Similaridade - Convecção Térmica

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Similarity Solution for Laminar Forced
Convection over a Horizontal Plate
Isabela Florindo Pinheiro, Universidade Federal Fluminense
T
he effects of a external flow over a hori-
zontal plate are investigated by the use
of the Similarity Theory. The classical
technique is applied to develop the velocity
profile and to find the thickness of the bound-
ary layer, whereas the concept of the no-slip
condition, introduced into fluid mechanics by
Ludwig Prandtl (1875-1953). A heat trans-
fer analysis is also studied and the isother-
mal and uniform heat flux conditions are ap-
plied. Dimensionless quantities are used to
minimize the computational effort employed
by the Mathematica [14] platform with a sim-
ple NDSolve routine. The results of this work
are presented considering different scenarios
with variations of the Prandtl Number (Pr),
Eckert Number (Ec), Reynolds Number (Re)
and Modified Brinkman Number (Br*) for a
laminar forced convection.
1 Introduction
The analysis of external forced convection flows
has an important role in engineering applications,
as heat exchangers, turbine blades, aircrafts and
automobiles, electronic devices, etc. The study of
this particular topic was directly influenced by the
pioneering work of Ludwig Prandtl and Paul Richard
Heinrich Blasius. Their work laid the foundation of
the fluid dynamic studies of boundary layers over
a solid surface [10]. Over one century later, many
experimental, analytical and numerical investigations
are still based in their studies.
Lin and Lin [8] proposed a similarity solution
method that provided very accurate solutions for
laminar forced convection heat transfer from either
an isothermal surface or an uniform-flux boundary
to fluids of any Prandtl Number. Different boundary
conditions were also analysed in both analytical and
numerical work by Shu and Pop [12]. Other investiga-
tions presented by Schneider [11], Ping [9] and Wang
[13] studied a mixed convection in different types of
solid surfaces, in other words, they took into consid-
eration a combination of free and forced convection
flows. Another way to use different solid surfaces in
combination of another heat transfer method can be
seen in [2], where a numerical study was conducted in
order to analyse the Magnetohydrodynamics (MHD)
boundary layer over an exponentially stretching sur-
face with thermal radiation.
There is also a matter of laminar and turbulent
flow analysis. While in this work, the velocity and
temperature profiles were developed based in a lam-
inar flow, some works focus on turbulence models,
in which velocity and temperature fluctuations in
the turbulent thermal convection must be considered.
For turbulent flows, studies proposed by Castillo
and George [5] and Chien [6] can be presented as
examples.
The effects of viscous dissipation are also presented
in this article using variations of the Eckert and
Brinkman Number, in order to explore the signif-
icance of the energy dissipation on the Boundary
Layer Heat Transfer. Among some studies of those
effects, one should mention the work presented by Ab-
dul Hamid et al. [1], in which the effects of Joule Heat-
Page 1 of 15
ing and Viscous Dissipation on the MHD Marangoni
convection boundary layer flow are studied and the
behaviour of the flow with variations of the Eckert
and Prandtl Numbers is explored, and the study
developed by Choudhury et al. [7], who analysed
the visco-elastic MHD Boundary Layer Flow in a
inclined surface also with the consideration of energy
dissipation.
In this work, a similarity method is applied to the
development of the solution for the external forced
convection over a horizontal flat plate, considering
four sets of boundary conditions for the heat transfer
solution. The concept of similarity means that cer-
tain features (e.g., velocity profiles) are geometrically
similar. Analytically, this amounts to combining the
x and y spatial dependence on a single indepen-
dent variable η. The velocity components u(x, y)
and v(x, y) are expressed by a single dimensionless
stream function f(η), and the temperature T (x, y)
into a dimensionless temperature θ(η). Recently,
Castillo [4] extended the similarity theory to include
boundary layers with imposed pressure gradient.
Based on the analysis of the laminar forced con-
vection over a flat plate, this work investigates the
similarity theory to reach the velocity profile and the
temperature distribution. Using this technique, the
Nusselt Number is provided, regarding its relation
with the Reynolds Number. Even though different
scenarios are explored, the transformations needed
are the same as presented years ago by the classic
Blasius solution.
2 Velocity Profile Formulation
Figure 1: Laminar and Turbulent Flows in a Flat Plate.
Figure 1 shows the transition between laminar
and turbulent flows in a flat plate, in which the
Momentum and Energy Equations will be developed
for the first part (xc).
