Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 3, pp. 603-619, Warsaw 2007

TURBULENT HEAT TRANSFER IN THIN LIQUID FILMS AT

LOW AND HIGH HEAT FLUXES

Dariusz Mikielewicz

Gdańsk University of Technology, Faculty of Mechanical Engineering, Poland

e-mail: Dariusz.Mikielewicz@pg.gda.pl

Jarosław Mikielewicz

The Szewalski Institute of Fluid-Flow Machinery PAS, Gdańsk, Poland

e-mail: jarekm@imp.gda.pl

In the paper presented are considerations of turbulent heat transfer in thin
liquid films at low and high heat fluxes. Postulated have been simplemodels
of heat transfer for laminar and turbulent liquid films formed by impinging
jets and exposed to nucleate boiling, namely under high heat fluxes, as well
aswithoutnucleateboiling, at lowheatfluxes, as a simplified case.Turbulen-
ce in such case is stronglymodified and difficult to bemodelled. Turbulence
model due toPrandtl has been appliedwhere, in the case of high heat fluxes,
the mixing length is strongly modified. In the case of high heat fluxes, in-
corporated into themodel is a blowing velocity, whichmodels the transverse
transport of momentum caused by departing bubbles. Calculated have been
the velocity and temperature distributions in the liquid film, which enabled
determination of the corresponding heat transfer coefficient and the Nusselt
number.

Key words: blowing velocity, impinging jet, nucleate boiling

Notations

cp – specific heat
δ – boundary layer thickness
h – film thickness
λ – thermal conductivity
ν – kinematic viscosity
ρ – density
q – heat flux density
qvap – heat flux due to evaporation of film
Q – volumetric flow rate
T – temperature



604 D. Mikielewicz, J. Mikielewicz

τ – shear stress
ϑ,w – radial and vertical velocity components
r,z – radial and vertical co-ordinates
uτ – friction velocity, uτ =

√

τw0/ρ
Subscripts

tr – border between laminar and turbulent sublayers
i – interface
l – liquid
m – mean value
w – wall
0 – initial value, nozzle outlet

Superscripts

+ – non-dimensional

1. Introduction

Surface cooling bymeans of thin liquid films hasmany practical applications.
The demand for effective cooling schemes increases the need for the deve-
lopment of appliances incorporating high-heat flux convective heat transfer
mechanisms such as change of phase. Apart from the areas, where up to da-
te the efficiency of cooling was highly desirable, just to mention metallurgy
and nuclear power, there is rapidly developing a new branch of applications
inmicroelectronics, where seemingly steady trend of achieving ever larger sca-
les of circuit integration is straining the capabilities of existing high-heat flux
technologies. In such case there are numerous limiting restrictions such as ava-
ilable space, choice of coolant, local environmental conditions and maximum
allowable surface temperatures. In relation to the range of plate temperatures
wemay face differentmodes of convective heat transfer, namely forced convec-
tion, nucleate boiling, transitionboilingorfilmboiling.Liquidfilmevaporation
can significantly increase the rate of heat removal from the solid surface and
renders this kind of heat transfer very efficient. Heat transfer rates are even
greater in the case of nucleate boiling. In the paper considered has been a
problem of nucleation in the film formed by impinging jet on a hot plate,
as well as a simplified case when nucleate boiling is not present. In case of
high heat fluxes, temperature of the plate exceeds the saturation temperature,
which causes nucleation of bubbles. Several works can be found,where similar
topic is investigated, for example Buyevich and Mankevich (1996), Deb and
Yao (1989), Liu andWang (2001), Mikielewicz andMikielewicz (2005), Webb



Turbulent heat transfer in thin liquid films... 605

and Ma (1996), Wolf et al. (1994). Despite numerous studies regarding that
topic, there is still a lack of complete understandingof the phenomenonaswell
as theoretical models, which would enable calculation of hydrodynamics and
heat transfer in liquid films produced by impinging liquid jets in the nucleate
boiling regime.

