THERMAL TRANSFER THROUGH THERMAL CONDUCTIVITY
Onelayer flat wall
The
case regards an established flat parallel field in onelayer flat wall
generated by a thermal source located in its vicinity. The field is
geometrically determined by the dimensions of the wall – internal surface S1,
external S2 and thickness d, where it is
assumed as accepted that the areas of the two surfaces many times exceed d. Figure 1
(Table 1) illustrates schematically the thermal transfer through onelayer flat
wall and substitution thermal chain.
Table 1

t_{1}
– temperature to the side of thermal influence upon the wall t_{2}
– temperature to the cold side of the wall t_{0}
– temperature of environment t_{cp}
– average value of the temperature in the center of the wall q_{c1},
q_{c2}
– temperature differences at the two sides
of the wall S_{1}, S_{2} – areas d 
thickness of the wall q – thermal flow l 
coefficient of thermal conductivity R_{l}
, G_{l}
 thermal resistance and thermal conductivity C_{T} – thermal capacity (it does not influence at an
established mode) 
Designations
used in the graph 
According
to the regarded case of established mode, the field is described in time with
the equation:
The
limit conditions are: x = 0, t = t_{1} (for x = d and t = t_{2}) with which
for the thermal deviations is obtained q_{C1} = t_{1}  t_{0} and q_{C2} = t_{2}  t_{0}. From the
dependencies thus expressed for q_{C1} and q_{C2} deduced is the equation
of the thermal deviation q_{R} = q_{C1}  q_{C2} = t_{1}  t_{2}. Through the
wall with thermal resistance R_{l} passes the
thermal flow q_{l} which carries on the
quantity of heat Q_{l}. The flow q_{l} is the nominal flow of thermal losses (losses
from idle move). According to Fourier’s law the density of the thermal flow q_{0}
is proportionate to the temperature gradient dt/dx according to the equation:
The sign minus shows that the flow is against the direction
of the field. Solving the latter equation of the placed limiting
conditions leads to its transforming in the following form:
For
the total thermal flow when determining S – median surface of the wall – used is the
criterion:
is obtained:
The
thermal resistance R_{l} depends on
the geometric sizes of the wall and on the coefficient of thermal conductivity
for the chosen material. Since l depends on
the temperature, then R_{l} too has a
variable character. In most part of the literature on the discussed question
about the accounting of the functional dependency l = ¦(t) the following approach
is recommended – the coefficient is chosen as a constant from the formula in
accordance with the average temperature for the wall or the process:
Where:
l_{0} –
coefficient of thermal conductivity at 273 Ê;
b – temperature
coefficient;
t_{cp} = 0,5.(t_{1} + t_{2}) – average temperature
of the wall (Ê);
The
described dependencies of thermal transfer through onelayer wall liken the
principal laws of electrical engineering. The thermal flow q_{l} can be
regarded as electric current I running through electric resistances R under the influence of
The
accounting of the momentary temperature in the transition process of heating
enables usage of temperature dependency of each value when making a
mathematical model of the process and its realizing as computer simulation. The
established mode in this case is obtained as final value from solving the
system of DE with variable coefficients (l = ¦(t) and C = ¦(t)). The procedure of
making and solving the thus obtained equations is presented with the analysis
of the transition processes.
Multilayer flat wall
The
process of thermal transfer in a multilayer wall is regarded where the deduced
equations for a onelayer wall are applied to each layer with the relevant
conditions. Table 2 shows a thermal scheme for a multilayer wall, a
substitution scheme, and a system of equations describing the process of
thermal transfer. The accepted designations are the same as these used in
onelayer wall.
Table 2

In the graph the designations are according to these used
in the files of the system of differential equations used in Matlab. R_{mn}_{ }– thermal
resistance; m – consecutive layer, n – consecutive resistance in the chain. C_{mn} – thermal
capacity; m – consecutive layer, n – consecutive capacity in the chain. R_{kln} – thermal
resistance to the environment, n – consecutive wall. RklSn –
resistance of radiation from the heater
to (n) consecutive wall 


