HEAT TRANSFER FROM IMPINGING
FLAME JETS
, HEAT TRANSFER FROM IMPINGING
V FLAME JETS
HEAT TRANSFER FROM IMPINGING
FLAME JETS
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus, prof.dr. J . M . Dirken, in het openbaar
te verdedigen ten overstaan van een commissie door het College van Dekanen
daartoe aangewezen, op 10 september 1987 te 14.00 uur
door
Theodorus Hendrikus van der Meer
geboren te Zoetermeer natuurkundig ingenieur
Dit proefschrift is goedgekeurd door de promotor prof.ir. C.J. Hoogendoorn
aan mijn ouders aan Funny
CONTENTS
1 . INTRODUCTION
1 .1 Background 11 1.2 Aims of this study 12
1.3 Outline of the investigation 13
2. LITERATURE SURVEY
2.1 Hydrodynamics 17 2.1.1 Turbulent free jets 18
2.1.2 The stagnation flow region 20 2.1.2.1 A bluff body in a uniform cross flow 21
2.1.2.2 The stagnation flow region of an impinging
jet 25 2.1.3 The wall jet region 28
2.2 Heat transfer of impinging flows 29 2.2.1 Stagnation point heat transfer 29
Influence of the turbulent length scale on
stagnation stagnation point heat transfer 35 2.2.2 Heat transfer from cold impinging jets 36
2.2.2.1 The laminar impinging jet 37 2.2.2.2 The turbulent impinging jet 40 2.2.3 Heat transfer from flame jets 50
3. THEORY
3.1 The governing equations 53
3.2 Turbulence models 55 3.2.1 The k-£ model of turbulence 56
3.2.2 A low Reynolds number model 58 3.2.3 Drawbacks of the k-e model 60 3.2.4 The anisotropic model 62 3.3 The energy equation 67
4. THE NUMERICAL METHOD
4.2 The hydrodynamic solver 73
4.3 The grid 74 4.4 The boundary conditions 76
4.5 Determination of the heat transfer coefficient 79
THE EXPERIMENTAL METHOD
5.1 Heat transfer from the isothermal jet 81
5.1.1 Experimental set-up 81 5.1.2 Temperature measurements with liquid crystals 83
5.2 Heat transfer from the flame jet 85
5.2.1 The experimental set-up 85 5.2.2 The Gardon heat flux transducer 88
5.3 The laser Doppler anemometer 91 5.3.1 The optical configuration 91 5.3.2 The electronic equipment 93 5.3.3 T h e s e e d i n g o f t h e f l o w 9 5
R E S U L T S O F T H E E X P E R I M E N T S
6.1 Introduction 97 6.2 Flow structure 97 6.2.1 Velocity and turbulence on the axis of the
free jet 97 6.2.2 The radial velocity gradient in the vicinity
of the stagnation point 1 07
6.2.3 The radial velocity profiles 110
6.2.3.1 The isothermal jet 110
H/d = 2 111 H/d = 6 113 The boundary layer thickness 116
6.2.3.2 The flame jet 117
H/d = 2 119 H/d = 6 119 6.2.4 Axial temperature distribution 122
6.3 Heat transfer 123 6.3.1 Stagnation point heat transfer 123
6.3.1.1 Isothermal jet 124 6.3.1.2 The flame jet 128
Radiation heat transfer 128 Convective heat transfer 130 6.3.2 Radial heat transfer distributions 137
6.3.2.1 The impinging isothermal jets 137 6.3.2.2 The impinging flame jets 141
Temperature distributions 141 The heat flux distributions 143 The Nusselt number distributions 145
7. RESULTS OF NUMERICAL SIMULATIONS
7.1 The laminar impinging jet 149 7.1.1 Comparison with literature data 154
7.2 The turbulent impinging jet 157 7.2.1 Comparison of results on different grids 159
7.2.2 Comparison of numerical with experimental
results 162 7.2.2.1 H/d = 6 162
7.2.2.2 H/d = 2 165 7.2.2.3 The stagnation point heat transfer 171
8. DISCUSSION AND CONCLUSIONS
8.1 The flow structure 173 8.2 Heat transfer 174 8.3 The simulated laminar impinging jet 175
8.4 The simulated turbulent impinging jet 176
8.5 Main conclusions 176
LIST OF PRINCIPLE SYMBOLS 179
LIST OF REFERENCES 183
SUMMARY 191 SAMENVATTING 1 93
CURRICULUM VITAE 195
1. INTRODUCTION
1.1. Background
Heating, cooling and drying processes are often used in industry. In most applications high heat transfer rates leading to short processing times are required. The high heat transfer rates are especially needed in circumstances where the energy consumption of the process is relatively high. Obtaining short processing times is often needed for reasons of product quality. A very well-known technique for heating or cooling purposes is the application of impinging jets. The high heat transfer due to turbulent forced convection by impinging one or more jets of hot air or one or more flames on the object to be heated makes a relatively short exposure time possible. In the metallurgical industry this technique is called rapid heating. In cooling and drying a similar situation occurs; one or more jets of cold (dry) air impinge to cool (dry) a product.
Rapid heating of products in furnaces is a common process in, for instance, the glass and steel industry. To obtain a uniform heat transfer rate to the object in most cases radiative heat transfer is preferred over convective heat transfer. Radiative heat transfer can be achieved by firing gas, coal or oil in a radiation furnace. The walls of the furnace are heated and in its turn the object is heated by radiation heat transfer from these walls. Also often an electrically heated wall is used. In this way the control of the radiation temperature over the hot surface can easily be obtained. When a short exposure time of the object to the high temperatures is needed, it can be advantageous to enlarge heat transfer by impinging gas flames directly on this object. For this purpose high velocity burners are used, the major heat transfer is by convection. There are several other advantages of using these so-called impinging burners in rapid heating furnaces:
radia-tion furnaces, giving lower wall heat losses. Starting up and cooling down periods are much shorter which also result in an energy saving.
- Energy can be saved by switching on the burners only when heat is demanded.
- Compared to heating electrically the primary fuel demand is smaller.
- It is possible to heat locally.
The energy savings compared to a conventional radiation furnace can be more than 50%. A major disadvantage of rapid heating furnaces can be nonuniformity of the heat flux distribution. With convective heat transfer it is much more difficult to obtain uniform heating of an object than with radiation heat transfer. It is possible that hot spots are created and overheating at such spots (for instance, at a stagnation point) can occur. For this reason it is important to know the heat flux distribution of a flame jet impinging on an object.
1.2. Aims of this study
The main purpose of the investigation presented in this thesis was to study the nonuniformity of the heat flux and to find out the influence of turbulence on the heat transfer for a single
Ufflk/
1 I
''•Ml
or
flame jet impinging on a flat plate. The flow configuration is given in figure 1.1. The flame jets were produced by modified commercial rapid heating tunnel burners. The highest heat transfer rates applying impinging jets can be achieved for distances between the nozzle exit and the plate of 2 to 12 nozzle diameters. In this region for a turbulent jet the shape of the velocity profile and the turbulence intensity profile change with the distance from the nozzle. The jet shape also depends on the shape of the nozzle from which it originates. For these reasons one simple expression for the heat transfer to a plate on which the flame jet impinges cannot be given from literature. In this study both heat transfer and flow structure of impinging flame jets and of isothermal air jets from the same burners are thoroughly examined.
1.3. Outline of the investigation
From literature much data on stagnation flows and impingement heat transfer are available. In chapter 2 this literature is discussed. Since the flow around bodies of revolution has its analogies with the impinging jet on a flat plate these flows are discussed in the first place. Here an important parameter is defined: the gradient of the radial velocity near the stagnation point just outside of the boundary layer: aR =
(3v/3r)r_,.0 A similar parameter can be defined in impinging jet
flow: the gradient of the maximum radial velocity near the stagnation point: B = (3v x/ 3 r )r . These parameters appear to
depend on the shape of the body of revolution and the shape of the impinging velocity profile, respectively. The influence of the shape of the body or the shape of the impinging velocity profile on the heat transfer at the stagnation point can be expressed using the radial velocity gradients aR or g.
