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Hydrodynamic entrance length of non-newtonian liquid films
Chrmicol Printed
Engineering
Science.
in GreatBritain.
Vol. 45, No.
2, pp. 537-541.
1990. c
HYDRODYNAMIC ENTRANCE LENGTH NON-NEWTONIAN LIQUID FILMS HELGE Division of Applied Mechanics, (First
I. ANDERSSON and FRIDTJOV The Norwegian Institute of Technology,
received 2 January
M)o9-2509/90 33.00 + 0.00 1990 Pergamon Press plc
OF
IRGENS N-7034
1989; accepted in revisedform
Trondheim-NTH,
26 May
Norway
1989)
Abstract-Accelerating laminar thin-film flow of power-law fluids along vertical walls is considered. The momentum integral method approach is used to predict the flow development in the hydrodynamic entrance region, and two different assumptions are employed for the streamwise velocity profile. It is demonstrated that dilatant films develop more rapidly than films of pseudoplastic fluids. The entrance length predictions compare favourably with experimental data for Newtonian films.
A
INTRODUCTION
In spite of the advent of powerful personal computers and efficient numerical solution algorithms for partial differential equations, the integral method approach is still popular for various reasons: (i) the integral formulation tions is fairly simple and even sometimes be obtained; (ii) the integral solutions ameters which are normally applications; (iii) the effects of different bulence models and boundary be compared.
of the governing equaanalytical solutions may provide the flow parrequired in engineering rheological models, turconditions may readily
Nakayama (1988), for instance, recently described a rapid and accurate calculation procedure for nonNewtonian boundary layer problems, which was based on the integral method. The momentum integral approach is applicable to the developing flow in the hydrodynamic entrance region of gravity-driven thin liquid films, and has been extensively employed in the analyses of Newtonian films over the past 25 years. Although the majority of these investigations are purely theoretical, the comparisons of integral method solutions with experimental data by StiicheIi and ijzisik (1976), Tekic et nl. (1984) and Andersson (1984a) support the continued use of this simple yet fairly accurate predictive method. In spite of their obvious technological importance, for istance in polymer and plastics fabrication, food processing and in coating equipment, only modest attention has been devoted to film flow of nonNewtonian fluids. However, gravity-driven thin-film flow of substances which obey the so-called power-law rheological model al4 “-‘au
TxY =K-
II aY
is the profile. Unfortunately, neither the sinusoidal profile considered by Narayana Murthy and Sarma (1977) nor the second-degree polynomial used by Yang and Yarbrough (1973, 1980) and Narayana Murthy and Sarma (1978) reduce to the exact analytic solution for the velocity as the flow becomes fully developed. The velocity profile proposed more recently by Tekic et al. (1986) does exhibit this important feature, and it is readily demonstrated that their profile reduces to a second-degree polynomial for n = 1. The comparative study of Newtonian films by Andersson (1987) revealed that the application of the second-degree profile leads to underestimation of the predicted extension of the hydrodynamic entrance length. The objective of the present work is therefore to investigate how the flow development and, in particular, the hydrodynamic entrance length, depend on the assumptions made for the velocity profile. More specifically, predictions obtained with the profile assumed by Tekic et al. (1986) will be compared with results obtained using the more flexible velocity profile proposed by Andersson and Irgens (1988). Some new analytical solutions for the streamwise variation of the local film thickness will also be provided. Unfortunately, the few experimental investigations of non-Newtonian film flow which the present authors are aware of [see, for example, Andersson and Trgens (1989)], deal with hydrodynamically fully developed films only. Until experimentalists have provided data for accelerating non-Newtonian films, theoretical entrance length predictions represent the only available results for practicing engineers. To be able to compare the present predictions of the entrance region flow with experimental observations, we have to resort to data obtained for Newtonian films, i.e. for the special case n = 1. crucial
approximation
point
in the integral
analyses
made for the streamwise
velocity
-
8Y
has been considered by Yang and Yarbrough (1973, 1980) and Narayana Murthy and Sarma (1977, 1978), and more recently by Tekic et al. (1986) and Andersson and Irgens (1988, 1989).