System of
equations depending on the number of the walls 
thermal parameters
flow q_{l}, tension Q_{R}
and resistance R_{l} 
The
deduction of the equation for q_{l} shows
dependency of the flow only on the initial and final temperature. Such an
assertion would be wellgrounded only if the coefficient of thermal
conductivity is accepted as a constant value. As it has been noted l = ¦(t) which makes the thermal flow dependent of the temperature
of the intermediate layers. So as to obtain the thermal picture in multilayer
walls numerical solving of differential equations is applied. The basic
numerical method used in this project is the RungeKutta’s method
of fourth and fifth order for solving equations by MatLab program.
Onelayer and
multilayer cylindrical wall
The
process of thermal transfer in the walls with cylindrical form does not
principally differ from this in flat walls the influence only being exerted by
the geometric form. When analyzing the wall we assume that the axial size l
many times exceeds the radial sizes r_{1}(d_{1})
and r_{2}(d_{2}), i.e. that the isotherms are concentric
circumferences with decreasing inward surface, and the thermalpower lines
develop radially. Table 3 contains figures showing a
onelayer and multilayer cylindrical wall as well as the specific for the case
designations.
Table 3


R_{1}
– radius of the internal circumference R_{2}
– radius of the external circumference dr – differential element 
Designations
used 
For
the differential element dr of the wall located at
distance r from the center of the circumference, the following expression of
the thermal flow can be deduced:
When
solving the last equation for initial conditions r = r, t = t_{1}, r = r_{2},
t = t_{2} obtained is
the equation of the thermal flow passing through the cylindrical wall. The
dependencies expressing the laws in onelayer and multilayer cylindrical walls
are shown in Table 4:
Table 4


Onelayer
cylindrical wall 
Multilayer
cylindrical wall 
Similarly
to the regarded case of thermal transfer through a flat wall, the dependency l = ¦(t) remains valid for cylindrical walls, too.
THERMAL TRANSFER THROUGH
CONVECTION
Convection
is thermal transfer in liquid or gas in which the individual components,
particles and separate elements within the volume of the substance, transfer
their relevant supply of thermal energy through their movement. In the process
of convective thermal exchange, exchange of thermal energy and mass is done at
the same time. Depending on the way of its initiation it can be natural or
forced. Figure 1 (table 1) presents a graph of convective thermal exchange on the
surface of a heated vertical wall and the substitution thermal scheme.
Table 1

t_{1}
– temperature of the wall t_{2}
= t_{0}
– temperature of the environment q_{k} – thermal
flow of convection S_{k} – the surface on which the thermal transfer through
convection takes place. Thermal section AB – laminar movement of particles with
temperature gradient dt/dx = const. Section CD – turbulent movement in which dt/dx = 0 and t
= ¦(x)
= const R_{k} – thermal
resistance of convection. 
Designations
used 
The
thermal exchange through convection is described with the
q_{k}
– thermal flow
F_{k} – area
of the surface of the convective thermal exchange
t_{cp} – temperature of the medium
t_{ct} – temperature of the wall
a_{k}
– coefficient of thermal transfer in convection
The
complex nature of the regarded case of thermal transfer necessitates its description
through a system of equations: thermal transfer, movement of the medium,
uniformity and homogeneity of the medium. The analytical solving of these
equations presents great difficulties due to which the design of thermal
exchanges in concrete conditions is carried out on the basis of experimental
results. By the help of differential equations the convective thermal exchange
is reduced to a few criteria according to which different coefficients of
computation of the process are determined (Table 2).
Table 2
1. Nusselt’s
criterion 

a_{k} – coefficient
of thermal transfer through convection. l
– determining size (length of the wall, diameter of the pipe and others) l_{cp.}  the thermal
conductivity of the medium 
2. Grasshoff’s
criterion 

b – temperature
coefficient of linear expansion of the medium. n – kinematic
viscosity of the medium. g
– acceleration in free fall. Dt = t_{cp.} – t_{ct.} – temperature drop. 
3. Pecle’s
criterion 

w – speed
of movement of the medium. a_{cp}_{.} – coefficient of temperature conductivity. 
4. Raynold’s
criterion – expresses the hydrodynamic similarity of
the processes 