The governing equations for the flow and heat transfer are presented in chapter 3. These are the continuity equation, the Navier-Stokes equations, the energy equation and equations forming a model to calculate the turbulent viscosity of the
flow (the k-e model). Since the turbulence in a stagnation flow will be anisotropic due to the deceleration in axial direction and acceleration in radial direction, the commonly used k-e model has been extended with a third parameter, which takes the anisotropy into account.
In chapter 4 the numerical technique used, the finite volume method, is given. Together with the appropriate boundary conditions we have all ingredients to be able to solve the governing equations from chapter 3 numerically. The results from these numerical calculations are discussed in chapter 7. At first some experimental methods and set-ups for determining heat transfer and flow characteristics are given in chapter 5. The heat transfer measurements for the isothermal impinging jets are performed with a liquid crystal technique. Also a Gardon heat flux transducer is used for the determination of the heat transfer distributions of the impinging flame jets. Temperatures in the flame jets are measured with thin wire Pt-Rh thermocouples. At last in chapter 5 the laser Doppler anemometer for velocity and turbulence intensity measurements is discussed.
The next two chapters deal with the actual results from our study. The experimental results in chapter 6 and the numerical results in chapter 7. At first in chapter 6 results of the flow structures of the impinging jets are given. Important characteristics of the jets are:
- the axial velocity decay and the axial turbulence development as a function of the distance from the burner. Comparisons between isothermal jets and flame jets can give insight into the effects of combustion on the turbulence.
- the radial velocity gradient near the stagnation point (6). The impact velocity profile will have its influence on this parameter. With (3 a first estimation of the heat transfer at the stagnation point can be made.
- the radial velocity profiles close to the plate.
Stagnation point heat transfer coefficients, determined with the radial velocity gradient Q, are compared with stagnation
point heat transfer coefficient, calculated from heat flux measurements. The influence of turbulence is examined. Heat transfer results from flame jets and from isothermal jets are compared and described as much as possible in a similar way. At last the radial heat transfer distributions of the impinging isothermal jets and impinging flame jets are discussed in this chapter.
Chapter 7 contains results on numerical calculations. Laminar impinging jets with three different impact velocity profiles (flat, parabolic and Gaussian) are simulated. With these calculations the influence of the impact velocity profile on the heat transfer distribution on the plate can be determined. Besides this the computer code can be validated by comparing the results with results from other investigators. Results of simulations of turbulent impinging jets are also given in this chapter. Calculations have been performed with the standard k-e model of turbulence with modifications for low Reynolds numbers and with a k-e model including a parameter for the anisotropy of the turbulence. The computed flow fields and heat transfer are compared with the measurements for validity for the two H/d values: H/d = 2 and H/d = 6.
Finally, in chapter 8 the conclusions from this study and their consequences for the practical use of impinging flame jets are discussed.
2. LITERATURE SURVEY
2.1. Hydrodynamics
Extensive studies on the hydrodynamics of stagnation flows have been done in the past. They will be reviewed in this chapter. Before entering into details a brief description will be given of the flow pattern of an impinging round jet on a flat plate. This flow can be divided into three regions (see figure 2.1): the free jet region, the stagnation flow region, and the wall jet region.
stagnation
zone wall let
U *\ ►
Fig. 2.1. Flow regions of a jet impinging on a flat plate.
In the free jet region the flat plate has no perceptible influence on the flow. According to Schrader (1961) this region extends to a distance of 1.2 times the nozzle diameter (1.2d)
from the surface. '-■ In the stagnation flow region the axial flow strongly
decelerates and the radial flow accelerates giving rise to an increased pressure in this region. The characteristics''of the
stagnation flow region depend strongly on the dimensionless nozzle to plate distance (H/d). It extends from 1.2d in axial distance from the plate to about 1.1d in radial direction for small nozzle to plate distances (H/d < 12).
In the wall jet region the fluid spreads out radially over the surface in a decelerating flow.
In the following paragraphs the three regions will be discussed in more detail.
2.1.1. Turbulent free jets
The free circular turbulent jet has been studied thoroughly in the past. In this paragraph only a brief description will be given of the results of these investigations. More detailed information can be found in the handbooks of Rajaratnam (1976) and Abramovich (1963).
The free circular turbulent jet can be divided into three zones. Referring to figure 2.2 we have:
developed zone developing zone potential core zone F i g . 2 . 2 . The f r e e j e t .
1 . The potential core zone immediately downstream of the nozzle. In this zone the potential core is the flow region where the velocity remains constant and equal to the velocity at the nozzle exit. Turbulence is being generated by the large shear stresses at the jet boundary and diffuses towards the axis. The length of the potential core depends on the initial velocity profile and on the turbulence intensity in the nozzle exit. According to Gauntner, Livinggood and Hrycak (1970) the potential core length varies from 4.7d to 7.7d.
2. The developing zone in which the axial velocity starts decaying. The velocity profile develops into a profile which
is independent of the nozzle geometry.
3. The zone of fully developed flow, where the velocity profile has reached its final shape. Tolmien (1948) and Gortler
(1942) calculated a radial velocity profile from boundary layer type equations with the use of Prandtl's mixing length theory. Reichardt (1942) performed measurements and found that a Gaussian velocity distribution fitted his results best.
It is shown by Rajaratnam (1976) that in the fully developed jet flow the jet broadens linearly and the velocity at the axis decays linearly. This has been justified by experimental results. For the axial velocity decay Hinze and v.d. Hegge-Zijnen (1949) and Schrader (1961) give the correlations:
u 6.39
Hinze and Zijnen: — = — (x/d i 8) (2.1)
uQ x/d + 0.6
u 8.0
Schrader: — = — (x/d i 8) (2.2)
uQ x/d + 3.3
If the jet has a different density than the surrounding fluid, a correction is required. Based on the conservation of momentum flux Thring and Newby (1953) find an equivalent
nozzle-diameter, :de, for non-constant density jets. Due to the high
rate of entrainment the density within the jet will approach the density of the surroundings (Ps) within a short distance
from the nozzle. The momentum flux is:
ird2 ird*
G = — P o V = — f "
Ps
uo' <
2-
3>
which leads to
d_ = d (^2) (2.4) Ps
The relationships for isodensity jets can be used for non-isodensity jets using this equivalent diameter. Chen and Rodi (1978) come to the same equivalent diameter by dimensional considerations. Due to density differences also a buoyancy effect can occur. Chen and Rodi also give the limits within which a hot round jet will be non-buoyant, being:
Fr- 2 (-°-)_4 - < 0.5 (2.5)
Ps d
Here Fr is a densimetric Froude number D u 3
Fr = 1°^°- (2.6) g(Ps - P0)d
This densimetric Froude number in our experiments was high enough to obey criterion (2.5). .
2.1.2. The stagnation flow region
In the stagnation flow region the axial flow strongly decelerates and the radial flow accelerates giving rise to an increased pressure. The characteristics of this region depend strongly on the dimensionless nozzle to plate distance H/d. The limits of the stagnation flow region too are determined by H/d. According to Schrader (1961) for nozzle to plate distances H/d < 10 the stagnation flow region extends to 1.2d from the
impinged plate in axial direction and up to about 1.1d from the stagnation point in radial direction. Before entering into details of stagnation of an impinging jet, the simpler flow around a bluff body will be discussed.
2.1.2.1. A bluff body in a uniform cross flow
The first solutions of the boundary layer equations for a two-dimensional shear layer along a cylindrical body, which is perpendicular to a uniform cross flow, were given by Blasius
(1908), Hiemenz (1911) and Howarth (1935) (see Schlichting, 1968). They supposed the flow outside of the boundary layer to be a potential flow. The velocity along , the body can be expressed as:
V(x) = v.] z + V3Z + V5Z + . . . (2.7) Here z is the coordinate along the surface of the body. The
velocity profile in the shear layer was calculated as a similar polynomial in the coordinate perpendicular to the surface.
In the vicinity of the stagnation point the velocity decay due to stagnation; and the acceleration of1 the fluid flow along
the surface just outside of the boundary-layer are given for axisymmetric flow by:
U = - 2 aRy and V = aRz (2.8)
and for plane flow by:j • ■ ■
U = - axy and V = axz' ' (2.9)
, Homann (1936) solved the boundary layer equations for the case of axisymmetric flow with assumption (2.8)..