ANALYSIS
The basic idea of the integral method approach is to assume a form of the streamwise velocity distribution, and then integrate the governing equations in the 537
538
HELGE
I. ANDERSSON
cross-stream direction: h(x) u(x,_~)d~ = Q I0
(2)
Here, eq. (2) represents the conservation of volumetric flow rate Q, while eq. (3) is the integrated form of the streamwise momentum equation for thin-film flow along a vertical wall. It is readily demonstrated that the velocity profile u(x,y)=~[~][l-~l-~~+‘)“]
(4)
and FRIDTJOV IRGENS
(1988). By neglecting the influence of surface tension included in the analysis of Tekic et al. (1986), their integrated momentum equation reduces to eq. (7) for film flow along vertical walls. While eq. (7) has to be integrated numerically for general n values, exact analytic solutions exist if 2n is equal to an integer. The particular solutions for n = 0.5 and n = 1.5 can be written as A- 1, =f(B) where
B’ f(B) =7In82 _11 2
L
and
assumed by Tekic et al. (1986) reduces to the exact analytic profile as the flow becomes fully developed and the film thickness h(x) reaches its asymptotic value
hm=[yJLp).]+ The same important more flexible profile
feature is also exhibited
Q ““-][l_ lb(x) u(x3y)=h(x) [ l +tn+
(5)
by the
(l_&)%q
proposed by Andersson and Irgens (1988). Now, if one of the approximations (4) or (6) are substituted into the integrated momentum eq. (3), we obtain either
(11)
for n =0.5
(12a)
for n = 1.5. (12b) The integration has been carried out subject to the upstream condition /3= /I0 at 1= il,. A corresponding solution of eq. (7) in the Newtonian case n = 1 was provided earlier by Stilcheli and &isik (1976). The inclusion of surface tension in the analysis of Tekic et al. (1986) prohibited the existence of such simple analytic solutions for the film flow development. RESULTS
(6)
-LlPo)
AND
DISCUSSIONS
While the velocity profile (4) assumed by Tekic et al. (1986) exhibits the same form for all film thicknesses, the shape of the more flexible velocity profile (6) proposed by Andersson and Irgens (1988) crucially depends on the flow depth ratio 8, as shown in Fig. 1. It is observed that the profiles become more uniform, i.e. flatter, as the flow depth ratio increases from unity. Moreover, for a given flow depth ratio the velocity distribution in the pseudoplastic film (n = 0.5) is more uniform than in the dilatant case (n = 1.5).
or 4(1+
dR hp=
[2(1+
l/n)Z/P+8(1+
l/n)P+
l]‘([(l
l/n)/?2”-1+2p2”-2
+ l/n)P+
l]“-(2+
l/n)./P”>
Here, A and p are the dimensionless streamwise coordinate and the local flow depth ratio defined as
and Re=
Q
K/P is a generalized Reynolds power-law fluids.
(hc>n--l &
number for film flow
(10) of
SOLUTIONS
The numerical integration of the ordinary differential eq. (7) or (8) can easily be carried out on any programmable desk calculator. In the particular case of a Newtonian fluid, i.e. n = 1, an exact analytic solution of eq. (8) was provided by Andersson and Irgens
(8)
The variation of the local film thickness along the wall are displayed in Fig. 2 for two different power-law fluids. Both set of solutions converge asymptotically to /? = 1, i.e. to the correct downstream film thickness h, given in eq. (5). A somewhat slower acceleration of the film is obtained when the velocity profile is approximated by eq. (6) instead of by eq. (4), thereby indicating that the least flexible profile leads to the fastest flow development. This is consistent with the recent findings of Andersson (1987) for Newtonian films. Figure 2 moreover indicates that liquid films of dilatant fluids are subject to a more rapid development than are the pesudoplastic films. Van der Mast et al. (1976) observed experimentally that the heat transfer in the inlet section of a falling film evaporator contributes significantly to the average overall heat transfer coefficient. The ability to accurately predict the flow development in the hydro-
Hydrodynamic
entrance length of non-Newtonian
liquid films
539
Fig. 1. The velocity profile (6) for a pseudoplastic film n = 0.5 (broken lines) and a dilatant film n = I.5 (solid lines). Different values of the dimensionless film thickness /7.