In order to develop the solution of the velocity pro-
file for the laminar forced convection, the Boundary
Layer Momentum Equation is taken into considera-
tion (1).
u
∂u
∂x
+ v
∂u
∂y
= −1
ρ
∂Pδ
∂x
+ ν
∂2u
∂y2
(1)
As mentioned before, the studied problem involves
a laminar flow over a flat plate. Because the pressure
gradient is constant in this case, the new Momentum
Equation is:
u
∂u
∂x
+ v
∂u
∂y
= ν
∂2u
∂y2
(2)
And the Continuity Equation:
∂u
∂x
+
∂v
∂y
= 0 (3)
Similarity for boundary layer flow follows from the
observation that while the boundary layer thickness
at each downstream location x is different, a scaled
normal distance η can be employed as a universal
length scale. Presence of natural length scales (such
as a finite-length, plate, cylinder, or sphere) generally
precludes the finding of similarity solutions. Using
the scaled distance, the similarity procedure finds
the appropriate normalized stream and temperature
functions that are also valid at all locations [3]. With
that considered, the similarity variables are presented
as below:
η = η (x, y) (4a)
δ =
x√
Rex
=
x√
xU∞
ν
(4b)
η =
y
δ
=
y
√
xU∞
ν
x
(4c)
The derivation of η by the spatial coordinates x
and y are represented by equations 5a and 5b.
∂η
∂x
= −
y
√
xU∞
ν
2x2
= − η
2x
(5a)
∂η
∂y
=
√
xU∞
ν
x
=
η
y
(5b)
To facilitate the solution of the boundary layer
problem in a flat plate, the Stream Function Ψ can
be applied (6a and 6b).
u (x, y) =
∂ψ
∂y
(6a)
Page 2 of 15
v (x, y) = −∂ψ
∂x
(6b)
The velocity component u (x, y) related to the
similarity variable has to be determined:
u =
∂ψ
∂y
= g(η)U∞ (7a)
u =
∂ψ
∂η
∂η
∂y
(7b)
ψ = δ f(η)U∞ = ν f(η)
√
xU∞
ν
(7c)
∂ψ
∂η
= ν f ′(η)
√
xU∞
ν
(7d)
u = U∞ f ′(η) (7e)
And the velocity component v(x, y):
v = −ν ∂η
∂x
√
xU∞
ν
f ′(η)− U∞ f(η)
2
√
xU∞
ν
(8a)
∂η
∂x
= −
y
√
xU∞
ν
2x2
(8b)
v =
1
2
U∞
y f ′(η)
x
− f(η)√
xU∞
ν
 (8c)
In order to develop the ODE system from the
Momentum Equation, some derivatives are needed:
∂u
∂y
= U∞
∂η
∂y
f ′′(η) =
U∞
√
xU∞
ν f
′′(η)
x
(9)
∂2u
∂y2
=
U∞ ∂η∂y
√
xU∞
ν f
′′′(η)
x
=
U2∞ f ′′′(η)
ν x
(10)
∂u
∂x
= −
y U∞
√
xU∞
ν f
′′(η)
2x2
(11)
Replacing the derivatives in the momentum equa-
tion, the final ODE for the velocity profile is obtained
(12).
2 f ′′′(η) + f(η) f ′′(η) = 0 (12)
And the boundary conditions:
u (x, 0) = 0 (13a)
u (x,∞) = U∞ (13b)
v (x, 0) = 0 (13c)
u (0, y) = U∞ (13d)
The dimensionless ODE system is then represented.
2 f ′′′ + f f ′′ = 0 (14a)(
∂f
∂η
)
η=0
= 0 (14b)
(
∂f
∂η
)
η=∞
= 1 (14c)
f (0) = 0 (14d)
3 Temperature
Distribution
Analysis
The same concept as before is applied in the Energy
Equation. The equation to obtain the temperature
distribution for a laminar flow over a horizontal plate
can be seen in 15:
u
∂T
∂x
+ v
∂T
∂y
=
µΦ
c ρ
+ α
∂2T
∂y2
(15)
The viscous heating rate is determined in equation
16.
Φ =
(
∂u
∂y
)2
(16)
The dimensionless temperature depends on the
boundary condition in the solid surface.
3.1 Isothermal Plate, with Ts 6= T∞
If there is a laminar flow over a isothermal flat
plate with a temperature that is different than the
far field temperature (Ts 6= T∞), the characterization
of the dimensionless quantity is:
θ(η) =
T − Ts
T∞ − Ts (17a)
T = ∆T θ(η) + Ts (17b)
And, the derivatives of the temperature for each
of the spatial coordinates:
∂T
∂y
= ∆T
∂θ
∂η
∂η
∂y
(18)
∂2T
∂y2
=
∆T η
√
xU∞
ν θ
′′(η)
x y
(19)
∂T
∂x
= −∆T η
2x
∂θ
∂η
(20)
Page 3 of 15
The same procedure followed for the velocity field
is done in this case to reach an ODE system that
can be easily solved in the Mathematica platform.