The constants aR and ax in equations* 2.8 and 2.9;' depend on-'the
shape and size of the body of impingement and on the. uniform flow velocity.' For three different bluntr bodies it is found from potential flow solutions (see Kays; 1966 and Kottke,
c i r c u l a r sphere c y l i n d e r d i s c aR aR ax = 4 = 3 = 4 U^/dTr U „ / d Uoo/d (2.10) (2.11) (2.12) where d is the diameter of the body involved and U^ the uniform flow velocity. More accurate experimentally determined values of aR and ax are given by:
Kottke, Blenke and Schmidt (1977) for a disc:
aR = Ujö (2.13)
Newman, Sparrow and Eckert (1972) for a sphere:
aR = 2.66 U^/d (2.14)
and Hiemenz (1911) for a cylinder:
ax = 3.63 Ujd (2.15)
Compared to the infinitely extended laminar flow around a body, the turbulent flow is far more complex. Let us consider the influence of turbulence.
Due to the experimentally found strong sensitivity of stagnation point heat transfer of cylinders and spheres to small changes in the intensity of free stream turbulence (see Kestin and Maeder, 1957; Kestin, Maeder and Sogin, 1961; Kestin, Maeder and Wang, 1961), Sutera, Maeder and Kestin (1963) and Sutera (1965) did a theoretical investigation into the vorticity amplification in stagnation point flow. In a basically two-dimensional flow vorticity was distributed periodically over the third dimension. The normal velocity far from the stagnation surface had a periodic waviness along the direction normal to the plane of the basic flow (see figure 2.3). They showed that such vorticity, having a sufficiently large scale, can enter the boundary layer and significantly alter the heat transfer at the wall. Vorticity with a scale larger than the neutral wave length Xm±n = 21I/(aT}/v)i or X •
Fig. 2.3. The distorted stagnation flow studied by Sutera, Maeder and Kestin (1963).
2Tr/(ax/v)5 will be amplified. The distortion of the velocity
field seemed to be small. Figures 2.4 and 2.5 show the distortions of velocity and temperature of the mean flow along the surface compared to the undisturbed case for Pr = 0.74. The shear stress increased by 4.85%, the temperature gradient by 26%. Experimental verification of this theory is presented by Sadeh, Sutera and Maeder (1970). They conclude that turbulence, and hence vorticity, is being amplified by the deceleration of the stagnation flow and by stretching of fluid elements. The amount of amplification seems to depend on the direction of the vorticity as was predicted by the theory. For natural turbulence on a stagnation streamline the turbulence intensity they found is given in figure 2.6. The dependence of amplification on scale was also found to be in accordance with the predictions of the vorticity amplification theory.
0.8
0.6
0.4
0.2
1—r
Fig. 2.4. The distorted stagnation velocity after Sutera, Maeder and Kestin (1963).
J L
v '
Fig. 2.5. The distorted temperature field on a stagnation streamline after Sutera, Maeder and Kestin (1963).
V7~
2.0 f.5r.o
0.5 o 0 0.1 0.2 0.3 yFig. 2.6. The turbulence intensity on a stagnation streamline for natural turbulence (after Sadeh, Sutera and Maeder, 1970).
2.1.2.2. T h e stagnation flow region of an impinging jet
In the case of a bluff body in a uniform cross stream we have seen that the average flow field in the stagnation region is dependent of the shape of the body. So, it can be expected that in the case of an impinging jet on a flat plate, the average flow field in the stagnation region depends on the oncoming velocity profile. For a free turbulent jet the velocity profile changes from a flat profile into a fully developed one with a Gaussian shape. Thus the character of the centreline velocity decay in the stagnation region also changes. Similar to the definitions of aR and ax (equations 2.8 and 2.9) for a bluff body in a uniform cross flow, this same parameter ■ can bé defined for an impinging jet:
axisymmetric flow u = - 2 aRy and vm a x = aRr (2.16)
plane flow u = - axy and vm a x = axz (2.17)
Here v is the maximum velocity along the plate. The analogy
between a bluff body in a cross flow and an impinging jet only exists in the direct vicinity of the stagnation point. Where for a bluff body in a cross stream the value of aR is
determined by the shape of the body, this value for an impinging jet is determined by the shape of the oncoming velocity profile.
For an inviscid uniform impinging jet Strand (1964) calculated for the velocity along the deflecting surface (H/d = 1 ):
V = 0.9032 ^ ° - + . . . . (2.18) d
Scholtz and Trass (1970) obtained a similar expression for an inviscid parabolic impinging jet. They found for H/d = 0.25:
V = 2.322 ^ ° — + . . . . (2.19) d
From experiments it appears that the value of aR for a
disc with diameter d in a uniform cross flow is the same as for a uniform jet with diameter d impinging on a flat plate. Schrader (1961) and Dosdogru (1974) found for aR in the case of
small nozzle to plate distances of a uniform turbulent impinging jet (1 1 H/d S 10): H un Schrader aR = (1.04 - 0.034 -) -°- (2.20) d d Dosdogru aR = (1.02 - 0.024 -) -°- (2.21) d d
Giralt, Chia and Trass (1977) correlate their radial velocity gradient in the stagnation zone with the impact velocity and the jet half radius at the beginning of the impingement region, where the axial velocity in the impinging jet becomes 98% of the axial velocity in the undisturbed jet. The impact velocity from measurements by Giralt (1976) being:
H H u„ = u^ (1.004 - 0.003 -) - S 5.5 (2.22) c ° d d H H u„ = u„ (1.35 - 0.066 -) 5.5 < - S 10.0 (2.23) "- ° d d 7.37 H u_ = u„ - > 10.0 (2.24) c ° 0.67 + H/d d
The length scale at the beginning of the impingement region is characterized by them as:
rni H H -iz- = 0.493 + 0.006 - 1.2 s - S 6.8 (2.25) d d ■ d r, 1 H H - i * = 0.069 (1 + -) - > 6/8 (2.26) d d d
For the value of aR can then be found:
aR = U 1 . ^ ° - (2.27)
15
u-| is a function of H/d expressing the influence of the shape of the velocity profile. For H/d = 1 .2, where uc = u0 and r^j. =
id, they find:
aD = 0.916 ^2 (2.28) d *R For H/d > 10, however, aR 1.852 ^ (2.29) d i 2
Like in equations 2.18 and 2.19 one can see the strong influence of the shape of the oncoming velocity profile on the flow characteristics in the vicinity of the stagnation point.
The role of the turbulence in the flow field has been visualized by Yokobori, Kasagi and Hirata (1977). They showed that when the plate was positioned in the developing region of
the jet (4 < H/d < 10) large scale eddies existed in the stagnation zone. The eddies seemed to be much larger than the thickness of the laminar boundary layer, and appeared to be generated by the large *shea_r at the jet boundary'upstream. For H/d < 4 the stagnation zone looked laminar-like and for H/d > 12 the eddies seemed to be distorted and accompanied by small scale turbulence.
2.1.3. The wall jet region
Where the velocity essentially is parallel to the plate the wall;,jet region starts. Schrader (1961) gdves a correlation for the radius r at which the velocity along the wall starts decaying. This he defines as the beginning of the wall jet
. v ■ . 'i
region. The correlation fór r is:
y
-2 = 1.09 (-)0-034 ' ' (2.30)
d H
For the maximum velocity in the wall jet he finds:
U, : — — — : L- + K,(H/d - 1 . 2 ) ( — - 1 )
J0.. 1 +0.1.8. (H/d.- 1 .2);l ..2 , -.rgl
r -1.17
( — ) - ' • " ' (2.31)
rg
with K1 = 1.10 and K2 = 0.27 for H/d S 4.7
and K1 = 1.45 and K2 = 0.09 for H/d > 4.7.
For the wall jet velocity profile Schrader found that already at r/d 6 2 the profile was similar to the profile calculated by Glauert (1956) for a fully developed wall jet (see figure 2.7). The validity of Glauert's calculations is shown by Bakke (1957) and Poreh, Tsuei and Cermak (1967) who did measurements for larger distances from the stagnation point (r/d > 10). Because our experiments 'are "restricted to small values of H/d (H/d i
12) and" r'/d (r/d s 5) no' details of their measurements are given.
