4.Energy Saving Technologies: in public and private sector

Anastasija Zhiravecka, associate professor

Riga Technical University

4.1.Building: fundamental physical processes in buildings and building envelopes. Reduction of heat losses. Heating and conditioning. Heat pumps.

Short introduction

In today's world, energy costs for maintenance and operation of buildings accounts for a significant share of total energy consumption. In the total operating costs of energy in buildings more than half is spent for heating, more than 10 % – for lighting of nearly a quarter – for hot water supply and other purposes. Insufficient resistance of the building envelopes to a heat transfer can be a source of high heat loss.

The building envelopes are intended to create the required temperature and humidity conditions in residential, public and industrial buildings. The temperature regime of spaces is crucial at operation of buildings. It determines the feeling of thermal comfort of people, the normal flow of production processes, the state and durability of the building structures and its equipment. The main processes taking place in the envelopes are heat transfer processes, moisture transport and air filtration.

The task of providing the necessary thermal properties of the outer envelopes is solved to give them the required thermal stability and heat resistance.

Thermal resistance should be large enough to ensure the allowable temperature conditions in the room in the coldest period of the year. Thermal stability of structures is related to their ability to maintain a relatively constant temperature in rooms under periodic oscillations of the outdoor air temperature. The thermal stability of the whole structure is largely determined by the physical properties of the material of its outer layer, which senses temperature oscillations.

Spontaneous irreversible processes of heat propagation in space are referred to as heat exchange. Internal energy exchange between the individual elements and between the regions of the considered medium is understood as the process of the heat propagation. The heat transfer is carried out in three main ways: conduction, convection and thermal radiation.

Heat conduction is the process of heat transfer by atoms, molecules, electrons, phonons, which is determined by the presence of a temperature gradient in the volume under consideration.

Convection of heat is the process of its transfer by the moving of the fluid or gas volume in the space between the regions with different temperatures. In this case, the heat transfer is inseparably linked with the transfer of the medium volume itself.

Heat radiation is the heat propagation by electromagnetic waves. It is associated with the temperature and the optical properties of the emitting body. The internal energy of the body (medium) goes into the energy of the radiation. The processes of transformation of the internal energy of matter into the radiation energy, radiation transfer and absorption are called as the radiation heat transfer.

In the nature and technology the elementary processes of heat distribution (heat conduction, convection and thermal radiation) often occur together.

Thermal conduction in pure form usually occurs only in solids.

Heat convection is always accompanied by heat conduction. Joint process of the heat transfer by convection and heat conduction is called convective heat exchange.

In engineering calculations one study frequently a convective heat transfer between the fluid flow and a solid surface; this process of convective heat exchange is called convective heat transfer.

Heat conduction and convection may be accompanied by the heat transfer by radiation. Heat transfer caused by radiation heat transfer and thermal conductivity jointly is called radioactive-conductive heat exchange. If the heat transfer is carried out additionally by convection, such a process is called a radioactive-convective heat transfer. Sometimes radioactive-conductive and radioactive-convective heat transfers are called a complex heat exchange.

In engineering and life the processes of heat exchange between the different liquids (or gases) separated by a solid wall often occur. The process of heat transfer from the hot medium to the cold one through a wall separating them is called heat transmission.

The differential equation of heat conduction

In natural conditions the heat transfer through the building envelopes is always a non-stationary. The outdoor temperature, intensity of solar radiation, wind strength and direction are continually changing. Indoor temperature also varies. Its fluctuations occur around some average values. Therefore, despite the some complexity, for the correct calculation of the thermal regime of rooms the heat transfer through the envelopes should be considered as non-stationary.

In the derivation of the differential equation of heat conduction will use the following assumptions:

1. The studied body is isotropic.

2. The relative deformation of the volume under consideration due to temperature change is very small.

3. The heat sources can be located within the body.

The derivation of the differential equation of the heat conduction is based on the energy conservation law: the quantity of heat introduced into the volume element by the heat conduction and the action of internal sources is equal to the change of the internal energy or enthalpy (depending on the process under consideration: an isobaric or isochoric) of the substance contained in this volume element.

, (1)

where dQ₁ is the quantity of heat introduced into the elementary volume,

dQ₂ – the quantity of heat released in the elementary volume as a result of internal heat sources,

dQ – change in the internal energy (or enthalpy) of the elementary volume.

Select in considered sample a volume element with sides equal to dx, dy, dz (Fig. 1).

Fig. 1 – To derivation of differential equation of heat conduction

The amount of heat passing through the left edge per unit time:

,

where q_x is projection of the heat flux density vector q (its dimension is W/m²) on the X axis.

The amount of heat escaping through the opposite face per unit time:

.

When q_x  q_x+dx, the elementary volume will be heated.

Expand q_x+dx in a Taylor series:

If we restrict the first two members of the series,

,

and the amount of heat supplied by heat conduction to the considered volume per unit time in the direction of the X axis:

.

The amount of heat supplied by heat conduction to the considered volume per unit time in all directions:

. (2)

The amount of heat generated by internal sources in an elementary volume per unit time:

, (3)

where w_v is the power density of the internal sources ([w_v] = W/m³).

When isochoric process, all the heat supplied to the elementary volume goes to the change in internal energy of the matter:

(4)

Using expressions (2 – 4), we obtain:

, (5)

or

. (6)

The equation (6) is called an energy equation for the isochoric process.

For isobaric process:

where H is an enthalpy of the system (Н= U+рV), h is enthalpy per unit volume.

The differential energy equation for the isobaric process can be represented as:

(7)

or

. (8)

In solids, heat transfer is carried out in accordance with the Fourier law:

, , .

For solids, it can be taken с_р = с_v = с. Then we obtain from (5)

, (9)

or

. (10)

Equation (10) is called the differential equations of heat conduction. They establish a connection between the temporal and spatial changes in temperature at any point in the medium, wherein the process heat conduction occurs.

If the thermal characteristics of the sample under consideration are constant, then the differential equation of heat conduction is transformed into:

.

Introducing the thermal diffusivity and using the notation , we obtain

. (11)

If the body has no internal heat sources, i.e. w = 0,

. (12)

If the body has internal heat source, but the heat conduction process is stationary, the differential equation of heat conduction becomes a Poisson equation:

. (13)

At steady-state heat conduction and the absence of internal heat sources, the differential equation of heat conduction is the Laplace equation:

. (14)

Thermal diffusivity is essential for non-stationary processes and characterizes the rate of the temperature change. The heat conduction coefficient describes the ability of bodies to conduct a heat; the thermal diffusivity is a measure of the thermal inertia of the body. The temperature change with time at any point in space is proportional to the thermal diffusivity coefficient. Therefore, the alignment of the temperature at all points will be faster in the body, which has higher а coefficient. The coefficient a depends on the nature of the substance. For example, liquids and gases have a high thermal inertia and thus small a coefficient. Metals have low thermal inertia and large а.

Dimension of a is

Define the physical meaning of the thermal diffusivity.

Denote the internal energy of the body per unit volume as . One can write

,

where n is coordinate in direction of unit vector .

The heat conductivity equation:

At steady-state one-dimensional heat flow

(15)

Thus, the coefficient а is equal to the quantity of heat that passes per unit time through unit area for a unit gradient of the volume concentration of the internal energy.

At a constant pressure, the thermal diffusivity characterizes transfer of the body enthalpy:

From the above equations it is seen that the coefficient a is the diffusion coefficient of internal energy or enthalpy depending on the coupling body and the environment.

Suppose there is an isothermal surface whose equation is

The total differential of this function is:

, or

where is the velocity of isothermal surface movement.

It follows from the differential equation of heat conduction and the last equation that

(16)

The value is the ratio of the first differential parameter to the second one. Denote =А. Then

(17)

Consequently, the thermal diffusivity is proportional to the velocity of propagation of an isothermal surface.

The a coefficient depends on the temperature, and for porous bodies also on the density and moisture content. Therefore, only with an approximation the coefficients а and  can be considered as constant.

Boundary conditions

The differential equation describes whole class of heat conduction phenomena. Particular peculiarities describing together with the differential equations the particular heat conduction process are called the terms of uniqueness or boundary conditions.

Terms of uniqueness include:

1. Geometric conditions that characterize the shape and size of the body.

2. Physical conditions that characterize the physical properties of the medium and the body (, с, , etc.).

3. Temporal (initial) conditions that characterize the temperature distribution in the studied body at the initial time:

T(x,y,z,0)=f(x,y,z). (18)

4. The boundary conditions that characterize interaction of the body with the environment.

The boundary conditions can be specified in various ways.

The boundary condition of the first kind consists in assigning the temperature distribution on the body surface:

(19)

In the particular case, , i.e. the temperature on the surface is constant throughout the entire heat transfer process. This can be realized by artificially maintaining a constant temperature or under special conditions of heat transfer between the environment and the body surface.

The boundary condition of the second kind consists in assigning the heat flux density for each point on the body surface as a function of time:

(20)

In the simplest case, the heat flux density is constant: . For example, such a case occurs when heating metal products in high temperature furnaces, where heat transfer takes place mainly by radiation according to the Stefan-Boltzmann law.

The boundary condition of the third kind characterizes the heat exchange between the body surface and the environment. For a description of this process the Newton-Richman law is used:

(21)

where T_n is the ambient temperature, T_c is the body surface temperature, is heat exchange coefficient (heat transfer coefficient). It characterizes the intensity of heat exchange between the body surface and the environment. The latter equation can be derived from Fourier's law, assuming that the heat transfer from the flowing fluid (gas) to the body near its surface follows the Fourier's law:

,

where is the thermal conductivity of liquid (gas), is conditional thickness of the boundary layer, .

Conventional thickness of the boundary layer depends on the liquid velocity and its physical properties. Therefore, depends on , temperature and varies along a body surface in the movement direction. As an approximation, the heat transfer coefficient can be considered constant, independent of the temperature and the same for the whole surface of the body.

