda and A. II Idrissi Laboratoire d’Energetique Faculte des Sciences Ben M’Sik B.P. 7955 Sidi Othmane Casablanca, Morocco

(Communicated

by E. Hahne and K. Spindler)

ABSTRACT An analysis is performed to study the laminar flow and heat transfer of non-Newtonian falling liquid film on a horizontal tube for the case of variable surface heat flux. The inertia and convection terms are taken into account. The governing boundary layer equations are solved numerically using an implicit finite difference method. Of particular interest are the effects of the mass flow rate F, the concentration C of carboxymethylcellulose (CMC) solutions, the exponent m for the power-law surface heat flux, and the tube diameter D on the film thickness profiles, as well as on the local and average Nusselt numbers. It was found that an increase in the mass flow rate I and exponent value m increases the local and average heat transfer rates. Finally, the present simulation is found to be in good agreement with previous experimental and numerical results for Newtonian films. Ca2001 Elsevier Science Ltd Introduction Heat transfer to liquid films flowing on horizontal tubes is of great importance for many industrial applications

such as chemical engineering

systems, petrochemical experimental

operations, food and polymer processing

industries, cooling

and distillation plants, and in ocean thermal energy conversion systems. Several

and theoretical

investigations

are reported in the literature concerning the heat transfer in

falling liquid films on horizontal tubes both for laminar and turbulent flow conditions evaporating and non-evaporating

liquid films. In all these studies, the fluid considered

[l-8]

with the

was Newtonian

with the thermal boundary condition of constant wall heat flux or constant wall temperature. Relatively modest attention has been devoted to thin films of non-Newtonian their Newtonian

counterpart.

concerned essentially

liquids as compared with

In the main, the studies related to non-Newtonian

plane geometry’s. The heat transfer from a constant temperature

heat flux wall into a hydrodynamically

fully developed non-Newtonian 1125

liquid films were wall or constant

films was considered by Yih and

1126

D. Ouldhadda and A. II Idrissi

Vol. 28, No. 8

Lee [9] and Gorla [IO]. The problem of heat transfer from an isothermal inclined or vertical plate in accelerating non-Newtonian

power-law films has been studied by Murthy and Sarma [ 111, and recently

by Shang and Andersson [ 121. In a very recent paper, Rao [ 131 measured experimentally in a fully developed non-Newtonian

the heat transfer

fluid films falling down a vertical tube with uniform wall heat flux.

On the other hand, the studies devoted to the horizontal cylindrical configurations

are not received

attention, except for the simplified one undertaken by Sarma and Saibabu [14]. In this reference, the authors used an approximate be expressed difference

integral method assuming that the velocities and temperatures

as a power series. More recently,

Ouldhadda

et al. [ 151 employed

method to analyze the problem of heat transfer in non-Newtonian

profiles can

an implicit finite

falling liquid film on a

horizontal circular cylinder for both constant surface heat flux and constant wall temperature. From the above discussion Newtonian

it is clear that no study has been conducted

falling liquid film on horizontal tubes that have nonuniform

for heat transfer in non-

surface heat flux. This has

motivated the present study in which the surface of the tube is subjected to a power-law variation in the surface heat flux, q,(x) = qo(l+ruc)“. The governing equations are solved numerically using an implicit finite difference

scheme. The effects of the concentration

value of the exponent

C of CMC solutions, mass flow rate r, and

m on the film thickness profiles, as well as on the local and average Nusselt

numbers are examined.

Formulation and Analvsis Pbvsical Model A liquid sheet of non-Newtonian

fluids without splashing impinges on the upper stagnation line of

horizontal tube which is heated under variable surface heat flux. The physical model and coordinate system used are shown in Fig. 1. The surface of the tube is maintained at a power-law variation in surface heat flux, qw(x) = qo (l+ux)“‘, where a is a constant and m is the exponent. The liquid impinging on the tube with an inlet temperature To and an initial mass flow rate of 2lY. The coordinate x is measured from the stagnation point of the tube and they one is measured normally from the surface wall to the fluid. The corresponding convection

velocity components

in the x and y directions are u and v, respectively.