In the paper the issues of turbulence in modelling of thin films featuring
nucleate boiling have been addressed in a greater detail. That issue is by no
means simple, since the presence of generated bubbles significantly modifies
velocity field and hence the turbulence. The theory known from single phase
flows has been applied to the cases where strongmodifications of velocity field
are found. Modelling of turbulence in such cases is a very challenging task.
Standard models of turbulence are usually perceived as not capable, without
seriousmodifications, of revealing the phenomena occuring in the flow. In the
paper turbulence is modelled using the Prandtl mixing length model, which
resolves the problem quite well. Themodel of the thin liquid film formed as a
result of jet impingement, scrutinised in the paper, is a simplemodel of boiling
heat transfer within such liquid films just outside the stagnation region. The
case with nucleate boiling is regarded as a case where large rates of heat are
present. In such amodel considered is a change of film thickness due to inertia
forces and friction. The transverse component of bubble velocity, in the study,
is regarded as constant and has beenmodelled bymeans of the theory of ”blo-
wing”. The case taken into consideration consists of two layers, i.e. laminar
sublayer and turbulent core in the bulk of the flow. The model is based on
the consecutive solution of the conservation equation ofmass,momentumand
energy. The obtained solutions are approximate, but in the analytical form,
which enables further qualitative examination of solutions. Thedevelopedmo-
del forms an extension of the earlier models by the authors (Mikielewicz and
Mikielewicz, 1999, 2001, 2002), where the laminar and turbulent films have
been considered and the solutions were obtained by means of the ”thin layer
theory”. These cases will also be presented here, derived on the basis of sim-
plification of the model featuring the nucleate boiling. In the present work,
according to procedures developed for respectivemodels, calculated have been
shear stresses, velocity and temperature distributions in the liquid film, which
enabledalso determination of theheat transfer coefficient andNusselt number,
corresponding to the relevant cases.

2. Governing equations

Figure 1 shows a schematic diagram of the case under scrutiny applicable
to both cases of high and low heat fluxes respectively. The following non-



606 D. Mikielewicz, J. Mikielewicz

dimensional quantities have been introduced into the analysis

ϑ+ =
ϑ

uτ
z+ =

zuτ
ν

r+ =
ruτ
ν

δ+ =
δuτ
ν

T+ =
T

T0
τ+ =

τ

τw
τ+0 =

τw
τw0

q+ =
q

qw

(2.1)

Fig. 1. Schematic diagram of a single phase jet

Equations describing the behaviour of non-compressible two-dimensional
turbulent liquid films written in the boundary layer approximation in cylin-
drical co-ordinates yield (Deb and Yao, 1989; Mikielewicz and Mikielewicz,
2005):
— continuity equation

∂(r+ϑ+)

∂r+
=0 (2.2)

—momentum equation

ϑ+
∂ϑ+

∂r+
+
w0
uτ

∂ϑ+

∂z+
= τ+0

∂τ+

∂z+
(2.3)

— energy equation

∂(ϑ+T+)

∂r+
+
w0
uτ

∂T+

∂z+
=

qw
ρcpT0uτ

∂q+

∂z+
(2.4)

In the above equations the transverse velocity component, w, is assumed
constant, namely w ≈ w0 = const, which corresponds to implementation of
the so-called ”blowing velocity”. In the discussed case it models the contribu-
tion from nucleate boiling. Value of that velocity can be estimated from the
rate of evaporation, i.e. w0 = qvap/hfgρv. The total heat flux consists of two



Turbulent heat transfer in thin liquid films... 607

components, namely the heat flux due to convection in the liquid and the heat
flux due to evaporation

q= ql+ qvap (2.5)

The heat flux due to evaporation in considered case is very small compared to
the heat flux through liquid, however bubbles significantly distort the velocity
profile modifying in such a way the turbulence. In the expression (2.5) the
radiative heat flux contribution could also be considered, but in the present
study that contribution has been omitted due to the fact that in cases to be
studied, there are no significant surface temperatures involved.

The momentum and energy equations enable determination of general
forms of the shear stress and heat flux distributions in the liquid film

τ+ =1+
1

τ+0

(

z+
∫

0

ϑ+
∂ϑ+

∂r+
dz++

w0ϑ
+

uτ

)

(2.6)

q+ =1+
ρcpT0uτ
qw

z+
∫

0

∂(ϑ+T+)

∂r+
dz++

w0
uτ
(1−T+w )

These distributions are valid both for the laminar and the turbulent flow
conditions. The boundary conditions for the considered case yield: z = 0 ⇒
τ+ = τ+0 and q

+ = 1. The challenge now is in appropriate resolution of the
shear stress and heat flux in respective sublayers.