5. Prandtl’s criterion
– reflects the physical properties of the medium 


The
process of thermal transfer through convection consistent to the described
criteria and investigated with the help of Matlab
program is shown here. The file presents description of the cooling of a heated
body through convection and radiation to the environment.
In
natural convection:
In
gases with identical atomness (the air in particular)
i.e. with Pr = idem:
The Prandtl’s number for most gases
changes a little depending on the change of the temperature. In making further
estimates of the convective thermal exchange the following data can be used
(Table 3):
Table 3
Atomness of the gas 
1 
2 
3 
4 
Pr 
0,67 
0,72 
0,8 
1 
From
the description thus made it can be concluded that the exact accounting of the
convection when designing electrotechnological devices relates to solving of
some problems. Exact data are needed about the condition of the medium which
are hard to collect in view of its changeable character. The substantial
quantity of equations leads to additional loading of the computing process. To
solve these problems when designing resistance furnaces data gathered in
practical investigations are sometimes used. In this project the convective
thermal exchange of the walls of the furnace to the environment is analyzed
through stepbystep approximation as shown in the table here.
Convective thermal exchange in natural
convection
The
thermal exchange between bodies with vertical cylindrical or flat surfaces for
any medium with a laminar mode of movement, within the range of 10^{3}
< (GrPr)_{cp} < 10^{9} is expressed by:
For
a turbulent mode of movement at (GrPr)_{cp} > 10^{9}:
The
height of the surface h is pointed at
in its capacity of a determining size, and in the capacity of a determining
temperature at which the physical properties of the medium are selected and the
values of Gr_{ cp} and Pr_{ cp} calculated is chosen the
temperature of the medium t_{cp}
at a
distance from the wall.
The
convective thermal exchange depends on the geometric parameters of the concrete
medium. In thermal release of “a horizontal pipe” in conditions of a free movement
of the medium used is the dependency:
As
a determining size is taken the diameter of the pipe d.
The
thermal exchange in the inside of a closed volume (between the layers of
thermoisolation) characterizes with circulation of gas in medium. The process
depends on the character of the medium, the temperature, the difference between
the temperatures of the walls and their location to each other. The description
of the thermal exchange of a homogenous flat wall is given by:
Where:
t_{ct1} and t_{ct2} – are the
temperatures of the hot and the cold wall.
S_{np} – thickness of the interspace.
l_{ekb}
= e_{k}.l_{cp}–
equivalent thermal conductivity of the layer.
e_{k}
– corrective coefficient reflecting the influence of
the convection:
e_{k}
= f(G_{r}P_{R})_{cp}
At (G_{r}P_{r})_{cp} < 10^{3} the coefficient of
convection e_{k}
= 1; in this case the process of thermal exchange is effected only with thermal
conductivity. At 10^{3} < (G_{r}P_{r})_{cp} < 10^{6} for e_{k}:
At
10^{6} < (G_{r}P_{r})_{cp} < 10^{10}:
Approximately,
for all values of G_{r}P_{r} > 10^{3}
can be assumed:
Convective thermal exchange in forced convection
The
convective thermal exchange in forced movement of the medium is regarded
depending on the physical and geometric properties of the medium. This
necessitates discussion of a few private cases:
Laminar movement of the
medium (Re_{cp} £ 2300) –
Table 4:
Table 4



1. Lamina Criterion Nu depending on criterion Pr 
2. Round
straight canal (pipe) with
length l and diameter D criterion Nu depending on criterion Pe 
3. Flat slot
with thickness d and length
l – depending on
criterion Pe 
Turbulent
movement of the medium (Re > 5000) – Table 5 and Table 6:
Table 5



Straight
pipe with thickness D and length l: 
Cylindrical
slot with sizes d^{’} = D_{2}
– D_{1} 
Bent pipe 
Table 6


Lamina 
Cross
streamlining of one base 
LAWS OF THERMAL TRANSFER –
RADIATION
Radiation
– transfer of heat in the invisible (infrared) and visible band of the
spectrum. The
radiation of thermal energy is in the form of electromagnetic waves with a
wave length of 0,4 – 400 mm.
Table 1




