* f t - o >-co o •-• ro O u * 00 ch BIBLIOTHEEK TU Delft P 1256 6168 C 351896
MOMENTUM. HEAT AND
MASS TRANSFER TO A MOVING
CONTINUOUS CYLINDER
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN
OP WOENSDAG 8 OKTOBER 1969 TE 16.00 UUR
DOOR
JOOST WILLEM ROTTE
scheikundig ingenieur geboren te Zundert
3
Bij het tot stand brengen van het werk dat in dit proef-schrift resulteerde zijn velen betrokken geweest. De studen-ten wil ik hiervoor met name bedanken, te westuden-ten:
A.L. Berg F.C. Dijkstra J.K. Eisenloeffel F.V. Engelenburg H. Kimmels P.D, de Levita J.J. Leyen P. Noordsij M.P. Noordzij C.M.S. Raats B.J.M.S. van Rooden J.W.M, van Rijnsoever F.W. Sevenstern M. de Steenwinkel G.L.J. Tummers D.W. Wink
CONTENTS Summary
Introduction
1.1 Introduction 13 1.2 Considerations of practical interest lU
1.3 Formulation of the scientific approach ^k
^.k A closer view of the problem; its characteristics 15
1.5 Structure of the thesis l6 Forced convection fluid flow along a moving
con-tinuous cylinder
2.1 Introduction 19 2.2 Formulation of the laminar flow problem 20
2.3 An exact solution for the moving continuous flat
plate 21 2.U Solutions for the moving continuous cylinder 23
1 General considerations; the Sakiadis solution 23 2 An improved logarithmic velocity profile 25 3 Test of the solutions when the influence of
curvature is small 25
h Comparison of the solutions at larger values
of the curvature parameter 26 2.5 Experimental work in the laminar flow regime 29
1 Introduction 29 2 Review of earlier experimental work 29
2.6 Measurements of the velocity profile in the
laminar boundary layer 30 1 Principle of the experimental method 30
2 Description of the experiments 31
3 Results 33
k Discussion and conclusions 37
2.7 Turbulent momentum transfer 39 1 Transition to turbulence at the flat plate 39
2 The turbulent velocity profile for the flat
plate 39 3 The friction factor for the cylinder at
turbulent flow UO
F o r c e d c o n v e c t i o n h e a t a n d m a s s t r a n s f e r t o a m o v i n g c o n t i n u o u s c y l i n d e r
3.1 I n t r o d u c t i o n k3
3.2 Theoretical solutions for laminar flow 1+3
1 B a s i s o f t h e s o l u t i o n p r o c e d u r e ; t h e O - p a r a m e t e r
s o l u t i o n U 3 2 E f f e c t s o f P r a n d t l n u m b e r , c u r v a t u r e a n d f i n i t e
h e a t c a p a c i t y ; t h e 1-parameter s o l u t i o n 1+5 3 C o m b i n e d e f f e c t s ; t h e g e n e r a l s o l u t i o n 1+9
3.3 Experimental work on forced convection heat t r a n s
-fer 51
1 E a r l i e r e x p e r i m e n t s 51
2 Experiments b a s e d on S a k i a d i s ' conception 52 3.1+ Electrochemical measurements of m a s s transfer
coefficients 53 1 General description of t h e technique 53
2 Description of t h e experimental set-up 55
3 Results 56 1+ Discussion 59 3.5 Conclusions 62 Free convection heat and mass transfer at vertical
cylinders
l+.l Introduction 63
1+.2 Theoretical solutions for laminar flow 6k
1 T h e flat p l a t e s o l u t i o n 61+ 2 The solution for the vertical cylinder 65
1+.3 Experiments on free convection heat trans-fer to
vertical cylinders 67
k.k Application of the electrochemical method to the
measurement of free convection mass transfer
co-efficients 70 1 Introduction 70 2 T h e e v a l u a t i o n o f c o n c e n t r a t i o n d i f f e r e n c e s 71
3 Calculation of the density difference 72 lt.5 Electrochemical measurement of free convection
mass transfer coefficients 7!+ 1 D e s c r i p t i o n o f t h e e x p e r i m e n t a l s e t - u p ^k
2 Results 76 3 Discussion and conclusions 78
7
Contents (cont.)
5_ Combined free and forced convection along moving con-tinuous cylinders
5.1 Introduction 83 5.2 Preliminary discussion on combined flow 83
5.3 Heat transfer meas\irements with a moving
thermo-couple technique 81+
1 I n t r o d u c t i o n 81+
2 Experimental technique 87 3 Results and discussion 88
5.1+ Conclusions 93 _6 Final considerations and conclusions
6.1 Final considerations 95
6.2 Conclusions 95 I A Pohlhausen method for the calculation of boundary
layer parameters at a moving continuous cylinder 97 II A perturbation method for the calculation of the
veloctiy distribution around a moving continuous
cylinder 99 III Physical properties of aqueous solutions, used at the
electrochemical determination of mass transfer
coef-ficients 103
References 105 Notation 109
MOMENTUM HEAT AND MASS TRANSFER TO A MOVING CONTINUOUS CYLINDER
The incentive of the present investigation is to enhance knowledge of the many industrial processes in which continu-ous bodies are handled. These operations are often accompanied by a variety of complex transport processes to groups of threads or sheets. Being a good starting point for scientific research, momentum, heat and mass transfer to one individual thread was selected as the object of this study. A few excep-tions excluded, both theory and experiments deal with the laminar flow regime.
Chapter 2 is devoted to forced convection fluid flow along a moving continuous cylinder. The simplest case, the flat plate situation, has been investigated experimentally by other workers (§2.5). Both theory and experiment give for this
For a stationary flat plate (Blasius flow) the value of C^/Re^^ is 25^ lower. This difference is caused by the more rapid transport of momentum in the part of the boundary layer close to the moving continuous flat plate, where longitudinal veloci-ties are relatively high compared with the velociveloci-ties close to a stationary flat plate (§2.3).
In §2.6 a description is given of an experimental arrange-ment for measuring local liquid velocities near a moving con-tinuous cylinder. The experiments were designed so that the hydrodynamic boundary layer thickness was not much smaller than the cylinder radius, in other words: with conditions when the curvature of the cylinder might be expected to affect the velocity profiles. The quantity 6/R was a measure for the de-viation from the flat plate behaviour (in this context 6 stands for the boundary layer thickness in case of zero curvature). Velocity profiles near the cylinder were visualised by sus-pending small particles in the liquid through which the cylin-der passed. Stroboscope photographs of the particle trajecto-ries provided values of local liquid velocities in the bounda-ry layer. The results reflected the relationship
9
which was envisaged a priori. Sakiadis' approximate theory
on this relationship was reconsidered on the basis of the ex-perimental results: it could be improved to some extent.
In practical situations threads move through a vessel with a diameter which is not infinitely large compared with the boundary layer thickness. This may result in hindering the boundary layer development. This effect was investigated experimentally: it was found that the deviation from the ide-al situation (no wide-all effect) could be correlated with the ratio of boundary layer thickness and vessel diameter.
Both the results of other workers and those presented here lead to the conclusion that free convection plays an im-portant role in heat and mass transfer processes to moving continuous cylinders. Prior to the exploration of the combined flow situation (chapter 5) forced and free convection were studied individually and this work is discussed in chapter 3 and 1+ respectively.