1.2 0.8 0.4 O O 0.4 0.8 1.2 1.6 2.0 y/y*
Fig. 2.7. Wall jet velocity profiles measured by Schrader (nozzle diameter of 50 mm, H/d =• 2) and the profile calculated by Glauert (a).
2.2. Heat transfer of impinging flows
The heat transfer characteristics of impinging flows will be discussed in the next three paragraphs. Of course the heat transfer is determined by the hydrodynamics treated in the previous paragraph. Firstly, results from literature, of heat transfer at a stagnation point will be discussed, mainly for cylinders in a uniform cross flow. The next paragraph concerns local heat transfer distributions of impinging round jets with almost constant fluid properties. In the last paragraph impinging flame jets will be discussed where, due to the large temperature differences, the fluid properties (like dynamic viscosity, specific density and thermal conductivity) vary strongly.
2.2.1. Stagnation point heat transfer
Heat transfer at a stagnation point of a body of revolution has
I
y-*
/ i i i i vo*L, i i i i o V 0 A a l 1 1 r=S0 mm r=60 r=80 r=100 r = 150 a £? o I 0 V I ■ I o I-been studied extensively in the past. Knowledge of the heat transfer at this point is of importance because here the heat transfer will be at a maximum. From literature we know the solutions in expansion series from Pohlhausen (1921), Eckert (1942) and Merk (1958). Sibulkin (1952) solved the boundary layer equations for laminar heat transfer to a body of revolution near the forward stagnation point. His solution can be regarded as the basis of all other experimental and theoretical results. For the Nusselt number in the stagnation point of a body of revolution he found:
Nu = 0.763 d (-)5 Pr0'4 (2.32)
v
In this equation 3 is equal to the velocity gradient just outside the boundary layer:
3r
= < — > y = 6,r+o <2-33>
For a two-dimensional stagnation point flow Kays (1966) gives a similar equation, which comes to:
Nu = 0.57 d (-)5 Pr0*4 - - (2.34)
v
For a sphere, cylinder and disc the values of 3 are known (see paragraph 2.1.2), leading to the corresponding stagnation point heat transfer results:
2.66 u n c r\ A
sphere 6 = aR = Nu = 1.2 4 Re^^Pr"*4 (2.35)
cylinder 3 = a„ = — Nu = 1.09 Re0*5Pr0-4 (2.36)
disc 3 = aR = — Nu = 0.763 Re°-5Pr0-4 (2.37)
be applied. When an impinging jet has a nonuniform velocity profile (e.g. parabolic or Gaussian) the value of 8 will in general be higher leading to a higher heat transfer rate at the stagnation point. This in analogy with the heat transfer to the stagnation point of a cylinder or a sphere.
Much experimental and theoretical work has been done on the heat transfer of a cylinder in a turbulent cross stream in the past. From these studies much can be understood from the influence of turbulence on stagnation point heat transfer. In this paragraph a review of these studies will be given.
Kestin, Maeder and Sogin (1961) and Kestin, Maeder and Wang (1961) showed that the influence of free stream turbulence on the heat transfer rate on cylinders in cross flow was important. From experiments on heat transfer to a plate at zero incidence it was concluded that only in the presence of a pressure gradient the free stream turbulence had large effects on heat transfer coefficients. The biggest enhancement in heat transfer occurred at relatively low turbulence levels. Kestin, Maeder and Wang (1961) found that the local Nusselt number increased by amounts of 25%-50% when the turbulence intensity increased from 0.5% to 2%.
Sutera, Maeder and Kestin (1963) and Sutera (1965) presented a mathematical model for a steady plane stagnation point flow. They showed that probably the dominant mechanism of heat transfer enhancement by turbulence is vorticity amplification by stretching (see paragraph 2.1.2). Computations done by them showed that a certain amount of vorticity in the oncoming flow caused an increase of the wall shear stress of 4.85%, while the heat transfer was - augmented by 26% (at Pr = 0.74) .
Smith and Kuethe (1966) performed experiments in low-turbulence wind tunnels. They found that the influence of free stream turbulence increased with increasing Reynolds number. At high Reynolds numbers (Re > 105) a phenomenological theory for
experimental, results,. The. theoretical- curve they found for the •heat transfer at the stagnation point on a cylinder is:
Nu
- — = 1 + 0.0277,Tu/Re (2.38) /Re
The assumption was made that the eddy viscosity is proportional to the free stream turbulence and to the distance from the wall;. From their theory; Smith and Kuethe then concluded that Tu/Re; iwoyld be the single correlation parameter .to describe stagnation point heat transfer. From, their experiments at Re-number.s lower than 105 they found that Tu/Re was not the only
parameter, instead there.- was another dependency on the Re
number : -'! . - . ! ■ • ■
Nu " " '" c " - — = 1 + 0.0277 Tu/Re (1 - exp (- 2.9 10~bRe)) (2.39)
/Re
-j. . ,. - , . . . , . . 1
■ -i Many ..-investigato.rs , later used the parameter Tu/Re in their correlations.. Some, of them used, the theoretical calculations ,of Frossling (1940). as a -basis for their turbulent heat transfer correlation. Frossling gives for the; laminar heat transfer ,at the stagnation point on a cylinder:
Nu
. — - . = 0-34.45,. (2.40) /Re
which 'is the; same..as; .equation 2.36 with Pr =. 0.7 for air. J<estin- and Wood (1971) presented their experimental results-using the Smith-Kuethe parameter Tu/Re and the result from-..Frossling. Thus they found- for the turbulent- heat transfer at ; the stagnation' point of a..cylinder: (7.5 lO1* < Re, ' < 1 . 2 5 1 05) Nu / R e Tu/Re Tu/Re 0.945 +. 3.48 — — — - 3.99 ( ,-);* (2.41 ) 100 100
:. Sikmanovic, Oka and-. Koncar-Dj'urdj evic (1974) found, thati.at a
heat transfer was absent for Tu < 2%. They found: Nu
Tie
0.945 + 1 .94 Tu/Re Tu/Re 2.41 ( — — - ) • 100 100 (2.42)This correlation comes very close to equation 2.39 for Re = 19,000.
Lowery and Vachon (1975) on the contrary did not find the dependency on the Reynolds number as in equation 2.39. This is not surprising because their Reynolds numbers vary from 1.10s
to 3.10s, where with the exponent in equation 2.39 the effect
hardly counts. Lowery and Vachon found that a turbulence intensity of 14% gave a maximum increase of the laminar heat transfer of 60%. Raising the turbulence intensity more did not seem' to increase the local heat transfer' anymore, however, it needed more data to justify this statement. They found the correlation: Nu / R e T u / R e T u / R e = 1 . 0 1 + 2 . 6 2 4 — 3 . 0 7 ( . . ): 100 100 (2.43)
The results expressed in equations 2.38, 2.41, 2.42 and 2.43 are gathered in figure 2.8.
Apart from the experimental results discussed so far, some investigators studied theoretically the stagnation point heat transfer on a cylinder in a crossflow. Already mentioned is the phenomenological theory of Smith and Kuethe (1966).
Galloway (1973) formulated a roll cell model, which has been simplified into an eddy viscosity. He used the findings of Sadeh et al. (1970) who showed the formation of roll cells in a two-dimensional stagnation flow. Galloway found a strong amplification for high Prandtl number flows.
Traci and Wilcox (1975) used the Saffman turbulence model in their partly analytic, partly numerical solution of stagnation point heat transfer. They considered three regions: the free stream flow, the still inviscid body distorted flow and the viscous wall region flow. Solutions of the three regions were matched to each other. In agreement with Sadeh et al. (1970) they find amplification of turbulent energy in the stagnation region, while their heat transfer calculations do agree with the known experimental results.
Miyazaki and Sparrow (1977) constructed a model for the eddy viscosity on the basis of measured turbulent velocity fluctuations. It contained a single unknown parameter which was determined from experimental heat transfer results. Their numerical calculations showed that the Nusselt number increased with the free stream turbulence but to a lesser extent as the turbulence intensity increases. The effect of turbulence on the friction factor was much less than on the heat transfer, which also was shown by previous investigators.
