One-layer flat wall


The case regards an established flat parallel field in one-layer 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 one-layer flat wall and substitution thermal chain.


Table 1

t1 – temperature to the side of thermal influence upon the wall

t2 – temperature to the cold side of the wall

t0 – temperature of environment

tcp – average value of the temperature in the center of the wall

qc1, qc2  temperature differences at the two sides of the wall

S1, S2 – areas

d - thickness of the wall

q – thermal flow

l - coefficient of  thermal conductivity

Rl , Gl - thermal resistance and thermal conductivity

CT – thermal capacity (it does not influence at an established mode)

One-layer wall and thermal substitution scheme

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 = t1 (for x = d and t = t2) with which for the thermal deviations is obtained qC1 = t1 - t0 and qC2 = t2 - t0. From the dependencies thus expressed for qC1 and qC2 deduced is the equation of the thermal deviation qR = qC1 - qC2 = t1 - t2. Through the wall with thermal resistance Rl passes the thermal flow ql which carries on the quantity of heat Ql. The flow ql is the nominal flow of thermal losses (losses from idle move). According to Fourier’s law the density of the thermal flow q0 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 Rl 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 Rl 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:




l0 – coefficient of thermal conductivity at 273 Ę;

btemperature coefficient;

tcp = 0,5.(t1 + t2) – average temperature of the wall (Ę);


The described dependencies of thermal transfer through one-layer wall liken the principal laws of electrical engineering. The thermal  flow ql can be regarded as electric current I running through electric  resistances R under the influence of voltage U. The thermal equivalents of R and U are the thermal resistance Rl and the thermal tension QR. The thermal capacity CT has the character of electric capacity in electric chain. The electro-technical analogy thus made helps building of substitution schemes and also enables the description of the processes of thermal transfer. Thus it can be used as basis for making the systems of differential equations. In fact, the processes after establishing of the temperature are the same as the electric processes in a chain with identical parameters. The difference is in the transition mode of establishing the temperature when valid is the dependency l = ¦(t). The change of the values for Rl and CT determines the impossibility for usage of linear equations.

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.


Multi-layer flat wall


The process of thermal transfer in a multi-layer wall is regarded where the deduced equations for a one-layer wall are applied to each layer with the relevant conditions. Table 2 shows a thermal scheme for a multi-layer 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 one-layer 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.

Rmn – thermal resistance; m – consecutive layer, n – consecutive resistance in the chain.

Cmn – thermal capacity; m – consecutive layer, n – consecutive capacity in the chain.

Rkln – thermal resistance to the environment, n – consecutive wall.

RklSn – resistance of radiation from the heater  to (n) consecutive wall


Multi-layer flat wall and substitution scheme

Thermal substitution scheme with presentation of the heater and the resistance of radiation to the wall

System of equations depending on the number of the walls

thermal parameters flow ql, tension QR and resistance Rl


The deduction of the equation for ql shows dependency of the flow only on the initial and final temperature. Such an assertion would be well-grounded 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 multi-layer walls numerical solving of differential equations is applied. The basic numerical method used in this project is the Runge-Kutta’s method of fourth and fifth order for solving equations by MatLab program.


One-layer and multi-layer 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 r1(d1) and r2(d2), i.e. that the isotherms are concentric circumferences with decreasing inward surface, and the thermal-power lines develop radially. Table 3 contains figures showing a one-layer and multi-layer cylindrical wall as well as the specific for the case designations.


Table 3

R1 – radius of the internal circumference

R2 – radius of  the external circumference

dr – differential element

One-layer cylindrical wall

Multi-layer cylindrical wall

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 = t1, r = r2, t = t2 obtained is the equation of the thermal flow passing through the cylindrical wall. The dependencies expressing the laws in one-layer and multi-layer cylindrical walls are shown in Table 4:


Table 4

One-layer cylindrical wall

Multi-layer cylindrical wall


Similarly to the regarded case of thermal transfer through a flat wall, the dependency l = ¦(t) remains valid for cylindrical walls, too.




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


t1 – temperature of the wall

t2 = t0 – temperature of the environment

qk – thermal flow of convection

Sk – 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 BC – transition from laminar to turbulent movement.

Section CD – turbulent movement in which dt/dx = 0 and t = ¦(x) = const

Rk – thermal resistance of convection.



Thermal transfer through convection

Designations used


The thermal exchange through convection is described with the Newton’s formula:



  qk – thermal flow

  Fk – area of the surface of the convective thermal exchange

  tcp – temperature of the medium

  tct – temperature of the wall

akcoefficient 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

akcoefficient of thermal transfer through convection.

l – determining size (length of the wall, diameter of the pipe and others)

lcp. - the thermal conductivity of the medium

2. Grasshoff’s criterion

btemperature coefficient of linear expansion of the medium.

nkinematic viscosity of the medium.

g – acceleration in free fall.

Dt = tcp.tct.   temperature drop.

3. Pecle’s criterion

wspeed of movement of the medium.

acp.coefficient of temperature conductivity.