Forced convection mass transfer coefficients were mea-sured electrochemically (chapter 3). In this electrochemical system the object to which mass transfer is measured acts as a cathode (in this case a moving wire). The electric current that passes through the system is recorded. This current is proportional to the rate of the mass transfer controlled cath-ode reaction, thus providing an excellent means to measure mass transfer rates and (§3.1+) coefficients. The Schmidt number of the electrochemical liquid is large (about 2000) so that only processes in the innermost part of the hydrodynamic boundary layer play a role. Therefore this technique is very adequate to illustrate the characteristic nature of transfer processes to moving continuous objects: for this case mass transfer rates are roughly three times greater than for an analogous situation at a stationary object placed in a homo-geneous outer flow. In the former case liquid velocities and convective transport rates in the mass transfer boundary layer are far larger than in the latter. The experimental results agree with the equation derived theoretically:
— — — ^ = f5(— , Sc) , which is the logical mass /Péx transfer analogy of Eq.(S-l). These experiments covered an exceptional case in so far as the concentration along the wire did not change during the transfer process. In general a situation may occur in which
the thread has a small and the boundary layer a relatively large (heat or mass) capacity. Transfer (of heat or mass) will then cause a large decrease of the temperature composition of the thread, while the adjacent part of the boundary layer has retained the higher temperature or concentration previously acquired. This may have the effect of lowering the transfer coefficients. The parameter which governs this effect is in heat transfer terms the ratio of the heat capacities of bound-ary layer and thread:
^ =
7 7 ^
R *
A general solution for the laminar forced convection problem is therefore (§3.2):
^ ^ ^ ^ ,6 ^P=)f 6 , .
^ s
In chapter 1+ free convection along vertical cylinders was investigated. As this subject is not a particularly new
one, many results from other workers are available in the literature (§l+.3). All experiments reported had been per-formed in air. They could be correlated with the help of a theory (§l+.2) in which again the Prandtl number and the curva-ture parameter 6/R appear as variables:
Nu
1.99 - ^ = fi2(Pr, 6/R) . (S-2)
For the present experimental work the electrochemical method was adapted in order that free convection correlations
could be determined. To that end the concentration differences of each individual solute between the solution at the cath-ode and in the bulk were evalueated from mass balances. The resulting density difference was then computed using an empir-ical correlation (§1+.U). These experiments can be considered as a first step towards a future research programme for the experimental investiation of combined flow situations.
The results confirmed Eq.(S-2) for large Schmidt numbers; they also demonstrated that free convection flow at high Schmidt numbers is more stable than is generally assumed.
11
For combined flow (chapter 6) no quantitative theoretical approach is available. The problem was attacked experimentally by developing a new moving thermocouple technique: a hot
moving thermocouple junction but welded as a wire drawn through a cold liquid, was made to indicate its own temperature conti-nuously, so providing the necessary heat transfer data.
Both the lack of experimental accuracy and the complexi-ty of the physical phenomena studied has inhibited a reliable quantitative description. Qualitatively however, the prelimi-nary exploration has contributed to the understanding of the process.
§ 1 . 1 13
Chapter 1 INTRODUCTION
1.1 Introduction
Transfer processes to continuous surfaces have attracted increasing interest during the past decade. This is mainly due to the rapidly expanding number of industrial processes in which continuous bodies are handled. The most striking ex-amples can be found in textile industries (spinning) and in the manufacture of wires.
The fabrication of cellulose acetate film is an example of a process that fundamentally involves momentum and mass transfer to a moving continuous sheet: The continuous acetate passes through a series of liquid baths. It is led into and removed from each bath by rollers. The boundaries of the sys-tem are fixed in space by the location of these rollers, and any transfer processes are invariant with time at any fixed location. However, because the film is moving and continuous, analysis of any particular element of the film material must accomodate the instantaneous unsteady state conditions that apply as the film moves through the system.
During the passage of the sheet through the bath, the adjacent liquid is entrained, which means that momentum is transferred. Meanwhile, chemicals, dissolved in the liquid, move towards the sheet, where they are removed by reaction. The transfer of these reactants to the sheet is a mass trans-fer process to a moving continuous sheet. In order to design this film manufacturing process properly, it is necessary to know the momentxam and mass transfer rates as a function of the process variables such as film velocity, the length of the bath and the properties of the fluid.
Another example of a moving continuous object is the spinning of nylon filaments from a melt. These filaments are formed by the downward spinning of jets of molten nylon in a box in which they solidify and cool off. In this box fairly large quantities of air are entrained by the filaments (momentum transfer). The filaments loose heat during this pro-cess, so the heat and momentum transfer rates are clearly im-portant parameters in this process. In this case also
knowl-edge of the transfer rates will be of great help in finding optimal process conditions.
Generally, many spinning processes and other continuous operations (such as leaching, dyeing, washing and cooling of filaments or sheets and the manufacturing of wires) have common hydrodynamical features. In all cases momentum trans-fer from the continuous surface plays a role; in addition to that heat or mass transfer is often encountered.
1.2 Considerations of practical interest
Although the processes mentioned above exhibit a close similarity at first sight, they reveal, when looked upon in detail, important differences. It will be shown that actu-al practicactu-al transfer situations can be quite complex and differ from each other.
Filament processes usually involve several strands treated simultaneously in bundles. The flow, concentration and temperature patterns associated with an isolated thread will be considerably modified in the presence of such an
as-sociated more or less parallel moving group of similar fibres. The fluid through which the continuous material is led is, in many cases, not a quiet one: it is forced by external means to flow relative to the continuous surface.
In spinning processes, the formation of the filament is often accompanied by a decrease in diameter and, consequently, an increasing velocity.
If the velocity, temperature or concentration in the body is not uniform over a cross section, then the rates of
penetration of momentum, heat or mass into the body may play
an important role in the overall transfer rates. Evaluation of this internal resistance however, requires an approach which is quite different from the approach required for the evaluation of the external resistance.
A further complication that occurs quite often is vi-bration of the continuous body; this might seriously affect transfer rates.
1.3 Formulation of the scientific approach
In view of the preceeding we shall now formulate the subject of this study such that it comprises those features only that are characteristic for transfer processes to
§1.3 15
an externally forced convection of the fluid, nor with bun-dles, nor with vibration and a non uniform velocity of the body. The fluid flow around the continuous material will be limited to that caused only by the body itself: either as a result of its uniform motion (forced convection) or by free convection arising from temperature or concentration gradi-ents in the fluid.
Only the external resistance to transfer will be con-sidered. This iirplies that over a cross section the velocity, temperature and concentration are assumed to be uniform.
We shall select the circular cylinder as the shape of the body. Sheets can be considered as degenerated cylinders with negligible curvature.
In free convection to cylinders the orientation of the cylinder with respect to the gravity force is, of course, important. Because this would introduce another parameter, we shall restrict ourselves to one orientation: the vertical cylinder. The fluid flow which arises in this case is more similar to the forced convection flow than free convection flow around horizontal cylinders would be. Another argument for this choice is that free convection around horizontal
cylinders has been studied extensively already (1).
The problem can now be formulated as the determination of the rates of momentum, heat and mass transfer to a verti-cal continuous cylinder moving along its axis through quiet fluid, without axial or lateral vibration. The velocity of the cylinder is the same everywhere (no stretching) and the concentration and temperature are uniform at any cross sec-tion.
1.1+ A closer view of the problem; its characteristics In §1.1 we mentioned some industrial processes in which moving continuous bodies are handled. These complex situa-tions were abstracted to the scientific problem formulated in
§1.3. One may ask: why is this problem worth a special study and what makes it differ from the vast quantity of transport problems already solved? In other words: what specific fea-tures have the industrial processes mentioned in common?