-
profile
(6)
---
profile
(4)
h
Fig. 2. Streamwise variation of the dimensionless film thickness for two different power-law fluids. Solid and broken lines correspond to the velocity profiles (6) and (4), respectively.
dynamic entrance region of the thin-film flow is therefore of crucial importance for the estimation of heat or mass transfer rates. Figure 3 shows the predicted entrance length, defined as the streamwise distance from the inlet A,, to the position A, at which the film thickness has reached its asymptotic value (5) to within 2%. Some finite-difference solutions of the twodimensional streamwise momentum equation for Newtonian film flows by Andersson (1984b) have been included for comparison. It is readily observed that results obtained with the flexible profile (6) is closer to the experimental data point due to Bertschy et al. (1983) and the accurate finite-difference solutions for n
= 1, than are the results obtained with the velocity profile (4). In accordance with the present results displayed in Fig. 3, the finite-difference calculations of Yilmaz and Brauer (1973) for Newtonian films indicated that /2,- ;LO= constant
(13)
for the larger values of the dimensionless upstream film thickness &. In order to compare the present predictions with the experimental data of Fulford (1962) for n = 1, eq. (13) is rewritten as (& - &)(Q/v)~/~ = constant x (Q/v)~‘~
(14)
HELGE 1. ANDERSSON and FRIDTJOV IRGENS
540 2 01
1
n= 1.5 ____-profile
------l
profile
- - -. (66) (4)
Andersson
0 Bertschy
.lP 1.0
CONCLUSIONS
Two different assumptions for the streamwise velocity profile have been used to predict the flow development in the hydrodynamic entrance region of power-law films. In both cases it is found that a dilatant film develops more rapidly than a film of a pseudoplastic fluid. However, the velocity profile (4) due to Tekic et al. (1976) leads to a more rapid flow development than the profile (6) assumed by Andersson and Irgens (1988). Since the entrance length predictions obtained with the latter assumption compare more favourably with experimental data for Newtonian films, it can be anticipated that the profile (6) also
et al. 10
Fig. 3. Predicted hydrodynamic entrance length for arbitrary upstream flow depth ratio p,,. Solid and broken lines correspond to the velocity profiles (6) and (4), respectively. The filled symbols denote finite-difference results by Andersson (1984b) while the open symbol denotes experimental data by Bertschy et al. (1983) for n = 1.
represents
the development
of power-law
films
more realistically than eq. (4). No experimental data for developing power-law films is known to the authors, and it is therefore suggested that more effort should be directed in laboratory studies of the entrance region flow of nonNewtonian liquid films.
NOTATION
(ME _ ~,) CQ,vj/3
-
profile
(61
- --
profile
(4)
Fulford
A,/
_) //’
I
f 9 h
h, K
; Re
u x Y Greek a
1 V
P z
dimensionless gravitational
function acceleration
film thickness asymptotic film thickness (5) coefficient of consistency power-law index volumetric flow rate per unit width generalized Reynolds number (10) streamwise velocity component streamwise coordinate cross-stream coordinate letters
local flow depth ratio (9) dimensionless streamwise coordinate kinematic viscosity density shear stress
(9)
REFERENCES Fig. 4. Predicted hydrodynamic entrance length for Newtonian films with large upstream flow depth ratio PO. Solid and broken lines correspond to the velocity profiles (6) and (4), respectively. The symbols denote experimental data of Fulford (1962) obtained from Fig. 6 in Yilmaz and Brauer ( 1973).
where v is the kinematic viscosity of the Newtonian fluid. The left hand side of eq. (14) is plotted versus the Reynolds number Q/V in Fig. 4, and it is observed that the somewhat larger entrance length obtained with the velocity profile (6) compares more favourably with the experimental data points than do results obtained with the profile (4) due to Tekic et al. (1986).
Andersson, H. I., 1984a, On integral method predictions of laminar film flow. Chem. Engng Sci. 39, 1005-1010. Andersson, H. I., 1984b, Numerical solutions of a TSL-model for free-surface flows. Notes on Numerical Fluid Mechanics, Vieweg Verlag, 7, 9-16. Andersson, H. I., 1987, The momentum integral approach to laminar thin-film flow. ASME Symp. Thin Fluid Films, FED 48, 7-13. Andersson, H. 1. and lrgens, F., 1988, Gravity-driven laminar film flow of power-law fluids along vertical walls. J. NonNewtonian Fluid Mech. 27, 153-172. Andersson, H. I. and Irgens, F., 1989, Film flow of power-law fluids, in Encyclopedia of Fluid Mechanics (Edited by N. P. Cheremisinoff), Vol. 9, Gulf Publishing, Houston, TX (in preparation). Bertschy, J. R., Chin, R. W. and Abernathy, F. H., 1983, Highstrain-rate free-surface boundary layer flows. J. Fluid Mech. 126, 443461. Fulford, G. D., 1962, Gas-liquid flow in an inclined channel. Ph.D. thesis, University of Birmingham, U.K.