T_{1}, T_{2} – absolute temperatures of the two
surfaces. T_{m} – intermediate temperature R_{l1}, R_{l2} – thermal resistances of the two
surfaces. e_{1}
e_{2}
– coezfficients of blackness of the two surfaces S_{1}, S_{2} – areas 
Thermal chain presenting the transition resistances of
radiation of the two surfaces 
Thermal chain presenting the accounting the intermediate
temperature with a measurement body 
Substitution
scheme presenting the resistance to radiation toward a onelayer wall by a
heater 
Designations
used 
The thermal transfer through radiation in the strict sense of the
concept is not thermal but electromagnetic transfer. In the source the thermal energy
is transformed in electromagnetic which is
transferred to the receptor where it turns into thermal again. Due to this the
transferred radial flow is neither “hot” nor “cold” i.e. is not thermal “flow”.
The laws to which the radial thermal transfer submits are very different from the
laws of thermal transfer through convection and thermal conductivity.
The vector of radiation determines the direction of transfer of
the radial thermal transfer in the spot of the most intensive thermal exchange
in the field. Numerically it equals the flow of the resultant radiation,
transferred in a unit of time through a unit of surface orthogonal to a random
direction of a transfer of radiation.
The elementary flow passing through the landing
dF is expressed through the
scalar product of the vector of radiation _{} of _{}
q_{n}
– projection of the vector of radiation of the normal to the surface.
The flow of radiation falling onto the landing dF under angle y_{1}<p and dQ_{pd1}=E_{pd1}dF
has a positive sign, and under angle y_{1}>p and dQ_{pd2}
it has a negative sign. A flow equal to the resultant one passes through the
landing dF:
To obtain the flow of the resultant radiation with accounting of cosy_{1}=
– cosy_{2}:
The full flow of radiation is comprised of the compounds along the
three coordinate axes Ox, Oy, Oz by which the vector
can be defined as integral of the intensiveness of radiation by spherical
angles:
The latter dependency is integral form of the vector of radiation;
apart from it gradient form can also be expressed. The latter is used mainly in
the cases of transfer of discrete particles – photons.
s_{0}
– Stephan – Boltzman’s constant
Thermal
transfer through radiation between parallel walls.
If the temperature T1 of wall 1 is higher than wall 2 there is
thermal tension q_{R}
= T_{1} – T_{2} = t_{1}
 t_{2}.
The radial energy Q_{0} falling on the body is divided into partially
absorbed (Q_{A}), partially reflected (Q_{R}) and partially
passing through the body (Q_{D}):
Table 2





A –
absorbed energy 
R – the
reflective ability of the body 
D – the
ability of the body to let the radial flow through 
At A = D = 0 and R = 1, the body reflects the thermal rays thoroughly,
i.e. it is an absolutely white body. For the absolutely black bodies which
absorb the thermal energy totally, it is known: R = D = 0 and A = 1. With the
real bodies it is always A < 1 and R < 1. The
spectral density (or the spectral intensiveness) for the absolutely black body
with a wave length l
(mm) and temperature T (K) is described by the Plank’s law:
For the constants C_{1} and C_{2}: C_{1} =
3,74.10^{16} [W.m^{2}], C_{2}
= 1,44.10^{2} [m.K]. The full radiated
energy from an absolutely black body q_{s}:
Balance (or black) is called radiation in which all bodies in a
certain radiating system have the same temperature, i.e. they are in
thermodynamic balance. Therefore the thermal radiation has dynamic character
due to which bodies let out or absorb separate quantities of heat.
According to the Plank’s law, to any wave length there is
correspondent value q_{l}_{s}.
The density of the radiated flow characterizes with separate isotherms passing
through the maximum. At l ® 0 and l ® µ is inclined
to zero.
The Plank’s law has two boundary cases. One of them manifests when
the product lT
is much larger than the constant c_{2};
in this case the line can be written:
If the members of a higher rank are ignored the RelayJins’s law is obtained:
The second boundary case corresponds to a small difference between
the product lT
and c_{2}. In this case the Vin’s law is in force:
at a derivative made equal to zero
from the last equation is deduced:
l_{max} –
wave length corresponding to the maximal density of the radiation. The last
dependency expresses Vin’s
law of mixing.
The maximal density of the flow of radiation from a black body can
be found out from the Plank’s by assuming l = l_{max}:
Vin’s
law of mixing enables writing of Plank’s law in unsizeable
form:
StephanBoltzman’s law defines the
dependency between the density of the flow and the temperature which can be
obtained from the Plank’s law:
For convenience, in practical estimates the following equations
are used:
c_{0}
= 5,6703 »
5,67 [W/(m^{2}.K^{4})] – radiative
ability of an absolutely black body
C_{12 } is called
coefficient of mutual irradiation:
In the latter formula e
and a are
the coefficients of absolute radiation (or absorption of rays), for an
absolutely black body e
= a
= 1. The coefficient a_{l}
is the coefficient of radial thermal transfer:
The total radial thermal resistance R_{l}
consists of two transition contact resistances R_{l1} and R_{l2},
expressed respectively when the flow is coming out of the source and when it
goes into the receptor:
StephanBoltzman’s law which
characterizes the radial thermal exchange in vacuum does not depend on the
coordinates or direction of the vector. The volume density of radiation is
found from:
The same law is also applicable for grey bodies. The fact should
be accounted that with grey bodies, just as with black, the own radiation is
proportionate to the absolute temperature at fourth power but the energy of
radiation is smaller than that of the black bodies at the same temperature.