The inertia and

terms are retained in this analysis. In this paper, the flow behavior of the aqueous CMC

solutions was characterized by the power-law model of Ostwald-de Waele: r = Kv"

(1)

where r is the shear stress, K is the fluid consistency index, + is the shear rate and n is the flow behavior index. The rheological model represents pseudoplastic case n = 1 represents a Newtonian fluid.

fluid if n < 1 and dilatant for n > 1. The particular

Vol. 28, No. 8

NON-NEWTONIAN FALLING LIQUID FILM

1127

2r I

110 T,

filmsurface lg

FIG. 1 Physical model and coordinate system. For Newtonian fluids, the film Reynolds number is defined by Re = 4I1~ with I and p denoting, respectively, the mass flow rate per unit length and the dynamic viscosity. Concerning the Newtonian liquid, several experimental observations indicated that the film flow is laminar if Re 2 3 15 [4,16], and the transition from laminar to turbulent flow correspond to Re =” 4000-6000 [2,17]. Consequently, in the majority of investigations, the flow is considered laminar when the film Reynolds number Re is contained between 400 and 4000. For water film at To= 20 ‘C, the value: 400 and 4000 of Reynolds number correspond to the values 0.1 and 1 (kg/m s) of the mass flow rate I. Since nonNewtonian liquids have higher viscosities than most Newtonian liquids, the assumption of laminar flow is realistic in many circumstances. From the bibliographical data above-mentioned, we adopt that the flow in the film is laminar for 0.1 I r (kg/m s) Il. In addition to these indications, the following simplifying assumptions are made: (1) The problem is two-dimensional and the flow is steady-state; (2) the liquid film thickness is considered to be thin compared to the tube radius; (3) the film surface is smooth; (4) at the free surface of the film, the shear stress is negligible; (5) the effects of interfacial waves and surface tension at the free surface of the liquid are assumed to be negligible; (6) the density of the gas or vapor outside the film is negligible. Governine Eouations for the Liauid Film

Considering the preceding hypotheses, the equations of continuity, momentum and energy, for an incompressible non-Newtonian fluid, may be respectively written as

1128

D. Ouldhadda and A. 11Idrissi

Vol. 28. No. 8

(2)

uaU*vau ax

= gsin(()

ay

uz + &T ax ay

+

(3)

_ --- 1

d2T

= q,(x)

= qo(l+ax)”

(4)

PC, ay2

The associated boundary conditions are at y=O:

dT -L--ay

u = v = 0,

au

aty=&

ay

dT =0,

0,

ay= T = T,,

(54

for heating

(5b)

for evaporation

The conservation of the film mass flow rate per unit length is expressed by

(6)

I- = djp,rdy 0

The evaporation case is studied in this work to compare our numerical results with those found in the literature for Newtonian films. The system of equations (2)-(6) can be transformed following dimensionless

into a dimensionless

form by introducing the

variables:

6, = aRei(“+‘) ,

A=aR

R UC

,,Re”(“+‘)

I_,

m

v=

u_

,

V-To 1

(q=

a,R

f(* 2

VW

0)

Re’;/(“+‘) m

The transformation

yields: au *

uau 5

?#lda - _LU6, d4

- Lds,uBU 6, W ti --L!SuE+__=__ Use a4 6, d4

+

vav

au

+_

4

=()

6, 4~ =

6, q

(9)

Fr v de

ti

1 av

6, q

i Pr,6,2

a*e **

(10)

where Fr, Re,, Pr,,,, and U, are the Froude number, the modified film Reynolds number, the modified Prandtl number, and the reference velocity, respectively.

Vol. 28, No. 8

8

NON-NEWTONIAN

2

F,.=

2,

prm-

FALLING LIQUID FILM

Kbn 7u,= 1

a

R”r

1129

1/(2-n)

&-$(“+‘),

[ P

R”

(11)

The transformed boundary conditions are at q=O:

atq=l:

30 *

(I= v=o,

86

BU -= ti

0,

- 0,

e = 0, I-*

and

zj--

=-s,(l+h#)m

where I’* = r /p u,S is the dimensionless

(W

for heating

Wb)

for evaporation

= i”dq 0 mass flow rate.

(13)

Heat Transfer Coefficient The Nusselt number Nu characterizing the heat transfer in dimensionless

form is defined by

WL,

Nu(+) = -

where h(#) = q,/(T,” - Tc)represents characteristic

(14)

L

the local heat transfer coefftcient,

L, = Rl Arz-2-“Y(2+n) is a

length with Ar,,, = [gR’*+“H*_“‘]l[(K/p)“‘2~ “)] is the modified Archimedes number.