In themodelpresentedbelow, thermal-hydraulicanalysis of thephenomena
in high heat fluxes case followed by the low heat flux case just outside the
stagnation region is presented.

2.1. Hydrodynamics of thin liquid films at high heat fluxes

2.1.1. Developing film

The boundary layer in the developing region grows as presented in Fig.1.
The analysis below pertains to the case where the boundary layer, δ+, is
smaller than the liquid film thickness, h+. Such situation is present within
the distance of approximately eight nozzle radii from the stagnation point. In
such a case the problem needs to be considered in two zones, in the region of
boundary layer and beyond, in the region of undisturbed liquid between the
boundary layer border and the film border. As the first approximation the
velocity profile for a turbulent boundary layer flow on a flat plate has been
assumed

ϑ+ =
ϑ0
uτ

(z+

δ+

)

1
m

(2.7)



608 D. Mikielewicz, J. Mikielewicz

Such profile satisfies the conditions of zero velocity at the wall and a value
of undisturbed velocity at the border of the boundary layer, ϑ0. Beyond the
boundary layer the velocity does not vary and is equal to ϑ0. The flow in the
boundary layer will be analysed in two regions, namely in the laminar and
turbulent sublayers. In the case of high heat fluxes, the border of the laminar
boundary layer is assumed to fall at z+ =11.6. That fact should be properly
recognised as bubbles strongly modify the velocity field and still the usual
assumptions for the division of boundary layer are acceptable. Substitution of
(2.7) into the momentum equation gives a relation describing the shear stress
distribution in the laminar sublayer

τ+ =1+
1

τ+0

[

a(r+)(z+)
2
m
+1+

w0ϑ0
u2τ

(z+

δ+

)

1
m

]

(2.8)

where

a(r+)=−
ϑ20δ
−
m+2
m

(m+2)u2τ

dδ+

dr+

Solving (2.8) in the laminar sublayer,where τ+ = dϑ+/dz+,weobtainvelocity
distribution in that region, i.e. z+ ¬ 11.6

ϑ+ = z++
ma(r+)

2(m+1)
(z+)

2(m+1)
m +

w0ϑ0
u2τ

m(δ+)
1
m

m+1
(z+)

m+1
m (2.9)

In the turbulent sublayer (11.6<z+ ¬ δ+), the Prandtlmixing lengthmodel
has been applied

τ+ =κ2(z+)2
(dϑ+

dz+

)2

(2.10)

The resulting differential equation has the following form

dϑ+

dz+
=
1

κz+

√

1+
1

τ+0

[

a(r+)(z+)
2
m
+1+

w0ϑ0
u2τ

(z+

δ+

)

1
m

]

(2.11)

Knowledge of laminar and turbulent velocity profile distributions enables de-
termination of the boundary layer thickness, which can be determined from
the condition that at the border of boundary layer, the following condition
must hold

dϑ+

dz+

∣

∣

∣

z+=δ+
=0 (2.12)

This condition enables determination of the change of the liquid film thickness
as a function of the nozzle outlet velocity ϑ0

dδ+

dr+
=(2+m)

u2τ
ϑ20

(

τ+0 +
w0ϑ0
u2τ

)

(2.13)



Turbulent heat transfer in thin liquid films... 609

By integration of (2.13), using the boundary condition that for r+ = 0 ⇒
δ+ ≈ 0⇒C2 =0we obtain the ratio of thickness of the boundary layer with
respect to the radius. Next, from the mass balance equation, we perform the
liquid film flow rate balance

Q+ =
Quτ
2πν2

= r+
[

δ+
∫

0

ϑ+ dz++
ϑ0
uτ
(h+− δ+)

]

(2.14)

Relation (2.14) enables determination of the liquid film thickness at any ra-
dius. The velocity profile integral is to be determined from integration of the
respective velocity profiles in laminar and turbulent sublayers