Hydrodynamic
entrance length of non-Newtonian
Nakayama, A., 1988, Integral methods for forced convection heat transfer in power-law non-Newtonian fluids, in Encyclopedin of Fluid Mechanics (Edited by N. P. Cheremisinoffi. Vol. 7. DD. 3055339. Gulf Publishina. ’ 1Houston, TX. ” Naravana Murthv. V. and Sarma, P. K., 1977, A note on hydrodynamic _ ‘entrance lengths of non-Newtonian laminar falling liquid films. Chem. Engng Sci. 32, 566567. Narayana Murthy, V. and Sarma, P. K., 1978, Dynamics of developing laminar non-Newtonian falling liquid films with free surface. ASME J. nppl. Mech. 45, 19-24. Stiicheli, A. and &isik, M. N., 1976, Hydrodynamic entrance lengths of laminar falling films. Chem. Engng Sci. 31, 369-372. Tekic, M. N., Posarac, D. and Petrovic, D., 1984, Entrance region lengths of laminar falling films. Chem. Engng Sci. 39, 165-167.
liquid films
541
Tekic, M. N., Posarac, D. and Petrovic, D., 1986, A note on the entrance region lengths of non-Newtonian laminar falling films. Chem. Engng Sci. 41, 323&3232. Van der Mast, V. C., Read, S. M. and Bromley, L. A., 1976, Boiling of natural sea water in falling film evaporators. Desalination 18, 71-94. Yang, T. M. T. and Yarbrough, D. W., 1973, A numerical study of the laminar flow of non-Newtonian fluids along a vertical wall. ASME J. appl. Mech. 40, 290-292. Yang, T. M. T. and Yarbrough, D. W., 1980, Laminar flow of non-Newtonian liquid films inside a vertical pipe. Rheol. Acta 19, 432436. Yilmax, T. and Rrauer, H., 1973, Beschleunigte Striimung von Fliissigkeitsfilmen an ebenen Wanden und in Fiillktirperschichten, Chemie-Ingr-Tech. 45, 928-934.
Engineering
Science.
in GreatBritain.
Vol. 45, No.
2, pp. 537-541.
1990. c
HYDRODYNAMIC ENTRANCE LENGTH NON-NEWTONIAN LIQUID FILMS HELGE Division of Applied Mechanics, (First
I. ANDERSSON and FRIDTJOV The Norwegian Institute of Technology,
received 2 January
M)o9-2509/90 33.00 + 0.00 1990 Pergamon Press plc
OF
IRGENS N-7034
1989; accepted in revisedform
Trondheim-NTH,
26 May
Norway
1989)
Abstract-Accelerating laminar thin-film flow of power-law fluids along vertical walls is considered. The momentum integral method approach is used to predict the flow development in the hydrodynamic entrance region, and two different assumptions are employed for the streamwise velocity profile. It is demonstrated that dilatant films develop more rapidly than films of pseudoplastic fluids. The entrance length predictions compare favourably with experimental data for Newtonian films.
A
INTRODUCTION
In spite of the advent of powerful personal computers and efficient numerical solution algorithms for partial differential equations, the integral method approach is still popular for various reasons: (i) the integral formulation tions is fairly simple and even sometimes be obtained; (ii) the integral solutions ameters which are normally applications; (iii) the effects of different bulence models and boundary be compared.