According to the energy conservation law, the amount of heat eliminated from a surface due to heat transfer must be equal the amount of heat applied to the surface from the internal volume of the body by the heat conduction:

(22)

where n is coordinate normal to the body surface, is unit vector normal to the surface.

In non-stationary heat transfer process between the surface of the body and the medium flow, conditional thickness of the boundary layer will depend not only on the velocity of the medium and its physical properties but also on the thermo-physical properties of the body and will continuously change over time. Therefore, the relation is used only in special problems of convective heat transfer.

Boundary conditions of the third type can also be used when considering the body heating or cooling by a radioactive heat transfer. According to the Stefan-Boltzmann law, the radioactive heat flux between two surfaces

where is reduced emissivity, Т is absolute surface temperature of the body, which receives a heat. At a low temperature difference one can write:

(23)

where is radioactive heat transfer coefficient.

Equation (23) is an expression of the Newton-Richman law for cooling or heating of the body. Although equation (23) is similar to the equation of convective heat transfer at a constant heat flux, its physical meaning is different. The radioactive heat transfer coefficient depends on the temperature and the properties of the surfaces of bodies involved in the radioactive heat transfer. If the temperature changes slightly, the coefficient can be considered as approximately constant.

If the ambient (air) temperature and the temperature of the body that receives a heat Т are the same and the absorption coefficient of medium is very small, it is possible to write a Т_с instead of Т. In this case,

Coefficient will be the total heat transfer coefficient

,

where α_c is convective heat transfer coefficient.

According to the energy conservation law

,

where for generality in the problem formulation Т_с is considered to be variable, and the heat transfer coefficient approximately constant: .

Usually, the boundary conditions of the third kind are written as

(24)

From the boundary conditions of the third kind, as a special case one can get the boundary condition of the first kind. Indeed, if ,

, and

.

i.e. the temperature of heat emitting body surface equals to the ambient temperature.

Similarly, if , we obtain particular case of the second kind boundary condition, namely, a zero heat flow through a body surface.

The boundary condition of the fourth kind. This boundary condition corresponds to the heat exchange between the surface of the body and the environment (convective heat transfer between the body and the fluid) or heat exchange between contacting solids, when the temperatures of contacting surfaces are the same:

. (25)

Stationary heat transfer through the building envelope

Consider stationary heat conduction through the uniform wall with thickness h. We assume that the wall thickness is many times smaller than its length and width and on the surfaces of the wall the constant temperatures T₁ and T₂ are maintained (the boundary conditions of the first kind). To find the temperature distribution it is necessary to solve the one-dimensional steady-state heat conduction problem:

.

The solution of this equation gives:

. (26)

The heat flux density

, (27)

where h/ is thermal resistance of the wall.

As is seen from (27), at the given temperature difference Т₁-Т₂ the value of the heat flow through the wall is decreasing with the growth of its thermal resistance.

For a multilayer planar wall consisting of n layers (assuming that at all contacts between layers the boundary conditions of the fourth kind are realized), the temperature distribution over the cross section is defined by the following expression:

(28)

where Т_x is the temperature in an arbitrary section of envelope; Т₁ and Т₂ are the temperatures on the inner and outer surfaces, respectively; R_1-x is the thermal resistance from the inner surface to the section x, R_o is the total thermal resistance.

It follows from (28) that the temperature changes over the wall are proportional to the corresponding thermal resistances.

In a steady state the heat flux through any isothermal surface of multilayered wall is the same.

At the given temperatures on the outer surfaces of such wall, layer thicknesses h_i, and conductivity coefficients of layer materials _i, the heat flux density is

(29)

The parameter equal to the sum of thermal resistances of all layers is called the total thermal resistance of a multilayered wall. It determines the thermal insulation properties of the envelope.

The house can be considered as a limited volume dissipating into the environment the heat flow, which depends on the power of the internal heat sources and thermal resistance of the envelopes.

In real conditions the convective heat transfer is realized on the surfaces of the building envelopes (the boundary conditions of the third kind). With known indoor and outdoor air temperatures, the corresponding heat flux densities in the steady state for the wall separating the inner region from the outer environment can be expressed as:

(30)

where Т_с1 and Т_с2 are the air temperatures, Т₁ and Т₂ are temperatures on the wall surfaces, ₁ and ₂ are heat exchange coefficients on the inside and outside surfaces of the envelope.

Solving the system of equations (30) it is possible to find the heat flux passing through the wall:

(31)

The heat flux density through the multilayer wall consisting of n layers will be equal to:

(31^/)

Thus, the heat loss through the building envelopes depends on the thermal resistance to heat conduction of the envelope layers and thermal resistance to heat transfer from the wall to the medium (1/₂) or from the environment to the wall (1/₁):

Therefore, one can increase the thermal resistance of the building envelope by increasing its thickness, or the use of materials with low thermal conductivity.

Heat transfer through the cylindrical wall

At movement of hot water or steam in the pipes of heating mains, the significant heat loss can take a place. Consider the heat transfer through the cylindrical wall (Fig. 2) and the methods of heat loss minimizing.

The differential equation of heat conduction written in a cylindrical coordinate system has the form:

(32)

Match the Z axis with the pipe axis.

Fig. 2 – The cylindrical pipe with the inner diameter d₁ = 2r₁

and the outer diameter d₂ = 2r₂

We assume that in the considered temperature range the thermal conductivity of the wall material is constant. It is necessary to find the temperature distribution over the cylinder wall and the heat flow there through.

If the pipe diameter is much smaller than its length, on the given boundary conditions of the first type (constant temperatures T₁ and T₂ on the wall surfaces) the temperature is changed only in the radial direction and the temperature field is one-dimensional, therefore

(a)

Furthermore, since the temperatures at the outer and inner surfaces of the pipe are fixed, isothermal surfaces are cylindrical and have a common axis with the pipe. Then, the temperature should not depend on , i.e.

(b)

With regard to (a) and (b), equation (32) becomes:

(33)

We assume here that the boundary conditions of the firth kind are ttue:

, (34)

where T_m1 and T_m2 are the temperatures of the inner and outer media, respectively.

Solving the equation (33) with (34), we obtain the temperature field in the cylinder wall. We introduce a new variable

(c)

then

(d)

Substituting (c) and (d) into equation (32), we obtain:

(35)

Integrating (35), we get:

(e)

Potentiating the expression (e) and passing to the original variables, we obtain:

(f)

After integration we obtain:

(36)

Constant c₁ and c₂ can be determined by substituting the boundary conditions in equation (36):

(g)

Solving of equations (g) relative to c₁ and c₂ provides:

Substituting the values ​​of c₁ and c₂ in equation (36), we obtain:

. (37)

To determine the amount of heat passing through the cylindrical surface area S per unit time, we can use the Fourier law:

(38)

Substituting the value of the temperature gradient into (38), we obtain (taking into account that S = 2rl):

(39)

It follows from (39) that the amount of heat passing through the cylinder wall per unit time is independent of the radius.

The heat flow through unit length of pipe, W/m

(40)

This value is called a linear density of heat flow. The linear density of the heat flow is not dependent on the surface area of ​​the cylindrical wall. Heat flux densities q₁ and q₂ through the inner and outer surfaces are not the same (q₁ > q₂).

The boundary conditions of the third kind. Consider a homogeneous cylindrical wall (pipe) with a constant thermal conductivity coefficient . Temperatures of the media inside and outside the pipe are T_c1 and T_c2, respectively; the values ​​of the heat transfer coefficients on the inner and outer surfaces of the pipe are, respectively, ₁ and ₂ (Fig. 3).

Fig. 3 – Heat transfer through a homogeneous cylindrical wall

when the boundary conditions of the third kind are specified

It is necessary to find q_l and T. Assume that the length of the pipe is large as compared with its thickness. Then the heat losses from the pipe ends can be neglected.

In the steady thermal state the heat flow given by a hot medium to the inner wall will be equal to flow through the wall and the heat flow given to a cold environment by an outer wall. Therefore, we can write:

(41)

We represent these equations as follows:

(41')

Adding the equations appearing in the system (41'), we get:

and, hence:

. (42)

Introducing the parameter k_l

, (43)

We can rewrite equation (42) as

(42')

The parameter k_l is called a linear heat transfer coefficient and is measured in W/(mK). It characterizes the heat transfer intensity from one environment to another through a wall separating them. The value of k_l is numerically equal to the quantity of heat which passes through 1 meter of the wall per unit time from one medium to another at the temperature difference between them equal to 1 K.

The value R_l = 1/k_l inverse to linear coefficient of heat transfer is called a linear thermal resistance of heat transfer. It is equal to:

(44)

where 1/(₁d₁) and 1/(₂d₂) are thermal resistances of the heat transfer on the corresponding surfaces (denote them as R_l1 and R_l2, respectively); is thermal resistance to heat conduction of the wall (denote it as R_lс).

In practice, often the pipe wall thickness is small compared to the diameter. In this case, the calculation can be done using the simplified formulas.

Indeed, expand the value in a series:

If the ratio d₂/d₁ → 1, then such a series converges rapidly and we can limit oneself with sufficient accuracy to the first term:

where  is the thickness of the cylindrical wall.

Substituting this value in equation (43), we obtain

(45)

Therefore, if the pipe wall is thin, in practical calculations we can use the formula

(46)

where k (Wm^-2К^-1) has a form similar to the equation for a flat wall. When d₂/d₁ < 2, the calculation error does not exceed 4 %. For many technical calculations, this error is quite acceptable. Usually in engineering calculations for d₂/d₁ ≤ 1.8 one uses the formula (46).