δ+
∫

0

ϑ+ dz=

11.6
∫

0

ϑ+
l
dz++

δ+
∫

11.6

ϑ+t dz
+ (2.15)

Expression (2.14) permits to derive the limiting value of radius, r+gr, where
the boundary layer will reach the film thickness, i.e. h+ = δ+

r+gr =
Q+

δ+
∫

0

ϑ+ dz+
(2.16)

2.1.2. Fully developed film

The region we are about to consider now extends from the point where the
boundary layer has reached the film thickness.The following analysis is similar
to the one presented above and therefore only themost important points will
be highlighted. In the considered case the thickness of the film consists of two
sublayers, namely the laminar and the turbulent ones and the analysis will be
carried out in both sublayers respectively. Similarly as in the latter case, the
velocity profile corresponding to turbulent flow in the boundary layer flow on
a flat plate has been assumed but in the present case, instead of the boundary
layer thickness, the film thickness is applied in its distribution

ϑ+ =
ϑ0
uτ

(z+

h+

)

1
m

(2.17)

In the case of high heat fluxes the border of the laminar boundary layer is
assumed to fall at z+ = 11.6. Substitution of (2.17) into the momentum
equation gives a relation describing the shear stress distribution across the
film thickness

τ+ =1+
1

τ+0

[

a(r+)(z+)
2
m
+1+

w0ϑ0
u2τ

(z+

h+

)

1
m

]

(2.18)



610 D. Mikielewicz, J. Mikielewicz

where

a(r+)=−
ϑ20(h

+)−
m+2
m

(m+2)u2τ

dh+

dr+

Solving (2.18) in the laminar sublayer, where τ+ = dϑ+/dz+, we obtain the
distribution of velocity in that region, i.e. z+ ¬ 11.6, where nucleation of
bubbles is again modelled bymeans of the blowing velocity

ϑ+ = z++
ma(r+)

2(m+1)
(z+)

2(m+1)
m +

w0ϑ0
u2τ

m(h+)
1
m

m+1
(z+)

m+1
m (2.19)

In the turbulent sublayer (11.6<z+ ¬ δ+), the Prandtlmixing lengthmodel
has been applied. The resulting differential equation has the following form

dϑ+

dz+
=
1

κz+

√

1+
1

τ+0

[

a(r+)(z+)
2
m
+1+

w0ϑ0
u2τ

(z+

h+

)

1
m

]

(2.20)

Next, using the mass balance equation, we obtain the liquid film flow rate
balance

Q+ =
Quτ
2πν2

= r+
(

h+
∫

0

ϑ+ dz+
)

(2.21)

The solution of relation (2.21) enables determination of the liquid film thick-
ness at any radius in the fully developed case of spreading film.

2.2. Hydrodynamics of thin liquid films at low heat fluxes

In the case to be considered below, the term of low heat fluxes corresponds to
the case where no bubble nucleation is present. This means that the blowing
velocity is equal to zero, w0 =0. Equations (2.2)-(2.4) describing the problem
in Section 2.1 are valid also in the present case, but they assume a slightly
simpler form. Let’s first consider the flow in the developing range of the film.

2.2.1. Analysis of the developing film

The boundary layer in the developing region, in the case of low heat fluxes,
develops also as presented in Fig.1. The analysis below pertains to the case
when the boundary layer δ is smaller than the liquid film thickness. In such
a case the problem needs to be considered in two zones, namely in the region
within the boundary layer and beyond, in the region of undisturbed liquid
between the boundary layer border and the filmborder.The regionwithin the
boundary layer consists of laminarandturbulent sublayers.Thevelocity profile
typical of a turbulent boundary layer flow on a flat plate has been assumed
as the first approximation. In such case, a form presented by Equation (2.7)



Turbulent heat transfer in thin liquid films... 611

has been used. Such profile obeys the condition of zero velocity at the wall
and a value of undisturbed velocity at the border of the boundary layer, ϑ0.
Beyond the boundary layer the velocity does not vary and is equal to ϑ0.
The flow in the boundary layer can be analysed in two regions: in the laminar
and turbulent sublayers. The border of the laminar boundary layer is assumed
conventionally to be z+ = 11.6. Note that here we are not dealing with the
presence of bubbles. Substitution of (2.7) into the momentum equation gives
the expression for the distribution of shear stress

