Conductive heat transfer. Conductive heat transfer Conductive heat transfer in a flat wall

Conductive heat transfer (lat. conduce, conductum to reduce, connect) T. by conducting heat to (or from) the surface of any solid body in contact with the surface of the body.

Large medical dictionary. 2000 .

See what “conductive heat transfer” is in other dictionaries:

Heat transfer due to the combined transfer of heat by radiation and thermal conductivity... Polytechnic terminological explanatory dictionary

radiation-conduction heat exchange- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Topics: energy in general EN heat transfer by radiation and conduction ... Technical Translator's Guide

The Vernon sphere spherical thermometer is a hollow, thin-walled, metal (brass or aluminum) sphere with a diameter of 0.1-0.15 m. The outer surface of the sphere is blackened so that it absorbs ε ≈ 95% of the thermal... ... Wikipedia

Thermal properties of materials- Rubric terms: Thermal properties of materials Humidity state of the enclosing structure Operating humidity ... Encyclopedia of terms, definitions and explanations of building materials

- (a. survival suit, protective gear; n. Schutzanzug, Schutzkleidung; f. costume de protection; i. traje protector) in the mining industry, special clothing to protect mine rescuers, firefighters, etc. from the harmful effects of the environment… … Geological encyclopedia

Books

• Heat transfer and thermal testing of materials and structures of aerospace technology during radiation heating, Viktor Eliseev. The monograph is devoted to the problems of heat transfer and thermal testing of materials and structures of aerospace technology using high-intensity radiation sources. The results are given...

PREFACE

“Hydraulics and Heat Engineering” is a basic general engineering discipline for students studying in the direction of “Environmental Protection”. It consists of two parts:

Theoretical foundations of technological processes;

Typical processes and devices of industrial technology.

The second part includes three main sections:

Hydrodynamics and hydrodynamic processes;

Thermal processes and apparatus;

Mass transfer processes and apparatus.

For the first part of the discipline, lecture notes by N.Kh. Zinnatullina, A.I. Guryanova, V.K. Ilyina (Hydraulics
and Heat Engineering, 2005); for the first section of the second part of the discipline - textbook N.Kh. Zinnatullina, A.I. Guryanova, V.K. Ilyina, D.A. Eldasheva (Hydrodynamics and hydrodynamic processes, 2010).

This manual outlines the second section of the second part. This section will discuss the most common cases of conductive and convective heat transfer, industrial methods of heat transfer, evaporation, as well as the operating principle and design of heat exchange equipment.

The textbook consists of three chapters, each of which ends with questions that students can use for self-control.

The main objective of the presented textbook is to teach students to carry out engineering calculations of thermal processes and select the necessary equipment for their implementation.

PART. 1. HEAT TRANSFER

Industrial technological processes proceed in a given direction only at certain temperatures, which are created by supplying or removing thermal energy (heat). Processes, the rate of which depends on the rate of heat supply or removal, are called thermal. The driving force of thermal processes is the temperature difference between the phases. The devices in which thermal processes are carried out are called heat exchangers; heat is transferred into them by coolants.

The calculation of heat transfer processes usually comes down to determining the interphase heat transfer surface. This surface is
from the heat transfer equation in integral form. The heat transfer coefficient, as is known, depends on the heat transfer coefficients of the phases,
as well as from the thermal resistance of the wall. Below we will consider methods for determining them, finding the temperature field and heat flows. Where possible, the required quantities are found from solving the equations of conservation laws, and in other cases simplified mathematical models or the method of physical modeling are used.

Convective heat transfer

During convection, heat transfer occurs by macrovolume particles of the coolant flow. Convection is always accompanied by thermal conductivity. As is known, thermal conductivity is a molecular phenomenon, convection is a macroscopic phenomenon, in which
Whole layers of coolant with different temperatures are involved in heat transfer. Heat is transferred much faster by convection than by conduction. Convection near the surface of the apparatus wall decays.

Convective heat transfer is described by the Fourier-Kirchhoff equation. The patterns of medium flow are described by the Navier-Stokes (laminar regime) and Reynolds (turbulent regime) equations, as well as the continuity equation. The study of the patterns of convective heat transfer can be carried out in isothermal and non-isothermal formulations.

In the isothermal formulation, the Navier-Stokes and continuity equations are first solved, then the obtained velocity values ​​are used to solve the Fourier-Kirchhoff equation. The values ​​of heat transfer coefficients obtained in this way are subsequently refined and corrected.

In the non-isothermal formulation, the Navier-Stokes, continuity and Fourier-Kirchhoff equations are solved jointly, taking into account the dependence of the thermophysical properties of the medium on temperature.
As experimental data show, the dependences with p(T), l( T)
and r( T) are weak, and m( T) - very strong. Therefore, usually only the dependence m( T). It, this dependence, can be presented in the form of an Arrhenius dependence or, more simply, in the form of an algebraic equation. Thus, so-called coupled problems arise.