So, the system is represented by the equation and
the boundary conditions described below:
2Ec
(
f ′′(η)
)2
+ f(η) θ′(η) +
2 θ′′(η)
Pr
= 0 (21a)
θ (0) = 0 (21b)
θ (∞) = 1 (21c)
The Eckert Number and the Prandtl Number are
shown in equations 22 and 23.
Ec =
Br
Pr
=
U2∞
c∆T
(22)
Pr =
ν
α
(23)
3.1.1 Without viscous dissipation
A different temperature transformation can be
used to simplify the solution of the problem without
viscous dissipation.
φ (η) = θ′ (η) (24)
So, in this particular case, the temperature trans-
formation is described as:
2Ec (f ′′(η))2 + f(η)φ(η) +
2φ′(η)
Pr
= 0 (25a)
φ (0) = 1 (25b)
If the viscous dissipation is not considered, the
simplified equation becomes:
φ′(η) = −Pr
2
f(η)φ(η) (26)
The solution of the equation 26 can be easily done
analytically:∫
1
φ
dφ = −1
2
Pr
∫
f(η) dη (27a)
φ
φ0
= e−
1
2
Pr
∫ η
0 f(η
′) dη′ (27b)
∫ η
0
θ′(η) dη = θ′(0)
(∫ η
0
e−
1
2
Pr
∫ η′
0 f(η
′′) dη′′ dη′
)
(27c)
θ(η) = θ′(0)
(∫ η
0
e−
1
2
Pr
∫ η′
0 f(η
′′) dη′′ dη′
)
+ θ(0)
(27d)
Due to boundary condition θ(0) = 0:
θ(η) = θ′(0)
(∫ η
0
e−
1
2
Pr
∫ η′
0 f(η
′′) dη′′ dη′
)
(28)
As for the condition in η = ∞, the boundary
condition is equal to 1. Thus, to calculate θ′(0):
1 = θ′(0)
(∫ ∞
0
e−
1
2
Pr
∫ η′
0 f(η
′′) dη′′ dη
)
(29a)
θ′(0) =
1∫∞
0 e
− 1
2
Pr
∫ η′
0 f(η
′′) dη′′ dη
(29b)
3.2 Isothermal Plate, with Ts = T∞
For the second set of boundary conditions, an
isothermal plate has the same temperature as the
far field temperature (Ts = T∞). The following
dimensionless quantity is then introduced:
θ(η) =
c (T − T∞)
U2∞
(30)
With this new consideration, the derivatives are
transformed.
∂T
∂y
=
η U2∞
c y
∂θ
∂η
(31)
∂2T
∂y2
=
η U2∞
√
xU∞
ν θ
′′(η)
c x y
(32)
∂T
∂x
= −η U
2∞
2 c x
∂θ
∂η
(33)
The ODE system is given by equations 34a, 34b
and 34c:
2
(
f ′′(η)
)2
+ f(η) θ′(η) +
2 θ′′(η)
Pr
= 0 (34a)
θ (0) = 0 (34b)
θ (∞) = 0 (34c)
3.3 Adiabatic Plate
The third set of boundary conditions is regarded
an adiabatic horizontal plate. The dimensionless
quantity is expressed by equation 30, same as the
second case with an isothermal wall.
So, with that said, the ODE system is similar
to the one expressed before, but the dimensionless
Page 4 of 15
boundary conditions are differently represented.
2
(
f ′′(η)
)2
+ f(η) θ′(η) +
2 θ′′(η)
Pr
= 0 (35a)
(
∂θ
∂η
)
η=0
= 0 (35b)
θ (∞) = 0 (35c)
3.4 Uniform Heat Flux
Figure 2: Uniform heat flux condition in a flat plate.
The last set of boundary conditions is related to
an uniform heat flux in the flat plate (figure 2). This
case is similar to the Adiabatic Plate, but with a
different dimensionless temperature transformation
(36).
θ(η) =
k (T − T∞)
q˙′′ L
(36)
The new transformation generates the following
ODE system:
2 Br*
(
f ′′(η)
)2
+ Pr f(η) θ′(η) + 2 θ′′(η) = 0 (37a)(
∂θ
∂η
)
η=0
= −1 (37b)
θ (∞) = 0 (37c)
The Brinkman Number (38) shows the effect of the
viscous dissipation in the temperature distribution.