1
black body 2
grey body 3
selective radiation 
Density
of the flow of radiation depending on the wave length 
Designations
used 
For grey bodies the law obtains the following form:
Kirhoff’s
law establishes the quantitative link between the energy of radiation and
absorption, between the surfaces of grey and absolutely black body. This law
can be obtained form the balance of the radial energy for a radiating system,
consisting of a relatively large closed volume with thermoisolated walls and
bodies placed in it. For each of these bodies, in established thermodynamic
balance, the energy of radiation equals to the absorbed energy:
According to this law, the ratio of the energy of radiation to the
energy of absorption does not depend on the nature of the bodies and equals to
the energy of radiation of an absolutely black body at the same temperature.
The energy, radiated from a body in certain direction is
established by Lambert’s law according to which the flow radiated from an
absolutely black body is proportionate to the flow radiated in direction of the
normal toward the surface and the cosinus of the
angle between them.
I_{y}
and I_{n} – density of the flow of the
integral radiation, respectively in the direction determined by angle y and in
direction along the normal toward the surface.
Lambert’s law expresses an important consequence related to the
brightness of radiation of absolutely black body:
Therefore if the radiation submits to Lambert’s law the brightness
does not depend on the direction i.e. it appears a constant value:
The
angle dw
is defined as
dF–elementary landing, expressing the body angle
of the surface of the sphere with the radius r. This landing can be presented
by two elements rdy
and rsinydw, and by
this:
After the last equations of flow density of semispherical
radiation obtained is:
E – density of integral semispherical
radiation of absolutely black body, determined along StephanBoltzman’s law, I – brightness of radiation. In the final
form the law of the cosinuses or the law of the
direction of Lambert is obtained:
The surface density of the own radiation of a random body is
determined by:
The purpose of the regarded laws of thermal transfer is to deduce
the key dependencies by which the mathematical model of operation of
electricresistance furnaces is made. The model is based on a system of
differential equations whose numerical solution presents the development of the
thermal picture in the transition process of establishing of the temperature in
the furnace. The radiation is the principal thermal transfer between the heater
and the heated body in the chamber of the electroresistance furnace which
determines the key role of its correct description. The equations, describing
the process of thermal transfer through radiation in the given form are not
convenient for operation with application of numerical methods and their
realization through mathematical software (in this case Matlab
program). The reason for this is that in the beginning of the process, for the
coefficient of radial thermal transfer a it is possible to obtain a value which cannot
be determined, which would lead to suspension of the computing process. In case
that the temperatures T_{1} and T_{2} are assigned at zero
degree, in the nominator of the expression zero is obtained. In order to avoid
division to zero certain transformations are needed:
Deduced in this form, the equations can be written according to
the Matlab syntax. The record given above of the
coefficient a_{l}
and the resistance of radiation is used everywhere further on in this project being
updated according to the concrete requirements of the substitution scheme.
Private
cases of thermal transfer through radiation.
The
analysis done so far describes the process of thermal transfer through
radiation between parallel bodies which can be regarded as the most trivial in
this case. Some private cases present interest; these which
find application in making the mathematical model of operation of the
electroresistance furnaces.
Table 3
The smaller surface has not recesses 


The
smaller surface has recesses 


Two
parallel walls whose sizes are much larger in comparison with the distance
between them 
j_{12} =
j_{21} =
1 

Two
parallel walls and a body between them whose sizes are much smaller 
j_{12} = j_{21} = 1 j_{23} = j_{13} = 0 j_{31} = j_{32} = 0,5 