Gorla and Nemeth (1982) constructed a mathematical model in which the momentum eddy diffusivity depended on the free stream turbulence and the length scale. Available experimental data were used to find the eddy viscosity as a function of Tu/Re.
enhancement around the stagnation point of a cylinder was performed by Hijikata, Yoshida and Mori (1982). They added an extra equation to the k-e model of turbulence to take into account the production of turbulent energy due to anisotropy between the longitudinal and lateral Reynolds stress components in the free stream (see paragraph 3.2.4). A reasonable agreement with reported experimental data was found.
Influence of the turbulent length scale on stagnation point heat transfer
Next to the influence of the turbulence intensity on the heat transfer investigations were done to the role of the turbulent length scale. For the definition of a characteristic scale most investigators use the turbulent macroscale or integral scale being the scale of the energy containing eddies.The macroscale of turbulence can be found by integrating the area under the space correlation function:
Lv = ƒ R(x) dx (2.44)
x o
with
u(X)2
Van der Hegge-Zijnen (1958) found a rapidly increasing heat transfer with increasing macroscale. He suggested the existence of an optimum value of the ratio between scale and cylinder-diameter which corresponds to a condition of resonance. Then, some frequency of turbulence coincides with the frequency of the eddies shed by the cylinder. The work done by Sutera, Maeder and Kestin (1963) and by Sutera (1965) has already been mentioned before. From their mathematical model it follows that wavelengths shorter than a so-called neutral wavelength:
Xmin =-2ir/(av) (2.46)
cannot satisfy the governing equations. The physical signifi cance of this is that vorticity with a scale smaller than the neutral scale is dissipated more rapidly due to viscous action than it is amplified by stretching. Recent experiments varied the length scales of the flow to measure its influence. Sikmanovic, Oka and Djurdjevic (1974) found that the Nusselt number slightly decreased with an increase of the turbulent macroscale in the region Lx/d = 0.05 to Lx/d = 0.182. In the
region 0.015 < Lx/d < 0.095 Lowery and Vachon (1975) did not
find a noticeable effect of the macroscale of turbulence. Neither did Katinas, Zhyugzhda, Zhukauskas and Shvegzhda (1976) in the region Lx/d = 0.16 to 0.36. Yardi and Sukhatme (1978)
examined the effect of turbulent macroscale on the heat transfer . explicitly. They varied the macroscale over the wide range of Lx/d = 0.03.to Lx/d = 0.38. They found that the heat
transfer coefficient at the front stagnation point increases by about 15% as Lx/d is reduced from 0.4 to 0.05. The effect seems
to diminish as Tu/Re is increased. At the value of the parameter (Lx/d)/Re of about 10 the effect of the macroscale is
at a maximum.
More recently Gorla and Nemeth (1982) presented a mathema tical model to predict heat transfer from a cylinder in crossflow. They used an eddy viscosity model in which Tu/Re and
(Lx/d)/Re are parameters. The dependence of the turbulent
viscosity on Tu/Re was determined by fitting the results to the experimental data available. The measurements done by Yardi and Sukhatme (1978) were used by them to find the expression in the eddy viscosity for the length scale parameter.
2.2.2. Heat transfer from cold impinging jets
In the paragraphs 2.1.2 and 2.1.3 the flow structure in the stagnation flow region and in the wall jet region of an impinging jet has been described. The heat transfer as related
to this flow structure will be treated in the following paragraph, where we restrict ourselves to flows of fluids with constant fluid properties. Firstly, results from literature on laminar impinging jets will be discussed, followed by results on turbulent impinging jets.
2.2.2.1. The laminar impinging jet
Most of the results reported in literature on heat and mass transfer from laminar impinging jets are from theoretical studies, although some experimental works are also available. From these studies influences of some parameters on the transfer of mass and heat could be determined without the existence of turbulence in the flow. The effect of the Reynolds number on the Sherwood or Nusselt number in the stagnation region (a), the influence of the velocity profile of the impinging jet (b), the influence of the separation distance H/d between nozzle and plate (c) and the dependency of the transfer coefficient on the radial distance along the plate (d) will be discussed. Because of the sparsity of results on axisymmetric jets, also results on two-dimensional (slot) jets are considered.
a. As can be seen in paragraph 2.2.1 the Nusselt number at the stagnation point of a body of revolution in a uniform flow depends on Re 5 This same dependency has been derived by
Scholtz and Trass (1970)' for a parabolic impinging round jet and by Sparrow and Lee (1975) for a nonuniform impinging slot jet. In both studies a solution for the inviscid flow field was obtained. This solution was employed as a boundary condition for the viscous flow along the impingement surface. Another way of predicting thé flow field and heat transfer from a laminar impinging jet is by solving the full Navier-Stokes equations with the appropriate boundary conditions. This is done using' a finite difference representation of the equations by Saad (1975), also
published by Saad, Douglas and Mujumdar (1977) for an impinging round jet and by Van Heiningen (1982), also published by Van Heiningen, Mujumdar and Douglas (1976) for an impinging slot jet. A conclusion for the round jet study was that Nu ~ Re for a parabolic velocity profile in the range of 900 < Re < 1950. For a flat velocity profile in the same Re-range they do find the 0.5 power of Re, as the boundary layer theory predicts. For the slot jet Van Heiningen et al. (1976) find for a flat velocity profile again agreement with the boundary layer theory: Nu ~ Re • . For a parabolic velocity profile, however, they find Nu ~ R e0 , 6, which differs from the similar axisymmetric case.
Finally, two references give experimental results on mass transfer in the stagnation region. Scholtz and Trass (1970) confirm their theoretical results and find experimentally the Re • -dependency of the Sh-number in the stagnation region of a nonuniform impinging round jet. Sparrow and Wong
(1975) experimentally confirm the results of Van Heiningen et al. (1976) for a slot jet with a parabolic velocity profile: Sh - R e0-6.
The influence of the velocity profile.
In paragraph 2.1.2 from equations 2.18 and 2.19 we can see the influence of the shape of the impinging velocity profile on the radial velocity gradient near the stagnation point. According to the theory of Sibulkin (1952) this velocity gradient (6) determines the heat transfer coefficient in the stagnation point. In the case of a uniform flow over a body of revolution the stagnation point heat transfer strongly depends on the shape of the body (see paragraph 2.2.1). In the same way the shape of the velocity profile of a jet impinging on a flat plate will have its effect on the stagnation point heat transfer. This influence has been shown by several authors. From the boundary layer theory Scholtz and Trass (1970) find for H/d = 0.5:
0.8242 S c0*3 6 1 + 0.1351 (-) 3 S c0-3 8 6
-R
0.0980(-)" S c0"4 0 8 + . .' (2.47)
R
This relation holds for a parabolic velocity profile. With the inviscid solution of a uniform impinging jet from Strand (1964) they calculate (H/d = 1.0):
0.3634 S c0-3 6 1 + 0.03441 (-)2 S c0 - 3 8 6
-R
0.002531 (-)" S c0'4 0 8 + . . (2.48)
R
These two equations hold for 1 < Sc < 10.
The numerical computations done by Saad et al. (1977) also show the importance of the velocity profile. Not only in the stagnation region, but also in the wall jet region the heat transfer from a parabolic impinging jet is higher than that from a uniform impinging jet, according to their calcula tions .
c. The influence of the separation distance between nozzle and plate has been studied by" Saad, Douglas and Mujumdar (1977). They found from their numerical calculations on axisymmetric impinging jets with a parabolic velocity profile in the range 1 .5 < H/d < 12 a decrease in stagnation point heat transfer of 15% with increasing H/d at Re = 450. At Re = 950 they found no perceptible decrease of the Nusselt number in this range. At the same separation distances Sparrow and Wong (1975) measured mass transfer from a laminar impinging slot jet with the naphthalene sublimation technique. They found no influence of the separation distance on the heat transfer for H/d < 5 (277 < Re < 1700). At higher values of H/d they find turbulence effects.
d. The radial variation of the transfer rate is given by Scholtz and Trass (1963) from their theory by:
Sh /Re
Sh /Re
Sh = 0.4264 R e3 / 4( - ) "5 / 4 g(Sc) (2.49)
d with
g(Sc) = 0.3733 S c1 / 3 (2.50)
' for high Sc-numbers (Sc > 10).