Also of interest is the average Nusselt number &

defined as -

where i is the average heat transfer coefftcient defined by ?J = $(&dm

(16)

Numerical Resolution The conservation

equations (8)-(10) with the appropriate

boundary conditions

(12a-b)

are solved

numerically by using a fully implicit finite difference scheme. The angular step used for calculations one degree in the ( direction and 90 intervals are used in the q direction. The diffusion

is

and normal

convective terms are approximated by central differences and longitudinal convective terms by upstream differences.

The resulting system of algebraic equations, for U and 0, is then written in a tridiagonal

matrix form that can be solved efftciently by de Thomas algorithm [ 181. At each ) angular position, the V

1130

D. Ouldhadda and A. 11Idrissi

Vol. 28, No. 8

normal velocity is directly calculated from the continuity equation (Eq. 8). The integral representation

of

the overall mass conservation

of

(Eq. 13) is evaluated by the Simpson numerical method. Convergence

the solution for each ( location is considered achieved when the relative errors in the film thickness and the longitudinal velocity between two successive

iterations are less than lo- 4. The maximum relative

error is adopted for the longitudinal velocity.

Results and Discussion Phvsical Prooerties of Studied Fluids Several aqueous solutions of carboxymethylcellulose fluids have a pseudoplastic

(CMC) are studied here. These non-Newtonian

behavior (n c 1). For these fluids, the thermophysical

properties, C,, p and It,

are supposed to be constant and equal to those of water. According to Brewster and Irvine [19] their evaluation at the temperature To= 20 “C is C, = 4185 J/kg K,

p = 998.2 kg/m’,

/z = 0.598 W/m K

(17a)

The viscous properties for water from reference [ 193 at To= 20 ‘C are K = 1.071 lO”Pas,

n = 1.0

(17b)

Also from reference [ 193, the viscous properties for CMC solutions in water at To= 20 “C are K = 1.045 10-4Co.g4g2, n = 1.193 C-oo”82 C is the concentration

(18)

of CMC expressed in parts per million by weight (ppm): 50 s C (ppm) I 1000.

Typical numerical results for the case of power-law variation of surface heat flux were obtained for To = 20 “C, A = 0.3 and tube diameter D of 0.06 and 0.12 m with F ranging from 0.1 to 0.6 kg/m s, and the concentration

C from 50 to 1000 ppm. The values of the exponent m were limited to -0.5 I M I 1.

Evolution of the Film Thickness Figure 2 shows the evolution of the film thickness &of non-Newtonian

fluid as a function of the polar

angle 4 for different values of the mass flow rate I with C = 1000 ppm. We observe that the film thickness increases with increasing values of I’. Furthermore,

the variations of the film thicknesses

are

more rapid in the vicinity of the stagnation point. This result can be explained by the dominance of the inertial forces in the conservation solution with the analytical

of momentum equation. In addition, Fig. 2 compares the numerical

solution, which ignores the effects

of inertia terms. As expected,

predictions of film thickness by the numerical and analytical methods had negligible differences

the

at small

flow rates, while at increasing mass flow rate the numerical solution gradually departs from the analytical solution. This indicates that inertia effects are important and cannot be ignored at higher values of mass flow rate I. The effect of the concentration

C on the variation of the film thickness is illustrated in Fig. 3

for F = 0.4 kg/m s and D = 0.06 m. It is seen from the figure that the local film thickness increases as C increases. In addition, the variation of the film thickness is more rapid for larger concentration

C.

NON-NEWTONIAN

Vol. 28, No. 8

1131

FALLING LIQUID FILM

0.005 -with -.

-C= 1000 ppm . .. . . . . . . . . c x 300 ppm

inertia terms

-

wthout

Inertia terms

-.-C-

0.004

. . . . Water

II

IOOppm lilm

r =04kg/ms D=006m

0.000 0

60

YO

150

ILTU

30

60

b(degrees)

I20

150

$(degrees) FIG. 3 Effect of C on the film thickness profile.

FIG. 2 Effect of I- on the film thickness profile.

Heat Transfer Coeffkients variation of and

Nusselt number

for different

flux. We

of C, that the

the development accompanied an

by

the variation local

C and m. It

increasing

Nu. Also,

exponent m approximately

of

exponent m

be seen

% when

of D,

this figure

average Nusselt

exponent m

the increasing

to I

in

are shown

I’ =

as (

larger tube. transfer when Fig. 6

various significant

and the

of of Nu

large. The kg/m s

the

smaller [6,15].