Three
surfaces forming a closed system 


Four
surfaces forming a closed system 


Two
endless parallel flat surfaces of equal width and at a distance from each
other 


Two
parallel circles with diameters d_{1} and d_{2} with centers
located on one axis and at distance h from each other 
d_{1}
º d_{2} 

Two
equal rectangulars (a X b) in parallel surfaces at
distance h from each other 


Two
mutually perpendicular rectangulars with common
side 


Two
parallel cylinders with one diameter d 


Unlimited
surface and cylinders placed in one line 

REFLECTORS
When a thermal screen between two bodies with
different temperatures is available, the radial flow between them changes
depending on the number of the screen, the coefficient of radial emission, the
materials of the bodies and the area of the surfaces.
Table 4


T^{’}_{e},
T^{’’}_{e} – temperatures of the
two sides of the screen. T^{’}_{m},
T^{’’}_{m} – temperatures in the
space between the screen and each wall. e^{’}_{e}
, e^{’’}_{e}
– coefficients of blackness of the two sides of the screen S(F) – areas 
Designations
used 
Where C_{12}, C_{23,} ...
C_{(n+1)(n+2)} are the reduced coefficients of radiation, and S_{12},
S_{23,} ... S_{(n+1)(n+2)} are the reduced areas. The
temperatures of the screens are obtained from the equations:
When accounting the coefficient of thermal absorption between the
two bodies (A_{1,2}), for a random number of
screens the following system is made:
In
the private case of equal coefficients:
For
the reduced ability of radial emission:
The
temperature of the individual screens is obtained from the system of equations:
Where q_{(}_{1,2)}_{e} is each
obtained value coming next.
MIXED AND COMPLEX THERMAL EXCHANGE
Despite
their different nature, in most cases the radial and convective thermal exchange manifest jointly and cannot be regarded in
separate. This necessitates, in order to achieve a full description of the
cases of thermal exchange in electrothermal devices, that a formal analogy be
sought in analyzing the
two types of thermal transfer. So as to facilitate the analysis
it is accepted that the surfaces of convective and radial thermal exchange
coincide. The coefficients of the two processes are regarded as common
coefficient of mixed thermal transfer.
Table 1


The
substitution thermal scheme of the complex thermal exchange is similar to
this presented when discussing the convective thermal exchange; the accepted designations
for points A,B,C,D are the same. q_{kl} – thermal flow of
convection and radiation The
thermal scheme of convective – radial thermal transfer through a radiator for
heating presents two parallelly connected thermal
resistances of convection (R_{k}) and
radiation (R_{l}). S_{l}, S_{k }– surfaces of
radial and convective thermal exchange. Rkl. – transitional contact (external) thermal
resistance. Gkl. – thermal conductivity for convective –
radial thermal transfer. 
Convective – radial thermal transfer through a vertical
wall 
Designations
used 
With these assumptions for the thermal flow
the following equation can be written:
For the cases
when it cannot be accepted that the surfaces of the two processes coincide
there is no way to determine a common coefficient a_{kl}. This necessitates
that the convective and radial thermal exchange be discussed in separate.
According to the electrothermal analogy
the description is reduced to two parallelly
connected transitional contact resistances R_{l}
and R_{k} which can be replaced by one equivalent resistance
R_{kl}.
The
thermal transfer in electrotechnological devices is always complex i.e.
manifested are the three types of thermal exchange, which makes the forming of
equations a complex task too. In order to facilitate the analysis it is
convenient to discuss a private case: onelayer wall of certain parameters and
thermal influence at the one side. Formally, it can be regarded as a
threelayer wall in which
t_{1} » t_{n} and the
thermal tension q_{R} = t_{n} – t_{0} is fully
known. The equations are deduced by analogy with the multilayer walls as
substitution thermal schemes and the equations are presented in Table 3:
Table 3


Onelayer
flat wall: Onelayer
cylindrical wall: 
Complex thermal transfer through a onelayer cylindrical
wall 
Equations 
In
a similar way can be deduced the equations for the complex thermal transfer
through a multilayer flat
and a cylindrical wall – Table 4:
Table 4


Multilayer
flat wall: Multilayer
cylindrical wall: 
Complex thermal transfer through a multilayer cylindrical
wall 
Equations 