They find agreement of this correlation with experiments obtained with a liquid jet at a high Schmidt number
(1000-4000). Later the same authors find agreement also for a Schmidt number of 2.45 (at r/d > 1.5) (Scholtz and Trass, 1970). Also Kapur and Macleod (1974) found agreement between their measurements and equation 2.49. They determined local mass transfer coefficients by holographic interferometry. Scholtz and Trass used for their solution of the boundary layer equation for the mass concentration the analysis of Glauert (1956). He obtained a solution of the boundary layer equations for the motion of axisymmetric wall jets on the basis of self-preservation of the form of the velocity profile. The theory of Scholtz and Trass, therefore, cannot predict the difference in wall jet heat transfer originating from a parabolic or a uniform impinging jet as observed by Saad et al. It gives a higher power of Re in the wall jet region than is found in the stagnation point region. However, the fully developed region may not yet be reached in the calculations by Saad et al. Their results for the wall jet region do not seem to agree with the theory of Scholtz and Trass (1963) and the experiments by them and Kapur and Macleod.
2.2.2.2. The turbulent impinging jet
In most practical applications of heat or mass transfer from impinging jets the flow will be turbulent. Exact solutions of the problem are then no longer possible. Because of the practical importance of this flow many investigators performed
experiments and tried to correlate the heat or mass transfer rate to the flow parameters. Also numerical studies with the help of turbulence models were performed. In this paragraph only results from axisymmetric impinging jets will be discussed.
There are several ways in which the heat transfer can be correlated to the flow parameters. One approach is correlating the Nusselt number (ad/X) to the relevant parameters by the
Reynolds number in the nozzle Re = uQ d/v, the turbulence in
the nozzle exit (Tu = / u0'3/ u0) , the separation distance
between nozzle and plate (H/d), the radial distance from the stagnation point (r/d) and the fluid properties. Thus a correlation would have the form:
H r
Nu = f(Re, Tu, -, -, Pr) (2.51) d d
Correlations in'this form have been used in the past. The development of jet velocity, turbulence and jet velocity profile with x/d is accounted for by a single parameter H/d in this equation. The disadvantage of this method lies in the fact that results on heat transfer from impinging jets with jets from different orifices do not agree. Especially the turbulence level at the jet origin and the initial velocity profile influence the jet development and subsequently the transfer rates.
Because of the complexity of a result in the form of equation 2.51 the heat transfer at the stagnation point is often separated from the radial dependency.
Another way to describe the stagnation point heat transfer is to use local (impact) parameters of the flow. Parameters which describe the free jet at the plane of impact when the plate is not inserted. In this way a correlation can be found of the form:
Here the Reynolds number is based on the impact velocity; the turbulence grade Tuc is based on the impact turbulence
intensity; y is a parameter" which'is a function of the shape of
the impact velocity profile. Now all parameters in equation 2.52 are a function of x/d.
A review will be given of the most important contributions to literature on heat transfer from axisymmetric turbulent impinging jets.
Smirnov, Verevochkin and Brdlick (1961) correlated their heat transfer measurements together with results from Perry (1954) and Schmidt, Schuring and Sellschopp (1930) into one equation for the stagnation point heat transfer:
Nu = 0.034 d0-9 R e1 / 3 pr 0-4 3 exp (-0.037 -) (2.53)
d
The range of variables where this formula holds is: 0.5 < H/d < 10, 1600 < Re < 50,000 and 0.7 < Pr < 10. The dependence on the non-dimensional nozzle diameter d (in mm) (which varied from 2.5 mm to 16.5 mm) in this correlation is rather surprising and is not confirmed by later experimentalists,
Huang (1963) used the impact velocity measured by a pressure probe on the spot of impingement to correlate the heat transfer rate. He finds for the stagnation point (1 < H/d < 10 103 < Re < 10"):
Nu = 0.0233 R ec 0-8 7 pr 0-3 3 (2.54)
Surprisingly he did not find any other dependency on H/d than that of the impact velocity alone.
It is difficult to verify these and former results because little is known of the characteristics of the jets that were used.
The first extensive experimentalists who studied the influence of turbulence on the heat transfer were Gardon and Gobonpue (1962) and Gardon and Akfirat (1965). They showed that in contrast to a laminar impinging jet the stagnation point
Fig. 2.9. Radial heat transfer distribution for a round impinging jet on a flat plate at H/d = 2 (from Gardon and Akfirat, 1965).
r/d
Fig. 2.10. Heat transfer for a round impinging jet on a flat plate for Re = 28,000 (from Gardon and Cobonpue, 1962).
heat transfer from a turbulent impinging jet increases when H/d increases from 0 to 5. This is due to an increasing turbulence level on the axis of a jet in this range where the velocity remains constant. Several peaks were found in the local radial heat transfer distributions as can be seen in figures 2.9 and 2.10. At small separation distances (H/d < 4) the maximum heat transfer rate was situated at r/d - 0.5. This can be explained by the existence of a minimum of the boundary layer thickness at this place as was predicted by Kezios (1956). At a higher radial distance (r/d = 1.9) an outer peak was distinguished which at low Reynolds numbers separated into two outer peaks at r/d = 1.4 and' at r/d . = 2.5. Two reasons for the possible existence of an outer peak were mentioned:
1) Penetration of turbulence into the boundary layer coming from the mixing layer of the jet.
2) Transition from a-laminar to.a turbulent boundary layer. At higher values of H/d the inner as well as the outer peaks disappeared due to the higher turbulence level of the impinging jet for higher H/d. Experiments with turbulence promoters in the nozzle exit showed that turbulence indeed had an enormous influence:- at H/d =. 2 -the stagnation point heat transfer was augmented and the outer peaks disappeared. The results from this study were confirmed by Schliinder and Gnielinski (1967). Measurements of the turbulence intensity very close to the impingement surface (0.15 mm) showed a qualitative agreement between this turbulence intensity and the heat transfer coefficients. From this could, be', concluded that the outer peak in the radial heat transfer distributions at r/d ^ 1.9 is due
to turbulent eddies penetrating the- boundary layer.
From mass transfer measurements Jeschar and Potke (1970) concluded that, for .1 ... s H/d. S 20 and for 5 s r/d é 40 the
Nusselt number can be correlated with (r/d + 1 ) ~1*1 pr 0-4 2
f(Re). For the stagnation point mass transfer in the ranges 10 < H/d S 20, 8,000 < Re < '30,000, they found:
Sh = 1.2 Re°-7(-)"1-1 S c0'4 2 (2.55)
d
It was possible to find a correlation without Tu as a parameter, because in this range of H/d the turbulence leyel does not vary significantly anymore.
Nakatogawa, Nishiwaki, Hirata and Torii (1970) made an attempt to correlate the heat transfer rate to local flow parameters. Their starting point is the correlation for heat transfer in a plane laminar stagnation point flow (see equation 2.34), however, they consider an axisymmetric flow. For the velocity gradient near the stagnation point, the axial velocity decay and the jet half width diameter, they use empirical relations. In spite of some poor assumptions, the experimental heat transfer results for small separation distances (H/d < 5) agreed quite well with their predictions. For higher distances H/d the experimental results were 1.25 to 1.5 times larger than the predicted values probably due to turbulence effects which were at H/d = 8 at the highest level. The dependency on the shape of the velocity profile was not accounted for by them. For the wall jet region theoretical solutions, obtained by assuming a velocity profile according to the 1/7th power law, agreed well with the experimental values.
The quantitative influence of turbulence has been studied by Donaldson, Snedeker and Margolis (1971a). They applied a correction factor to the laminar stagnation point heat transfer which is a function of the free stream turbulence level. For the theoretical description of the laminar heat transfer they, used a correlation from Lees (1956):
C_ dv j.