One thing becomes clear directly: the first common characteristic is associated with the special kind of bounda-ry layer flow generated by a moving continuous object in a quiet fluid. A second feature is the influence of the slen-derness of the cylinder: whether the cylinder is thin or thick
compared with the entrained boundary layer can be expected to affect the velocity profile.
These first two characteristics have been formulated in terms of momentum transfer; they hold equally well for heat and mass transfer because there is a very strict similarity between these three transport processes.
The third common characteristic applies for practical situations only to heat and mass transfer. During the heat transfer process the moving cylinder cools off and the ad-jacent and accompanying boundary layer is warmed up. This may seriously affect the temperature distribution in the boun-dary layer and, accordingly, affect the heat transfer coeffi-cient. As this situation exists in all applications involving heat and mass transfer with continuous bodies, it will be con-sidered as a third common characteristic.
The influence of the slenderness of the body (2nd charac-teristic) and of its limited heat or mass capacity (3rd char-acteristic) are also observed in other situations, e.g. in free convection transfer with vertical cylinders, which pro-blem has received much attention in the recent past. This si-tuation will also be dealt with because often both forced and free convection influence the transfer rates to moving fibres and sheets.
1.5 Structure of the thesis
The thesis will be divided into two separate parts: the forced convection problem and the free convection problem.
The simplest case, that of momentum transfer in the forced convection situation, will be analysed first, in chap-ter 2, Chapchap-ter 3 deals with heat and mass transfer in forced convection. In chapter 1+ free convection around vertical cy-linders will be investigated. The complex situation of com-bined free and forced convection flow will be discussed in
chapter 5»
In principle the fluid flow can be in either case lami-nar or turbulent. Most experiments were performed with a laminar flow regime. The momentum, heat and mass transfer rates will be deduced theoretically for this condition: this will show the influences of the three characteristics men-tioned. The expressions to be derived for laminar free con-vection flow along vertical cylinders will show analogies with those for laminar forced convection flow.
§1.5 17
the formulation of experimentally or theoretically sound state-ments. This flow situation will be treated with a more or less qualitative discussion.
Little work was done on turbulent flows. For the sake of completeness some results from other workers will be reviewed. The experimental techniques which we used in laminar flow si-tuations are also suitable for experiments with turbulent con-ditions.
§2.1 19
Chapter 2
FORCED CONVECTION FLUID FLOW ALONG A MOVING CONTINUOUS CYLINDER
2.1 Introduction
The motion of a continuous cylinder will exert drag on the fluid in the neighbourhood of the cylinder by means of viscous forces in the fluid. Because of inertia forces the in-fluence of the moving cylinder is restricted to a relatively small boundary layer around the cylinder.
Sakiadis (2) was the first to recognise clearly the boun-dary layer problem associated with transfer to a moving con-tinuous cylinder. Not only did he formulate the equations and boundary conditions governing this flow, but he also gave
so-lutions to this problem in a series of three papers (2, 3, 4).
This type of boundary layer flow has two important cha-racteristics that distinguish it from other boundary layer flows. The first characteristic is inherent to boundary layer flow around any moving continuous body. It arises from the fact that the boundaries of the system are not defined solely by the boundaries of the continuous body itself. The condi-tions at the places where the body enters and leaves the ves-sel in which the process takes place are also of importance
(the place where the body leaves the vessel is, in the scope of this thesis, thought to be so far away that it does not in-fluence the main part of the boundary layer).
The case of a moving continuous body is different from the well-known description of the boundary layer flow along a flat, stationary plate of finite length. Although this dif-ference might, at first sight, seem as trivial as a choice of a coordinate system, it has important consequences for the transport processes. In the inner part of the boundary layer the longitudinal convection of momentum (and also of heat and mass) is important for moving continuous surfaces because the longitudinal velocity is quite large there; this is not so, however, for stationary surfaces of finite length.
The second characteristic is typical for boundary layer flow parallel to slender objects. It originates from the fact that the thickness of the boundary layer may be of the order of magnitude of the radius of the cylinder - or even larger
Fig. 2-1 Laminar boundary layer flow around a moving continuous cylinder
than that.
With the help of Sakiadis '
theoretical treatment of the laminar boundary layer flow not only will some important boundary layer parameters be computed, but also the two characteristic fea-tures of this will be illustrated
(§2.1+).
Measurements of the velocity profile will confirm and adjust the theory for laminar flow and some of the requirements which are necessary if the flow is to
be described by Sakiadis' theory
will be found (§2.6). 2.2 Formulation of the laminar flow problem
First the boundary layer flow neglecting the curvature of the cylinder will be discussed; this case of zero curva-ture will hereafter be referred to as the "flat plate".
This laminar boundary layer flow is sketched in Fig. 2-1. The development of the flow can be considered as a process starting at x = 0, the point where the continuous body enters the system. As this process continues, the amount of entrained fluid increases steadily: the fluid is accelerated in the x-direction. This acceleration demands momentum, which diffu-ses from the suface of the plate into the y-direction. The larger the distance y from the surface, the more momentum has been consumed by fluid acceleration. Therefore the longitu-dinal velocity, u, decreases with increasing distance from the surface.
How is this picture changed for a cylinder instead of a flat plate? In that case there is a diffusion of momentum in the y- (or r-) direction to ever increasing circumferences: the momentum will be distributed over ever growing areas. This will give an extra contribution to the decrease of the velocity in the y- (or r-) direction.
Our aim is now to find a description for the velocity profile, to compare it with that on a stationary surface of finite length (first characteristic).
It is reasonable to question whether the situation visu-alised is physically feasible in the entrance region where
§2.2 21
the boundary layer begins to develop (x ^ O). It should be re-membered that boundary layer theory is an approximation that
never holds at the leading edge (see for instance Sahliahting
(5)). Therefore the solution from boundary layer theory in the entrance region is important only in so far as it is in-dispensable in order to construct the solution at "moderate " values of x. The experiments to be described later will sup-port the approach as providing a satisfactory model for the velocity field that is established.
2.3 An exact solution for the moving continuous flat plate This solution offers an opportunity to illustrate the first characteristic of boundary layer flow around moving continuous bodies by comparison with the solution for a sta-tionary flat plate of finite length.
Sakiadis (3) provided the solution for a moving continu-ous flat plate; it will be summarized briefly. It belongs to the class of so called similar solutions to boundary layer problems. This means that the dimensionless longitudinal and transverse velocities are functions of one dimensionless group only. This group is
FT
11 = y Y — > which is proportional to the ratio of the distance to the surface, y, and the boundary layer thick-ness or depth of momentum penetration
Sakiadis solved this boundary layer problem numerically. The result is given in Fig. 2-2.
Some illustrative differences with the result for the stationary flat plate of finite length are:
1. The slope of the line u/Ug versus n at n = 0 equals - O.l+UU for the moving continuous plate. For the stationary flat plate of finite length this value is - 0.332. This means
that the friction factor on a moving continuous plate is 3kfj
higher than on a stationary plate of finite length. At the outer edge of the boundary layer however, the velocity gradi-ent is greater in the case of the stationary surface of fi-nite length.
2. The line that represents the relative longitudinal velocity, u/Ug in Fig. 2-2 starts to deviate markedly from a straight line at a certain value of n. This value of n is
1.0 08 06 0.4 as 1 2 3 4 5 6
Figure 2-2 Dimensionless longitudinal velocity at the flat plate of finite length (right axis) and the moving continuous flat plate (left axis)
larger for the stationary surface of finite length than for the moving continuous surface.