τ+ =1+a(δ+)z
2
m
+1 (2.22)

where

a(δ+)=−
ϑ20(δ

+)−
m+2
m

2(m+2)u2τ

dδ+

dr+

Solving (2.22) in the laminar sublayer, where τ+ = dϑ+/dz+, we obtain the
velocity distribution in that region, i.e. z+ < 11.6

ϑ+
l
= z+−

ma(δ+)

2(m+1)
(z+)

2(m+1)
m (2.23)

In the turbulent sublayer, the Prandtl mixing length model can again be ap-
plied. Neglecting the blowing velocity in the equation of motion allows us also
to obtain the simple analytical form of velocity distribution in that region.
The resulting differential equation has been solved using the following appro-
ximation

√

1+a(r+)(z+)
m+2
m ≈ 1+

1

2
a(r+)(z+)

m+2
m (2.24)

Application of (2.24) enables to obtain an analytical solution to velocity di-
stribution in the turbulent flow regime in the form

ϑ+t =
1

κ

[

ln
z+

11.6
−
1

2

m

m+2
a(δ+)

(

(z+)
m+2
m −11.6

m+2
m

)]

(2.25)

Knowledge of laminar and turbulent velocity profiles enables determination
of the boundary layer thickness, which can be determined from the condition
(2.14). This condition expresses the change of the liquid film thickness δ+ as
a function of the nozzle outlet velocity ϑ0. The result is similar to relation
(2.16), if ”blowing velocity” w0 would be assumed to be zero. Integration of
(2.14) using the boundary condition that for r+ = 0 ⇒ δ+ ≈ 0 ⇒ C2 = 0,
yields the ratio of thickness of the boundary layer with respect to the radius.
Similarly as in the case of high heat flux, we can find the location where the
boundary layer meets the film thickness, i.e. h+ = δ+, namely relation (2.16).



612 D. Mikielewicz, J. Mikielewicz

2.2.2. Fully developed film flow

Similar task has been considered inMikielewicz andMikielewicz (2002, 2004).
The solution is sought for in two sublayers where turbulent velocity profile
(2.17) has been assumed. The final distributions of velocity in laminar and
turbulent sublayer give forms similar toEquations (2.19) and (2.20) where the
blowing velocity has been assumed to equal 0.

3. Heat transfer during the film development

Solution of the heat transfer problem is based on the analysis of energy equ-
ation (2.4). As apartial differential equation, it is difficult to be solved in other
than a numerical manner. However, using the assumptions presented below it
is possible to develop analytical relations describing the temperature field in
the liquid film.
In the case of the approximate analysis of developing film, we should con-

sider two cases:
1. Themotion is laminar in the entire film, i.e. δ+ <z+tr.

2. Themixedmotionwhere the boundary layer thickness δ+ >z+tr. In such
a case in the region 0 < z+ < z+tr there exists the laminar flow and in
the region 11.6<z+ <z+tr – the turbulent flow.

In the case when the boundary layer thickness is greater than the lami-
nar sublayer thickness, we can assume that the undisturbed temperature T0
extends from the filmborder to the border of the boundary layer, i.e. Tδ =T0.
For that reason, the total temperature drop in the liquid film in the boundary
layer consists of two drops, in the laminar and turbulent sublayers. In the case
when the boundary layer thickness is smaller than the laminar sublayer thick-
ness, we are dealing only with one laminar temperature distribution in the
boundary layer. In the present paper, a boundary layer is considered where
temperature drop exists both in the laminar and turbulent sublayers.
First term of the energy equation (2.4) will be modelled in the present

work as a mean product of instantaneous velocity and temperature, namely

∂(ϑ+T+)

∂r+
∼=
∂(ϑ+T+)m
∂r+

(3.1)

Next, let us define the mean film temperature in the form

Tm =

h
∫

0

ϑT dz

h
∫

0

ϑdz

=
2πr

Q

h
∫

0

ϑT dz (3.2)