C implies I- and

when C from -

value of

is foreseeable

increases with is more

a given

tube than

function of

causes

Figure 5

For a

enhancement

C

noted from

diameter tube

obtained for

a given

average Nusselt

D and

D. This

results Nuas

average Nusselt

increasing of

the tube

rapidly on

transfer coefficients

concentration

value of

increasing tube

results show,

increase of

can also

various values

layer increases

downstream due

transfer rate.

flux rate

4

constant surface

number. Indeed,

a larger

number decreases the thermal

increases.

shows that

of the

number Nu

presented on

entrance and

in the K and

angle 4,

corresponds to

high near

increases with of m

a higher

m=

layer. Figure

consistency

the local

addition, the

The case

very significant

a higher

of the

transfer coefficient

local Nusselt

of C.

values

I, and

the thermal

of the

figure that

as a

C=

ppm.

Vol. 28, No. 8

D. Ouldhadda and A. 11Idrissi

1132

2.0

r

-m-l

.. .. . _._

m

=

0

=04kglms

0 =006m

1.5

m=.cls

.

-r=Odkgims r=0.4wmr

.

-.-

c =loooppn m-l

r=o.)Lg/ms

3 1.0

0

60 +@egrees) 90 120

30

150

t

J 180

0.01"""""' 30 0

60

90

120

150

180

4@egnW

FIG. 4 Effect of C and m on local Nusselt number.

FIG. 5 Effect of Iand D on local Nusselt number.

1.5 -m-l

D=OMm

... ...... ” = 0 1.2 - -.mc.05

C=lOCOppm

3coppm

0.6

0.3

....... _._., ,,,,.... ~~..~sY~ ..rss..~:;._._._._.___ Y.i.. ~..YY~i.Y.~.YI-,.I.,.~~... ,~~w,.~,‘,~YYYT 1oOppm

-

0.01 ’ 0.1

’

n ’

’

0.2

0.3

0.4

Wdm

s)

’

’ 0.5

1

400

0.6

800

1200

1600

2000

2400

Re

FIG. 7 Corrected average Nusselt number versus Re for water film evaporation and comparison with literature results (qw= const).

FIG. 6 Average Nusselt number versus I for different values of C and m.

Model Validation To validate the present model, we compared our numerical results with those found in the literature, concerning the evaporation of water films (n = 1). In this case, the both inlet temperature

and interface

liquid-gas temperature are equal to the saturation temperature z.. At the wall of the tube, a constant heat flux is imposed (m = 0). On the Fig. 7, we presented the evolution of the corrected average Nusselt

l

number, Nu , by the Zazuli’s empirical correlation (Eq. 19) [6,8] mentioned below. This to take account of the effects of the interfacial waviness in the range of Newtonian Reynolds numbers Re studied.

%’ = 0.687Re’”

%

(19)

Vol. 28, No. 8

NON-NEWTONIAN FALLING LIQUID FILM

1133

In this figure, the results of present work are compared with experimental results of Liu [3], and Chyu and Bergles [S]. Our results were also compared with numerical solutions of Kocamustafaogullari and Chen [6], and those of Lorenz and Yung [I] (reported in Ref. [6]). The conditions of comparison are given in this figure. It shows that the present results are found to be in very good agreement with previous experimental and numerical results.

Conclusion

This paper presents a numerical investigation of laminar flow and heat transfer of non-Newtonian falling liquid film on a horizontal tube for power-law variation in surface heat flux. The governing transfer equations were solved numerically by using an implicit finite difference method. Numerical results for the film thickness profiles, and the local and average Nusselt numbers were presented and discussed. Comparisons with previously published results for Newtonian films were performed and found to be in excellent agreement. The major findings of this study are: (1) The inertia terms cannot be neglected because its contribution is of the same order of magnitude as those of the viscous and gravity terms in the momentum equation, (2) the local heat transfer coefficients were found to decrease gradually from a large value near the entrance due to the development of the thermal boundary layer starting at the entrance section, and (3) the local and average heat transfer coefftcients are found to increase with increasing concentration C, increasing mass flow rate I, increasing value of the exponent m, and decreasing tube diameter D. Nomenclature a

constant in the power-law variation of the surface heat flux

Arm

modified Archimedes number

c

concentration of CMC solutions, ppm

CP

specific heat, J/kg K

D

diameter of the tube, m

Fr

Froude number defined in Eq. (11)

g

acceleration of gravity, m/s*

h

local heat transfer coefftcient, kW/m’ K

i

average heat transfer coefftcient, kW/m* K

K

fluid consistency index, Pa s

L

film characteristic length, m

m

exponent in the power-law variation of the surface heat flux

n

flow behavior index

D. Ouldhadda and A. 11Idrissi

1134

NU

local Nusselt number

Nu

average Nusselt number

Nu*

corrected average Nusselt number defined in Eq. (19)

wm

parts per million

Pr,

modified Prandtl number defined in Eq. (11)