-E-^r { p u ( - )r.n)2 (2.56)
, 2(PrT* ^ ^ d r 'r= °J
This relation is similar to Sibulkin's equation for the stagnation point heat transfer of a body of revolution (equation 2.32). For the radial velocity gradient (dv/dr)r = 0
Donaldson, Snedeker and Margolis (1971b) who assumed:
dv , 1 32P i
<^>r=o = t" ^ > r / o >a <2'57>
Because extensive measurements were done on the flow structure of the free jet, the ratio of the theoretical laminar heat transfer could be determined as a function of the average relative turbulent intensity in the free jet, defined by
k = -(ü71" + 2 V7 1" ) ^ (2.58)
In the range of 0.10 < k/u < 0.25 the ratio of turbulent to laminar heat transfer varied from 1.4 to 2.2. Although very much scatter was found they did not find any discernable effect of the Reynolds number on this ratio.
For the average heat transfer coefficients Subba Raju (1972) derived relations which fitted the experimental results of different authors. In the range of parameters 1 < H/d < 10, 2.10" < Re < 4.10s, 0.7 < Pr < 8.0 and 1 < D/d < 60 he found: Nu\(-)0-5 = 1.54 R e0'5 Pr1/3 - s 8 (2.59) d d Nu P r "1 / 3( - )3 = 35.0 R e0*5 + 0.28 Re°-8( 8) - è 8 d d d (2.60) This result gives an indication that for D/d s 8 the boundary layer along the impingement surface is laminar (Nu - R e0*5) ,
while it is turbulent (Nu = R e0 - 8) for D/d a 8.
Kataoka and Mizushina (1974) investigated the local enhancement of the heat transfer rate by free stream turbulence. A minimum in heat transfer is found by them in the stagnation point for H/d < 0.5 and a maximum at r/d = 0.6. Here the large eddies coming from the mixing region penetrate the boundary layer. The local skin friction showed a secondary peak at r/d = 2.2. In contrast to this, the local Nusselt number had
a secondary peak at r/d = 4 (for 6 < H/d < 8.5). It should be noted that their measurements were performed at high Prandtl numbers (2420 to 3300).
The necessity of using local parameters of the impinging flow to correlate heat transfer was observed by Chia, Giralt and Trass (1977). They adopted the already mentioned boundary layer solutions obtained by Scholtz and Trass (1970) to the velocity and length scales proposed by Giralt, Chia and Trass (1977), discussed in paragraph 2.1.2. These scales are the collision velocity at the stagnation point Uc and the jet half
width radius at the beginning of the impingement region. The result of this approach is a mass transfer rate for the stagnation region without the influence of turbulence:
Sh, 1 V-, r ( n-5>iam = "2 V1a { r,( S c ) + —Jd2' ( S c ) ( )3 + . . . }
R e iu . b lam I o v^ z r ^
(2.61) The functions c0'(Sc), d2'(Sc) etc. are tabulated by Scholtz
(1965). The coefficients V-j , V3 etc. are tabulated by Giralt et al. (1977) for different nozzle to plate distances. The coefficients V^ , V, etc. take into account the varying impinging velocity profile.
The effect of turbulence is taken into account by: Sh_- Sh,
7 i t T =
(1 + Yi » <7ie->la
m (2'
62)For Yf a form also used by Lowery and Vachon (1975) and Galloway (1973) for heat transfer to cylinders in a cross flow is assumed:
y± = a S c1 / 6 (TU;L Re^ - b) (2.63)
Experimental results showed that beyond - H/d = 11.0 the variation of Sh^/ZRe^ with r/r-j. is universal, although the turbulence free mass transfer (Sh^//Re^ ) -L a m is universal beyond
turbulence between H/d = 8.0 and H/d = 11.0, while the velocity profile does not change shape beyond H/d a 8.0. For the enhancement factor Chia et al. (1977) found:
y± = 0 Tuj/Re < 4.0
Yi = 0.0156 Sc1/,6 (Tui/Re - 4) 4<Tui/Re < 34.0 (2.64)
"yi = 0.468 S c1/6 Tu±/Re > 34.0
These results are in qualitative agreement with results obtained .for heat transfer from cylinders in a uniform cross flow discussed in .paragraph 2.2.1. It should be mentioned that the results found by Chia et a l . (1977) are based on measurements at a single Re-number (Re = 34,000) at Sc = 2 . 4 5 . However, the resultant mass transfer equations have been used to predict literature data and found to be consistent over a wide range of flow conditions.
' 'A similar approach of using the stagnation point heat transfer results of a, cylinder in a cross flow has been undertaken by Den Ouden and Hoogendoorn (1974).' They influenced the turbulence level at the nozzle exit by placing grids in the nozzle,.. It was found that for small separation. distances (H/d < 4) the experiments could be correlated with equation
N u ' ' Tu /Re Tu /Re
- — = 0.497 + 3.48 3.99 ( ) 2 (2.65)
/Re 100 100
Almost the same equation was found by Kestin and Wood (1971) for cylinders (see equation 2.41). At higher separation distances (H/d > 4) apparently the influence of the changing velocity profile was the cause that equation 2.65 did not hold anymore.
"Special attention to the radial distribution of the heat transfer coefficient has been given by Vallis, Patrick and Wragg (1978). They used an electrolytic mass transfer technique .in a Reynolds number range of 3.880 < Re < 23,000. Supposing a
Nu = 1 . 9 3 R e0 - 5 8 P r1/3 ( _ ) - ° -7 4 10 < - < 20 ( 2 . 6 6 ) d d
This result differs very much from that of Jeschar and Pötke (1970) discussed earlier. For the fully developed wall jet region Vallis et al. (1978) found:
Nu = 0.078 R e0-8 2 P r1 / 3 ( - ) "1-0 5 8 < r/d < 17 (2.67) d
or
N ur = 0.11 R er 0*8 2 P r1 / 3 (2.68)
with the distance from the stagnation streamline r as a characteristic length scale in Re and N ur.
A very detailed study of the influence of turbulence on heat transfer in the stagnation region of a two-dimensional, submerged, impinging jet has been done by Yokobori, Kasagi, Hirata and Nishiwaki (1978). They observed the stagnation flow field with the aid of a flow visualization technique. Results are already discussed in paragraph 2.1.2. The large vortex-like motions they observed in the region 4 < H/d < 10 enhance heat transfer considerably. By fixing a fine cylindrical rod at the nozzle exit, it seemed possible to create artificially a pair of large scale vortices on the impinging wall. Even when the wall was positioned in the potential core, the vortices were observed. The heat transfer in this region was enhanced by the artificial eddies to the same level as the maximum increase in heat transfer produced by the mixing induced large scale eddies. This study thus demonstrates that heat transfer predominantly is affected by large scale structures.
More recently Kataoka et al. (1987) also studied the mechanism of the enhancement of stagnation point heat transfer by large scale turbulent structures. They demonstrated the existence of vortex rings at x/d = 1 as already shown by Yule
(1978) and Strange and Crighton (1983). These coherent large scale structures are produced due , to the instability of the
laminar shear layer. At x/d = 2.2 two vortex rings pair into one before breaking up into large scale eddies at the end of the potential core region. Autocorrelation of centreline velocity fluctuations indicate for x/d S 4 periodicity. With the characteristic frequency fe and the centreline velocity u a
Strouhal number is defined as St = fgd/u. This number equals
about 0.6 for 1 < x/d < 2 and 0.3 for 2 < x/d < 4. For x/d > 4 the Strouhal number is defined with the frequency of large scale eddies, determined from the integral time scale. This resulted in a Strouhal number of about 2 at x/d = 6 decreasing to about 1 at x/d = 10. Kataoka et al. correlated the heat transfer enhancement with a surface renewal parameter being the product of a turbulent Reynolds number Re^ = / ug' 2 d/v and
this Strouhal number. The value of us , a has been obtained from
measurements 5 mm upstream of the stagnation point. In this way they also show that enhancement of stagnation point heat transfer is mainly due to turbulent surface renewal by large scale eddies.