Recently, methods have been developed for solving many problems of heat transfer in laminar fluid flows, taking into account the dependence of the fluid viscosity on temperature. For turbulent flows everything is more complicated. However, it is possible to use approximate numerical solutions using computer technology.

To solve these equations, it is necessary to set uniqueness conditions, which include initial and boundary conditions.

Heat transfer boundary conditions can be specified in various ways:

Boundary conditions of the first kind are specified by the wall temperature distribution:

; (19)

the simplest case is when T c t = const;

Boundary conditions of the second kind - the heat flow distribution on the wall is specified

; (20)

Boundary conditions of the third kind - the temperature distribution of the medium surrounding the channel and the heat transfer coefficient are specified
from the environment to the wall or vice versa

. (21)

The choice of the type of boundary condition depends on the operating conditions of the heat exchange equipment.

On a flat plate

Let us consider a flow with constant thermophysical characteristics (r, m, l, c p= const), performing forced movement along a flat semi-infinite thin plate and exchanging heat with it. Let us assume that an unlimited flow with a speed
and temperature T° runs into a semi-infinite plate coinciding
with plane X – z and having a temperature T st = const.

Let us distinguish the hydrodynamic and thermal boundary layers
with thickness d g and d t respectively (area 99% change in speed w x
and temperature T). In the thread core and T° are constant.

Let us analyze the continuity and Navier-Stokes equations. The problem is two-dimensional because w z, . According to experimental data, it is known that in a hydrodynamic boundary layer . In the thread core const, therefore, according to the Bernoulli equation , in the boundary layer the same

.

As is known " X» d g, therefore .

Therefore, we have

; (22)

. (23)

Write similar equations for the axis at doesn't make sense because w y can be found from the continuity equation (22). Using similar procedures, you can simplify the Fourier-Kirchhoff equation

. (24)

The system of differential equations (22)–(24) constitutes an isothermal mathematical model of a flat stationary thermal laminar boundary layer. Let us formulate the boundary conditions
at the boundary with the plate, i.e. at at= 0: for any X speed w x= 0 (no-slip condition). On the boundary and outside the hydrodynamic boundary layer,
those. at at≥ d g ( X), as well as X= 0 for any at: w x= . For the temperature field there are similar arguments.

So, the boundary conditions:

w x( x, 0) = 0, x > 0; w x (x, ∞) = ; w x(0, y) = ; (25)

T (x, 0) = T st, x > 0; T (x, ∞) = T ° ; T (0, y) = T°. (26)

An exact solution to this problem in the form of infinite series was obtained by Blasius. There are simpler approximate solutions: the method of integral relations (Yudaev) and the momentum theorem (Schlichting). A.I. Razinov solved the problem using the method of conjugate physical
and mathematical modeling. Velocity profiles were obtained
w x (x, y), w y ( x,y) and temperatures T, as well as the thickness of the boundary layers
d g ( x) and d t ( X)

; (27)

, Pr ≥ 1; (28)

Pr= ν/a.

Coefficient A in formula (27) for Razinov – 5.83; Yudaeva – 4.64; Blausius – 4; Sizing – 5.0. An approximate form of the found dependencies is shown in Fig. 1.3.

As is known, for gases Pr≈ 1, drip liquids Pr > 1.

The results obtained make it possible to determine the coefficients of momentum and heat transfer. Local values ​​γ( x) And Nu G, x

, . (29)

y

w x

T st

(T–T st)

d g ( x)

d t ( x)

x

Rice. 1.3. Hydrodynamic and thermal laminar boundary layers

on a flat plate

Average values ​​and along a section length l

,
, . (30)

Similarly for heat transfer

,
; (31)

, . (32)

In this case, the analogy of heat and impulse transfer is preserved (the initial equations are the same, the boundary conditions are similar). The criterion characterizing the hydrodynamic analogy of the heat transfer process has the form

P t-g, x = Nu T, x/Nu G, x = Pr 1/3 . (33)

If Pr= 1, then P t-g, x= 1, therefore a complete analogy of the processes of pulse and heat transfer.

From the resulting equations it follows

γ ~ , m; a ~ , l. (34)

As a rule, such a qualitative dependence holds
not only for a flat boundary layer, but also for more complex cases.

The problem is considered in an isothermal formulation, thermal boundary conditions of the first kind T st = const.

As you move away from the edge of the plate (increasing coordinates X) there is an increase in d g ( X). In this case, the inhomogeneity of the velocity field w x spreads to areas increasingly distant from the phase boundary,
which is a prerequisite for the occurrence of turbulence. Finally, when Rex, kp the transition from laminar to turbulent regime begins. The transition zone corresponds to the values X, calculated according to Rex from 3.5 × 10 5 ÷ 5 × 10 5.
At distances Rex> 5 × 10 5 the entire boundary layer is turbulized,
with the exception of a viscous or laminar sublayer with a thickness of d 1g. In the core of the flow, the speed does not change. If Pr> 1, then inside the viscous sublayer we can distinguish a thermal sublayer of thickness d 1m, in which molecular heat transfer prevails over turbulent heat transfer.