When heat passes through a multi-layered cylindrical wall, the system of equations (41) should be replaced by a system that takes into account the thermal resistance to heat conductance of all layers, and one can easily obtain

(47)

or

, (47')

where

(48)

is called the total linear thermal resistance of the multilayer cylindrical wall and is measured in mK/W. It follows from (47) that

(49)

As noted above, the boundary conditions of the first kind can be regarded as a limiting case of the boundary conditions of the third kind, when the heat transfer coefficients on the surfaces ₁ and ₂ tend to infinity, i.e. T_с1 and T_с2 become equal to T₁ and T_(n+1). Under these conditions, equation (47) becomes:

(50)

and the expression for the temperature calculation at the boundaries between the layers is the following:

(51)

Critical diameter of the cylindrical wall

Consider the effect of outer diameter on the thermal resistance of a uniform cylindrical wall. According to (48),

At constant values ​​of ₁, d₁,  and ₂ the total thermal resistance to the heat transfer for the cylindrical wall will depend on its outer diameter. Under these conditions, 1/(₁d₁)  R_l1 = const.

The thermal resistance to heat conductance will increase with d₂, and thermal resistance to heat transfer 1/(₂d₂) = R_l2 will decrease. Obviously, the total thermal resistance will be determined by the change of the components R_l2 and R_lс.

Let’s investigate R_l as a function of d₂; take the derivative of R_l on d₂, and equate it to zero:

Thus, a value of diameter equal to 2/₂ corresponds to the minimal thermal resistance to the heat transfer. This value is called the critical diameter and signed d_кр:

(52)

When d₂ < d_кр, the total thermal resistance to heat transfer decreases with increasing of d₂, as the increased area of ​​the outer surface has greater influence on a thermal resistance than the increase in wall thickness. When d₂ > d_кр, the thermal resistance to heat transfer increases with increasing d₂, indicating that the thickness of the wall has the dominant effect.

These considerations must be taken into account when selecting thermal insulation coatings for various cylindrical vessels and pipelines.

Consider the critical diameter of the insulation applied to the pipe (Fig. 4).

Fig. 4 – To determination of the critical diameter of the insulation

The thermal resistance for such pipe

It is seen from the last equation that when external diameter of insulation d₃ is increasing, q_l will first increase and at d₃ = d_кр will have a maximum. With further increase in the outer diameter of the insulation q_l will decrease (Fig. 5).

Fig. 5 – The dependence of the heat loss on the thickness

of the insulation applied to the cylindrical wall

Selecting a heat-insulating material to cover the cylindrical surface, it is necessary first of all to calculate the critical diameter according to (52) for the given _из and ₂. If value d_кр is greater than the outer pipe diameter d₂, then the use of the selected material as heat insulation is unreasonably. In the range d₂ < d₃ < d_{кр. из} the increase of heat loss will be observed with the insulation thickness. Only at d₃ = d_3eff heat losses will again be the same as for the original, non-insulated pipe. Consequently, a certain layer of thermal insulation will not meet its purpose. Hence, for efficient thermal insulation it is necessary d_{кр. из} ≤ d₂.

The solution of non-stationary heat conduction problem

Consider wall with thickness of 2. If the wall thickness is small compared to its length and width, then it can be considered unlimited.

With the specified boundary conditions the heat transfer coefficient is the same for all points on the surface of the wall. The temperature change occurs only in the x direction, in two different directions the temperature does not change (T/y = /z = 0), and hence the problem is one-dimensional in space. The initial temperature distribution is given by a function T(x, 0) = f(x). Cooling takes place in a medium with a constant temperature T_с = const. On both surfaces the heat removal is carried out at a time-constant heat transfer coefficient. Let’s introduce the notation T – T_с = . Since the problem in the space is one-dimensional, then the differential equation (10) takes the form:

(53)

Initial conditions:

(54)

Under given cooling conditions the problem is symmetric, so it is convenient to put the origin of coordinates on the axis of the plate. The boundary conditions on the axis and on the plate surface can be written as follows:

(55)

The differential equation (53) in conjunction with the initial (54) and the boundary (55) conditions completely articulate the task.

To solve the differential equation we will use the variables separation method:

(56)

After substitution of the last expression in the differential equation (53), we obtain:

or

In this equation, the variables are easily separated and it can be written as follows:

(57)

The left-hand side of equation (57) is only  function, and the right-hand one – only x function. Since the equation (57) must hold for all values ​​of x and , then its both sides must be equal to the same constant value:

and hence

(58)

(59)

The constant k will be determined from the boundary conditions; the sign “-” is chosen from the physical considerations, since the thermal processes always tend to the thermal equilibrium. The result is a system of the ordinary differential equations (3.33) and (3.34), which can be easily integrated.

The function satisfies the equation (58).

The function satisfies the equation (59).

Substituting these expressions for () and (x) in equation (56), we obtain a general solution:

(60)

Expression (60) satisfies the original equation (53) for all values ​​of the constants C₁, C₂, C₃ and k.

Equation (60) must comply with the initial and boundary conditions to be a solution of the problem. At x = 0

and hence:

or

and consequently C₂ = 0.

This means that a particular solution (x) = C₂ sin(kx) should be excluded as not satisfying the given boundary conditions.

When we consider that C₂ = 0 and denote C₁C₃ = A, then (60) can be written as

(61)

Using the boundary condition on the wall surface,

we obtain:

(62)

Transform this equation:

where / = Bi and is called the Biot number. If we denote k = , the last expression can be written as follows:

(63)

From the analysis of this trigonometric equation, called the characteristic, it follows that at each value of Bi there are infinitely many values ​​of _n and each subsequent is larger than the previous:

In this case, its own set of roots of the equation (63) corresponds to each value of the Bi number.

In particular, when Bi → ∞, the line y₂ = /Bi coincides with the horizontal axis, and the roots of the equation (63) are:

When Bi → 0, the line y₂ = /Bi coincides with the axis of ordinates and the slope tends to infinity, and the roots of (63) are:

where n = 1, 2, 3...

For other finite values ​​of the Bi number the parameters _n have intermediate values.

Consequently, the different particular temperature distributions will correspond to the different values of :

(64)

The obtained particular solutions (64) will satisfy the differential equation for all values ​​of the constants A₁, A₂, …, A_n, but none of these solutions will not match the actual temperature distribution at the initial time. However, an imposing of infinite number of such distributions with appropriate choice of A_n constants ​​can represent any actual temperature dependence at the initial time.

Consequently, the general solution can be represented as the sum of an infinite series:

(65)

It is known that if the individual distributions (64) satisfy the differential equation (53) and the boundary conditions (55), then their sum also satisfies the same conditions.

The constants A_n in (65) can be found from the initial conditions:

(66)

Equation (66) is an expansion of an even function in a Fourier series with the given parameters _n defined by the characteristic equation (63). For this sequence of numbers _n the following formula is valid:

,

which allows to determine all coefficients A_n in equation (66). To do this, multiply both sides of (66) by cos(x/)dx and then integrate the resulting function over the thickness of the plate, then

(67)

since all the other terms on the right side (for which m  n) vanish. The integral in the right-hand side of (67) is equal to , then

. (68)

It is seen from the equation (68) that A_n are functions only of the corresponding roots of the characteristic equation (63) and the initial temperature distribution F(x).

Substituting the expression for the constants A_n in equation (65), we obtain the final expression for the temperature field of the cooled uniform wall:

(69)

Equation (69) allows to get the temperature value at any point of infinite wall for any time  and for any initial temperature distribution F(x).

If at the initial time ( = 0) the temperature in the wall is uniform, i.e. T₀ – T_с = ₀ = const, the integral in equation (68) is equal to (₀2/_n) sin_n. Then the expression for the constants A_n takes the form:

(70)

Substituting the A_n values ​​derived for this case in equation (69), we obtain

(71)

or

, (72)

where /₀ =  is dimensionless temperature; x/ = X is dimensionless coordinate; a/² = Fo is the Fourier number, which is a dimensionless time.

Analysis of the obtained solution. Since ₁, ₂, …, _n are the series of growing numbers, the contribution of subsequent terms of the series compared to the previous ones is smaller. In addition, the larger the Fo number, the faster the terms of the series will decrease with the increase in the number n.

Numerous studies have shown that even with Fo ≥ 0.3 the series (72) becomes so rapidly convergent that the temperature distribution is quite accurately described by the first term of the series:

(73)

It follows from equation (72) that during the cooling (heating) of the wall for any time at the given boundary conditions the temperature field has the form of a symmetrical curve with a maximum on the axis of the wall (X = 0). For each subsequent point in time we will have a curve that is monotonically decreasing to wall surfaces.

The proven property of the temperature curves makes it possible to determine the temperature change in the body at the given Bi number. Consider with three cases.

1. Bi → ∞ (practically Bi > 100). If the number of Bi tends to infinity, the temperature of the wall surface becomes equal to the wall environment temperature. It follows from definition of the Biot number that at predetermined physical parameters and the wall thickness Bi → ∞, when → ∞, i.e. when there is a very high intensity of heat removal from the surface. In these cases the cooling process is determined by the physical properties and dimensions of the body. In so doing and the coefficients in the series (72) are

The solution for this case takes the form:

(74)

Then the temperature on the wall axis (X = 0)

(75)

When X = 1 , and, hence, _{X = 1} = 0.

As mentioned above, when Fo ≥ 0.3, series (72) converges fast and the error does not exceed 1 %, if we ignore all terms except the first. Under these conditions, equation (75) becomes:

(76)

If we find the logarithm in (76) and solve relative to Fo, we obtain:

(77)

Considering that Fo = a/², equation (77) can be written as

(77')

It is possible to determine by formula (77) the time required to heat the middle of the wall to a predetermined temperature.

2. Very small Bi numbers (practically Bi < 0.1). If the Bi number is small, then all the coefficients of the series D_n → 0, as now _n = (n – 1), except for D₁, which is equal to:

From the expression Bi = (/)/(1/) it is seen that small values ​​of the Bi number may occur at small values ​​of the thickness of the wall, for large values ​​of the thermal conductivity  and small values ​​of the heat transfer coefficient . For small ₁ values the functions tg ₁ and sin ₁ can be replaced by their arguments and then the characteristic equation (63) can be written as

Using this equation, we can rewrite (72) as follows:

(78)

Find the temperatures on the axis and the wall surface:

for X = 0

; (79)

for X = 1

. (80)

Ratio of the temperatures on the axis of the wall surface:

For small Bi values the temperature on the wall surface is slightly different from the temperature on the axis. This indicates that the temperature of wall thickness is distributed evenly and the function Θ(X) remains almost parallel to the X-axis for each time point.

