HEAT TRANSFER BY CONVECTION
AND CONDUCTION FROM THE FLUID MOVING
AT SOLID WALLS
Associate Professor Ph.D. Amado George STEFAN, Lt.Eng., doctoral student Constantin NISTOR
MILITARY TECHNICAL ACADEMY
Abstract. The paper shows the convective heat transfer in flat plate with similitude Nusselt criterion. The report
 Pr f / Prp 
0,25
takes account of changes in thermal properties of the boundary layer thickness. Thermal
conductivity is the only possible way to transmit heat in solids mass. If fluid transfer occurs only when it is at rest
or when macroscopic flow is laminar them, but in the last Heat Transfertransmite boundary layer of heat and fluid
flow, whether laminar or turbulent character's current. Size a   /    c  [m2/s] is called the temperature
diffusivity or temperature conductivity. The size of the report depends on the speed with which transmits a
temperature variation in substance considered. Heat transfer is modeled shoulders from a central jet of warm air
from outside coaxial jet of cold air.
Keywords: heat transfer, thermodynamics.
1. INTRODUCTION
The study of thermal phenomena using similarity
is very important. The first to use this method for
convection, otherwise known process beforehand
and successfully applied in hydrodynamics was
Nusselt. În this way he introduced order into the
multitude of existing experimental results at that
time in this area. Until then, each experimental
result was only valid for the case investigated by
the experimenter, without any method by which this
result can be generalized to all similar phenomena.
Nusselt applied in this case the most rigorous
way to determine the similarity criteria, based on
the phenomenon of differential equations.
very little on temperature. With very good accuracy
can be considered (Prf/Prp)0,25= 1. In equation (1) is
adopted:
Re  Recr  5 105 , 0, 6  Pr  1, 5 .
2. CONVECTIVE HEAT TRANSFER.
NUSSELT CRITERION. FLAT PLATE
In Figure 1 is the flat plate boundary layer
scheme. With the help of experimental corrections,
where rolling motion similarity relation to the
distance X  x / L , the flat plate is:
 Pr f
0,33
Nu f  0.33  X 0,5  Re0,5
Pr



f
f
 Prp





0,25
,
2/2013
Using relation (1) is calculated following
numerical values Nuf(Ref,X,Pr):
(1)
where the subscript "f" refers to the fluid in motion,
and "p" to the wall. Report (Prf/Prp)0,25 take account
of changes in the thermal properties of the
boundary layer thickness. Number Pr gas depends
TERMOTEHNICA
Fig. 1. Laminar and turbulent boundary layer on flat plate:
δd – dynamic boundary layer thickness; δT – thermal boundarylayer thickness; Tf – temperature of the fluid; Tp – temperature
of the wall.
Ref
1·105
2·105
3·105
4·105
5·105
Nuf(Ref)
Nuf(Ref)
Nuf(Ref)
X = 0.3, Pr = 0.6 X = 0.5, Pr = 0.6 X = 0.7, Pr = 0.6
48.291
62.343
73.765
68.294
88.167
104.32
83.642
107.982
127.765
96.582
124.686
147..531
96.582
139.404
164.945
91
Amado George STEFAN, Constantin NISTOR
Fig. 2. Variation of the numerical values of the number
Nuf(Ref,X,Pr).
Blasius's solution for laminar boundary layer:
Nu x 
x  x
1/3
 0.332  Re0.5
x  Pr , Pr  0.6

(2)
for:
Pr  0,05; Pe = Re  Pr > 100
Nu x 
x  x
 0.565  P e0.5
x

(3)
For any number Pr is valid correlation Churchill
– Ozoe:
Nu x 
1/3
x  x
0,3387  Re0,5
x  Pr