The thickness of the entire turbulent thermal boundary layer is usually determined from the condition ν t = a t, therefore d g = d t.

First, consider a turbulent hydrodynamic boundary layer (Fig. 1.4). Let us leave in force all the approximations made for the laminar layer. The only difference is the presence of ν t ( at), That's why

. (35)

Let us also preserve the boundary conditions. By solving the system of equations (35)
and (22) with boundary conditions (25), using the semi-empirical Prandtl wall turbulence model, the characteristics of the turbulent boundary layer can be obtained. In the viscous sublayer, where the linear law of velocity distribution is implemented, turbulent momentum transfer can be neglected, and outside it, molecular transfer. In the near-wall region
(minus the viscous sublayer), a logarithmic velocity profile is usually adopted, and in the outer region, a power law with an exponent of 1/7 (Fig. 1.4).

Rice. 1.4. Hydrodynamic and thermal turbulent boundary layers

on a flat plate

As in the case of a laminar boundary layer, it is possible to use length-averaged l impulse return coefficients

. (36)

Let us consider a thermal turbulent boundary layer. The energy equation is

. (37)

If Pr> 1, then inside the viscous sublayer we can distinguish a thermal sublayer, where molecular heat transfer

. (38)

For the local heat transfer coefficient, the solution of the mathematical model has the form

Average value over the plate length defined like this

Below are the formation of a turbulent boundary layer (a) and the distribution of the local heat transfer coefficient (b) during longitudinal flow around a flat semi-infinite plate (Fig. 1.5).

Rice. 1.5. Boundary layers d g and d t and local heat transfer coefficient a

on a flat plate

In the laminar layer ( X ≤ l kr) heat flow only due to thermal conductivity; for a qualitative assessment, the relation a ~ can be used.

In the transition zone, the total thickness of the boundary layer increases. However, the value of a increases in this case, because the thickness of the laminar sublayer decreases, and in the resulting turbulent layer, heat is transferred not only by thermal conductivity, but also by convection together
with a moving mass of liquid, i.e. more intense. As a result, the total thermal resistance of heat transfer decreases. In the zone of developed turbulent regime, the heat transfer coefficient again begins to decrease due to an increase in the total thickness of the boundary layer a ~ .

So, the hydrodynamic and thermal boundary layers on a flat plate are considered. The qualitative nature of the obtained dependencies is also valid for boundary layers formed during flow around more complex surfaces.

Heat transfer in a round pipe

Let us consider stationary heat exchange between the walls of a horizontal straight pipe of circular cross-section and a flow that has constant thermophysical characteristics and moves due to forced convection inside it. Let us accept thermal boundary conditions of the first kind, i.e. T st = const.

I.Areas of hydrodynamic and thermal stabilization.

When liquid enters the pipe, due to braking caused by the walls, a hydrodynamic boundary layer is formed on them.
As you move away from the entrance, the thickness of the boundary layer increases,
while the boundary layers adjacent to the opposite walls
won't close. This section is called the initial or hydrodynamic stabilization section - l ng.

Just as the velocity profile changes along the length of the pipe, the
and temperature profile.

II.Let's consider laminar fluid motion.

Earlier, in the section of the discipline “Hydrodynamics and hydrodynamic processes”, we examined the hydrodynamic initial section. To determine the length of the initial section, the following relationship was proposed

.

For liquid Pr> 1, therefore, the thermal boundary layer will be located inside the hydrodynamic boundary layer.
This circumstance allows us to assume that the thermal boundary layer develops in a stabilized hydrodynamic section and the velocity profile is known - parabolic.

The temperature of the liquid in the inlet section of the heat exchange section is constant over the cross section and is equal to T° and in the thread core it does not change. Under these conditions, the thermal boundary layer equation has the form

. (41)

Solving this equation under the above conditions gives:

for the length of the thermal initial section

; (42)

for local heat transfer coefficient

; (43)

for average heat transfer coefficient length

; (44)

· for local Nusselt number

; (45)

· for the average Nusselt number

. (46)

Let's consider equation (42). If , That .
For liquids Pr> 1, so in most cases, especially
for liquids with large Pr, heat exchange during laminar movement occurs mainly in the thermal stabilization section. As can be seen from relation (43), a for a pipe in the thermal stabilization section decreases with distance from the inlet (the thickness of the thermal boundary layer dt increases) (Fig. 1.6).

Rice. 1.6. Temperature profile at the initial and stabilized sections

with laminar flow of liquid in a cylindrical pipe

With a turbulent flow in a pipe, as on a flat plate, firstly, the thicknesses of the hydrodynamic and thermal boundary layers coincide; and secondly, they grow much faster than for laminar ones. This leads to a decrease in the length of thermal sections
and hydrodynamic stabilization, which allows in most cases to neglect them when calculating heat transfer

. (47)

III.Stabilized heat transfer with laminar movement of the medium.