In this case the process of body heating (cooling) is determined by heat transfer intensity on the wall surface. In other words, the temperature smoothing occurs essentially rapidly than the heat sink from the surface.

3. The Bi number is in the range 0.1 ≤ Bi < 100.

In this case _n are the functions of the Bi number, i.e. depend on the wall thickness. The heating (cooling) intensity is determined both internal and external thermal resistances.

Consider the problem of transient heat conduction through the multi-layered envelope. Fig. 6 shows a construction with designations of layers (1, 2, 3) and the characteristic boundaries corresponding to inner (I) and outer (IV) wall surfaces and contact surfaces between the layers (II, III) in the wall thickness.

Fig. 6 – Scheme to the problem of transient heat conduction through a multi-layered envelope

The task is to find the change in temperature T(x,,) and heat flux q(x,,) in time () and their distribution over the envelope thickness in space (x).

We can write the system of differential equations, each equation is written to a separate layer with additional conditions on the boundary of layers:

(81)

At the boundary II (at the boundary between layers 1 and 2, Fig. 6) the boundary condition of the forth type is specified, which is defined by the equality of heat flows

(82)

and temperatures

(83)

The same condition is at the boundary III (between layers 2 and 3):

(84)

(85)

The initial conditions are usually specified in the form of the temperature distribution at the beginning of the process, i.e., at  = 0. In general, it can be written as

(86)

In addition to the expressions (82) – (85), it is necessary to record the heat transfer conditions on the inside and outside envelope surfaces, which are in contact with the inner T_с1 and outer T_с2 air and surrounded by other surfaces having a temperature T_Ri. The convective and radiation heat transfer are defined by _к and _л coefficients, respectively. It is necessary to take into account also that the envelopes can be irradiated by heat sources (sun, heating elements in the room, etc.). The radiation heat flux absorbed by the surface is

(87)

where  is the absorption coefficient of the envelope surface for given radiation; q is the intensity of the incident radiation.

In the general case, the complex heat transfer on the envelope surfaces can be described by the II (the given heat flux) and III (given conditions of heat exchange with the environment) type boundary conditions. On the boundary I the condition has the form

. (88)

On the boundary IV:

. (89)

Usually, in rooms the conditions of radiative-convective heat transfer are taken into account by equivalent coefficient of heat exchange _в referred to the air temperature Т_в. In this case, the write of conditions at the inner boundary I is simplified and has the form

(90)

The similar simplification using heat exchange coefficient ₂ can be made ​​for the boundary condition on the outer surface for winter climate. When calculating the heat exchange in summer, it is necessary to consider a solar radiation, so the boundary condition on the surface IV (equation (89)) for the summer is usually written as

(91)

Selection of the equation is determined by the specific conditions of the task and accepted methods of its solving.

Consider the process of heat transfer through a two-layer wall consisting of layers with different thermal characteristics. The layers are represented as two unlimited plates with thickness h₁ and h₂, which are in contact. The initial temperatures of the plates are the same. At the initial moment one of the free surfaces is heated instantly to a temperature T_c, which is kept constant throughout the heating process. The opposite surface is maintained at an initial temperature Т_о=0. It is necessary to find the temperature distribution in the plates in the transient and steady-state regimes and heat flow through the plates.

Choose the origin of coordinates in the plane of the contact between the layers. Then the boundary conditions can be written as:

, (92)

(93)

,

(94)

.

The solution can be written as:

(95)

where

The steady state (Fo=) temperature distribution obeys a linear law. The heat flux is determined by differentiating (95).

Heat transfer in a limited volume

Rooms in the building are isolated from the external environment the protective envelopes, allowing them to create a certain microclimate.

Outdoor envelopes protect the premises from direct atmospheric influences, and special microclimate conditioning system supports certain parameters of the medium inside the volume. Under the influence of the outside and inside temperature difference, solar radiation and wind the rooms lose heat through the envelope in the winter and overheat in the summer. Effects of wind and ventilation, gravitational forces create pressure differences, resulting in the overflow of air between the communicating spaces and its filtration through the pores in material and leakages in envelopes. Precipitation, humidity and water generation in rooms, the difference between indoor and outdoor air humidity lead to moisture exchange through the envelope that can lead to materials moisturizing and the deterioration of their protective properties and durability of the exterior walls and coatings.

Processes forming the space thermal environment must be considered in close connection with each other because their mutual influence can be significant. For example, the air filtering and humidifying of constructions can increase by several times the heat loss in winter. At the same time, the creation of favourable air ambient in the room requires the organization of air and moisture exchange with the outer environment.

Radioactive heat exchange

The temperatures of individual surfaces in the room are not the same. Radioactive heat transfer between the surfaces occurs. It follows to general physical laws, which are difficult to use in engineering calculations. Radioactive heat exchange in the room occurs in a limited range of temperatures, the radiative properties of certain surfaces, the geometry of their location, etc.

Consider the radiation heat transfer in the room.

All surfaces in the room are sources of heat radiation. Heat radiation at sufficiently low temperatures characteristic of the surfaces in the room takes a relatively narrow wavelength range and hence can be considered as monochromatic. The wavelength _max corresponding to maximum radiation intensity can be taken as the average. Its value is defined by the Wien law

(96)

where a = 0.2910^-2 mK is the radiation constant.

For the room surface temperatures from 0 to 150 ^oC the wavelengths _max are in the range from 7 to 11 μm. Thus, replacement of polychromatic radiation by monochromatic is the first possible simplification in the calculation of heat radiation by surfaces in the room.

The intensity of the radiation emitted by the heated surface for the whole wavelengths spectrum can be determined by integration of the Planck radiation formula. The value of the integral intensity E₀ (W/m²) of the radiation is determined by the Stefan – Boltzmann law:

(97)

where C₀=5.77 W/(m²K) is the emissivity of the black body.

Surfaces in the room are gray bodies. Slate, dark linoleum are examples of gray bodies. In contrast to the absolute black, the gray bodies emit less heat and the radioactive flux falling on them is not completely absorbed and partially reflected.

There is a significant difference between the radioactive properties of conductors and insulators in respect to infrared radiation. Conductors reflect and dielectrics absorb most of the incident infrared radiation. At the same time, as the emission and absorption coefficients of monochromatic radiation by material are equal (the Kirchhoff's law), the first emit less heat than the latter. The white surface reflects well the visible light, but it absorbs the heat (infrared) radiation as well as red. The emission coefficients of gray bodies C are always less than C₀. The emissivity of a gray body determined by relation

(98)

where  is the blackness degree (dimensionless value).

Absorption and reflection properties depend not only on the kind of material but also on the surface state, the spectral range, the incidence angle. In a narrow wavelengths range for integrated radiation it can be taken that the average values ​​of the emission and absorption coefficients of surfaces in the room are equal.

Heat radiation of surfaces in the room is considered as monochromatic infrared diffuse radiation of gray bodies obeying the Stefan – Boltzmann, Lambert and Kirchhoff laws.

Thus, the rather complex properties of polychromatic radiation of bodies can be greatly simplified with respect to the problem under consideration.

The air in the calculation of radioactive heat transfer between the surfaces can be considered as a transparent medium. It consists mainly of diatomic gases (nitrogen, oxygen), which are transparent to thermal radiation and do not emit thermal energy. Low content of polyatomic gases (water vapour and carbon dioxide) at small thickness of the air layers does not change practically this property.

Consider the heat exchange between the two parallel walls. Suppose two gray surfaces spaced much smaller than their size. Assume that the temperature of the first wall is higher than the temperature of the second one: Т₁ Т₂. The unit surface area of the first wall radiates per unit time the energy Е₁. The colder surface of the wall 2 absorbs energy а₂Е₁ (where а₂ is absorption coefficient of the second wall) and reflects energy (1-а₂)Е₁. From this reflected energy surface 1 absorbs а₁(1-а₂)Е₁ and reflects (1-а₁)(1-а₂)Е₁. The surface 2 from energy reflected by surface 1 absorbs а₂(1-а₁)(1-а₂)Е₁, etc.

Thus forming a series, we find that the surface 2 will absorb

,

where р=(1-а₁)(1-а₂)1.

Through similar reasoning concerning the radiation of the surface 2, which also radiates, we obtain a similar expression for the total amount of heat absorbed by the wall 1:

.

Determine the quantity of heat received by the colder surface 2 per unit time due to the heat emitted by the body 1. The surface 2 holds the heat emitted by the body 1 and spends for its own radiation . The rest of this radiation is returned to the body 2. Thus, the final balance shows a surplus of heat in the body 2:

. (99)

This value is the amount of heat received by the body 2 per unit time and given by the body 1. As

(100)

Taking into account that the absorption coefficient of the body а is equal to and its emissivity  at the same temperature, as well as (97) and (98), one can write:

Thus, the amount of heat received by the body 2 of the unit surface area per unit time from the body 1 is equal

(101)

(102)

The energy absorbed by the whole surface area 2 per unit time is expressed by the formula

, (103)

where is equivalent (reduced) emissivity.

Consider the radioactive heat transfer between two absolute black surfaces 1 and 2 arranged at an angle to each other (Fig. 7). In accordance with the Stefan – Boltzmann law, elementary area dF₁ on the surface 1 radiates per unit time in all directions within a hemisphere the energy equal to

(104)

Fig. 7 – Radiative heat transfer between two surfaces

The intensity of radiation in the direction normal to the surface dF₁ is π times less than dq₁, i.e.