3. The transverse velcotiy, v, is directed towards the moving continuous surface but away from a stationary surface of finite length. This requires no further explanation.
These differences can be explained by the typical behav-iour of boundary layer flow around moving continuous bodies. In a volume element of a boundary layer a part of the momen-tum carried on by molecular transport (by "viscous forces") is taken away by convection ("inertia forces"). This is especial-ly important in the region where the shear stress is still relatively large, i.e. not far from the wall. Near the wall of a continuous surface this convection of momentum is more important than near the wall of an object of finite length, because in the former case the longitudinal velocity is larg-er. At the outer part of the boundary layer the opposite is true; this is a less important effect though, as the shear stress is smaller there. It will be shown later that similar reasoning can be applied if heat or mass instead of momentum is transported (one can already anticipate that the typical
§2.3 23
behaviour at moving continuous bodies will be most significant at large Prandtl numbers).
The main reason why the shear stress at the wall of a moving continuous surface must be larger than in a similar situation at a stationary surface of finite length is there-fore, in qualitative terms: because momentum is convected away faster. It will also be clear that the profile of the longitudinal velocity near the wall is less linear in the case of a moving continuous surface: because the shear stress decrease is larger.
2.1+ Solutions for the moving continuous cylinder
2.l+.l General considerations; the Sakiadis solution
In these solutions the second characteristic of boundary layer flow around a moving continuous cylinder (the influence caused by the fact that the boundary layer is not much thin-ner than the cylinder) is also relevant.
Firstly we shall consider Sakiadis' approximate solution
of the problem. The increase of the total momentum flux in a cross section of the boundary layer must be equal to the mo-mentum flux from the wall:
T
u 2 2 7 r r d r = 2TTR — ( 2 - 1 )
P
Using a Pohlhausen method an expression for the velocity is
inserted into this integral momentum balance. In this expres-sion the velocity is given as some function of the distance to the surface relative to the boundary layer thickness. The
wall shear stress, T^, may be expressed as a function of the
boundary layer thickness. Working out Eq. (2-1) results ul-timately in an expression for the boundary layer thickness as a function of, among other things, the distance x. When this expression is obtained, the whole velocity distribution is known approximately.
The Pohlhausen method has been successfully applied to many boundary layer problems. The great advantage of this method is that the resulting expression for the wall shear
stress is not very sensitive to the velocity profile chosen. Usually a tentative velocity profile is constructed that satisfies certain boundary conditions at the wall of the ob-ject and at the outer part of the boundary layer. At the
periphery of the boundary layer the condition is applied that the first n derivatives of the velocity, u, with respect to the transverse distance, y, should be zero. The higher this number, n, the more complicated the velocity profile and the smoother its adaption to the flow outside the boundary layer.
At the wall of a cylinder the boundary condition is: r = R , u = Ug .
Close to the wall the fluid velocity is virtually equal to the velocity of the cylinder, so no momentum is "lost" by accelerating the fluid. Hence, the shear force exerted on a cylindrical volume element of fluid cannot decrease with in-creasing distance to the object:
^ = « . ^ (r 1^) = ° •
In order to satisfy this second condition at the wall, the velocity profile has to be logarithmic, i.e.
^ = f ( m |) . I (2-2) s
Sakiadis chose the simplest of all possible logarithmic
pro-files:
^ = 1 - •!• ln(|) for r i R + 6
u = 0 for r = R + 6 . (2-3) Apart from the obvious condition u = 0 a t r = R + 6 , this
profile does not satisfy any boundary condition at the outer part of the boundary layer.
The coefficient 6 is related to the boundary layer thick-ness: ^ p _^ ^
6 = In (—r—) , and can be found by Pohlhausen's
I
method. It proves to be a function of 5 = 1+ WTT ^^ > 5 ^^^ be looked upon as the ratio of the boundary layer thickness in case of zero curvature ('^ Vrr"^ ^'^^ ^^^ radius of the
cy-S P 4- A
linder, R. Clearly, the relationship between g = l n ( — = — ) and C (- 6 at zero curvature/R) will reflect the influence of the second characteristic of boundary layer flow around moving continuous cylinders: the influence of the curvature of the boundary layer.
§2.1+.2 25
2.1+.2 An improved logarithmic velocity profile
Sakiadis' profile, 77- = 1 - In (r/R)/ln (—r—) ,
degene-Uc. R u y ° .
rates to -rr- - '^ - '^ fo^ the moving continuous flat plate (R -»• °°). These profiles are simple but not very accurate, especially in the outer part of the boundary layer. Fig. 2-2 illustrates this argument for the moving continuous body. In order to improve the description of the flow field in the outer part of the boundary layer an alternative formulation will be suggested. Because the profile has to be logarithmic, a logical choice is:
S p p P
This satisfies two more boundary conditions at the periphery of the boundary layer, i.e.
„ ^ . 3u a^u .
r = R + 6, -r- = -r-~r = 0 . ' 3r dr'^
If one applies Pohlhausen's procedure a relationship between
.X
and C is obtained (see Appendix l). This completes the description of the velocity profile, expressed in Eq. (2-1+).
2.1+.3 Test of the solutions when the influence of curvature IS small
In order to test these approximate solutions they should be compared preferably with an exact solution. Seban^ Bond and Kelly (6, 7) showed how the flow parallel to cylinders may be computed exactly with the help of a perturbation method. Unfortunately their method is quite elaborate. It is based on a series expansion of the dimensionless stream function:
f(n,?) = f^di) + Cfi(n) + ?2f2(n) + . (2-5) Details are given in Appendix II. The result for the velocity gradient at the wall of a moving continuous cylinder is:
3(u/Ug) IITTRT" r=R 1:11^ - 0.3802 + O.OI85I+C - 0.002321^2 . ^ (2-6) I t i s d o u b t f u l whether t h i s s e r i e s converges f o r C > 1
and even if ? is smaller than unity one needs many terms that can only be found after lengthy calculation. The series (2-5) is even more awkward when a description of the flow at larger distance from the wall is wanted: the curvature effect and thus the higher order terms of (2-5) are relatively more
im-portant there. This means that the Seban-Bond-Kelly (S.B.K.)
solution can be used to compute the friction factor exactly only at very small 5-values (very small curvature effects); this is done in Table 2-1.
? 0.01 0.1 0.2 0 . 3 0.1+ 0.5 S a k i a d i s 1.636 1.666 1.699 1.732 1.765 1.797 Author 1.713 I.7I+8 1.788 1.827 1.866 I.90I+ S.B.K. ( e x a c t ) | 1.779 1.813 1.850 1.887 I.92I+ 1.961 1 - C„/5e"„ as a function of C r=R
It shows that both Pohlhausen solutions yield satisfactory
results for the friction factor and that the influence of the curvature is accounted for in the right way. It illustrates
the fact that Pohlhausen solutions give unexpectedly good
re-sults in many cases.