Let us consider stationary heat transfer in a round pipe, when the thermophysical properties of the liquid are constant (isothermal case), the velocity profile does not change along the length, the temperature of the pipe wall is constant and equal to T st, there are no internal heat sources in the flow,
and the amount of heat released due to energy dissipation is negligible. Under these conditions, the heat transfer equation has the same form as for the boundary layer. Therefore, the initial equation for studying heat transfer is equation (41).

Border conditions:

(48)

The solution to this problem was first obtained by Graetz, then by Nusselt, in the form of a sum of an infinite series. A slightly different solution was obtained by Shumilov and Yablonsky. The resulting solution is correct
and for the thermal stabilization section, subject to preliminary hydrodynamic stabilization of the flow.

For the region of stabilized heat transfer, the local heat transfer coefficient is equal to the limiting one

or (49)

As can be seen from the figure (Fig. 1.7), with increasing number Nu decreases, asymptotically approaching in the second section of the curve
to a constant value Nu= 3.66. This occurs because for stabilized heat transfer the temperature profile along the length of the pipe
does not change. In the first section, a temperature profile is formed. The first section corresponds to the thermal initial section.

10 –5 10 –4 10 –3 10 –2 10 –1 10 0

1

3,66

Nu

Nu

Rice. 1.7. Change local and average Nu along the length of a round pipe at T st = const

IV.Stabilized heat transfer during turbulent movement of the medium.

Original equation

. (50)

Border conditions:

(51)

When solving the problem, the problem of choosing a speed profile arises w x. Some for w x use the logarithmic law (A.I. Razinov), others use the 1/7 law (V.B. Kogan). The conservatism of turbulent flows is noted, which consists in the weak influence of boundary conditions and the velocity field w x on heat transfer coefficients.

The following formula is proposed for the Nusselt number

. (52)

As for laminar movement in the region of stabilized heat exchange with turbulent flow of the medium Nu does not depend on the coordinate X.

We considered above special cases of heat transfer, namely: with an isothermal formulation of the problem and thermal boundary conditions of the first kind, heat transfer in smooth cylindrical pipes and flat horizontal plates.

In the literature there are solutions to thermal problems for other cases. Note that the surface roughness of the pipe and plate leads to
to increase the heat transfer coefficient.

Heat supply

To solve this problem, various coolants are used.
TN are classified by:

1. By purpose:

Heating HP;

Cooling HP, coolant;

Intermediate TN;

Drying agent.

2. According to the state of aggregation:

· Single-phase:

Low temperature plasma;

Non-condensing vapors;

Liquids that do not boil and do not evaporate at a given pressure;

Solutions;

Grainy materials.

· Multi-, two-phase:

Boiling, evaporating and gaseous liquids;

Condensing vapors;

Melting, solidifying materials;

Foams, gas suspensions;

Aerosols;

Emulsions, suspensions, etc.

3. By temperature and pressure range:

High-temperature HP (smoke, flue gases, molten salts, liquid metals);

Medium temperature heat pumps (water vapor, water, air);

Low temperature HP (at atmospheric pressure T kip ≤ 0 °C);

cryogenic (liquefied gases - oxygen, hydrogen, nitrogen, air, etc.).

As pressure increases, the boiling point of liquids also increases.

Industrial enterprises use flue gases and electricity as direct sources of thermal energy. Substances that transfer heat from these sources are called intermediate heating elements. The most common intermediate TN:

Water vapor is saturated;

Hot water;

Overheated water;

Organic liquids and their vapors;

Mineral oils, liquid metals.

Requirements for TN:

Big r, with p;

High heat of vaporization;

Low viscosity;

Non-flammable, non-toxic, heat resistant;

Cheapness.

Heat removal

Many industrial technology processes take place under conditions where there is a need to remove heat, for example, when cooling gases, liquids or during condensation of vapors.

Let's look at some cooling methods.

Cooling with water and low-temperature liquid refrigerants.

Water cooling is used to cool the medium to 10–30 °C. River, pond and lake water, depending on the time of year, has a temperature of 4–25 °C, artesian water – 8–12 °C, and circulating water (in summer) – about 30 °C.

Cooling water flow determined from the heat balance equation

. (83)

Here is the flow rate of the coolant; N n and N k – initial
and the final enthalpy of the coolant being cooled; N nv and N kv – initial
and final enthalpy of cooling water; – losses to the environment.

Achieving lower cooling temperatures can be achieved
using low temperature liquid refrigerants.

Air cooling. Air is most widely used as a cooling agent in mixing heat exchangers - cooling towers, which are the main element of the water circulation cycle equipment (Fig. 2.5).

Rice. 2.5. Cooling towers with natural (a) and forced (b) draft

The hot water in the cooling tower is cooled both by contact with cold air and by so-called evaporative cooling,
in the process of evaporation of part of the water flow.