(105)

The intensity of radiation in direction at an angle ₁ to the normal is equal

(106)

Within the solid angle d₁ defined by dF₂, the heat flux is

(107)

The solid angle d₁ is equal

(108)

where R is the distance between the elementary areas dF₁ and dF₂; dF₂cos₂ is projection of the area dF₂ on the sphere of radius R, when the angle between the direction of the radioactive heat exchange and the normal to the dF₂ surface is equal to ₂

Equation (107) can be written as

(109)

Equation similar to (109) can be written for the heat flux d²q₂ emitted by the area dF₂ towards dF₁. It is assumed that the surfaces are of black colour so that all radiation is absorbed by them. As a result of radioactive heat transfer, the heat transferred from the area dF₁ to the area dF₂ is equal

(110)

It is convenient to use the irradiance coefficient , which is a geometric characteristic. Denote the irradiance coefficient from the area dF₁ to the area dF₂ as . It equals to the ratio of the radioactive flux incident from the area dF₁ on the area dF₂ (9) to the whole flux emitted by the area dF₁, i.e.

(111)

Substituting this value in (110) we obtain

(112)

Radioactive heat transfer between two surfaces 1 and 2 can be calculated by double integration of the equation (12) on their areas F₁ and F₂. The first integration gives the value of the heat emitted by the elementary area dF₁ to the whole surface 2:

(113)

where is irradiance coefficient from the elementary area dF₁ on the whole surface 2, which is equal to

(114)

The second integration determines the total heat Q_1-2 transmitted by radiation from the surface 1 to the surface 2:

(115)

Irradiance coefficient _1-2 from the surface 1 to the surface 2 equal to

(116)

shows the percentage of the radiant flux incident on the surface 2 from all flux emitted by the surface 1.

Surfaces in the room are different from the black body. This complicates the problem, since the radiation incident on the gray surface is partly reflected. Some of that energy can be repeatedly reflected from the mutually irradiated gray surfaces until the full absorption. From the theory of radioactive heat transfer it is well known that during the heat exchange by monochromatic radiation between two gray surfaces, which satisfy the Kirchhoff and Lambert laws, the heat Q_1-2 is defined by the formula

(117)

In this formula, in contrast to (115), the value _пр.1-2 is reduced (relative) radiation coefficient for the heat transfer between the two gray surfaces.

To determine _пр we can consider the three simplest cases:

1) For two parallel surfaces when the distance between them is small compared to their dimensions the irradiance coefficient _1-2 is unity as practically all the radiation emitted by one surface falls on the another one. The reflected rays are fully returned to the emitting surface, and so on until the complete absorption. Reduced emissivity of two surfaces for this case is

(118)

where ₁ and ₂ are the emissivities of the surfaces.

2) One surface is fully surrounded by the other one (a sphere in the sphere, a cylinder in the cylinder or just nonconcavity surface surrounded by a larger area of ​​the same geometry). In this case, if the smaller surface has an area F₁ and the larger F₂, the value _пр.1-2 equals to

. (119)

3) The surfaces are small or the distance between them is large; in this case the part of the reflected radiation returning to the emitting surface becomes negligible. In this case, the coefficient _пр.1-2 is equal

(120)

For an arbitrary arrangement of surfaces it is difficult to determine _пр accurately. _пр will be between the values ​​defined by (118) and (120). Surfaces in the room have large emissivity values. For them, the difference in the determination of _пр according to (118) and (120) does not exceed 5%.

In the case when the temperature is varying over the surface, one can divide it into several parts with some medium temperatures and calculate heat exchange for each of them separately and then sum the results.

The recorded relations for the configuration coefficient are valid when there are no other surfaces that reflect radiation near the surfaces S₁ and S₂. With the existence of such surfaces the heat obtained by the surface S₂ will be different and the value of  should be corrected.

When studying the radioactive heat exchange inside the closed space, typically one try to highlight the heat radiating surface, the absorbing surfaces and, finally, the walls, which only reflect radiation remaining the constant temperature. For simplicity it is assumed that reflecting radiation such walls do not conduct a heat.

The radiation heat transfer occurs indoor between the surface of the heating elements and the walls, ceiling and floor. Radioactive heat exchange is dependent on the design of heating devices.

For example, the radiators have close located ribs shielding each other's. Most of the radiation from their external surface does not reach the room. Only up to 30 % of the heat emitted by the surface of the radiator is transmitted to the room. The coefficient of irradiance  for the radiators in this regard is 0.2 – 0.3. For devices with strongly ribbed surfaces (such as ribbed pipes) the fraction of the radiation that enters the room from entire external heated surface is only 5 – 10 %.

If device in the room is closed, its heat radiation decreases. At a certain position of the closing flap the reduction of radiation heat transfer is accompanied by a noticeable increase in the convective one and the overall heat emission of the device. This effect of the heat transfer increasing is used in modern devices (convector with cover).

To reduce heat exchange between the emitting surfaces the screens are used. Assume that between the two parallel surfaces with temperatures T₁ and T₂ there is a screen having a temperature equal to T_э. For simplicity, we assume that the emissivities and light emission coefficients of the surface and screen are the same.

In the absence of the screen the heat flow by radiation is

(121)

in the presence of the screen we have:

(122)

In the steady state q_1э = q_э2, i.e.

(123)

from which we determine the unknown temperature T_э:

(124)

Consequently, we have

(125)

Comparing the above formula (125) with equation (121), we find

(126)

i.e. the presence of a single screen reduces the heat flow twice. It can be shown by a similar calculation that under these assumptions the presence of two screens reduces the heat flow in 3 times, etc., i.e., in the presence of n screens the radiative flux decreases in the (n + 1) times.

Significantly greater effect is obtained if the screen has a small light emission coefficient. When _1э = _2э = _э and reduced light emission coefficient ^* = ₁, then with a single screen

(127)

Then at _э = 0.310⁸, ₁ = 4.510⁸ the ratio q_э/q₁₂ equals 0.033, i.e. the heat transfer rate is reduced by 30 times. The use of screens made ​​of aluminium foil allows the use of the air interlayers as thermal insulation.

Convective heat exchange

In total heat exchange in the room convection plays a significant role along with radiation. The air exchanges by heat with the cooled and heated surfaces of the envelopes, heating devices and cooling systems. The heated air flows climb upward, chilled ones fall down causing the overall mobility and mixing of air in the room. The air inflow and exhaust by ventilation enhances this process.

In most areas as a result of air mixing a relatively uniform temperature distribution is observed, which allows accept the same temperature value t_в in the calculation of heat exchange on all surfaces. Sometimes there is a non-uniform distribution of temperature in height and also in the plane of the room (at local arrangement of the heat sources). Over the warm sources the convective air flows arise. They form under the ceiling a layer of heated air ("thermal pad") going to the top. Thus, various forms of convective heat transfer can be indoors. In most cases, the heat exchange between the air and heated or cooled surfaces occurs in the mode of free (natural) convection. Under the conditions of a forced air flow along the surfaces the heat exchange is determined by regularities of forced convection. All these processes are rather complicated and their passing in a limited volume has certain specificity. Consider the main results of theoretical and experimental study of the convective heat transfer at the free surfaces and the surfaces located in the room.

Free convection

Around the heated and cooled surfaces the convective flows occur. They cause the heat exchange between the air and surfaces. The air flow caused by the difference in its density in the field of external forces is commonly called the free convection. Such external forces are the force of gravity and the density difference in the simplest case is determined by the temperature difference between the solid surface and the gaseous medium. The thickness of the boundary layer under conditions of free convection is much higher than in a forced movement, inasmuch as velocities of free motion are small.

In the air flow around a vertical surface the boundary layer with a thickness increased in the direction of motion is formed (Fig. 8). In the initial zone of movement the boundary layer is laminar. At some distance from the bottom of the heated surface when the boundary layer thickness reaches a certain value, the flow regime becomes turbulent.

Fig. 8 – Boundary layers at the free convection

A similar pattern is observed near the cooled surface, but the flow of free convection is directed downwards.

Heat transfer on the surface in the areas of laminar and turbulent flow occurs different. Within the boundary layer thickness the changes of the temperature and air velocity occur. A noticeable change in the temperature occurs within the thermal boundary layer a thickness of _t, and the damping velocity – within the hydrodynamic boundary layer a thickness of . In general, the thicknesses of these layers are not equal.

The intensity of the free convective flow for any form of surfaces and environments in a generalized form is determined by the Grashof criterion (Gr) or the product of the Grashof on Prandtl criteria (GrPr). For room air at 20 °C the product of these criteria is

(128)

where  is coefficient of thermal expansion of the air equal to 1/T = 1/293 1/°C; g is acceleration of free fall equal to 9.81 m/s²; v is the kinematic viscosity equal at 20 °C to 15.0610^–6 m²/s; l is the characteristic size of a surface in the direction of air flow, m; t is the difference between the surface temperature and the room temperature; Pr is the Prandtl criterion (for the air at 20°C Pr = 0.709).

Consider the regularities of the heat transfer at free convection. For the laminar mode of convective heat transfer, there are analytical solutions, which are in good agreement with the experimental studies.

For the laminar mode the thicknesses of hydrodynamic  and thermal _t boundary layers are almost equal. The distribution of temperature and velocity within the boundary layer has the form shown in Fig. 9. The maximum speed _макс in the flow is reached in the cross-section located at a distance of about 0.38 from the surface. The thickness of the boundary layer , which increases in the flow direction, at a distance x from the surface boundary is determined by the relation

(129)

where Gr is the Grashof criterion determined for the characteristic size x.

Fig. 9 – The distributions of velocity and temperature in the boundary layer for free convection

The heat exchange intensity depends on the thickness of the boundary layer. A minimal  value is at the beginning of the surface, where the heat exchange is greatest. Forward facing  increases and the heat transfer intensity decreases.

Criterial equation that determines the heat transfer intensity in an arbitrary section x for Pr=0.709 has a form

(130)

The local value of the Nusselt criterion Nu_x referred to an arbitrary cross section x is equal

(131)

where _кx is the local value of the convective heat transfer coefficient in the section x;  is the thermal conductivity coefficient of air.