,
1/4

1   0,0468 / Pr 2/3 


Pe x  100 .
(4)
For flat plate turbulent boundary layer
1/3
Nu f  0.0296  Re4/5
x  Pr f , 0,6 < Pr < 60 .
(5)
3. CONVECTIVE HEAT TRANSFER
IN CIRCULAR CYLINDERS
Fig. 3. Circular cylinder turbulent boundary layer and the
variation qualitative factor.
Intrinsic properties of substances are included in
the coefficient of thermal conductivity  . The
value of  depends on the substance and its mode
of presentation and phase. Coefficient  has the
dimension  W m  K  in S.I.
Thermal conductivity is the only possible way to
transmit heat in solids mass. In the case of fluids, it
occurs only when they are in their rest macroscopic
or when the flow is laminar, and in the latter case
only in the transverse direction to the direction of
flow. All heat is transmitted by conduction in the
boundary.
Coefficient of thermal conductivity. For gases
the coefficient of thermal conductivity is a function
of temperature and pressure. We can write:
  f  t, p  .
Temperature dependence is written by the
approximate formula:
   0 · (1  · t ),
Figure 3 shows the flow in a cylinder and the
variation of convection coefficient depending on
boundary layer:
In the case of steady movement
(7)
(8)
4. THERMAL CONDUCTIVITY
where λ0 depends on the material and temperature.
This conductivity is a linear function of the specific
heat of the gas at constant volume and viscosity of
the gas flow.
Pressure dependence of the thermal conductivity
is very similar to the pressure dependence of the
specific heat.
The amount λ0 of the various gases of the technical at various temperatures is shown in Table 1.
The heat transfer phenomena are functions of
time. On the other hand, the heat transfer is
irreversible phenomenon, which occurs because the
temperature difference can never be infinitely small,
such as thermodynamic reversibility require.
Metals and alloys. The metals are the best of all
heat conductive materials, in particular in the pure
state. In the case of high specific conductivity of
metals (Ag, Cu, Au, Al etc.), even lower than the
conductivity of the traces of impurities.
Nu f 
92
0.021  Re0f ,8
 Pr 0f ,43
 Pr f

 Prp





0 ,25
.
(6)
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HEAT TRANSFER BY CONVECTION AND CONDUCTION FROM THE FLUID MOVING AT SOLID WALLS
Table 1
The values of 103·λ0 for gas, in [W/m·K]
Gas
H2
He
NH3
(H2O)C
O
N2
O2
Aer
CO2
CH4
–50
146
–
13,6
–
17,8
19,7
19,7
19,7
9,8
15,5
0
267
141
17,4
–
21,7
23,6
23,6
23,6
14,1
24
100
215
167
24,8
23,3
–
30,6
30,6
30,6
19,5
–
Temperature, °C
150
200
250
250
272
254
–
–
–
29,0
–
–
26,6
25,0
33,2
–
–
–
33,8
31,8
39,9
33,8
31,8
39,9
33,8
31,8
39,9
21,1
26,7
–
–
–
–
300
316
–
–
36,6
–
42,8
42,8
42,8
–
–
400
357
–
–
43,2
–
48,4
48,4
48,4
–
–
500
397
–
–
49,8
–
53,8
53,8
53,8
–
–
Table 2
The values of λ0 for gas, in [W/m·K]
Pure gold
Pure nickel
Technical nickel
Bronze (86 Cu, 7 Zn, Sn6)
Pure iron
Technical metals
Iron - Ni (5% Ni)
310
93
69,7
60,4
73,2
46,4… 58
34,8
5. THE GENERAL EQUATION OF
CONDUCTIVITY
2
Using the Laplace operator  and noting

, we can write:
a
c
t
q
 a   2t 

c
(10)
Equation (10) includes the case of stationary,
and if it is considered that this phenomenon occurs
in one direction x, we can write:
d 2t
0
dx 2
(11)
in which by integration gives:
t
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Q
 x  t1
  A
2/2013
18,6
11
14,5
32,5
22
82,4
159
where t1 is the highest temperature of the substance.
From equation (12) it can be seen that the
temperature of a flat wall varies linearly with x
(Fig. 4). The higher value, the difference between t1
and t is less than the same x.
(8)
this is the general differential equation of thermal
conductivity Fourier.

[m2/s] is called the diffusivity of
Size a 
c
temperature and conductivity on temperature. The
size depends on the speed with which this report is
transmitted through the substance temperature
variation considered.
If q = 0 there is considered that the body heat,
Fourier's equation is:
t
 a   2t