Mixing heat exchangers

In mixing heat exchangers (MHE), heat transfer from one coolant to another occurs when they are in direct contact or mixing, therefore, there is no thermal resistance of the wall (separating the coolant). Most often, SRT is used for condensing vapors, heating and cooling water and vapors. Based on the design principle, service stations are divided into bubbling, shelf, packed and hollow (with liquid splashing) (Fig. 2.18).

steam

water

V

air

water

water

water

steam

G

steam

heated liquid

A

air

water

steam

water + condensate

b

liquid

Rice. 2.18. Service station diagrams: a) bubbling mixing heat exchanger for heating water;

b) packed heat exchanger-condenser; c) shelf barometric capacitor; d) hollow

PART 3. EVAPORATION

Evaporation is the process of concentrating solutions of non-volatile solids by removing the volatile solvent in the form of vapor. Evaporation is usually carried out at boiling. Typically, only part of the solvent is removed from a solution, since the substance must remain
in a fluid state.

There are three evaporation methods:

Surface evaporation is carried out by heating the solution on the heat exchange surface due to the supply of heat to the solution through the wall from the heating steam;

Adiabatic evaporation, which occurs by flashing a solution in a chamber where the pressure is lower than the saturated vapor pressure;

Evaporation by contact evaporation - heating of a solution is carried out by direct contact between a moving solution
and hot coolant (gas or liquid).

In industrial technology, the first evaporation method is mainly used. Next about the first method. To carry out the evaporation process, it is necessary to transfer heat from the coolant to the boiling solution, which is only possible if there is a temperature difference between them. The temperature difference between the coolant and the boiling solution is called the useful temperature difference.

Saturated water vapor (heating or primary) is used as a coolant in evaporators. Evaporation is a typical heat exchange process - the transfer of heat due to the condensation of saturated water vapor to a boiling solution.

Unlike conventional heat exchangers, evaporators consist of two main units: a heating chamber or boiler and a separator. The separator is designed to catch drops of solution from steam that forms during boiling. This steam is called secondary or juice. The temperature of the secondary steam is always less than the boiling point of the solution. To maintain a constant vacuum in the condenser, it is necessary to suck out the vapor-gas mixture with a vacuum pump.

Depending on the pressure of the secondary steam, evaporation is distinguished at R atm, R hut, R vac. In case of evaporation at R vac the boiling point of the solution decreases, with p hut - secondary steam is used for technological purposes. The boiling point of a solution is always higher than the boiling point of a pure solvent. For example, for a saturated aqueous solution
NaCl (26%) T kip = 110 °C, for water T kip = 100 °C. Secondary steam taken from the evaporation plant for other needs is called extra ferry.

Temperature losses

Usually in single-shell evaporation plants the pressures of the heating and secondary vapors are known, i.e. their temperatures. The difference between the temperatures of the heating and secondary vapors is called the total temperature difference of the evaporators

. (96)

Total temperature difference is related to the useful temperature difference by the relation

Here D¢ is the concentration temperature depression; D¢¢ - hydrostatic temperature depression; D¢ is determined as the difference in boiling point of the solution T kip. p and pure solvent T kip. chr at p = = const

D¢ = T kip. R - T kip. chr, T kip. chr, D¢ = T kip. R - T vp. (98)

The temperature of the secondary vapors formed during boiling of the solution is lower than the boiling point of the solution itself, i.e. some temperatures are lost uselessly; D¢¢ characterizes the increase in the boiling point of a solution with increasing hydrostatic pressure. Usually, the average pressure is determined by the height of the boiling pipes, and for this pressure the average boiling point of the solvent is determined T Wed

Here p a is the pressure in the apparatus; r pz - density of the vapor-liquid mixture
in boiling pipes ; H- height of boiling pipes.

D² = T Wed - T ch, (99)

Where T cp is the boiling point of the solvent at p = p Wed; T VP - temperature of secondary steam at pressure p A.

Multi-effect evaporation

In a multi-effect evaporator installation, secondary steam (Fig. 3.2, 3.3) from the previous body is used as heating steam
in the subsequent building. This organization of evaporation leads
to significant savings in heating steam. If we accept for all buildings, then the total consumption of heating steam for the process decreases in proportion to the number of buildings. In practice, in real conditions this ratio is not maintained; it is usually higher. Next, we will consider the equations of material and heat balances for a multi-vessel evaporation plant (see Fig. 3.2), which are a system of equations written for each vessel separately.

Lecture 4. CONDUCTIVE HEAT TRANSFER.