But, in this particular case, the Modified Brinkman
Number is used with the same principle and is given
by equation 39.
Br =
µU2∞
k (T∞ − Ts) (38)
Br* =
µU2∞
q˙′′ L
(39)
As can be seen from all the cases, the temperature
distribution depends on the results for the velocity
profile in any of the boundary conditions.
4 Results and Discussion
The results regarding the velocity profile are the
same for all the different cases of the temperature
distribution. Because of that, the first set of results
to be commented is about the velocity field and then
the temperature distribution is going to be carefully
analysed with each of the boundary conditions.
4.1 Velocity Profile
The ODE system explained in 14 is solved using
the NDSolve routine in Mathematica. An high finite
number is selected to be η∞ in order to obtain the
numerical solution of the problem. A sufficient num-
ber to apply for the η∞ is 10, so the velocity field is
presented in figure 3.
2 4 6 8 10
η
0.2
0.4
0.6
0.8
1.0
u*
Figure 3: Dimensionless Velocity in terms of η.
With the velocity profile calculated, it’s possible
to obtain the boundary layer thickness according to
equation 40.
u (δ) = 0.99U∞ (40)
η99% = 4.90999 (41)
The best way to obtain the value of δ is with the
FindRoot routine in Mathematica. This command
obtains the δ from the surface to where the velocity
u reaches some fixed percentage (in this case, 99%)
of the stream value.
With the boundary layer thickness and the velocity
profile over the horizontal plate, the dimensional
Page 5 of 15
velocity field can be visualized in figures 4 and 5,
varying the kinematic viscosity (ν) and the far field
velocity (U∞).
0 2 4 6 8 10
x0
2
4
6
8
y
(a) Dimensional Velocity with ν = 0.01m2/s, in
x0 = 0.5m
0 2 4 6 8 10
x0
2
4
6
8
y
(b) Dimensional Velocity with ν = 0.05m2/s, in
x0 = 4m
0 2 4 6 8 10
x0
2
4
6
8
y
(c) Dimensional Velocity with ν = 0.1m2/s, in
x0 = 8m
Figure 4: Dimensional velocity ranging the kinematic
viscosity.
0 2 4 6 8 10
x0
2
4
6
8
y
(a) Dimensional Velocity with U∞ = 1m/s, in
x0 = 1m
0 2 4 6 8 10
x0
2
4
6
8
y
(b) Dimensional Velocity with U∞ = 5m/s, in
x0 = 4m
0 2 4 6 8 10
x0
2
4
6
8
y
(c) Dimensional Velocity with U∞ = 10m/s, in
x0 = 8m
Figure 5: Dimensional velocity ranging the far field ve-
locity.
As can be seen in figure 4, when the kinematic
viscosity increases, the boundary layer thickness in-
creases as well. The same can’t be said about the
far field velocity (figure 5), which has an indirect
relation with the boundary layer thickness. Both
instances can be explained by equation 4b where
the boundary layer thickness increases with an low
Reynolds Number.
4.1.1 Friction Factor
Using the definition of shear stress at the wall, τw,
and the friction factor, Cf , the following equation
can be obtained:
τw = µ
(
∂u
∂y
)
y=0
= Cf
ρU2∞
2
(42)
So, using the transformed variable for the velocity,
the friction factor coefficient is now defined by:
Cf = 2 f
′′ (0)Rex−
1
2 (43)
The friction factor is then calculated based on the
numerical solution for the velocity profile. The nu-
meric value is expressed in 44 and the Cf in relation
with the Reynolds Number is visualized in figure 6.
Cf =
0.664115√
Rex
(44)
Page 6 of 15
0 20 40 60 80 100
Rex
0.1
0.2
0.3
0.4
0.5
Cf
Figure 6: Friction Factor vs Reynolds Number.
4.2 Temperature Distribution for
Isothermal Plate Ts 6= T∞
With the ODE system described in 21, the tem-
perature can be calculated considering or not the
viscous dissipation. In this first instance, the values
considered for the Prandtl Number and the Eckert
Number were 1. The same η∞ is used and the results
of the velocity and temperature profiles are shown
in figure 7.
2 4 6 8 10
η
0.2
0.4
0.6
0.8
1.0
u* , θ
Figure 7: Velocity (blue) and temperature (red) profiles.
In other part of the solution, different values for
the Prandtl Number and the Eckert Number are
used, in order to visualize the behaviour of the curve
(figure 8).