Turbulent heat transfer in thin liquid films... 613

The denominator in (3.2) can be determined from the continuity equation
in the jet. The integral appearing in the denominator is the sought term in
the first term of energy equation. Then, following the conversion of Equation
(3.2) into the non-dimensional variables, we arrive at the relation for themean
temperature in the film

T+m =
r+

Q+

h+
∫

0

(ϑ+T+) dz+ (3.3)

In order to apply the assumption (3.1) into the energy equation (2.4) we need
the define themean product of velocity and temperature (ϑ+T+)m. That can
be obtained in the following form

(ϑ+T+)m =
1

h+

h+
∫

0

(ϑ+T+) dz+ (3.4)

Using the definition of mean temperature (3.3) we can determine its radial
distribution from single integration of the energy equation (2.4) in the limit
from 0 to the film thickness h+

Q+
∂

∂r

(T+m
r+

)

=
qw

ρcpT0uτ
(q+i −q

+
w)+

w0
uτ
(T+w −1) (3.5)

where q+w is the density of heat flux at thewall and q
+
i is the heat flux density

at the liquid-gas interface. Integrating (3.5) at the boundary condition r=0,
Tm =T0, we obtain the mean temperature distribution in the film dependent
on the radial co-ordinate

T+m =
1

Q+

[ qw
ρcpT0uτ

(q+i −q
+
w)+

w

uτ
(T+w −1)

]

(r+)(r+−r+0 )+
r+

r+0
(3.6)

In order to use the assumption (3.1) we explore the relation (3.4). Then the
mean product of velocity and temperature yields

∂

∂r+
(ϑ+T+)m =Q

+ ∂

∂r+

( T+m
r+h+

)

(3.7)

The energy equation (2.4) is now transformed to the form:

f(r+)+
w0
uτ

∂T+

∂z+
=
1

ρcpν

∂

∂z+

(

λeff
∂T+

∂z+

)

(3.8)

where

f(r+)=Q+
∂

∂r+

( T+m
r+h+

)



614 D. Mikielewicz, J. Mikielewicz

The first term in (3.8) can only vary in the radial direction, hence is not
dependent the z direction, thanks to the assumption (3.1). In (3.8) λeff
is the effective thermal conductivity, which consist of molecular and turbu-
lent components. Some attention will now be devoted to the way in which
effective thermal conductivity has been determined. In the turbulent flow
molecular effects can be neglected and the effective thermal conductivity
yields

λeff =
cpµt
σt

(3.9)

In (3.7) σt represents the turbulent Prandtl number and in the present
study we have assumed a constant value of σt = 0.85 (Mikielewicz and Ih-
natowicz, 1996). Considerations will, in the first instance, be conducted for
the case in which the turbulent dynamic viscosity µt will be determined from
one of the simplest models, namely the Prandtl mixing length model. In the
case of single phase flowmodels, that model plays a very useful role as a tool
enabling to obtain analytical solutions in the case of velocity field for simple
flow cases, such as flows in boundary layers or in tubes. Additionally, themo-
del is very simple in application and effective enough in numerous cases. In
the case considered here, the velocity profile does not attain maximum at any
other location than the core of the flow. If that would be the case then the
model would not be applicable in turbulence modelling. In authors’ opinion,
only in cases when simple models fail to reveal appropriate behaviour of the
phenomenon, themore complexmodels should be considered ormore complex
models are capable of leading tomore accurate calculations of turbulence.The
turbulent dynamic viscosity, in the case of Prandtl mixing length model, has
the form

µt
µ
=(l+m)

2dϑ
+

dz+
=κ2(z+)2

dϑ+

dz+
(3.10)

where lm represents themixing length. In calculations carried out in the pre-
sent work it turned out that a classical definition of the mixing length is
appropriate. That enabled us to obtain realistic velocity and temperature pro-
files.