4.1 Fourier equation for three-dimensional nonstationary

temperature field

4.2 Thermal diffusivity coefficient. Physical meaning

4.3 Uniqueness conditions – boundary conditions

4.1 Fourier equation for three-dimensional nonstationary

temperature field

The study of any physical process is associated with the establishment of a relationship between the quantities characterizing it. To establish such a dependence when studying the rather complex process of thermal conductivity, methods of mathematical physics were used, the essence of which is to consider the process not in the entire space under study, but in an elementary volume of matter over an infinitesimal period of time. The connection between the quantities involved in the transfer of heat by thermal conductivity is established by a differential equation - the Fourier equation for a three-dimensional non-stationary temperature field.

When deriving the differential equation of thermal conductivity, the following assumptions are made:

There are no internal heat sources;

The body is homogeneous and isotropic;

The law of conservation of energy is used - the difference between the amount of heat that entered the elementary volume due to thermal conductivity during the time dτ and that left it during the same time is spent on changing the internal energy of the elementary volume under consideration.

The body contains an elementary parallelepiped with edges dx, dy, dz. The temperatures of the faces are different, so heat passes through the parallelepiped in the directions of the x, y, z axes.

Figure 4.1 To derive the differential heat equation

According to the Fourier hypothesis, the following amount of heat passes through the area dx·dy during the time dτ:

https://pandia.ru/text/80/151/images/image003_138.gif" width="253" height="46 src="> (4.2)

where https://pandia.ru/text/80/151/images/image005_105.gif" width="39" height="41"> determines the temperature change in the z direction.

After mathematical transformations, equation (4.2) will be written:

https://pandia.ru/text/80/151/images/image007_78.gif" width="583" height="51 src=">, after abbreviation:

https://pandia.ru/text/80/151/images/image009_65.gif" width="203" height="51 src="> (4.4)

https://pandia.ru/text/80/151/images/image011_58.gif" width="412" height="51 src="> (4.6)

On the other hand, according to the law of conservation of energy:

https://pandia.ru/text/80/151/images/image013_49.gif" width="68" height="22 src=">.gif" width="203" height="51 src=">. (4.8)

Value https://pandia.ru/text/80/151/images/image017_41.gif" width="85" height="41 src="> (4.9)

Equation (4.9) is called the differential heat equation or Fourier equation for a three-dimensional unsteady temperature field in the absence of internal heat sources. It is the basic equation when studying the processes of thermal conductivity and establishes a connection between temporal and spatial temperature changes at any point in the temperature field.

Differential equation of thermal conductivity with heat sources inside the body:

https://pandia.ru/text/80/151/images/image019_35.gif" width="181" height="50">

It follows that the change in temperature over time for any point on the body is proportional to the value A.

Value https://pandia.ru/text/80/151/images/image021_29.gif" width="26" height="44">. Under the same conditions, the temperature of the body that has a higher thermal diffusivity increases faster. So gases have a small coefficient of thermal diffusivity, and metals have a large coefficient.

In non-stationary thermal processes A characterizes the rate of temperature change.

4.3 Uniqueness conditions – boundary conditions

The differential equation of thermal conductivity (or a system of differential equations of convective heat transfer) describes these processes in the most general form. To study a specific phenomenon or group of phenomena of heat transfer by conduction or convection, you need to know: temperature distribution in the body at the initial moment, ambient temperature, geometric shape and dimensions of the body, physical parameters of the environment and the body, boundary conditions characterizing the temperature distribution on the surface of the body or conditions of thermal interaction of the body with the environment.

All these particular features are combined into the so-called uniqueness conditions or boundary conditions which include:

1) Initial conditions . The conditions for temperature distribution in the body and the ambient temperature at the initial moment of time τ = 0 are specified.

2) Geometric conditions . They set the shape, geometric dimensions of the body and its position in space.

3) Physical conditions . Set the physical parameters of the environment and the body.

4) Border conditions can be specified in three ways.

Boundary condition of the first kind : the temperature distribution on the body surface is set for any moment in time;

Boundary condition of the second kind : Set by the heat flux density at each point on the surface of the body for any moment in time.

Boundary condition of the third kind : is set by the temperature of the environment surrounding the body and the law of heat transfer between the surface of the body and the environment.

The laws of convective heat exchange between the surface of a solid body and the environment are very complex. The theory of convective heat transfer is based on the Newton-Richmann equation, which establishes a relationship between the heat flux density on the surface of a body q and the temperature pressure (tst - tl), under the influence of which heat transfer occurs on the surface of the body:

q = α·(tst – tl), W/m2 (4.11)

In this equation, α is the proportionality coefficient, called the heat transfer coefficient, W/m2 deg.

The heat transfer coefficient characterizes the intensity of heat exchange between the surface of the body and the environment. It is numerically equal to the amount of heat given off (or perceived) by a unit of body surface per unit of time when the temperature difference between the body surface and the environment is 1 degree. The heat transfer coefficient depends on many factors and its determination is very difficult. When solving problems of thermal conductivity, its value is usually taken to be constant.