The transition from laminar to turbulent flow occurs at some distance l_кр from the beginning of the surface. Often for this section one take the value of the product (Gr·Pr)_кр  210⁷. For air, this transition occurs at a critical value of the Grashof criterion of about 10⁹.

Criterial equation of heat transfer in the turbulent regime which generalizes the numerous experimental data has the form

. (132)

The equation for _кx at an air temperature of 20 °C has the form

. (133)

As can be seen from (133), _к is independent of the geometrical parameter x and remains unchanged for the whole area of the turbulent regime.

Average values ​​of the convective heat transfer coefficient on vertical envelopes surfaces in the room can be determined without much error by formula (133), as the turbulent regime typically corresponds to the temperature changes and dimensions of heated and cooled surfaces having in reality.

All the above formula including (133) are recorded for the free vertical surface.

It was established experimentally that in case of the horizontal arrangement of the heated or cooled free surface one can also use the formula (133) for the calculation of the average convective heat transfer intensity, but the numerical value of the coefficient therein should be changed as shown in Table 1.

If the hot surface is facing up or cool one is facing down, the air flow around them occurs as shown in Fig. 10. At great surface sizes the air can be supplied to middle of its only normal. A so-called "Sahara-effect" occurs, when the air moves to the surface along the boundaries of hexahedrons, then is warmed and rises in the centers of the hexahedrons. To the boundaries of surfaces the air leaks unhindered. When the size of the horizontal surface is increasing, the average convective heat transfer coefficient decreases due to complicated air supply to the central part.

Table 1 – The values of numerical coefficient in (11) for different orientations of free surface

Surface orientation	The value of numerical coefficient
Vertical	1.43
Horizontal surface faced up:
- heated	1.86
- cooled	1.0
Horizontal surface faced down:
- heated	1.0
- cooled	1.86

In the case of a heated horizontal surface facing downward or cold facing upward the air movement is shown in Fig. 10b. Here, as in the first case, with the increase of the area the air supply becomes complicated and the average coefficient of convective heat transfer is reduced.

Fig. 10 – Air movement for free convection around the heated horizontal surface

facing upward (a) and bottom (b)

In all the above formulas the average temperature of the air and surfaces is usually taken as determining. If it is significantly different from 20 °C, the numerical coefficients in all the final formulas should be recalculated taking into account the tabular data.

For small changes of determining temperature, the numerical values ​​of the coefficients vary only slightly (with a temperature increase by 10 °C the numerical coefficient in the formula (133) is reduced by about 1%).

The main discussed peculiarities and equations of the free convection heat transfer are the result of experiments and theoretical conclusions, which are valid for ideal conditions. In real room, the violation of a perfect picture of free-convection heat transfer is possible due to impact of factors such as a closed and limited volume, the presence of several cold and hot surfaces, etc.

First, consider the heat transfer between two parallel plates with different temperatures (Т₁  Т₂). If the plates are disposed horizontally and the upper plate has a higher temperature (T₁> T₂) as compared with the bottom, the heat transfer from one plate to another through the medium layer will be by thermal conduction; this disturbance will occur only near the edges of the plate.

If the upper plate has lower temperature than the bottom one, then the heated layers will be collected at the upper plate. This leads to an unstable state, which turns into a state with closed convective medium flows in the space between the plates when Gr∙Pr > 1700.

In the case of vertical walls, the character of movement depends on the ratio between the wall height l and the distance between the walls L. Studies show that at value of the criterion

the temperature distribution in the medium layer along the normal to plates is linearly. Only near the lower and upper edges of the walls there is a deviation from linearity, this deviation is distributed inside the layer to a distance L. Heat transfer occurs by heat conduction.

The thickness of the boundary layer ₁ increases from the bottom upward on a heated wall; on the opposite less heated wall the thickness ₂ increases in the opposite direction.

In the central part of the medium the temperature is constant. The temperature distribution in the boundary layer is similar to that in the boundary layer of a single plate. In the case of two plates, heat transfer coefficients differ by no more than 20 % when calculation is made according to the formulas for a single plate; in so doing, the temperature of the fixed medium is assumed to be equal to the temperature of ambient between the two plates. This state is maintained as long as the two boundary layers do not merge into one that occurs with Gr∙Pr decreasing. However, as shown by the experiments, the temperature in the central region is not constant, but increases in the upward direction.

If the distance between the walls L is sufficiently large, the upward and downward flows exist without interference. If the value of L is small, due to the interference an internal circulation can occur.

With such a complex hydrodynamic pattern the calculation of heat transfer coefficients for a medium in a confined space is impossible. Therefore, the heat transfer between the plates through a layer of medium is considered as the transfer of heat by thermal conductivity introducing an equivalent thermal conductivity coefficient

The ratio of the equivalent thermal conductivity coefficient to the true thermal conductivity of the medium is called the convection coefficient _к, _к = _э/_с.

When Gr·Pr < 1000 convection coefficient is equal to one (_к = 1). Hence, when Gr·Pr < 1000 the heat transfer through the layer of medium takes place by thermal conduction.

Reducing of heat exchange intensity for large Gr·Pr values ​​can be explained by interference between the heated and cooled streams within the volume.

Consideration of the experimental data on the thermal regime of the boundary layer shows that the qualitative and quantitative the picture of the convective heat transfer on a heated vertical surface in the room is different from that considered for the free surface. The local convective heat transfer coefficients vary in height of the wall. In this case, there are three characteristic regions of the convective heat exchange: well-developed area with the laminar mode of heat exchange directly above the floor, area of ​​the turbulent mode in the middle of the surface, and the braking area at the top near the ceiling.

In confined spaces, the general intensification of the air flow around the surface occurs. Under the influence of the floor along which the air leaks to the surface at an angle of 90°, the flow rate increases and sooner than at the free surface the destruction of stable laminar flow occurs. The upper limit of the laminar region corresponds to the critical value

.

In the region of the laminar mode of air flow the heat exchange in general is more than the free surface. On upper boundary it is superior the value corresponding to the free surface by 25 %. The exception is a small part of the area bounded by 10 mm height from the floor, in which the heat transfer rate is less by 10 %.

In the turbulent region, the local convective heat transfer coefficient increases in the direction of motion. It is a feature of turbulent heat transfer in a closed space. Compared to the heat exchange on the free surface, _к is initially slightly less (5 %), then more (up to 25 %). Maximum and average values of _к in the turbulent region increase with the height of the room.

It is established experimentally the presence near the ceiling of the inhibition region, the height of which h_тор is equal to

, (134)

where h is room height, m; h_тор is inhibition region height, m; Gr_h is determined for the characteristic size h.

On the bottom boundary of the inhibition region the intensity of convective heat transfer is reduced. The average value _к in the inhibition region can be determined from the equation

(135)

where and are determined for the characteristic size h – h_тор.

The appearance of a braking region is a feature of the convective heat transfer in a closed space of the room. This phenomenon is explained by the influence of the ceiling and the air temperature stratification in height that results in a decrease in the temperature pressure and air mobility.

The generalized dependences of heat transfer for the three characteristic regions can be used for rooms with an arbitrary height h.

Forced convection

Assume that the medium moves along a surface in the horizontal direction (x axis). One can obtain the velocity distribution along the axis y normal to the surface. The velocity on the surface of a body is equal to zero.

In the y direction velocity increases and then beginning with some distance  from the surface almost no varies. Consequently, the main velocity drop from the value equal to zero to a value _c corresponding to the speed away from the wall (i.e., main medium flow) occurs in a  thick layer called the boundary layer. The reason for the gradual reduction of the velocity from _c to 0 is its viscosity, i.e., the ability of the gas medium to resist relative movement due to molecular random motion.

Consequently, the whole gas flow can be divided into two areas: the boundary layer, where the effect of the viscosity is manifested, and the main stream, where viscosity can be neglected, and a moving medium can be regarded as ideal.

In the boundary layer, the internal friction forces determined by the Newton law for internal friction act: shear stress (frictional force divided by the liquid (or gas) layer area) is directly proportional to the velocity gradient.

By the type of the fluid flow, the laminar and turbulent motions are distinguished.

Heat transfer in turbulent flow occurs more intensively as compared with a laminar one. The thickness of the boundary layer increases continuously along the surface of the plate, starting from zero at the front edge. At some distance from the front edge x_к the motion mode in the boundary layer changes transiting from laminar to turbulent flow. With increasing the speed _c the critical distance x_к decreases but the product _cx_к remains at this constant.

The transition from the laminar to turbulent motion occurs at a certain value, i.e. is characterized by the critical value of the Reynolds number (Re_к = ).

The transition from the laminar boundary layer to turbulent one does not occur immediately, but gradually, with the presence of some transition region.

Under the conditions of forced convection, the boundary layer thickness (Fig. 11) is much smaller than for free convection.

Fig. 11 – The boundary layer for forced convection

The transition to a turbulent boundary layer occurs at a Reynolds number of about 510⁵.

The Nu criterion for the mean convective heat transfer coefficient on the assumption that the boundary layer is turbulent over the entire length is equal:

(136)

When the average boundary layer temperature is different from 20 °C, a numerical factor in this equation is changed slightly. These changes correspond to the increasing of about 7 % when the temperature is lowered by 20 °C.

The condition of the mixed (free and forced) convection existence can be defined by conventional Reynolds number Re_усл, the value of which is equal

(137)

where Re is defined for the forced flow, Gr is criterion defined for the conditions of free convection.

The formula for determining the average convective heat transfer coefficient in the conditions of mixed convection is of the form

(138)

All the above discussed propositions and formulas for a convective heat transfer are strictly valid for the calculation of heat transfer on an isothermal surface. Often the surface temperature varies in direction of the air flow or the air flow changes its temperature. In the practice of engineering decisions one take that the temperature prehistory of the flow effects significantly greater in laminar flow than in turbulent. For turbulent flow it can be ignored.

Average heat exchange over the whole area can be calculated by summing the local values ​​on the assumption of their step change in the movement direction.