of the governing equaanalytical solutions may provide the flow parrequired in engineering rheological models, turconditions may readily
Nakayama (1988), for instance, recently described a rapid and accurate calculation procedure for nonNewtonian boundary layer problems, which was based on the integral method. The momentum integral approach is applicable to the developing flow in the hydrodynamic entrance region of gravity-driven thin liquid films, and has been extensively employed in the analyses of Newtonian films over the past 25 years. Although the majority of these investigations are purely theoretical, the comparisons of integral method solutions with experimental data by StiicheIi and ijzisik (1976), Tekic et nl. (1984) and Andersson (1984a) support the continued use of this simple yet fairly accurate predictive method. In spite of their obvious technological importance, for istance in polymer and plastics fabrication, food processing and in coating equipment, only modest attention has been devoted to film flow of nonNewtonian fluids. However, gravity-driven thin-film flow of substances which obey the so-called power-law rheological model al4 “-‘au
TxY =K-
II aY
is the profile. Unfortunately, neither the sinusoidal profile considered by Narayana Murthy and Sarma (1977) nor the second-degree polynomial used by Yang and Yarbrough (1973, 1980) and Narayana Murthy and Sarma (1978) reduce to the exact analytic solution for the velocity as the flow becomes fully developed. The velocity profile proposed more recently by Tekic et al. (1986) does exhibit this important feature, and it is readily demonstrated that their profile reduces to a second-degree polynomial for n = 1. The comparative study of Newtonian films by Andersson (1987) revealed that the application of the second-degree profile leads to underestimation of the predicted extension of the hydrodynamic entrance length. The objective of the present work is therefore to investigate how the flow development and, in particular, the hydrodynamic entrance length, depend on the assumptions made for the velocity profile. More specifically, predictions obtained with the profile assumed by Tekic et al. (1986) will be compared with results obtained using the more flexible velocity profile proposed by Andersson and Irgens (1988). Some new analytical solutions for the streamwise variation of the local film thickness will also be provided. Unfortunately, the few experimental investigations of non-Newtonian film flow which the present authors are aware of [see, for example, Andersson and Trgens (1989)], deal with hydrodynamically fully developed films only. Until experimentalists have provided data for accelerating non-Newtonian films, theoretical entrance length predictions represent the only available results for practicing engineers. To be able to compare the present predictions of the entrance region flow with experimental observations, we have to resort to data obtained for Newtonian films, i.e. for the special case n = 1. crucial
approximation
point
in the integral
analyses
made for the streamwise
velocity
-
8Y
has been considered by Yang and Yarbrough (1973, 1980) and Narayana Murthy and Sarma (1977, 1978), and more recently by Tekic et al. (1986) and Andersson and Irgens (1988, 1989).
ANALYSIS
The basic idea of the integral method approach is to assume a form of the streamwise velocity distribution, and then integrate the governing equations in the 537
538
HELGE
I. ANDERSSON
cross-stream direction: h(x) u(x,_~)d~ = Q I0
(2)
Here, eq. (2) represents the conservation of volumetric flow rate Q, while eq. (3) is the integrated form of the streamwise momentum equation for thin-film flow along a vertical wall. It is readily demonstrated that the velocity profile u(x,y)=~[~][l-~l-~~+‘)“]
(4)
and FRIDTJOV IRGENS
(1988). By neglecting the influence of surface tension included in the analysis of Tekic et al. (1986), their integrated momentum equation reduces to eq. (7) for film flow along vertical walls. While eq. (7) has to be integrated numerically for general n values, exact analytic solutions exist if 2n is equal to an integer. The particular solutions for n = 0.5 and n = 1.5 can be written as A- 1, =f(B) where
B’ f(B) =7In82 _11 2
L
and
assumed by Tekic et al. (1986) reduces to the exact analytic profile as the flow becomes fully developed and the film thickness h(x) reaches its asymptotic value
hm=[yJLp).]+ The same important more flexible profile
feature is also exhibited
Q ““-][l_ lb(x) u(x3y)=h(x) [ l +tn+
(5)
by the
(l_&)%q
proposed by Andersson and Irgens (1988). Now, if one of the approximations (4) or (6) are substituted into the integrated momentum eq. (3), we obtain either
(11)
for n =0.5
(12a)
for n = 1.5. (12b) The integration has been carried out subject to the upstream condition /3= /I0 at 1= il,. A corresponding solution of eq. (7) in the Newtonian case n = 1 was provided earlier by Stilcheli and &isik (1976). The inclusion of surface tension in the analysis of Tekic et al. (1986) prohibited the existence of such simple analytic solutions for the film flow development. RESULTS
(6)
-LlPo)
AND
DISCUSSIONS
While the velocity profile (4) assumed by Tekic et al. (1986) exhibits the same form for all film thicknesses, the shape of the more flexible velocity profile (6) proposed by Andersson and Irgens (1988) crucially depends on the flow depth ratio 8, as shown in Fig. 1. It is observed that the profiles become more uniform, i.e. flatter, as the flow depth ratio increases from unity. Moreover, for a given flow depth ratio the velocity distribution in the pseudoplastic film (n = 0.5) is more uniform than in the dilatant case (n = 1.5).
or 4(1+
dR hp=
[2(1+
l/n)Z/P+8(1+
l/n)P+
l]‘([(l
l/n)/?2”-1+2p2”-2
+ l/n)P+
l]“-(2+
l/n)./P”>
Here, A and p are the dimensionless streamwise coordinate and the local flow depth ratio defined as
and Re=
Q
K/P is a generalized Reynolds power-law fluids.