According to the law of conservation of energy, the amount of heat given off by a unit surface of a body to the environment per unit of time due to heat transfer must be equal to the heat that is supplied by thermal conductivity to a unit of surface per unit of time from the internal parts of the body:

https://pandia.ru/text/80/151/images/image023_31.gif" width="55" height="47 src="> - projection of the temperature gradient onto the direction of the normal to the site dF.

The above equality is a mathematical formulation of a boundary condition of the third kind.

Solving the differential equation of thermal conductivity (or a system of equations for convective heat transfer processes) under given conditions of unambiguity makes it possible to determine the temperature field in the entire body for any moment in time, that is, to find a function of the form: t = f(x, y, z, τ).

Thermal processes

And devices

HEAT EXCHANGE

Chemical technological processes proceed in a given direction only at certain temperatures, which are created by supplying or removing thermal energy (heat). Processes, the rate of which depends on the rate of heat supply or removal, are called thermal. The driving force of thermal processes is the temperature difference between the phases. The devices in which thermal processes are carried out are called heat exchangers; heat is transferred into them by coolants.

Calculation of heat transfer processes and devices usually comes down to determining the interphase heat transfer surface. This surface is from heat transfer equations in an integral form. Heat transfer coefficient, as is known, depends on the heat transfer coefficients of the phases, as well as on the thermal resistance of the wall. Below we will consider methods for determining them, finding the temperature field and heat flows. Where possible, the required quantities are found from solving the equations of conservation laws, and in other cases simplified mathematical models or the method of physical modeling are used.

Conductive heat transfer in a flat wall

Let us consider heat transfer in a stationary flat wall
from a homogeneous material whose thermophysical properties are constant
(with p, l, r = const) (Fig. 1.1).

Rice. 1.1. Temperature distribution in a flat wall

The general equation of non-stationary Fourier thermal conductivity has the form

(1)

The heat transfer process is stationary, then . We believe
that the height and length are much greater than the wall thickness d, therefore, there is no heat transfer in these directions, then the temperature changes only along one coordinate X, from here we have

Because the , we have

(2)

The obvious solution to this equation is

,

(3)

Border conditions:

at ;

at

We find And , , Then

. (4)

Distribution T by thickness d

. (5)

From the resulting equation (5) it is clear that in a flat wall the distribution T is straightforward.

Heat flow due to thermal conductivity is determined by Fourier's law

; (6)

. (7)

Here characterizes the thermal conductivity of the wall, and is the thermal resistance of the wall.

For a multilayer wall, the thermal resistance of the individual walls must be summed

. (8)

Let us determine the amount of heat transferred over time t across the square F

Real conditions of transfer of mass and energy in various types of thermal processes and natural phenomena are characterized by a complex set of interrelated phenomena, including the processes of radiation, conduction and convective heat exchange. Radiation-conduction heat transfer is one of the most common types of heat transfer in nature and technology

The mathematical form of the problem of radiation-conduction heat transfer follows from the energy equation, supplemented with appropriate boundary conditions. In particular, when studying radiation-conduction heat transfer in a flat layer of an absorbing and radiating medium with opaque gray boundaries, the problem is reduced to solving the energy equation

(26.10.2)

with boundary conditions

Here is the dimensionless flux density of the resulting radiation; - criterion of radiation-conduction heat transfer; - criterion for the dependence of the thermal conductivity of the medium on temperature; - dimensionless temperature in the section of the layer with thickness .

Equation (26.10.1) is a nonlinear integro-differential equation, since in accordance with equation (26.9.13) it is described by an integral expression, and the desired temperature value is presented in equation (26.10.1) both explicitly and implicitly through the equilibrium value of the radiation flux density:

In Fig. 26.19 gives the results of solving equation (26.10.1), obtained by N.A. Rubtsov and F.A. Kuznetsova by reducing it to an integral equation followed by a numerical solution on a computer using Newton’s method. The presented results on the temperature distribution in a layer of an absorbing medium with a frequency-averaged value of the volumetric absorption coefficient indicate the fundamental importance of taking into account the joint, radiation-conductive interaction in the transfer of total thermal energy.

Rice. 26.19. Temperature distribution in a layer of absorbing medium of optical thickness at

Noteworthy is the sensitivity of interaction effects to the optical properties of boundaries (especially for small values ​​of the radiation-conduction heat transfer criterion: .

A decrease in the emissivity of a hot wall (see Fig. 26.19) leads to a redistribution of the roles of the radiative and conductive components of the thermal energy flow. The role of radiation in the heat transfer of a hot wall decreases, and the surrounding medium is heated due to conduction from the wall. The subsequent transfer of thermal energy to the cold wall consists of conduction and radiation due to the medium’s own radiation, while the temperature of the medium decreases compared to the value that the medium would have in the case of conductive heat transfer alone. A change in the optical properties of boundaries leads to a radical restructuring of temperature fields.