2 4 6 8 10
η
0.5
1.0
1.5
2.0
θ
Pr=10,Ec=1
Pr=1,Ec=10
Pr=50,Ec=0
Pr=10,Ec=0
Pr=1,Ec=0
Pr=1,Ec=1
Figure 8: Dimensionless Temperature distribution for
different Prandtl and Eckert Numbers.
As can be seen by the red, orange and green
curves, with a null Eckert Number and ranging the
Prandtl Number between 1, 10 and 50, the curve
approaches the dimensionless temperature value 1
with less η. With the gray and purple curves, a high
Eckert and Prandtl Numbers generated a parabolic
curve before the temperature reaches the constant
number of 1.
The following dimensional temperature and ther-
mal boundary layer thickness are calculated with
Pr = 1 and Ec = 1. The result for the thermal
boundary layer thickness respects the same correla-
tion as for the boundary layer in the velocity profile.
The same 99% is used, as analysed in 45.
u (δT ) = 0.99 θ∞ (45)
ηθ99% = 4.5261 (46)
For the dimensional temperature, the profile will
vary if Ts is bigger or smaller than T∞. In figure 9,
this difference is shown.
0 2 4 6 8 10
x0
2
4
6
8
y
(a) Ts > T∞
0 2 4 6 8 10
x0
2
4
6
8
y
(b) Ts < T∞
Figure 9: Dimensional temperature profile for a isother-
mal plate with Ts 6= T∞.
4.2.1 Nusselt Number
The Nusselt Number (Nu) is the ratio of convec-
tive to conductive heat transfer across a surface. So,
Page 7 of 15
for the Nusselt Number in this problem, the following
relation must be developed:
Nu = θ′(0)
√
Rex (47)
So, for Ec = 1 and Pr = 1:
Nu = 0.498474
√
Rex (48)
And for Ec = 0 and Pr = 1:
Nu = 0.332057
√
Rex (49)
The Relation of the Nusselt Number and the
Reynolds Number is demonstrated in figure 10, for
different Prandtl and Eckert Numbers.
20 40 60 80 100
Rex
5
10
15
Nu
Pr=10,Ec=1
Pr=1,Ec=10
Pr=50,Ec=0
Pr=10,Ec=0
Pr=1,Ec=0
Pr=1,Ec=1
Figure 10: Nusselt Number vs Reynolds Number.
The variation of Nusselt with Prandtl Number is
observed by the graphic Nu
Rex
1
2
in figure 11.
20 40 60 80 100
Pr
-40
-20
20
40
60
Nu
Rex
Ec=10
Ec=0
Ec=1
Figure 11: The variation of Nusselt with Prandtl Num-
ber.
For Ec > 0, there is a oscillation of results for low
Prandtl Numbers, as demonstrated by the blue and
black curves. After a certain Prandtl Number, the
curve maintained a uniform variation, like in Ec = 0.
Other way to evaluate the Nusselt Number is to
consider the calculation for both limits of Pr.
Table 1: Nusselt Number for low and high Pr
Nusselt Number
Pr = 10−6 Pr = 106
Ec = 0 0.564
√
Pr
√
Rex 0.339
3
√
Pr
√
Rex
Ec = 1 0.564
√
Pr
√
Rex 32.21
3
√
Pr
√
Rex
4.2.2 Convective Heat Transfer Coefficient
Despite the Nusselt Number growth with Rex (and
therefore, growth with the x position), the Convec-
tive Heat Transfer Coefficient h has an opposite be-
havior, due to its relationship with Nu:
h =
k
x
Nu (50a)
h =
0.498474 k
√
Rex
x
(50b)
A function to calculate the Convective Heat Trans-
fer Coefficient according to dimensional parameters
is set and the variation of h with x for two different
fluids flows over the horizontal flat plate is produced
through figure 12:
0 2 4 6 8 10
x
500
1000
1500
2000
2500
3000
h
Figure 12: Variation of the h parameter with x for two
different flows with U∞ = 4m/s: water (red)
and air (black).
As can be analysed from figure 12, the water has
higher convective coefficient than the air, which is in
accordance with values presented in table 2.
Page 8 of 15
Table 2: Mean Values of the convection coefficient h.
Process h [ Wm2K ]
Free Convection Air 5− 30
Gas 4− 25
Liquids 120− 1200
Water, liquid 20− 100
Boiling Water 120− 24000
Forced Convection Air 30− 300
Gas 12− 120
Liquids 60− 25000
Water, liquid 50− 10000
Boiling Water 3000− 100000
4.2.3 Without Viscous Dissipation
The integral part of 29a can be represented by a
function that depends on the Prandtl Number. In
other words:
g(Pr) =
∫ ∞
0
e−
1
2
Pr
∫ η′
0 f(η
′′) dη′′ dη (51)
Different values of Pr generates different values
for the integral described in 51. Some examples are
described in table 3.