Heat transfer in hermetic air interlayer

Creation of air interlayer is a common technique for thermal insulation. Air interlayers are used in constructions of windows, stained glass windows, outer walls and floors, to shield for protection against radiation. In exterior walls and floors this technique is often used to prevent construction overwetting.

Air interlayer can be hermetic or vented.

Thermal resistance air interlayer R_в.п can not be defined as resistance to thermal conductance of the air layer, inasmuch as the heat transfer from one surface to the other through it happens in conjunction of heat conduction, convection and radiation.

In the simplest case of hermetic air interlayer, the heat flow from one wall to the other can be represented as a sum

(139)

where q_к is convective component, and q_л is radiative component of this flow.

q_к includes convective and conductive (by heat conduction) heat fluxes. The convective component of the heat flux q_к transferred from one surface to the other through an air layer is

(140)

where '_к is convective heat transfer coefficient referred to the temperature difference on the surfaces t = t₁ – t₂.

Convective heat transfer in a layer is associated with air circulation. In vertical air interlayer if the thickness is comparable with the height, ascending flows along the surface with the higher temperature and downflows on a cold surface can move without interference. In thin interlayers they are mutually inhibited and form internal circulation circuits, the height of which depends on the width of the gap.

At small thicknesses or small temperature differences the parallel-jet mode of air movement along the walls of the interlayer without mixing takes place. It was established experimentally that the laminar flow regime remains if the product of the criteria Gr∙Pr <1000. At an average air temperature of 0 °C

(141)

corresponds to the critical value ( is the thickness of the air layer, m).

Relation (52) provides a critical layer thickness _кр mm, which remains the laminar flow mode:

(142)

When this mode is in the interlayer, then

(143)

i.e. the heat transfer through air layer with a thickness  < _кр happens by a heat conduction.

When Gr·Pr > 1000, for the Nu criterion defined regarding '_к the following relationship is empirically established:

. (144)

In the horizontal interlayer, if the upper surface is heated more than the bottom, the air stirring will not be almost, therefore in all other conditions _к will be determined accurately enough by the formula (144). If the heat flow is directed upward, the ascending and descending flows forming the circulation circuits arise. The value _к in this case increases.

Radiative component of the heat flow density through the air interlayer:

(145)

where _л is radiative heat transfer coefficient.

The value of _л is markedly greater than _к, so to a greater extent the heat is transferred by radiation. One can reduce this flow and thereby increase the resistance of the interlayer to heat transfer due to the so-called "reinforcement", for example aluminum foil. Usually the foil coating is made on a warm surface to avoid condensation. This coating reduces the radiant flux of about 10 times. The foil pasting of the second surface will practically not change the result, since _пр with almost no change. The most effective method to increase the interlayer heat resistance is its screening, i.e. the separation by thin membranes-walls into a number of thin layers. With presence of screens _пр can be represented as

(146)

where n is number of screens; _экр is emissivity of the surface of the screens.

Even one screen of the same material as the main structure halves radiative flux when ₁ = ₂ = _экр. One screen of aluminum foil reduces the flow by 20 times. This effect increases with number of screens. This principle is the basis for insulation made in the form of flat foils which are placed at a distance of 10 – 15 mm from each other, or in the form of corrugated foil sheets. Conditional thermal conductivity coefficients of such insulating layers for smooth and corrugated sheets are equal to 0.31 and 0.28 W∙m^-1∙K^-1, respectively.

Usually in constructions the protective interlayers are in contact with the air as a result of infiltration and exfiltration through the material and the joints of individual structural elements. In so doing, thermal resistance decreases with decreasing tightness.

Heat transfer through the envelope in the presence of air permeability

In modern multistory buildings made ​​of large elements the air permeability affects significantly the thermal regime and heat loss, as well as the heat transfer through the particular envelopes.

During the filtration of air through a porous envelope the temperature field and heat transfer on surfaces are changed noticeably. This is due to the heat transfer by the filter percolating air flow. Flow rate of air coming through the envelope are usually small and are up to 10 m³/h per 1 m² of surface. Air moves through the pores and capillaries slowly (Reynolds number about 0.05) and its temperature in all sections of the envelope is almost equal to the temperature of the solid material. This simplifies significantly the mathematical consideration of the heat transfer through air-penetrable envelope.

Consider the heat balance of the elementary layer located at a distance x from the outer surface of the envelope, through which the infiltration of the outdoor air occurs. Denote the density of the mass flux as j.

Due to the thermal conductivity from the layer n+1 to the layer n the heat flow q_n+1,n will go; similarly, the heat flow q_n,n-1 will go from the layer n to the layer n-1. The difference between these flows can be written in the form of finite differences:

(147)

Outdoor air is filtered against the the thermal conductivity flow, flowing from layer n-l to the layer n and then passing from layer n to layer n+1. The difference in the quantity of heat brought by the air in layer n and blown out of it will be

(148)

The heat quantity, which will accumulate or disperse for the time interval  in a layer as a result of the thermal conductivity and heat transfer by air, will be equal to

. (149)

This heat will change the temperature of the layer. By the end of the time interval  the temperature of the material layer with a thickness of x and volume heat capacity c will change by _τt. In so doing,

. (150)

Equation (150) can be written in differential form

(151)

For the stationary temperature field equation (151) has the form

(152)

It is convenient to write the equation, using a thermal resistance scale. We multiply both terms of equation (152) by :

(153)

The solution of equation (153) is

(154)

The integration constants C₁ and C₂ can be determined from the boundary conditions:

at R = 0 t = t_н;

at R = R₀ t = t_в.

As a result, the formula for determining t in an arbitrary cross-section takes the following form:

(155)

The formula for determination of heat flux q in an arbitrary section of the envelope can be obtained differentiating equation (155):

(156)

When the exfiltration of air from the room through a wall takes place, j is taken as negative in the formulas (155) and (156).

In the absence of filtration j = 0 and the equation (155) becomes the uncertainty, which can by revealed using the L'Hospital rule:

. (157)

As a result, we obtain equation of a steady temperature field considered above:

(158)

When infiltration, as follows from (156), the heat flow at the inner surface of the envelope is the greatest. By approaching the outer surface q is decreased. This phenomenon occurs as a result of recovery (partial return) of a heat for heating the outside air, which is filtered through the envelope against the heat flow. From equation (156) it follows that the difference in heat flows on the inside q_в and the outside q_н thermal boundaries of the envelope is equal to

(159)

At high flow rates of air infiltration through the porous envelope the heat loss will not actually be, because the heat is almost entirely used for heating the outside air. As a result, it is possible to use the pore heating effect for the energy saving during ventilation. Heat consumption can be significantly reduced applying the airing through the outer porous envelope instead of conventional ventilation with outdoor air heating. In conventional ventilation, the heat consumption will be equal to the sum of the heat loss through the envelopes and heat loss for heating of the outside air. The considered heat effect is used to heat saving on ventilation.

The problem of the damping of temperature oscillations in the building envelopes

Outside air temperature is constantly changing, undergoing seasonal, diurnal and shorter in duration fluctuations. Heat generation and accumulation in the room are also constantly changing as a result of the functional processes and climate conditioning systems. In this connection the temperature of indoor air, surfaces, and the individual envelope layers is changed.

Building envelopes react differently to temperature variations. Some change the temperature after the outer and inner airs rapidly, others slowly. Therefore, temperature fluctuations of the outdoor air are transferred to the inner surface and the indoor air through one envelope rather than through others. This property of envelopes is related to their thermal stability.

Thermal stability is a property of envelope to keep the relative constant temperature at fluctuations of the heat flux.

It is most simply to consider the thermal stability of envelopes under harmonic oscillations. Periodic fluctuations in outdoor or indoor air temperatures, various types of heat gain in the room can be often presented without much error in the form of regular harmonic oscillations or the sum of a number of harmonics. The overall impact of the complex changes in the heat exchange conditions on the heat mode of envelopes or rooms can be represented as the sum of the particular effects of harmonics. Therefore, the study of the envelope properties under the action of regular periodic oscillation has a direct practical application.

To simplify the derivation of the basic calculation formulas of exact analytical solutions, first consider the problem of the damping of oscillations, when the average outside t_в0 and inside t_в temperatures are the same and equal to zero.

Assume that the outdoor temperature changes making the harmonic oscillations with a period T and amplitude A_iн near the average temperature t_н0 = 0 = const. Indoor temperature t_в = 0 = const. Heat transfer coefficients on the surfaces are _в and _н, thermal characteristics , c, ρ, and the thickness of the envelope layers  are specified. On the inner and outer envelope surfaces the heat transfer conditions of the third kind are given. At the boundaries between the material layers in the wall thickness the heat flows and temperatures are equal.

The problem is to determine the temperature oscillations damping in the bulk and on the surfaces of envelope, i.e to determine the temperature t(x, ) in any section x at an arbitrary time . From all variety of mathematical notation forms for the solution of the problem under consideration it is most convenient to use a decision with complex and hyperbolic functions.

The differential equation of heat conduction can be solved by separation of variables. The required dependence t(x, ) is represented as the product of two independent functions, one of which depends only on x and the other only on :

(160)

Differentiating and grouping the variables, we obtain

(161)

(162)

(163)

The solution of the differential equation (162) is a function

(164)

Constant ² in this task should be chosen so that the required dependences were periodic functions of time. Exponent (164) may be such a function if ² is an imaginary value equal

(165)

where  = 2π/T is angular frequency.

Write the expression (164) in the form

(166)

Function f₁(x) in the differential equation (163) should have a second derivative equal to itself. In particular, the hyperbolic sine and cosine may be such functions. For the simplest case, when f₁ depends only on one variable x, the general solution of (163) can be written as the sum of two particular solutions:

(167)

The solution of equation (160) can be written as:

(168)

or

The heat flux q(x, ) depending on time  and space x can be obtained by differentiating in x:

(169)

In equation (169)

(170)

S is the specific thermal absorptivity of the layer material and depends on thermal properties of the material  and c, density ρ and the oscillation period T.