In recent years, due to the widespread introduction of cryogenic technology, the problem of heat transfer by radiation at cryogenic temperatures (studies of optical properties, thermal insulation efficiency in superconducting devices and cryostats) has become fundamentally important. However, even here it is difficult to imagine the processes of radiative heat transfer in a refined form. In Fig. 26.20 shows the results of experimental studies carried out by N. A. Rubtsov and Ya. A. Baltsevich and reflecting the kinetics of temperature fields in a system of metal screens at temperatures of liquid nitrogen and helium. It also presents the calculation of the steady-state temperature field using equations (26.4.1) under the assumption that the main mechanism of heat transfer is radiation. The discrepancy between experimental and calculated results indicates the presence of an additional, conductive mechanism of heat transfer associated with the presence of residual gases between the screens. Consequently, the analysis of such a heat transfer system is also associated with the need to consider interconnected radiation-conduction heat exchange.

The simplest example of combined radiation-convective heat transfer is the transfer of heat in a flat layer of absorbing gas blown into a turbulent flow of high-temperature gas flowing around a permeable plate. Problem formulations of this kind have to be encountered both when considering the flow in the vicinity of the frontal point and when analyzing the displacement of the boundary layer by intense injection of absorbing gas through a porous plate.

The problem as a whole comes down to considering the following boundary value problem:

under boundary conditions

Here is the Boltzmann criterion, which characterizes the radiation-convective ratio of the components of the heat flow in a medium with constant thermophysical properties - the characteristic values ​​(in the undisturbed region or at the boundary of a nonequilibrium system) of speed and temperature, respectively; - dimensionless velocity distribution function in the region of boundary layer displacement.

In Fig. 26.21 presents the results of the numerical solution of problem (26.10.3) - (26.10.4) for a particular case: ; degree of emissivity of the permeable plate; free-stream emissivity for different values ​​of B0. As can be seen, in the case of low B0, which characterizes the low intensity of gas supply through the porous plate, the temperature profile is formed due to radiation-convective heat exchange. As B increases, the role of convection in the formation of the temperature profile becomes dominant. As the optical thickness of the layer increases, the temperature increases slightly at low Bo and correspondingly decreases as Bo increases.

In Fig. 26.22 the dependence characterizing the injection of absorbing gas, which is necessary to maintain the thermally insulated state of the plate, is plotted depending on the optical thickness of the displacement layer. There is a pronounced dependence of the B0 criterion on at small , when the insignificant presence of the absorbing gas component makes it possible to significantly reduce the flow rate of the injected gas. It is effective to create a highly reflective surface, provided that the optical thickness of the injected gas is small. Taking into account the selective nature of radiation absorption under the conditions under consideration does not introduce fundamental changes to the nature of the temperature profiles. This cannot be said about radiation fluxes, the calculation of which without taking into account optical transparency windows leads to serious errors.

Rice. 26.21. Temperature distribution in the curtain layer with optical thickness

Rice. 26.20. Calculated and experimental kinetics of temperature fields in a system of metal screens at temperatures of liquid nitrogen and helium ( - screen number; time, h)

Rice. 26.22. Dependence of B0 on ​​the optical thickness of the layer at and, respectively

The fundamental importance of taking into account the selectivity of radiation in thermal calculations is repeatedly noted in the works of L. M. Biberman, devoted to solving complex problems of radiation gas dynamics.

In addition to direct numerical methods for studying combined radiation-convective heat transfer, approximate calculation methods are of certain practical interest. In particular, considering the limiting law of heat transfer in a turbulent boundary layer under relatively weak influence of thermal radiation

(26.10.5)

We believe that it is a dimensionless complex of radiation-convective heat transfer, where is the total Stanton criterion, reflecting the turbulent-radiation heat transfer to the wall. In this case, Est, where is the total heat flow on the wall, which has convective and radiation components.

Turbulent heat flow q is approximated, as usual, by a polynomial of the third degree, the coefficients of which are determined from the boundary conditions:

where E is the dimensionless density of the hemispherical resulting radiation at the internal boundary points of the boundary layer.

The boundary conditions (26.10.6) include an energy equation compiled respectively for the conditions of the near-wall region and at the boundary of the undisturbed flow. Considering that , the dimensionless parameter required for the calculation is written as follows:

Note that the boundary conditions (26.10.6) were determined by the accepted condition for the formation of a thermal boundary layer near the surface flown around by the radiating medium. This significant circumstance allowed us to believe

What is done under prevailing conditions?

Convection.

The values ​​of and are determined from the analysis of solutions regarding the density of the resulting radiation in relation to the condition of a closed system that makes up the boundary layer. The turbulent boundary layer is considered as a gray absorbing medium with an absorption coefficient independent of temperature. The streamlined surface is a gray, optically homogeneous isothermal body. The undisturbed part of the flow, outside the boundary layer, radiates as a volumetric gray body that does not reflect from its surface and is at the temperature of the undisturbed flow. All this allows us to use the results of the previous consideration of radiation transfer in a flat layer of an absorbing medium with the significant difference that here only a single reflection from the surface of a streamlined plate can be taken into account.