(hc>n--l &
number for film flow
(10) of
SOLUTIONS
The numerical integration of the ordinary differential eq. (7) or (8) can easily be carried out on any programmable desk calculator. In the particular case of a Newtonian fluid, i.e. n = 1, an exact analytic solution of eq. (8) was provided by Andersson and Irgens
(8)
The variation of the local film thickness along the wall are displayed in Fig. 2 for two different power-law fluids. Both set of solutions converge asymptotically to /? = 1, i.e. to the correct downstream film thickness h, given in eq. (5). A somewhat slower acceleration of the film is obtained when the velocity profile is approximated by eq. (6) instead of by eq. (4), thereby indicating that the least flexible profile leads to the fastest flow development. This is consistent with the recent findings of Andersson (1987) for Newtonian films. Figure 2 moreover indicates that liquid films of dilatant fluids are subject to a more rapid development than are the pesudoplastic films. Van der Mast et al. (1976) observed experimentally that the heat transfer in the inlet section of a falling film evaporator contributes significantly to the average overall heat transfer coefficient. The ability to accurately predict the flow development in the hydro-
Hydrodynamic
entrance length of non-Newtonian
liquid films
539
Fig. 1. The velocity profile (6) for a pseudoplastic film n = 0.5 (broken lines) and a dilatant film n = I.5 (solid lines). Different values of the dimensionless film thickness /7.
-
profile
(6)
---
profile
(4)
h
Fig. 2. Streamwise variation of the dimensionless film thickness for two different power-law fluids. Solid and broken lines correspond to the velocity profiles (6) and (4), respectively.
dynamic entrance region of the thin-film flow is therefore of crucial importance for the estimation of heat or mass transfer rates. Figure 3 shows the predicted entrance length, defined as the streamwise distance from the inlet A,, to the position A, at which the film thickness has reached its asymptotic value (5) to within 2%. Some finite-difference solutions of the twodimensional streamwise momentum equation for Newtonian film flows by Andersson (1984b) have been included for comparison. It is readily observed that results obtained with the flexible profile (6) is closer to the experimental data point due to Bertschy et al. (1983) and the accurate finite-difference solutions for n
= 1, than are the results obtained with the velocity profile (4). In accordance with the present results displayed in Fig. 3, the finite-difference calculations of Yilmaz and Brauer (1973) for Newtonian films indicated that /2,- ;LO= constant
(13)
for the larger values of the dimensionless upstream film thickness &. In order to compare the present predictions with the experimental data of Fulford (1962) for n = 1, eq. (13) is rewritten as (& - &)(Q/v)~/~ = constant x (Q/v)~‘~
(14)
HELGE 1. ANDERSSON and FRIDTJOV IRGENS
540 2 01
1
n= 1.5 ____-profile
------l
profile
- - -. (66) (4)
Andersson
0 Bertschy
.lP 1.0
CONCLUSIONS
Two different assumptions for the streamwise velocity profile have been used to predict the flow development in the hydrodynamic entrance region of power-law films. In both cases it is found that a dilatant film develops more rapidly than a film of a pseudoplastic fluid. However, the velocity profile (4) due to Tekic et al. (1976) leads to a more rapid flow development than the profile (6) assumed by Andersson and Irgens (1988). Since the entrance length predictions obtained with the latter assumption compare more favourably with experimental data for Newtonian films, it can be anticipated that the profile (6) also
et al. 10
Fig. 3. Predicted hydrodynamic entrance length for arbitrary upstream flow depth ratio p,,. Solid and broken lines correspond to the velocity profiles (6) and (4), respectively. The filled symbols denote finite-difference results by Andersson (1984b) while the open symbol denotes experimental data by Bertschy et al. (1983) for n = 1.
represents
the development
of power-law
films
more realistically than eq. (4). No experimental data for developing power-law films is known to the authors, and it is therefore suggested that more effort should be directed in laboratory studies of the entrance region flow of nonNewtonian liquid films.