Table 3: Function g (Pr).
Pr g(Pr)
0.5 0.260
1.0 0.332
15 0.834
The Nusselt Number is then calculated based on
the simplification without viscous dissipation and a
Prandtl value of 1.
Nu = 0.332057
√
Rex (52)
This result is in agreement with 49, meaning that
despite using numerical methods to calculate the
Nusselt Number from the original problem, nothing
changes in comparison to the analytical solution that
considers a null Eckert Number.
If the borderline cases with Pr << 1 and Pr >> 1
are defined, then the following relations are gathered:
Pr << 1 : Nu = 0.564
√
Pr (53)
Pr >> 1 : Nu = 0.339
3
√
Pr (54)
4.3 Temperature Distribution for
Isothermal Plate Ts = T∞
For this case, the dimensionless quantity applied
to the temperature distribution changes the carac-
terization of the ODE system and the only variation
that is going to be explored is regarding the Prandtl
Number. Figure 13 stands for the velocity profile
with the temperature distribution for this case with
Pr = 1.
2 4 6 8 10
η
0.2
0.4
0.6
0.8
1.0
u* , θ
Figure 13: Velocity and Temperature Profiles for an
isothermal plate with Ts = T∞.
Different from the last problem, the temperature
curve approximates to 0 instead of 1. For different
Prandtl Numbers, the variation of the temperature
distribution can be seen in figure 14.
2 4 6 8 10
η
0.5
1.0
1.5
θ
Pr=100
Pr=50
Pr=10
Pr=1
Figure 14: Temperature Profiles for different Pr.
The maximum value of the temperature for Pr = 1
can be viewed below:
θmax = 0.150365 (55)
Page 9 of 15
4.3.1 Nusselt Number
The same procedure to develop the relation of
the Nusselt Number and Reynolds Number with
Pr = 1 (56) is applied to this case. Also, the same
relationship is printed for different Prandtl Numbers,
generating figures 15 and 16.
Nu = 0.332057
√
Rex (56)
20 40 60 80 100
Rex
10
20
30
40
50
60
Nu
Pr=100
Pr=50
Pr=10
Pr=1
Figure 15: Dependence of the Nusselt Number to the
Reynolds Number for different Pr.
20 40 60 80 100
Pr
1
2
3
4
5
6
Nu
Rex
Figure 16: The variation of the Nusselt Number with
Prandtl Number.
The same behavior as before is identified as the
variation of the Nusselt Number increases with higher
Prandtl Numbers. The same conclusion can be ob-
tained when analysing figure 16, for an isothermal
wall, with Ts = T∞.
4.3.2 Convective Heat Transfer Coefficient
In order to continue the observation developed in
table 2, the difference between the convective heat
transfer coefficient in two types of fluid flows (water
and air) is shown for an isothermal surface.
0 2 4 6 8 10
x
1000
2000
3000
4000
5000
6000
h
(a) Water
0 2 4 6 8 10
x
20
40
60
80
100
h
(b) Air
Figure 17: Convective Coefficient variation for a water
(red)
and air (black) flows over a horizontal
plate.
In figures 17a and 17b, the far field velocity used
was 2m/s.
4.4 Adiabatic Plate
In a adiabatic plate, there is no transfer of heat
through the solid surface. This case is similar to the
uniform heat flux plate in the matter that there is a
Neumann boundary condition in the surface, but in
a adiabatic plate, the condition is equal zero.
2 4 6 8 10
η
0.2
0.4
0.6
0.8
1.0
u* , θ
Figure 18: Velocity and Temperature Profiles for an adi-
abatic plate.
Page 10 of 15
The temperature distribution has a close behaviour
to the second isothermal plate case (figure 14),
mainly because the same dimensionless temperature
transformation is applied for the adiabatic plate.
2 4 6 8 10
η
1
2
3
4
θ
Pr=100
Pr=50
Pr=10
Pr=1
Figure 19: Temperature Profiles for different Pr.
With variations on the Prandtl Number, a main
difference between the second and third case is evi-
denced. For η = 0, the isothermal wall has a Dirichlet
boundary condition, which created curves originated
from zero. In the adiabatic plate, the Neumann
Boundary Condition in η = 0 generated a curve in
which the value of θ(0) has to be evaluated. In figure
19, θ(0) is shown for different scenarios.