In solving the Equation (3.8) it should be borne inmind that T+m and h
+

are known from solutions describing distributions of film thickness, either in
a developing flow (2.14), or in the fully developed case, relation (2.20) and
mean temperature (3.6). A good assumption is also constant value of the film
thickness. In order to obtain analytical formof temperature distribution in the
film,wewill evaluate themeanvalue of turbulent conductivity. Suchprocedure



Turbulent heat transfer in thin liquid films... 615

is fully justified since the film is rather thin and no significant variations of
turbulent conductivity are expected

λeff =
1

δ

δ
∫

0

λeff dz=
ρcpκ

2ϑ0ν

(2m+1)σtuτ
δ+ (3.11)

Determination of the mean value of turbulent thermal conductivity allows to
linearise Equation (3.8). The solution procedure of (3.8) will first require to
obtain the solution of a homogeneous equation and subseqently, the solution
of the non-homogeneous equation. The general solution to the homogeneous
Equation (3.8) yields

T+ =
(w0
uτ

ν

at

)

−1

C exp
(w0
uτ

ν

at
z+
)

(3.12)

where at = λeff/(ρcp). Subsequently, we seek a unique solution to Equation
(3.8) in the linear form. The final solution to temperature distribution yields

T+ =
(w0
uτ

ν

at

)

−1

C exp
(w0
uτ

ν

at
z+
)

+D−
(w0
uτ

)

−1

f(r+)z+ (3.13)

The constants C and D can be established from the boundary conditions

for z+ =0 T+ =T+w

for z+ =h+
∂T+

∂z+
=−

qiν

λeffT0uτ

(3.14)

Then the constant C has the form

C =
(w0
uτ

)

−1

f(r+)−
qiν

λeffuτT0
exp
(

−
w0
uτ

ν

at
h+
)

(3.15)

The constant D yields

D=T+w −C
(w0
uτ

ν

at

)

−1

(3.16)

The convective heat transfer coefficient α, as well as the Nusselt number, are
finally defined in the following way

α=
qw

Tm−Tw
=

qw

T0(T
+
m −T

+
w )

Nu=
αr

λ
(3.17)



616 D. Mikielewicz, J. Mikielewicz

4. Results

Calculations havebeenmadeusing theMathcad11 softwarepackage.The follo-
wing input has been assumed: Q+ =1.169 ·107, uτ =0.037m/s, qi/qw =0.8,
T0 = 20

◦C, properties of the film correspond to water properties at 20◦C,
the nozzle diameter 1mm. Thickness of the boundary layer in the considered
example corresponds to about δ+ ≈ 15 whereas when nucleation is present,
then δ+ ≈ 5.Calculated have been the distributions of the turbulent viscosity,
velocity and temperature profiles in both the fully developed and developing
regions, together with the calculations of heat transfer coefficient in both re-
gions for low heat fluxes, without bubble generation as well as for high heat
fluxeswhere nucleation is present. The results have been presented in Fig.2 to
Fig.5. It seems that proper qualitative trends are depicted by the postulated
model.

Fig. 2. Non-dimensional liquid film thickness distribution in the developing region

Fig. 3. Distribution of shear stress in the developing region in case of presence of
blowing and without it

Examination of the obtained results shows that the film thickness has
not been changed very much by the presence of nucleation, Fig.2. More pro-



Turbulent heat transfer in thin liquid films... 617

nounced changes are seen in Fig.3, where, as expected, presence of bubbles
strongly modifies the shear stress distribution. It should also be noted that
the boundary layer thicknesses are different in cases of presence of nucleation
and without it. The heat transfer coefficient (Fig.4) assumes relatively high
values, which are however typical for such cases, where these exceed the value
of 10000. Although the predicted heat transfer coefficients are very high, we
can observe some 20%enhancement of the heat transfer coefficient for the case
whennucleation is present. That happens despite a very small thickness of the
liquid film, which in the present case was about 0.4mm. Finally, temperature
distributions are shown in Fig.5.

Fig. 4. Distribution of heat transfer coefficient in the developing region in case of
presence of nucleation and without it

Fig. 5. Distribution of temperature in the developing region in case of presence of
nucleation and without it



618 D. Mikielewicz, J. Mikielewicz

5. Conclusions

Formulated has been a simplemodel of single-phase jet impinging on a surface
and forming a thin film on that surface. The model consists of conservation
equations of mass, momentum and energy. For such case the heat transfer
coefficients have beendetermined.Approximate analytical solutions have been
obtained in the developing and fully developed regions, allowing for a wide
discussion of the obtained results.