2.2.3. Heat transfer from flame jets
Knowledge of stagnation point heat transfer (see paragraph 2.2.1) and of heat transfer from impinging jets (paragraph 2.2.2) can be used when heat transfer from flame jets is studied. A large number of investigators have used the theoretical solution of Sibulkin (1952) for the boundary layer equations for heat transfer at the stagnation point of a body of revolution as a starting point for the prediction of heat transfer from flames. This theory leads for the heat flux density at a stagnation point to:
q" = 0.763 ( pfyfg )0-5 (hf - h ) Pr"0-6 (2.69)
where pf, yf and hf are the density, viscosity and enthalpy in
the flame just outside of the temperature boundary layer in the stagnation point, h is the enthalpy of the gasses at the wall. Fay and Ridell (1958) extended this theory by taking into
account the dissociation of air and recombination of radicals in the boundary layer along a cooled object. Their theory can be applied when the chemical reactions of the flame are still present in the boundary layer along the impinged surface. This resulted in:
q" = 0.763 (^W)0.1 (P U| 3 )0-5 (hf - h„) Pr"0'6 .
PfUf
{1 + (Le0-5 2 - 1) -ii£} (2.70) hf
where Le = D/a is the Lewis number (D being the diffusion coefficient) and h^_ D is the dissociation enthalpy.
Buhr, Haupt and Kremer (1976) found that for methane-air flames without preheating the radical concentrations are low. The heat coming free with the recombination of radicals was then found to be negligible. A number of studies concentrated on high temperature flames for which recombination of radicals in the boundary layer of a cooled surface is an important factor. Among these are studies from Conolly and Davies (1972), and from Kilham and Purvis (1971 and 1978).
Beer and Chigier (1968) reported results from an experimental investigation of a flame impinging at an angle of 20° on the hearth of a furnace. Their results show that heat transfer can be increased by a factor of 3 using direct impingement. The contribution of convection to the total heat transfer amounted 70%.
Milson and Chigier (1973) performed studies on methane and methane-air flames impinging on a cold plate. Both flames had a cool central core of unreacted gas giving rise to lower heat fluxes near the stagnation point than at some distance from this point (for 10 < H/d < 16). The heat transfer coefficient in the wall jet region was higher than in the impingement region due to the cool central core.
Horsley, Purvis and Tarig (1982) used impinging natural-gas-air flames from several types of burners. For the
stagnation point heat transfer they found that the results showed differences depending upon the turbulence structure of the free flames from the different burners. Yet all turbulent flames considered gave stagnation point heat transfer in the order of 1.2 to 1.6 times higher than calculated from Silbulkin's theory. This is in agreement with the findings of Giralt et al. (1977) for impinging isothermal jets discussed earlier.
3. THEORY
3.1. The governing equations
The flow of the impinging jet can be described by the full Navier-Stokes equations (the equations of motion) and the con tinuity equation. For a two-dimensional axisymmetric flow these equations in cylindrical coordinates read (see Bird, Stewart and Lightfoot, 1960): - continuity equation: 3p 1 9 3 — + (rpv) + — (pu) = 0 (3.1 ) 3t r 3r 3x - equations of motion: 3u „ 3u .. 3u 1 3 ~ 3 ? ^ , 3p p — + pu — + pv — = - { - — (rTrx) + — ^ } - — (3.2) 3t 3x 3r r 3r r x 3x 3x 3v : 8v _ 3v 1 3 _ 3 T _V TOO 3D p — + pu — + p v — = { (rt__) + —£*■ ^ } -3t 3x 3r r 3r r r 3x r .3r (3.3)
with u, v, T and p momentary values. The components of the stress tensor T Q and TQ do not appear in these equations because they are supposed to be zero due to the absence of a El-dependency in the problem. The non-zero stress-components are:
3 v 2 ->- ■+ u{2 (V.v)} (3.4) 3r 3 v 2 ->--»■ , g{2 (V.v)} (3:5) r 3 3 u 2 ■+ ->■ - g{2 (V.v)} (3.6) 3r 3
3u 3v
u ( — - — ) (3.7) dr 3x
Assuming incompressible flow which means that V.v = 0, the equations of motion can be reduced to:
3u „ 3u . _ 3u 1 3 3u 9 9u p — + p u — pv — = (pr — ) + — (u — ) + S~ (3.8) 3t 3x 3r r 3r 3r 3x 3x u 3v _ 9v .. 3v 1 9 9v 9 3v p — + pu — + pv — = (pr — ) + — (u — ) + S~ (3.9) 9t 9x 3r r 9r 3r 3x 3x v with 1 3 3v 3 3ü 3p S~ = - _ (ru — ) + — (w — ) - — (3.10) u r 3r 3x 3x 3x 9x 1 3 3v 3 3u 2uv 3p S~ = - — (ru — ) + — (U — ) - -r- - — (3.11) v r 3r 3r 3x 3r ra 3r
If the flow under study is turbulent a time averaging of the equations over a time larger than the biggest time scales of the turbulence is appropriate. For this reason at first the Reynolds decomposition of the variables will be executed: the momentary value of a variable is the sum of the averaged value and a fluctuating value
u = u + u'
v = v + v' (3.1 2) p = p + p'
Averaging of the equations for a stationary flow results in: - the continuity equation:
1 3 3
- — (rpv) + — (pu) = 0 (3.13) r 3r 3x
the equations of motion: 9u 9u 1 3 9u 9 9u 9 pu — + pv — = (ur — ) + — (y — ) - { — pu 9x 3r r 9r 3r 9x 9x 9x 1 9 + pru'v') + Sn (3.14) r 9r 3v 9v 1 9 9v 9 9v 9 pu — + pv — = (ur — ) + — (u — ) - { — pu'v' 9x 9r r 9r 9r 9x 9x 9x 1 3 Vfi'3 + prv'2 - p -2 } + Sv (3.15) r 9r r v with 1 9 9v 9 9u 9p Su = - — (ru — ) + — (u — ) - — (3.16) u r 3r 3x 3x 9x 9x 1 9 9v 3 3u 2yv 9p S„ = - — (ru — ) + — (u — ) - — (3.17) r 9r 9r 9x 3r r 3r v
The terms between brackets in equations 3.14 and 3.15 are called the Reynolds stresses. These express momentum transfer by turbulent motion, and will be treated as turbulent diffusion. Equations for the Reynolds stresses can be derived from the Navier-Stokes equations but the resulting equations contain higher order correlation terms which in their turn are unknown. This is the closure problem of turbulence. In the next paragraph it will be shown that by using turbulence models estimates are found for the unknown Reynolds stresses.
3.2. Turbulence models
The analogy between the Reynolds stresses and the viscous stresses is the basis for the Boussinesq hypothesis stating (Hinze, 1975):
3uH 9u^ 2
-pui'ui' = u. ( i + — 1 ) - - p k ó ^ (3.18)
Herein u^ is the turbulent viscosity and k is the kinetic energy of turbulent fluctuations:
k = i (u,a + v'2 + v6'2) (3.19)
Assuming that the turbulent viscosity is a scalar implies that nonisotropic effects of the turbulence cannot be taken into account. The k-e model of turbulence, discussed in the next paragraph, makes this assumption. This model is very widely used. From literature we learn that it also is applied to nonisotropic flows, however, that is not fully justified.
Applying the Boussinesq hypothesis to the time averaged Navier-Stokes equations (3.14 and 3.15) leads to:
3u 3u 1 3 9u 3 3u pv — + pu — = - — (ru f f — ) + — (u-ff — ) 9r 3x r 9r e r r 9r 9x e £ t 3x 9 2 - — (p + - pk) + S„ (3.20) 9x 3 3v 3v 1 3 3v 3 3v pv — + pu — = - — (rupff — ) + —- (P-ff — ) 3r 3x r 3r e r r 3r 9x e r r 3x 3 2 3r — (p + - pk) + Sv (3.21 )
with the source terms:
3 3u 1 9 3v Su = ^< Me f f ^ » + Ï J7 ( r Ueff ^ > (3.22) 3 9u 1 3 3v v S = — ( ue f f —-) + - — (rp-rr —-) - 2 u ff — (3.23) v 3x e r r 3r r 3r e t r 3r e r r r2 and with Ueff = U + Ut (3.24)
3.2.1. The k-e model of turbulence