The maximum value of the temperature for Pr = 1
is also calculated:
θmax = 0.501167 (57)
One interesting factor to be studied is regarded
the dimensional temperature. For this case, the
new profile can also vary with the heat capacity
because of the dimensionless quantity used for the
temperature distribution (equation 30). In figure
25, the behaviour of the dimensional temperature
distribution with changes in c is shown.
0 2 4 6 8 10
x0
2
4
6
8
y
(a) x0 = 1m, c = 0.5
J
kg·◦C
Those profiles can be explained because the ∆T
is equal to U
2∞
c , so the higher the heat capacity, the
lower the ∆T in the flat plate.
0 2 4 6 8 10
x0
2
4
6
8
y
(b) x0 = 4m, c = 1
J
kg·◦C
0 2 4 6 8 10
x0
2
4
6
8
y
(c) x0 = 8m, c = 3
J
kg·◦C
Figure 20: Dimensional temperature profile varying with
the heat capacity c.
4.4.1 Nusselt Number
As said before, the Nusselt Number (Nu) is the
ratio of convective to conductive heat transfer across
the boundary. So, the Nusselt Number can be rep-
resented in equation 58 for the case of a external
forced convection in a flat plate.
Nu = θ′(0)
√
Rex (58)
Because the flat plate is adiabatic, there is no flux
through the surface and the conductive component
is zero. This can be visualized by the boundary
condition expressed in 35b. So, in this matter, the
Nusselt Number is:
Nu = 0 (59)
The Nusselt Number zero doesn’t mean that there
is not any heat transfer between the plate and the
flow over the surface.
4.5 Uniform Heat Flux
Differently from the previous cases, the uniform
heat flux in a horizontal plate has a particular di-
mensionless temperature transformation, in which a
Page 11 of 15
new parameter is considered: the Modified Brinkman
Number, represented by Br*. In light of this new
information, the velocity and temperature profile are
shown in figure 21.
2 4 6 8 10
η
-2
-1
1
2
3
4
5
u* , θ
Figure 21: Velocity and Temperature Profiles for uni-
form heat flux on a flat plate.
The new temperature profile can be influenced in
different ways by the Prandtl and Modified Brinkman
Numbers. With that said, figure 22 illustrates the
temperature distribution in means of the Pr and
Br*.
2 4 6 8 10
η
2
4
6
8
θ
Pr=10,Br=1
Pr=1,Br=10
Pr=50,Br=0
Pr=10,Br=0
Pr=1,Br=0
Pr=1,Br=1
Figure 22: Temperature Profiles for different Pr and
Br*.
As can be seen in the blue and gray curves, the
higher the Modified Brinkman Number, higher is the
dimensionless temperature in η = 0. Also, by the
interpretation of the red, orange and green curves,
the higher the Prandtl Number with Br* = 0, the
lower θ(0) is.
As done before, the dimensional temperature can
be evaluated using the variable transformation ex-
pressed in equation 36. The behavior of the dimen-
sional temperature distribution can be visualized in
the matter of heat flux (q˙′′), plate length (L) and
thermal conductivity (k).
0 2 4 6 8 10
x0
2
4
6
8
y
(a) x0 = 4m, q˙
′′ = 0.5W/m2
0 2 4 6 8 10
x0
2
4
6
8
y
(b) x0 = 4m, q˙
′′ = 1.0W/m2
Figure 23: Dimensional temperature profile with varia-
tion of the heat flux.
0 2 4 6 8 10
x0
2
4
6
8
y
(a) x0 = 4m,L = 10m
0 2 4 6 8 10
x0
2
4
6
8
y
(b) x0 = 4m,L = 30m
Figure 24: Dimensional temperature profile with the
variation of the length L.
Page 12 of 15
0 2 4 6 8 10
x0
2
4
6
8
y
(a) x0 = 4m, k = 1
W
m.◦C
0 2 4 6 8 10
x0
2
4
6
8
y
(b) x0 = 4m, k = 10
W
m.◦C
Figure 25: Dimensional temperature profile with the
variation of thermal conductivity k.
Increasing the length of the surface and increasing
the heat flux has the same behavior, which is the
increase of the heat transfer in the plate, while the
increase of thermal coefficient imposes a reduction in
the heat transfer in the plate. This can be explained
by the equation 36, since there is a direct relationship
with q˙′′ and L and an indirect relationship with k.
4.5.1 Nusselt Number
The Nusselt Number depends on the Reynolds
Number, the Prandtl Number and also the Modi-
fied Brinkman Number. The relationship with the
Reynolds Number is expressed in equation 60. The
dependence with the others is implicit in θ(0).