NOTATION
(ME _ ~,) CQ,vj/3
-
profile
(61
- --
profile
(4)
Fulford
A,/
_) //’
I
f 9 h
h, K
; Re
u x Y Greek a
1 V
P z
dimensionless gravitational
function acceleration
film thickness asymptotic film thickness (5) coefficient of consistency power-law index volumetric flow rate per unit width generalized Reynolds number (10) streamwise velocity component streamwise coordinate cross-stream coordinate letters
local flow depth ratio (9) dimensionless streamwise coordinate kinematic viscosity density shear stress
(9)
REFERENCES Fig. 4. Predicted hydrodynamic entrance length for Newtonian films with large upstream flow depth ratio PO. Solid and broken lines correspond to the velocity profiles (6) and (4), respectively. The symbols denote experimental data of Fulford (1962) obtained from Fig. 6 in Yilmaz and Brauer ( 1973).
where v is the kinematic viscosity of the Newtonian fluid. The left hand side of eq. (14) is plotted versus the Reynolds number Q/V in Fig. 4, and it is observed that the somewhat larger entrance length obtained with the velocity profile (6) compares more favourably with the experimental data points than do results obtained with the profile (4) due to Tekic et al. (1986).
Andersson, H. I., 1984a, On integral method predictions of laminar film flow. Chem. Engng Sci. 39, 1005-1010. Andersson, H. I., 1984b, Numerical solutions of a TSL-model for free-surface flows. Notes on Numerical Fluid Mechanics, Vieweg Verlag, 7, 9-16. Andersson, H. I., 1987, The momentum integral approach to laminar thin-film flow. ASME Symp. Thin Fluid Films, FED 48, 7-13. Andersson, H. 1. and lrgens, F., 1988, Gravity-driven laminar film flow of power-law fluids along vertical walls. J. NonNewtonian Fluid Mech. 27, 153-172. Andersson, H. I. and Irgens, F., 1989, Film flow of power-law fluids, in Encyclopedia of Fluid Mechanics (Edited by N. P. Cheremisinoff), Vol. 9, Gulf Publishing, Houston, TX (in preparation). Bertschy, J. R., Chin, R. W. and Abernathy, F. H., 1983, Highstrain-rate free-surface boundary layer flows. J. Fluid Mech. 126, 443461. Fulford, G. D., 1962, Gas-liquid flow in an inclined channel. Ph.D. thesis, University of Birmingham, U.K.
Hydrodynamic
entrance length of non-Newtonian
Nakayama, A., 1988, Integral methods for forced convection heat transfer in power-law non-Newtonian fluids, in Encyclopedin of Fluid Mechanics (Edited by N. P. Cheremisinoffi. Vol. 7. DD. 3055339. Gulf Publishina. ’ 1Houston, TX. ” Naravana Murthv. V. and Sarma, P. K., 1977, A note on hydrodynamic _ ‘entrance lengths of non-Newtonian laminar falling liquid films. Chem. Engng Sci. 32, 566567. Narayana Murthy, V. and Sarma, P. K., 1978, Dynamics of developing laminar non-Newtonian falling liquid films with free surface. ASME J. nppl. Mech. 45, 19-24. Stiicheli, A. and &isik, M. N., 1976, Hydrodynamic entrance lengths of laminar falling films. Chem. Engng Sci. 31, 369-372. Tekic, M. N., Posarac, D. and Petrovic, D., 1984, Entrance region lengths of laminar falling films. Chem. Engng Sci. 39, 165-167.
liquid films
541
Tekic, M. N., Posarac, D. and Petrovic, D., 1986, A note on the entrance region lengths of non-Newtonian laminar falling films. Chem. Engng Sci. 41, 323&3232. Van der Mast, V. C., Read, S. M. and Bromley, L. A., 1976, Boiling of natural sea water in falling film evaporators. Desalination 18, 71-94. Yang, T. M. T. and Yarbrough, D. W., 1973, A numerical study of the laminar flow of non-Newtonian fluids along a vertical wall. ASME J. appl. Mech. 40, 290-292. Yang, T. M. T. and Yarbrough, D. W., 1980, Laminar flow of non-Newtonian liquid films inside a vertical pipe. Rheol. Acta 19, 432436. Yilmax, T. and Rrauer, H., 1973, Beschleunigte Striimung von Fliissigkeitsfilmen an ebenen Wanden und in Fiillktirperschichten, Chemie-Ingr-Tech. 45, 928-934.