The values ​​of the C₁ and C₂ coefficients can be determined by equations (168) and (169) by substituting to them the conditions on the internal envelope boundary. Assuming the origin of coordinates x = 0 at the inner surface, we obtain from the equation (168)

(171)

C₁ is the amplitude of the temperature oscillations at the inner surface of the envelope with regard to its initial phase. Assuming the initial time for the peak temperature in the section x = 0, C₁ is equal to the amplitude of the temperature oscillations at the inner surface A_t(x=0), and the harmonic oscillations of the surface temperature will be described by the equation

(172)

The heat flow on the inner surface according to equation (169) is

(173)

Modifier before an exponent in equation (173) corresponds to the oscillation amplitude of the heat flow taking into account the initial phase. Its real value is equal to the amplitude of the heat flux A_q(0).

Thus, one can obtain the values t(x, ) and q(x, ) when boundary conditions on the surface (x = 0) are given. On the inner envelope surface the condition of heat exchange with the indoor air can be written as

(174)

By the problem provisions, the indoor air temperature is constant and equal to t_в = 0. It follows from (174) that the ratio of the heat flux to the surface temperature at any time is equal to the heat transfer coefficient:

(175)

The ratio (175) is constant and equal to . It also follows from that there is no phase shift between the oscillations of q and t on the surface x = 0, and so the ratio of amplitudes also will be equal to :

(176)

Therefore, under normal heat exchange conditions on the inner surface not the temperature or heat flux change characteristics, but the ratio of these values is given.

In the theory of thermal stability the value called the heat absorption coefficient of surface Y is used. For an arbitrary envelope section x at any given time 

(177)

Heat absorption coefficient of surface is numerically equal to the heat flow per unit area of ​​the surface when its temperature is changing by 1 degree.

The value Y, as well as q and t, is a complex number and its module Y_м(x) is the ratio of the amplitudes:

(178)

and argument _y(x) is the phase advance by heat flow _q(x) of the temperature _t(x) in the section x:

(179)

Returning to the condition on the inner surface, it is possible to record the value of Y(x = 0) as

(180)

In so doing,

(181)

In this particular case, the heat absorption coefficient is equal to its modulus and is real number .

Using expressions (168) and (169) we can write the general expression for Y(x): (182)

In equation (182) based on (170) the replacement is made:

(183)

where R is the thermal resistance of the material layer of thickness x. Furthermore, the ratio C₂/C₁ is recorded regarding Y(x = 0). Based on the equations (171), (173)

(184)

and hence

(185)

In further calculations, the so-called damping factor of temperature oscillations v(x) is a convenient characteristic of the considered process. Its value shows how many times the temperature oscillations are damped by going a layer of material from the section x to the section x = 0. The ratio of temperature oscillations in the planes x (168) and x = 0 (171) is

(186)

The equation for v(x) can be found in a different form. Using equation (186) and the first record of the formula (182), we can write and transform identity

(187)

Based on the formula chx + shx = e^x we obtain from (187)

(188)

To determine v(x) and Y(x) in an arbitrary layer section x, as seen from the formulas (188) and (182), it is necessary to specify only one boundary condition in the form of heat absorption coefficient Y(0) at the boundary x = 0.

For the first layer of a multilayer envelope (which borders the room) the value of Y(0) =  is specified. Using equation (182), which includes the characteristics of only the first layer, it is possible to determine the coefficient of heat absorption Y₁ on the surface of the first layer, which borders with the second layer. Having Y₁ as a boundary condition for the layer 2 one can to calculate the oscillation damping v₂ in the layer 2 by (188) and the heat absorption coefficient Y₂ at the boundary with layer 3 by (182). Going consecutively from layer to layer, we can do the calculation for the last layer, which ends on the outer surface of the envelope. The values ​​of v and Y for all layers, including the last one, depends on only one boundary condition Y(0) on the inner surface of the envelope and on the geometry and thermal properties of the layers. One can write the formulas (188) and (182) in a general form for an arbitrary layer n, which is preceded by the layer n - 1:

(189)

(190)

Thus, having data about the outdoor temperature variation, heat transfer coefficients _н and _в, as well as all characteristics of the envelope, one can be calculated using damping coefficients the temperature variations at any section including the inner envelope surface.

Room heating

The heating device submits a heat from the heat-transfer agent of a heating system to the heated room. Its design, installation procedures, connection to the heating system should be thoroughly evaluated by the thermo-technical, economic, technical and aesthetic parameters.

For this it is necessary to determine the amount of heat expended on the room heating, the optimal design of the device, the ratio of convective and radiative heat given by them, as well as to estimate the microclimate created by the heating device.

Use of the device a particular design and its installation in various places of the room must not lead to a significant overexpenditure of heat.

The most common radiators usually mounted in recesses or near the outer surface of the wall. The surface behind the radiator overheats and much useless heat is lost through this part of the outer wall. Convectors spaced along the outer walls are better than radiators.

Indoor heat comfort depends not only on the quantity of heat supplied, but also on the location of the heater in the room, and its geometry. Heating devices compensating for heat loss should also serve as localizers of the cold in the room. Therefore, the heated surface of the device and the jet of warm air over it have to prevent the radiative supercooling and penetration of cold air flows in the occupied zone of the room.

Decision when all external fencings are uniformly heated and there are no cooled surfaces in the room is ideal in this respect.

Devices located under the windows along the exterior walls create good thermal conditions in the room. In this case, the service area and especially the area of ​​the floor of the room, which is particularly subject to overcooling by flowing streams of air, is thermally protected most efficiently.

In shallow rooms, where the distance from the outer wall to the opposite inner wall is small, the devices can be placed in the interior walls. The heating system in this case is compact. Estimation of the room heating efficiency at various heating devices can be approximately given by the temperature distribution over the height of the room. Formation of the thermal pads on the ceiling and the overheating of the upper zone of the room increases heat loss. Heating at uniform temperature distribution in height is the best.

Heat transfer of heating device. Through the heater walls a heat accumulated by the heat-transfer agent is transmitted into the rooms.

It is usually assumed that the heating device and the heat-transfer agent in it have a negligible thermal inertia, and therefore the amount of heat Q_т.н donated by heat-transfer agent at each time point is equal to the amount of heat transmitted to the room Q_пр:

(191)

The heat sulllied by the heating device Q_пр is determined by the fundamental equation of heat transfer in the form of

(192)

In this equation, it is assumed that the quantity of the given heat is proportional to the difference between the average heat carrier temperature in the device t_пр and the temperature of heated room t_пр. The heat carrier temperature in the device depends on the device construction and the method of its connection to the system, the type and heat carrier parameters. F_пр in (192) is external (usually artificially developed by thickening and ribbing) heat-transferring air washed device surface. Thermal resistance of heater R_пр is equal

(193)

Thermal resistance R_в to the heat transfer from the heat-transfer agent to the inner wall surface of the heating with area F_в.п should be divided to F_пр, so

(194)

Heat transfer coefficient _в between the heat-transfer agent and the inner surface of the device wall is determined mainly by the rate of flow in the device. depends on the flow and the diameter of the inner cavity of the device.

With the increase of water flow rate the heat transfer coefficient first increases considerably, and at high flow rate remains practically unchanged. In the cast iron radiators the water velocity in the column sections is usually small (about 0.01 m/s) and _в  58 Wm^-2K^-1. The ratio for them is about 1.3.

Thermal resistance of the device wall R_т can be written as

(195)

In a cast-iron radiator the wall thickness h is usually less than 0.01 m, and  is equal about 47 W∙m^-1∙K^-1.

For radiators R_в and R_т are negligible small. In concrete panels with grouted pipe coils the heat resistance R_т of array of concrete bulk from pipes to the panel surface has a significant amount and depends on the pipe diameter d, the distance (pitch) between them s, distance from the surface h, the thermal conductivity of the panel material , as well as the panel design.

Heat transfer resistance R_н at the outer surface of the device (all the components of R_пр are divided to its area F_пр) is

(196)

where _н – heat transfer coefficient on the heated surface of the device in the room.

Usually, the external surface of heating devices is ribbed. The ribbing influence on R_н can be taken into account as follows.

The quantity of heat Q transmitted by a single rib with a thickness of 2, width a and height from the ground l to the environment with a temperature t_в (t₀ is the temperature of rib) can be determined by the formula

(197)

where 2al = F_p — rib surface area.

When condition

is true for ribbing, the resistance to heat conduction of bulk of ribs can be ignored, and resistance to heat transfer at the ribbed outer surface can be determined by the formula (196).

For cast-iron devices at _н  8 and rib thickness of 2  0.005 m the heat transfer resistance on the outer surface R_н can be calculated by the formula (196) in almost all cases of actual designs.

When , the heat transfer resistance on the ribbed surface does not depend on l and its value is constant.

The coefficient _н for the surface of heating devices can be roughly determined as the sum of the coefficients for the convective _к and radiative _л heat transfer.

Convective heat transfer intensity _к is dependent on the height of the device. The lower part of the device is washed by a cold air and gives more heat than the top, which is washed by a warmer air. The average heat emission across the surface decreases with the height of device. In this regard, the best devices are low and represent for example horizontally arranged in one row smooth or ribbed pipes. By arranging the heating elements in several rows one above the other the convective heat transfer coefficient decreases. In the multilayered devices special deflector plates that lead the cold air to each series are set.

Radiative heat transfer depends on the design of devices. Radiators have closely located columns shielding each other. Most of the radiation from their external surface misses the room. Not more than 30 % of the heat emitted by the radiator surface is transmitted to the room. For devices with strongly ribbed surfaces (such as ribbed pipes) the radiation part that enters the room from all external heat-transferred surface is only 5 – 10 %.

If device in the room is closed, its heat emission by radiation decreases. At a certain position of the closing screen the reduction of radiative heat transfer is accompanied by a marked increase in the convective heat transfer from the device.