Iron - Ni (20% Ni)
Iron - Ni (35% Ni)
Iron - Ni (50% Ni)
Iron - Ni (80% Ni)
Monel metal (29 Cu, 67 Ni, two Fe)
Bronze (95%)
Dural (94.5% Al, 4 Cu, 0.5 mg)
(12)
Fig. 4. Variation of the thickness of a flat wall temperature,
for different values of λ.
6. THERMAL CONDUCTIVITY THROUGH
THE TUBE WALLS
1) Simple walled tubes. Of particular interest
for special presents, in addition to conduction
through plane wall, conduction through the tube
wall or simple compounds. Most of the time, the
technique is considered stationary phenomena, that
in steady state operation, without even periodic
variations in temperature.
If the tube has a constant cross-section length L,
then by a cylindrical layer of thickness dr in which
93
Amado George STEFAN, Constantin NISTOR
there is loss of temperature dt, the heat is transmitted
dt
(13)
W    2  r  L  .
dr
When transmitting stationary heat and if the
coefficient of thermal conductivity of the wall may
be admitted to the temperature constant, then
equation (13) is a differential equation ternperatura
loss depending on the radius of the tube. By separation of variables and the temperature variation
is obtained by integration along the radius of the
tube (inside wall):
W
 ln  r   C ,
  2  L
Integration constant is determined by the initial
and final temperatures conditions:
t 
ti  te 
r 
W
 ln  e 
  2  L
 ri 
7. CASE STUDY
Numerical modeling heat transfer in a central jet
hot air from external coaxial jet of cold air. The
numerical determination are made with Fluent
software.
Consider the case of a central air jet flow rate of
60 kg/s and a temperature of 1053 K flowing
through a pipe with an internal diameter of 880 mm
and a length of 1000 mm, pipe wall thickness is
2 mm.
The first duct is coaxial with a second pipe
internal diameter of 984 mm. Between the two
ducts is the second stream of air, at 6 kg/s flow rate
and a temperature of 300 K (Fig. 5).
(14)
the index e relates to the outside of the tube and i in
the interior.
Resistance to conduction, in the case of single
wall pipe, we can write:
r 
ln  e 
r
Rc   i  .
  2  L
(15)
2) Multi-walled tubes. In the case of a tube
wall composed of several layers of different materials, the total resistance is the sum of the partial
resistances:
r 
r 
r 
ln  1 
ln  e 
ln  2 
 ri  
 r1   ... 
 rn 
Rct   Rci 
1  2  L  2  2  L
 n 1  2  L
i 1
(16)
n 1
Fig. 5. The fluid-solid domain and boundary conditions.
For the mesh it use rectangles cell, with 10 mm
lengtht along the axis of revolution and 0.2 mm in
radial direction for the wall, and for the fluid
domain a variable dimension in radial direction,
from 0.1 mm in the vicinity of the wall with a
increase in geometrical progression with the ratio of
1.2 (Figure 6).
and the heat flow is:
W
ti  te
.
Rct
(17)
In all the above calculations it was assumed that
ti > te. But if ti < te, it changes the meaning of heat
transmission, since the flow calculations is always
positive. The temperature of the layer n and n + 1 is
determined by the relationship:
tn  ti 
 r1  1
r 
W 1
 ln  2   ... 
  ln   
2  L  1
 r1 
 ri   2
 r 
1
  ln  n  
n
 rn 1  
94
Fig. 6. Detail of the mesh in the walls area.
.(18)
Turbulence model chosen is k–ε the standard
wall position. At the inlet sections the turbulent
flow intensity is considered to be 10%. The air is
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HEAT TRANSFER BY CONVECTION AND CONDUCTION FROM THE FLUID MOVING AT SOLID WALLS
considered compressible, with variable specific
heat with temperature according to the law:
C p T   1161.482  2.368819  T 
0.01485511  T 2  5.034909  10 5  T 3 
9.928569  10 8  T 4 
1.111097  10 10  T 5  6.540196  10 14  T 6 
1.573588  10 17  T 7
9. CONCLUSIONS
Figures 7 and 8 present laws of variation for
velocity and temperature at the exit sections for the
hot and cold stream.
Figure 9 shows the temperature field in the
vicinity of the wall between the two flows in the
output. It is noted heat dissipation of the stream of
cold air downstream of the input section with the
decrease in temperature in the central stream of hot
air.
In figure 10 is presented the variation of
convection coefficient with axial distance (along
the wall).
Fig. 8. Velocity and temperature profile at
the cold air outlet section.
Fig. 9. Distribution of temperature (K) in the output
(detail of the partition).
Fig.7 Velocity and temperature profile at
the hot air outlet section.
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Fig. 10. Variation of convection coefficient.
95
Amado George STEFAN, Constantin NISTOR
BIBLIOGRAPHY
[1] Dobrovicescu, A., and others, Fundamentals of Technical
Thermodynamics. Elements of technical thermodynamics,
vol. I, Politehnica Publishing House, 2009.
[2] Leonăchescu, N., Thermodynamics, Didactic and Pedagogic Publishing House, Bucharest, 1985.
[3] Marinescu, M., Chisacof, A., Raducabu, P., Motoroga, A., O.
Basics of Technical Thermodynamics. Heat and Mass Transfer. Fundamental Processes, vol. II, Politehnica Publishing
House, 2009.
96
[4] Nistor, C., Heat transfer by conduction. Conduction
analysis of 1D, 2D and 3D scientific research report,
Military Technical Acacemia, Bucharest, 2010.
[5] Nistor, C., Heat transfer by convection. Research report,
Military Technical Acacemia, Bucharest, 2011.
[6] Olariu, V., Brătianu, C., Numerical finite element modeling,
Technical Publishing House, Bucharest, 1986.
[7] Ştefan A., G., Termohidrodinamice phenomena analysis by
finite element method, Ed Mirton, 2003
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