4. Raynold’s criterion – expresses the hydro-dynamic 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











From the description thus made it can be concluded that the exact accounting of the convection when designing electro-technological 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 step-by-step 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 103 < (GrPr)cp < 109 is expressed by:



For a turbulent mode of movement at (GrPr)cp > 109:



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 tcp 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 thermo-isolation) 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:




tct1 and tct2 – are the temperatures of the hot and the cold wall.

Snp – thickness of the interspace.

lekb = ek.lcp– equivalent thermal conductivity of the layer.

ekcorrective coefficient reflecting the influence of the convection:

ek = f(GrPR)cp

At (GrPr)cp < 103 the coefficient of convection ek = 1; in this case the process of thermal exchange is effected only with thermal conductivity. At 103 < (GrPr)cp < 106 for ek:



At 106 < (GrPr)cp < 1010:



Approximately, for all values of GrPr > 103 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 (Recp £ 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 = D2 – D1

Bent pipe


Table 6



Cross streamlining of one base




Radiation – transfer of heat in the invisible (infrared) and visible band of the spectrum. The radiation of thermal energy is in the form of electro-magnetic waves with a wave length of 0,4 – 400 mm.


Table 1


T1, T2 – absolute temperatures of the two surfaces.

Tm – intermediate temperature

Rl1, Rl2 – thermal resistances of the two surfaces.

e1 e2coezfficients of blackness of the two surfaces

S1, S2 – areas


radial thermal exchange

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 one-layer 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 electro-magnetic 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



qn – projection of the vector of radiation of the normal to the surface.

The flow of radiation falling onto the landing dF under angle y1<p and dQpd1=Epd1dF has a positive sign, and under angle y1>p and dQpd2 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 cosy1= – cosy2:



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.



s0 – 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 qR = T1 – T2 = t1 - t2. The radial energy Q0 falling on the body is divided into partially absorbed (QA), partially reflected (QR) and partially passing through the body (QD):


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 C1 and C2: C1 = 3,74.10-16 [W.m2], C2 = 1,44.10-2 [m.K]. The full radiated energy from an absolutely black body qs:



Balance (or black) is called radiation in which all bodies in a certain radiating system have the same temperature, i.e. they are in thermo-dynamic 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 qls. 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 c2; in this case the line can be written:



If the members of a higher rank are ignored the Relay-Jins’s law is obtained:



The second boundary case corresponds to a small difference between the product lT and c2. In this case the Vin’s law is in force:



at a derivative made equal to zero from the last equation is deduced:



lmax – 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 = lmax:



Vin’s law of mixing enables writing of Plank’s law in unsizeable form:



Stephan-Boltzman’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:



c0 = 5,6703 » 5,67 [W/(m2.K4)] – radiative ability of an absolutely black body



C12  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 al is the coefficient of radial thermal transfer:



The total radial thermal resistance Rl consists of two transition contact resistances Rl1 and Rl2, expressed respectively when the flow is coming out of the source and when it goes into the receptor:



Stephan-Boltzman’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 thermo-isolated walls and bodies placed in it. For each of these bodies, in established thermo-dynamic 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.



Iy and In – 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 semi-spherical radiation obtained is:



E – density of integral semi-spherical radiation of absolutely black body, determined along Stephan-Boltzman’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 electric-resistance 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 electro-resistance 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 T1 and  T2 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 al 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 electro-resistance 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

j12 = j21 = 1



Two parallel walls and a body between them whose sizes are much smaller

j12 = j21 = 1

j23 = j13 = 0

j31 = j32 = 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 d1 and d2 with centers located on one axis and at distance h from each other


d1 º d2



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






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

Te, T’’e – temperatures of the two sides of the screen.

Tm, T’’m – temperatures in the space between the screen and each wall.

ee , e’’e – coefficients of blackness of the two sides of the screen

S(F) – areas


radial thermal exchange through the screen

Location of n screens

Designations used



Where C12, C23, ... C(n+1)(n+2) are the reduced coefficients of radiation, and S12, S23, ... 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 (A1,2), 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.








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 electro-thermal 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.

qkl – 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 (Rk) and radiation (Rl).

Sl, Sk   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

Convective – radial thermal transfer through a radiator for heating – thermal chain and substitution scheme

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 akl. This necessitates that the convective and radial thermal exchange be discussed in separate.

According to the electro-thermal analogy the description is reduced to two parallelly connected transitional contact resistances Rl and Rk which can be replaced by one equivalent  resistance Rkl.

The thermal transfer in electro-technological 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: one-layer wall of certain parameters and thermal influence at the one side. Formally, it can be regarded as a three-layer wall in which  t1 » tn and the thermal tension qR = tnt0 is fully known. The equations are deduced by analogy with the multi-layer walls as substitution thermal schemes and the equations are presented in Table 3:


Table 3


In a similar way can be deduced the equations for the complex thermal transfer through a multi-layer flat  and a cylindrical wall – Table 4:


Table 4