2.1+.1+ Comparison of the solutions at larger values of the curvature parameter
In order to compare the author's solution with Sakiadis^
at larger values of 5, a different criterion will be chosen. There are many ways to characterise solutions of boundary layer problems. It seems obvious to consider the boundary
layer thickness as representative for the solution. In
Pohl-hausen solutions however, the definition of the boundary layer thickness depends upon the form of the assumed veloc-ity profile; therefore one needs other boundary layer para-meters to characterise the solution. For parallel flow around cylinders the momentum area and the displacement area are the most conventional boundary layer parameters. The reduced momentum area,
00
^ 2 = ^ ^ ƒ u22.rdr , (2-7)
° R m o K T ^ o 1 3(u/Us)
§2.1+.1+ 27
Figure 2-3 Reduced momentum area (—^) and reduced displace-ment area (rs?) computed using Pohlhausen's proce-dure when the curvature parameter, C, is small
is related to the average friction factor by <C^>
ïïR^ R
L (2-8)
The reduced displacement area, A
ÜR^
1
TTR^U ƒ u2iTrdr « (2-9)
is related to the pumping capacity of the moving cylinder by
q = AUg . (2-10) So the momentum area and the displacement area are not only
boundary layer characteristics, they are also of much practi-cal importance.
Figure 2-3 shows that for 5 ->- 0 the reduced momentum area and the reduced displacement area are proportional to C Bear-ing in mind that C ^ 6/R for 5 -*• 0, this agrees with the pre-diction of e and A for the moving flat plate.
For 5 -> CO all lines of Fig. 2-1+ display a proportionality of the boundary layer parameters with C^. In this situation the very large influence of the boundary layer c\irvature is clearly demonstrated: for very large 5 the friction factor is almost inversely proportional to the Reynolds number UgR/v and the volume flow rate of entrained fluid is nearly proportional to vL.
lofl-.=5(AuUior)
(Sakiadis)!
This can also be explained physically. If the situation not too far from the wall is considered, the ve-locity decay in the r-direction due to the diffusion of momentum to larger circumfer-ences dominates the velocity decrease caused by momentum con-sumption for fluid ac-celeration. Near the wall fluid acceleration is very unimportant and the simple logarithmic
shape of the Sakiadis
profile is realistic. Consequently, the mo-mentum flux from the wall must be nearly in-dependent of X and r in that situation. The shear force exerted on the curved surface of a cylindrical voltmie element (of unit length and concentric with the cylinder) will be almost in-dependent of both coordinates. Hence, this force per unit length must also be independent of R and therefore proportion-al to vpUg. This explains why for very large 5 the friction factor is almost inversely proportional to the Reynolds number UgR/v. A similar reasoning shows that the volume rate of en-trained fluid is proportional to vL.
Figure 2-3 and 2-1+ also show that the magnitude of the momentum area and hence of the average friction factor -is not very sensitive to the form of the chosen velocity
pro-file, even up to high values of £,. For the displacement area
- and thus the pumping capacity - the situation is entirely different. At high values of C the displacement area according to the author's velocity profile is many times greater than
that computed with Sakiadis' assumption. In order to explain
this we must consider the boundary layer cross sectional sur-face area perpendicular to the cylinder axis. Because of the large curvature, the surface area of the outer part of the Fig. 2-1+ Reduced momentum area
(—^) and reduced displacement area (~p7") computed using Pohl-hausen's procedure when the cur-vature parameter, C, is large
§2.1+.1+ 29
T
boundary layer is relatively large compared to that of the inner part. The author's profile has by definition a thicker boundary layer: the "tail" of the velocity profile is longer. Therefore the amount of entrained fluid is much larger than according to Sakiadis,
We may conclude that the Pohlhausen procedure gives good results for the friction factor, even with the very simple
Sakiadis profile. Calculation of the pumping action of the
moving cylinder with the help of a Pohlhausen method however, is quite tricky: experiments will be necessary. Sakiadis'
statement (4) that his velocity profile becomes more realistic when 5 increases can easily lead to false conclusions.
2.5 Experimental work in the laminar flow regime 2.5.1 Introduction
In Sakiadis' publications in I96I an entirely new kind
of boundary layer flow was described. The idea of "moving con-tinuous", the first characteristic, was quite new. Was Sakiadis^
conception physically feasible? For one thing, the description of the flow at x = 0, the plane through which the continuous object enters the system, does not look very plausible. If the theory does not apply there, how will this affect the si-tuation in other parts of the boundary layer? As for the sec-ond characteristic, these enormous effects of curvature were predicted only on the basis of an approximate boundary layer solution.
It is clear that experiments are necessary to confirm (or reject) these ideas in order to obtain a better under-standing of the flow around moving continuous objects. It is reasonable to start with experimental research on the simple "Sakiadis situation".
2.5.2 Review of earlier experimental work
Experiments, set up to verify Sakiadis' theories on mo-mentum transfer were performed by Griffith (8), Tsou, Sparrow
and Goldstein(9) and very probably by Sakiadis himself.
TsoUf Sparrow and Goldstein did very accurate
measure-ments on the velocity profile in the boundary layer at a moving continuous flat plate. They mounted a very small tube
in the boundary layer parallel to the plate. The impact pres-sure, sensed by this velocity probe, was read from a
manome-ter. The velocity profile obtained in this way was in excel-lent agreement with the exact boundary layer solution up to Re-r numbers of 1.5 "10^ (transition to turbulence).
Griffith used a flow visualisation technique in order to measure the velocity profiles in the boundary layer of a moving continuous cylinder. The velocities of particles, suspended in the liquid, were followed with the help of a travelling microscope. His experiments were less accurately
designed than those of Tsou et al.. The curvature parameter
C ranged from 1.5 to 165, the ReL number from 9.2 to 3000, the upper limit being determined by his experimental tech-nique.
His results show clearly that the simple logarithmic
form of Sakiadis' velocity profile is a good approximation
in the inner part of the boundary layer; taken as a whole however, the agreement with the profile described in § 2.1+.2 of this thesis is better. The shear stress at the wall is in fair agreement with its theoretical value at small values of
5; at larger values of E, the experimental shear stress was
significantly greater than the theoretical one. The deviation
between the theory and Griffith's experiments appeared to
in-crease with the ratio of boundary layer thickness to bath diameter, thus indicating a wall effect. It is not clear how-ever, to what extent the discrepancy may be attributed to this effect.
Therefore an experimental program was initiated with
the object of yielding an unambiguous verification of
Sakia-dis' theory. Another argument was to find out if the fourth
degree polynomial logarithmic profile could be justified ex-perimentally.
2.6 Measurement of the velocity profile in the laminar bound-ary layer
2.6.1 Principle of the experimental method
The experimental technique used was a flow visualisation method. A thread ran vertically through a cylindrical vessel, filled with liquid in which small particles were suspended. As the particle velocity everywhere can be assumed to be the same as the local liquid velocity, the method was based on the measurement of the particle velocities. By means of an optical arrangement a plane and narrow beam of light was thrown through the boundary layer onto the thread. The
geo-§2.6.1 31 I Focussing lens Condensing iense Light source -} metrical layout of the
experimental set-up is shown in Fig. 2-5.
A stroboscope method wap used: the
light flashed at accu-rately known time in-tervals. A long expo-sure photograph was made of the plane of the light bundle. From this photograph the length of the path that a particle cov-ered in known time in-tervals could be mea-sured and hence the local velocity
deter-mined. As many particles were photographed at the same time, each picture provided a complete velocity profile.
2.6.2 Description of the experiments
C 3 Camera ggga illuminatad araa
Fig. 2-5 Geometrical arrangement of the experiment
When we designed the experimental set-up much
informa-tion was supplied by Smith (W), who had used the same
tech-nique in the study of developing velocity fields. We used the same particles as he did: "Merlite". These particles have a very high reflectance coefficient and a density only slightly greater than the glycerol-water mixtures we used. The greatest dimension of a particle was <0.1 mm.