Nu = (θ(0))−1
√
Rex (60)
So, for Br* = 1 and Pr = 1,
Nu = 0.284682
√
Rex (61)
And for Br* = 0 and Pr = 1,
Nu = 0.332057
√
Rex (62)
20 40 60 80 100
Rex
2
4
6
8
10
12
Nu
Pr=10,Br=1
Pr=1,Br=10
Pr=50,Br=0
Pr=10,Br=0
Pr=1,Br=0
Pr=1,Br=1
Figure 26: Nusselt Numbers for different Pr.
As illustrated in figure 26, the higher the Prandtl
Number, higher is the variation of the Nusselt Num-
ber with the Reynolds Number with Br* = 0. The
same is not applied for the Modified Brinkman Num-
ber. Especially when comparing the blue and gray
curves, which demonstrated that the increase of Br*,
decreases the variation of Nu.
The variation of Nusselt with Prandtl Number is
described in figure 27.
20 40 60 80 100
Pr
0.5
1.0
1.5
Nu
Rex
Figure 27: Nusselt Numbers for different Pr.
The borderline cases are given in table 4, in which
a low and high Prandtl Numbers are evaluated.
Table 4: Nusselt Number for low and high Pr.
Nusselt Number
Pr = 10−6 Pr = 106
Br* = 0 0.564
√
Pr
√
Rex 0.339
3
√
Pr
√
Rex
Br* = 1 0.447
√
Pr
√
Rex 0.338
3
√
Pr
√
Rex
Page 13 of 15
4.5.2 Convective Heat Transfer Coefficient
As similar to previous cases, the convective heat
transfer coefficient h is studied with terms of the
Reynolds Number (63).
h =
0.284682 k
√
Rex
x
(63)
To complete the estimation of the convective coef-
ficient, the same procedure of comparison between
water and air flows is applied. For the far field veloc-
ity, U∞ is defined 4m/s and also the heat capacity
for both fluids must be considered.
0 2 4 6 8 10
x
1000
2000
3000
4000
5000
h
Figure 28: Variation of the h parameter with x for two
different flows with U∞ = 4m/s: water (red)
and air (black).
5 Summary and conclusions
This work presented the results for the velocity and
temperature profiles in a laminar forced convection
flow over a flat horizontal plate. The investigation
proceed with the use of the similarity theory and
the connection with dimensionless parameters were
defined. Although the calculation for the velocity
profile followed the same consideration, the determi-
nation of the temperature distribution were based
in four different sets
of boundary conditions. For
each set, the temperature analysis was based on the
study of the temperaure boundary layer thickness,
the Nusselt Number and the convective heat transfer
coefficient.
Once the results were established, a comparison
was carried out with a comprehensive literature
search, in order to confirm the obtained results.
With a comparison group of articles, studies and
books, it was identified the reliability of the relations
accomplished in this work.
Nomenclature
Br Brinkman Number
Cf friction coefficient
c heat capacity
Ec Eckert Number
f, g nondimensional functions
h convective heat transfer coefficient
k thermal conductivity
L reference length of the plate in x direction
Nu Nusselt Number
p pressure
q˙′′ heat flux
Re Reynolds Number
T temperature
u, v velocity
x, y spatial coordinates
Greek symbols
α thermal diffusity
δ boundary layer thickness
η similarity variable
φ nondimensional temperature transformation
µ dynamic viscosity
ν kinematic viscosity
ρ fluid density or specific mass
τ shear stress
θ nondimensional temperature
Φ viscous heating rate
Ψ stream function
Superscripts
* modified dimensionless parameter
’ differenciation with respect of η
Subscripts
∞ far field
s surface
max maximum value
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Page 15 of 15
	Introduction
	Velocity Profile Formulation
	Temperature Distribution Analysis
	Isothermal Plate, with Ts =T
	Without viscous dissipation
	Isothermal Plate, with Ts = T
	Adiabatic Plate
	Uniform Heat Flux
	Results and Discussion
	Velocity Profile
	Friction Factor
	Temperature Distribution for Isothermal Plate Ts =T
	Nusselt Number
	Convective Heat Transfer Coefficient
	Without Viscous Dissipation
	Temperature Distribution for Isothermal Plate Ts = T
	Nusselt Number
	Convective Heat Transfer Coefficient
	Adiabatic Plate
	Nusselt Number
	Uniform Heat Flux
	Nusselt Number
	Convective Heat Transfer Coefficient
	Summary and conclusions