40

surface heat flux at the entrance, W/m*

qw

surface heat flux, W/m*

R

radius of the tube, m

Re

Newtonian Reynolds number

Re,

modified Reynolds number for non-Newtonian

T

temperature, “C

TS

saturation temperature, T

To

inlet temperature, “C

TW

wall temperature, “C

l.4

velocity component in x-direction, m/s

ur

reference velocity (Eq. 11)), m/s

V

velocity component in y-direction, m/s

u

dimensionless

velocity component in x-direction

V

dimensionless

velocity component in y-direction

x

streamwise coordinate along the body surface, m

Y

normal coordinate to the surface, m

Greek symbols thermal diffusivity, m*/s mass flow rate per unit length, kg/m s dimensionless

mass flow rate

film thickness, m dimensionless

film thickness

polar angle, degree shear rate, se’ dimensionless

coordinate normal to the surface

dimensionless

temperature

thermal conductivity, W/m K coefficient defined in Eq. (7a) Newtonian dynamic viscosity, kg/m s

fluids defined in Eq. (11)

Vol. 28, No. 8

Vol. 28, No. 8

P

density of fluid, kg/m’

7

shear stress, Pa

NON-NEWTONIAN FALLING LIQUID FILM

1135

Subscripts s

saturation condition

W

wall condition

0

condition at entrance

References 1. J.J. Lorenz, and D. Yung, Combined boiling and evaporation of liquid films on horizontal tubes, Proc. 5th Ocean Thermal Energy Conversion Conf, Miami Beach, USA, vol. 3, pp. 46-70, (1978).

2. W.H. Parken, and L.S. Fletcher, Heat transfer in thin liquid films flowing over horizontal tubes, Proc. 7th Int. Heat Transfer Conf, Munich, Germany, vol. 6, pp. 415-420, (1982).

3. P. Liu, The evaporating falling film on horizontal tubes, Ph.D. thesis, University of Wisconsin, Madison, Wisconsin (1975). 4. J. Mitrovic, Influence of tube spacing and flow rate on heat transfer from a horizontal tube to a

falling liquid film, Proc. 8th Znt.Heat Transfer Conf, San Francisco, USA, vol. 4, pp. 1949-1956, (1986). 5. M.C. Chyu, and A.E. Bergles, ASME J. Heat Transfer 109,983 (1987). 6. G. Kocamustafaogullari, and I.Y. Chen, AIChE J 34, 1539 (1988). I.

J.T. Rogers, and S.S. Goindi, Can. J. Chem. Engng 67,560 (1989).

8. A. Agunaoun, A. DaIf, M. Grisenti, and R. Barriol, Can. J. Chem. Engng 72,961 (1994). 9. S.M. Yih, and M.W. Lee, Int. J. Heat Mass Transfer 29, 1999 (1986). 10. R.S.R. Gorla, J. Thermophysics Heat Transfer 5,444 (1991). 11. V.N. Murthy, and P.K. Sarma, Int. J. Multiphase Flow 4,413 (1978). 12. D.Y. Shang, and HI. Andersson, Int. J. Heat Muss Transfer 42,2085 (1999). 13. B.K. Rao, Int. J. Heat Fluid Flow 20,429 (1999). 14. P.K. Sarma, and J. Saibabu, W&me- und Stoj%bertragung 27,489 (1992). 15. D. Ouldhadda, A. II Idrissi, and M. Asbik, Heat and Mass Transfer, in press. 16. E.N. Ganic, and M.N. Roppo, ASMEJ. Heat Transfer 102,342 (1980). 17. T. Ueda, and H. Tanaka, Int. J. Multiphase Flow 2,261 (1975). 18. D.A. Andersson, J.C. Tannehill, and R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Hemisphere, New York (1984).

19. A. Brewster, and T.F. Irvine Jr., Int. J Heat Muss Transfer 32,95 1 (1989). Received August IO, 2001