The optical arrangement, sketched in Fig. 2-5 was de-signed so that the light intensity in the boundary layer was as high as possible. The second lens made an image of the vertical slit on the wire. Hence the bundle was a slightly converging one; its width was about 0.2 mm.
In order to avoid disturbing optical effects the cylin-drical vessel was equipped with two plane vertical walls: one on the side where the light came from and one on the side from which the photograph was taken. The space between the cylindrical and the plane walls was filled with the same liquid as that inside the vessel. Two different vessels were used: a small glass vessel (diameter 8 cm, height 38 cm) in which the majority of the experiments with small boundary layer thickness was performed. The other one was a larger
Perspex vessel (diameter 1+5 cm, height 110 cm) in which the experi-ments with thick boundary layers were done.
In both vessels an endless wire was used; it ran over two bicycle wheels, the lower one of which was driven on by a motor with variator (see Fig. 2-6). In the small vessel the wire ran downward; in the wide vessel, which was usually filled with a viscous liquid, it ran up-ward, thus avoiding entrainment of air bubbles that would not easily disappear.
A conventional electronic flash lamp was used as the light source, A double flash was produced by fit-ting the lamp with two condensors. Fig. 2-6 The wire The time interval between the double loop with accessories flash was accurately controlled
elec-tronically with a delay that could be measured by comparison with mains frequency. The time span
of one individual flash was very short (of the order of magni-tude of 1 millisecond), hence each particle was represented on the photograph by two small dots. The distance between two dots belonging to one particle is the product of the particle velocity and the known time interval of the flash lamp. Al-though the light intensity of the flash lamp was large, the total amount of light received by the liquid and the wire was
small. The advantages are obvious: a clear image on the photo-graph and little darkening of the negative in the region of the wire.
The camera used was a Hasselblad model 500 C, equipped with a lens of focal length of 80 mm. The film was Kodak Tri X Pan (27 DIN, 1+00 ASA). The exposure time was arbitrary as long as it was longer than the time interval between two flashes.
In all experiments the temperature of the glycerol-water mixtures was recorded. A rotation viscosimeter was used to measure the viscosity of the glycerol-water mixtures imme-diately after each experiment. When working with pure water
the viscosity was taken from Hodgeman (71).
§2.6.2 33
- a plastified copper conductor wire of 1.2 mm diameter - a plastic belt of circular cross section of 2.8 mm diameter - nylon filaments of diameters l.lU, 0.30, 0.l6 and 0.08 mm
diameter 2.6.3 Results
The independent variable in these experiments was the
curvature parameter E, ^ h \j • ^ • ^ wide range of values
of C was obtained by varying the kinematic viscosity, v, and the cylinder radius, R. This was achieved by working with glycerol-water mixtures of different mixing ratios and with wire and thread radii ranging from 1.1+ to O.OI+ mm. A review of the experimental conditions will be given, together with the results, in tables 2-2 through 2-1+.
The experiments can be divided into three groups: The first group comprises the experiments performed at small values of the curvature parameter. The resulting velocity profiles, when plotted as (Ug - u)/Us vs.ln ((y+R)/R), are curved lines and can therefore be compared best with the lo-garithmic fourth degree velocity profile. This is shown in table 2-2. R TniTi 1.1+ 1.1+ 0.6 i . U 0.6 1.1+ 1.1+ 1.1+ 0 . 6 0.6 0.6 0.6 1.1+ V 1 0 - ^ m ^ / s 36 Us m/s 0 . 6 6 6 0 . 6 6 0 1 . 0 7 0 0.1+1+0 0 . 5 1 0 0.232 0.132 0.133 0.276 0.292 0.182 O.II+I+ 0.186 i 1.78 1.81 2.12 2.27 2.87 3.11 1+.17 1+.20 I+.38 l+.l+O 5.1+1+ 9 . 2 5 20.1+ D mm I+U5 1+1+5 80 Ul+5 80 1+1+5 1+1+5 1+1+5 80 80 80 80 1+1+5 6 * exp 1.5 - 1.7 1.1+ - 1.8 1.6 - 1.8 1.7 - 2.1 2 . 0 - 2.2 2 . 0 - 2 . 2 2 . 2 - 2.7 2 . 6 - 2.7 2.1+ - 2.6 2 . 6 - 2.8 2 . 8 - 3.0 3.8 - 3.5 5.0 - 1+.5 ^ t h 1.61 1.51 1.69 1.76 2.06 2.18 2.56 2.57 2.61+ 2.65 2 . 9 6 3.76 1+.96
Table 2-2 Results for small values of the curvature parameter; comparison with theoretical predictions for a loga-rithmic fourth degree velocity profile
Figure 2-7 Results at C =2.27; comparison with logarithmic fourth degree polynomial profiles for g = 1 . 7 and 6* = 2.1
Figure 2-8 Results at C = 9.25; com-parison with logarithmic fourth
de-gree polynomial profiles for 3 = 3 . ^ and e = 3.5
Specimen of plots are given in Figs. 2-7 and 2-8. The shapes of the curves do not coincide exactly with those of the theoret-ical profiles. The ac-tual curves at small 5-values are somewhat more bended as can be
seen from Fig. 2-7. Therefore for each plot two experimental
6* numbers are given. The first one corre-sponds to the shape of velocity profile near the cylinder. The second one to the shape of the velocity profile at the outer part of the boundary
§2.6.3
35
layer. For the 5-values 9.25 and 20.1+ the trend is reversed, e.g. the actual curves are some-what less bended than the theoretical pro-files (Fig. 2-8).
The second group of experimental results those at large 4-values is represented in table 2-3. At these higher C-values the logarithmic velocity profiles are less bended and can be compared better with the
Sakiadis profile, as was done actually. Here too there is a tendency for these logarithmic profiles to become straighter when 5 in-creases. Figures 2-9 and
2-10 show two of these profiles.
For the third group of experiments the boundary layer thicknesses are of the order of magnitude of the vessel dia-meter. The ratio of the theoretical boundary layer thickness
(according to Sakiadis' profile) to the vessel diameter is
included in table 2-1+, together with the experimental condi-Fig. 2-9 Results at C = 25.7;
comparison with the Sakiadis
profile for 5 = 25.7 R mm 0.57 0.15 0.15 0 . 1 5 0.08 0.01+ V 10"^ m^/s 9.8 3.29 9.5 11.U 5.3 10.6 U s m/s 0.186 0.186 0.186 0.186 0.186 0.186 5 25.7 56.5 9U.1+ 1 0 3 . 0 135 381 B exp 3.1 l+.O li.5 ^ . 7 1+.9 5.9 6 ^ oh 3.28 It.15 h.7k 1+.70 5.10 6.20
Table 2-3 Results for large values of the curvature
para-meter; comparison with the predictions of Sakiadis'
Figure 2-10 Results at C = 135; comparison with the Sakiadis profile for C = 135 (6 = 5.10) V 10-^ m2/s 310 1230 1680 1230 2200 2380 U„ s m/s 0.186 0.188 0.067 0.039 O.OI+5 O.0I+I+
c
59.1+ I2I+.I+ 23I+ 266 1+58 580 3 exp 3.80 3.95 3.80 3.60 3.77 3.75 3a., t h 1+.08 5.01 5.69 5.83 6.1+2 6.67 3.^,.- 3 t h exp 0.28 1.06 1.89 2.23 2.65 2.92 <5^, /D t h 0.182 0.1+72 0.88 1.05 1.88 2.1+9Table 2-1+ Results for situations at which the wall effect is
important; comparison with the predictions of
Sakia-dis' theory (for these experiments R = l.U mm and
D = 1+1+5 mm)
tions and results. The logarithmic profiles for these condi-tions are almost linear; hence the results were compared with
those of Sakiadis' theory. In the outer part of the boundary
layer however, the velocity decrease with increasing distance from the cylinder is always smaller than the simple logarith-mic profile predicts. A specimen of such a velocity profile is given in Fig. 2-11.