Themodel incorporates the so-called ”blowing velocity” tomodel themo-
mentum transfer across the film. The heat flux consists of two components,
namely the convective heat flux through the liquid and the evaporative heat
flux. The latter component is approximately equal only to 1% for considered
case of the formeroneand sucha share in theheatbalance isusuallynegligible.
However, it changes the velocity profile to a significant extent. The boundary
layer, due to presence of the vapour bubbles, is very distorted, what means
also that the turbulence is also very vigorous as compared with single pha-
se films. In the present work the mixing length theory was applied to enable
calculation of the appropriate velocity. For the velocity profile determined in
such a way, the heat transfer coefficients have been determined allowing for a
wide discussion of the obtained results. Values of the latter agree qualitatively
with the heat transfer coefficients found in experiments for other fluids, what
confirms the presence of intense heat transfer under nucleate boiling regime.

References

1. Buyevich Y.A., Mankevich V.N., 1996, Cooling of a superheated surface
with a jet mist flow, Int. J. Heat Mass Transfer, 39, 2353-2362

2. Deb S., Yao S.-C., 1989, Analysis on film boiling heat transfer of impacting
sprays, Int. J. Heat Mass Transfer, 32, 11, 2099-2112

3. Liu Z.-H., Wang J., 2001, Study on film boiling heat transfer for water jet
impinging on high temperature flat plate, International Journal of Heat and
Mass Transfer, 44, 2475-2481

4. Mikielewicz D., Ihnatowicz E., 1996, Turbulence modelling using various
turbulent Prandtl number models,Trans. IFFM, 100, 141-154

5. MikielewiczD.,Mikielewicz J., 2005,Surface Cooling byMeans of Axially
Symmetric Liquid Jets, GdańskUniversity of TechnologyPublishers [in Polish]

6. Mikielewicz J., Mikielewicz D., 1999, New approaches to modelling of
impinging jets,Proc. 3rd Baltic Heat Transfer Conference, Gdańsk



Turbulent heat transfer in thin liquid films... 619

7. Mikielewicz J.,MikielewiczD., 2001,A ’thin layer’model of heat transfer
in a laminar liquid film formedby impinging jet,Proc. 5thWorldConference on
Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, 2, 1041-
1044, Thessaloniki

8. Mikielewicz J.,Mikielewicz D., 2002,Modelling of heat transfer in a lami-
nar liquid film formed by impinging jet,Proc. 12th International Heat Transfer
Conference, CD-ROM,Grenoble

9. Mikielewicz J., Mikielewicz D., 2004, Modelling of nucleate boiling heat
transfer in a film formed by impinging jet, Proc. 5th International Symposium
on Heat and Mass Transfer, CD-ROM,Minsk

10. Webb P., Ma C.F., 1996, Heat transfer to impinging jets, Advances in Heat
Transfer, 26

11. WolfD.H., Incropera F.P., ViskantaR., 1994, Jet impingement boiling,
Advances in Heat Transfer, 23

Turbulentna wymiana ciepła w cienkich filmach cieczowych przy małych

i dużych strumieniach ciepła

Streszczenie

W pracy przedstawiono rozwiązanie turbulentnej wymiany ciepła w cienkich fil-
mach cieczowych przy małych i dużych strumieniach ciepła. Zaproponowano proste
modelewymiany ciepła dla przypadków laminarnego i turbulentnegofilmucieczowego
wytworzonego uderzającą strugą w warunkach dużych i małych strumieni cieplnych.
W takich przypadkach turbulencja jest silnie modyfikowana i z tego względu trudna
domodelowania.Wpracyzastosowanomodel drogimieszaniaPrandtla iwprzypadku
dużych strumieni cieplnych droga mieszania jest szczególnie mocno zmodyfikowana.
Wprzypadkudużych strumieni ciepławprowadzonodomodelu tzw. prędkośćwzdmu-
chu, któramodeluje wymianę pęduwkierunku poprzecznymdoprzepływuwywołaną
odrywającymi się od ścianki pęcherzykami.Wyznaczono rozkłady prędkości i tempe-
ratury w filmie cieczowym, które umożliwiły wyliczenie współczynnika przejmowania
ciepła i liczby Nusselta.

Manuscript received February 5, 2007; accepted for print April 2, 2007