§2.6.1+
37
-- 1
-~ -~ -- .• • y/^ • • • • 1 •• • y — ' ' -1 • 1 ** J\* A
y^\
y^ \
\ — • '"<Y' J r 1 1 2.6.1+ Discussion and conclusionsThe results for negli-gible wall effect, e.g. those given in tables 2-2 and 2-3 will be dealt with first. Clearly neither
Sakiadis ' profile nor the logarithmic fourth degree profile gives a good re-presentation for the entire range of C-values. There is a general ten-dency for the logarithmic profiles to become straight-er as 5 increases. For very small C the actual profiles are more curved than the logarithmic fourth degree ones; this agrees with the flat plate limit
(C = 0 ) , where the exact profile is known theoreti-cally. Apparently the
larger 5, the greater the effect of curvature on the velocity
profile and the better the Sakiadis profile is approached.
The assumption on which the Sakiadis profile is based is
that the force on a cylindrical volume element around the thread is constant, e.g. ri^.^ = constant. This appears to be more realistic as the effect of curvature increases.
It is concluded therefore that the logarithmic fourth degree profile is better for values of C < 25 while the
Sakiadis profile is more satisfactory for values of C > 25. A reliable relation for the pumping capacity cannot be given though: even for high values of 5 the logarithmic profiles are curved in the outer part of the boundary layer. This region has a relatively large cross sectional area and will therefore contribute greatly to the total flow rate of en-trained fluid.
For the establishment of the magnitude of the wall ef-fect Griffith's (8) experimental results have also been used. The deviation from the "ideal" behaviour was characterised by ^th - ^exp ^^ which B.|.^ follows from the Sakiadis theory and
Fig. 2-11 Results at C = 266 and 6.^;h/D = 1.05; comparison
with the Sakiadis profile for
E, = 266 and no wall effect (3 = 5.83)
hh 1 — • y"^ •
•y
1 > • X y^ nl--"
1
• Griffith's resultsl X Our results " V D I I I3exp is the observed value of 3 from the experimental veloc-ity profile
U - u
s 1
In
i^)
Fig. 2-12 The effect of the wall of the vessel on the slope of the velocity pro-file around the moving cy-linder
U^ 3 "'' R s exp
To account for the wall effect attempts were made to corre-late 3t}i - 6p„ and S^j^/D for both Griffith's and the pre-sent results. It had been hoped that (S.tji/D would be an unambiguous yardstick for the wall effect, e.g. that its in-fluence on Bth - Bg^ would be independent of 5. Although there is an appreciable scat-ter in the points (Fig. 2-12) it was not possible to detect any significant influence of
5 on their location. It should be noted that Griffith's data
seem to be somewhat inconsistent: Two measurements, executed at the same C values, in the same vessel and with the same cylinder but with different velocities and x-values, yielded markedly different values of 3. This would imply that besides
such a quantity as ó^j/D and, eventually, C, another parameter, perhaps x/D, would influence the magnitude of the wall effect. This appears unlikely, and the reliability of his results is therefore somewhat in doubt.
The correlation based on Fig. 2-12, can be represented by th exp = 1.5
W D < 2.5
A . 0-76 ^ D ' 600 (2-11:It should be emphasized that this equation cannot be extra-polated into the region of established flow where 3 is a function of R/D only. For small wall effects and not^?oo large e-values however Safe-iadis'theory together with the correction Eq. 2-11 will give a quick and fairly reliable estimate of the flow profile.
§2.7.1
39
2.7 Turbulent momentum transfer
2.7.1 Transition to turbulence at a flat plate
The transition from laminar to turbulent flow for a moving continuous flat plate was analysed theoretically by
Tsou, Sparrow and Kuntz (12). Their conclusion was that for Rey niombers below 1+.96*10^ infinitesimal disturbances would die out, while at higher Reynolds numbers the flow would be-come unstable. For a stationary flat plate of finite length the corresponding theoretical Reynolds number is 0.91+9*10^. The higher stability of the flow at a moving continuous flat plate was attributed to the inward direction of the trans-verse velocity component. Disturbances are moved towards the wall where they are more readily damped out.
The accurate measurements of the velocity profile by
TsoUj Sparrow and Goldstein, reported in §2.5.2, were extended into the region of turbulent flow. They found experimentally
a transition to turbulence at Re^ ^ 1.5*10^. Tsou et al. made
no attempt to explain this with the help of their theoretical analysis of the transition. Two important conclusions may be drawn: First: the transition will occur at approximately Re^ = 1.5*10^, the accurate value depending on the magnitude of the disturbances. Second: the flow at moving continuous surfaces is more stable than at stationary surfaces. 2.7.2 Turbulent velocity profile for the flat plate
Tsou, Sparrow and Goldstein set up a theoretical model for the velocity distribution in turbulent flow:
IL ^ . 0.917 ^ e x p
erf
[S^m ^-^^^)\
/2 /T /P w 5 {0.109 (• U^^J^
•)Y U - u and w for V V p - = U.75 + 2.5 In ( ^ A„/p#) - f #
26 (2-12)with Tsou, Sparrow and
Goldstein's ments; these measure-ments were executed at Rer numbers between
10^ and 5'10^. The value of the shear stress at the wall can theoretically be de-duced from this veloci-ty profile by applying Pohlhausen's method.
ic? K? io' 10* The result is given in Fig. 2-13. The
experi-Figure 2-13 Average friction factor ments of Tsou et al.
at turbulent flow gave values of the friction factor that were ca. 6% higher.
Sakiadis too gave a theoretical description of the tur-bulent velocity profile and, accordingly, of the relation-between friction factor and Reynolds number for turbulent
flow. However, Tsou et al. 's description is a more refined
one; moreover, it agrees better with the experiments. There-fore Tsou et al. 's turbulent velocity profile and <Cf> - Re^ relationship are recommended.
2.7.3 The friction factor for a cylinder with turbulent flow The friction factor for a thin moving continuous cylinder tinder turbulent flow conditions has to be determined experi-mentally; no theoretical approach is known.
Selwood (13) presented measurements of the drag exerted on thin threads moving through air. The velocities ranged from 5 to 20 m/s, the threads ran through 20 m of air; the diameters were 0.027, 0.01+2 and 0.086 mm. The Reynolds^ num-bers were always greater than 6*10^. Under these conditions the fluid would be expected to be in turbulent flow and the length of the thread would have only a small influence on the friction factor. Accordingly attempts were made to correlate the average friction factor, <Cf>, with the ReynoldSjj number
(see Fig. 2-11+).
Apparently, the assumption that the length of the thread is unimportant is consistent with the experimental results. These can be correlated by the empirical equation:
§2.7.3
Ui
<c^>
.65 Re -o-d 65 8<Re^<112, 2.3«10-^<L/d<7.1f10-^ (2-13) Very probably, within the mentioned limits of 'Re^, theequation will be valid over a range of L/d ratios which is wider than the experimentally verified one.
The general correlation for the friction factor at a moving continuous cylinder under turbulent flow conditions is
<c,> \ ^ ^ N.
K
\%> 1X
X I 1 diameter ^ • 2.7.1Ö m X 0N
42.1^mJ 8.6.lif ml 1 1Fig. 2-11+ A dimensionless
cor-relation of Selwood's
experi-mental results
<C^> f(Rej^, Re^) .
For the flat plate this relation simplifies to: <C^> fl (Rej^)
which can be obtained from Fig. 2-13. For the cylinder,
Sel-wood' s experiments prove that - in a limited range of Rej^ and
B.e^ values - <Cf> is a function of Re^^ only. These relations cannot be matched because of the lack of experimental data, but in many cases they may help in making a reasonable guess of the friction factor at turbulent flow.
2.6 Conclusions
A new kind of boundary flow was identified and described by Sakiadis. The difference from the "usual" boundary layer flows can best be illustrated by investigating the velocity profiles near a stationary flat plate and near a moving con-tinuous flat plate. Essential differences from the Blasius profile (established near a stagnant flat plate) were
pre-dicted by Sakiadis; experiments showed that this is not
mere-ly a theoretical conception: the flow established in practice is predicted accurately by the theory of laminar flow near a moving continuous flat plate.
Flow around moving continuous cylinders has a second sig-nificant feature: the curvature; this is characterised by the ratio of boundary layer thickness and cylinder diameter. The curvature has a very large influence on boundary layer
flow characteristics such as the wall friction. The effect of curvature predicted theoretically was confirmed by the present experiments in the laminar flow regime. These experi-ments have, to some extent, refined the theory and set limits to the applicability of the boundary layer flow.
Velocity profiles and wall friction for turbulent flow around a moving continuous flat plate are known with
reason-able accuracy from experiments of Tsou et al. Little is known
about the friction factor for turbulent boundary layer flow around the moving continuous cylinder.
§3.1 k3
Chapter 3
HEAT AND MASS TRANSFER TO A MOVING CONTINUOUS CYLINDER; FORCED CONVECTION
3.1 Introduction
In cooling continuous hot threads moving in a cold fluid, a variety of complex situations can be met: the fluid may be a liquid or a gas, the thread may be thin or relatively thick and it may run vertically or horizontally. However, for the reasons given in the first chapter, discussion will be re-stricted to a number of simple situations.
Before proceeding to more complex situations an analysis will be made of the case of laminar flow heat transfer from cylinders, neglecting natural convection, with the additional assumption that the temperature across the cylinder is uniform at any given position along its length (i.e. the internal re-sistance to heat transfer is negligible).
As heat and mass transfer under such conditions are com-pletely analogous, they can be treated as one subject. In the theoretical paragraphs as well as in the experimental part of this thesis, equations, results and conclusions will be dis-cussed in terms of both heat and mass transfer.
The theoretical treatment of the laminar flow heat trans-fer will - of course - show again the two characteristics of flow around moving continuous cylinders: see §2.1. In addi-tion to that a third characteristic that is inherent to the cooling or the heating up of moving continuous bodies will be met. In the flow problem, the "driving force", i.e. the ve-locity difference between cylinder and fluid, remains
con-stant during the process. In the heat transfer problem,
how-ever, the "driving force", i.e. the temperature difference between cylinder and fluid, will change during the process as a result of the heat transfer. It will be proved that this may have a considerable influence on the local transfer coef-ficients.
3.2 Theoretical solutions for laminar flow
3.2.1 Basis of the solution procedure; the O-parameter solu-tion
Fig. 3-1 Hydrodynamic-al and thermHydrodynamic-al boundary layers around a moving continuous cylinder
The heat transfer rate to a moving continuous cylinder depends
on many factors; we will start with the simplest possible picture of the situation. Firstly, by ignoring the effects due to the curvature of the continuous surface, the flat plate limiting situation is considered. Secondly, the continuous body is assumed to have a constant temper-ature (infinite heat capacity). In the remaining "flat plate, constant temperature problem", examination of the situation at very high Prandtl numbers, as is sketched in Fig. 3-1, is used as the starting point.
As a result of the very high Prandtl number, the boundary layer for heat transfer is much thinner than the boundary layer for momentum transfer. Therefore the heat transfer takes place in a region where the longitudinal velocity is virtually equal to the cylinder velocity (this implies that the trans-verse velocity can be neglected); the heat transfer boundary layer is stagnant with respect to the moving surface. An ob-server, fixed on a material point of the continuous body now sees a process of non-steady penetration of heat into the ad-jacent part of the heat transfer boundary layer. This process
is governed by the well known Fourier equation:
3^T
a TT-T (a is the thermal diffusivity) _3T 3t 3y^ y = 0 1 t > 0 ^' (3-1: t = 0 y > 0
},
The solution provides the temperature profile from which the heat flux at the wall can be obtained: -X —
(A is the heat conductivity). y=0 By definition this heat flux equals a(TQ - Too) in which a is
the heat transfer coefficient. After the transformation X = U g t , the following expression for the heat transfer coef-ficient is obtained:
§3.2.1
U5
Figure 3-2 The influence of the Prandtl number on the heat transfer to a moving continuous flat plate of
finite heat capacity
/iT Nu ^ (Nu ax X Pe (3-2) U X s \
3.2.2 Effects of Prandtl number, curvature and finite heat capacity"; ~thi~T-paramëtir~ solution
The above solution (3-2) is valid for Pr -> «> only. If the Prandtl number has a finite value, we may expect a solu-tion of the form:
/iT Nu ,
= f5(Pr) ^ . (3-3)
X
An existing, exact solution of the boundary layer equation for a flat plate with infinite heat capacity produces this function f5(Pr) (see Fig. 3-2), From Fig. 3-2 we see that the function f5(Pr) does not deviate much from unity for most heat and mass transfer processes in gases or non-metallic
l i q u i d s . In the solution
1 l 2
.,, = f3(Pr) iSf^iM:
the "correction factor" f5 is important for small Prandtl numbers only.
The heat transfer to a stationary flat plate of finite length is represented, with good approximation, by
Nu = 0.332 Re ^Pr^/^ . (3-1+) Consequently, heat transfer to moving objects is much faster
than to stationary objects, especially at large Prandtl num-bers. Here, as with momentum transfer, the difference between these two solutions can be attributed mainly to the importance of convection (of heat in this case) near the surface of the continuous body. This explains why this difference becomes greater for large Prandtl numbers. This is the effect on the heat transfer process of what was defined in chapter 2 as "the first characteristic" of the flow around moving continu-ous cylinders.
The second characteristic refers to the influence of the curvature of the cylinder on the heat transfer coefficient. For momentum transfer, this effect proved to depend on the group 5 = 1+ y.. „2 > which is proportional to the ratio of boundary layer thickness at zero curvature to the cylinder radius. The boundary layer thickness for heat transfer is, according to the penetration theory solution (Pr -* <»), pro-portional to /ax/U . Hence, the group that - analogous to the group C for momentum transfer - represents the influence of
/ ax
curvature must be C = \L, p'> . So for infinite Prandtl numbers we may expect a solution of the form:
/TT N U
X
= f2(0 . (3-5)
A solution of the Fourier equation in cylindrical coordinates
(valid for Pr -* °=) provides this function f2(t)» which is displayed in Fig. 3-3. Here again we find a large influence of the curvature for relatively thin cylinders. For large
