Conductive heat transfer. Conductive heat exchange in a flat wall
Lecture 4. Conductive heat exchange.
4.1 Fourier Equation for three-dimensional nonstationary
temperature field
4.2 Coefficient of temperature ward. Physical meaning
4.3 Uniqueness Conditions - Edge Conditions
4.1 Fourier Equation for three-dimensional nonstationary
Temperature field
The study of any physical process is associated with the establishment of the dependence between its values \u200b\u200bcharacterizing. To establish such a dependence in the study of a rather complex thermal conductivity process, methods of mathematical physics were used, the essence of which is the consideration of the process not in the entire studied space, but in the elementary volume of the substance for an infinitely small period of time. The relationship between the values \u200b\u200binvolved in heat transmission with thermal conductivity is established by a differential equation - the Fourier equation for a three-dimensional nonstationary temperature field.
In the derivation of the differential equation of thermal conductivity, the following assumptions are accepted:
Internal heat sources are absent;
The body is homogeneously and isotropic;
The law of conservation of energy is used - the difference between the amount of heat, which included due to thermal conductivity into the elementary volume during the time Dτ and released from it during the same time, is consumed to change the internal energy of the elementary volume of the elementary volume.
In the body, elementary parallelepiped with DX, DY, DZ ribs are highlighted. The fabric temperatures are different, therefore it takes place through the parallelepiped heat in the directions of the x, y, z axes.
Figure 4.1 to the conclusion of the differential equation of thermal conductivity
Through the DX · DY platform during Dτ, according to the Fourier hypothesis, the following is the following heat:
https://pandia.ru/text/80/151/images/image003_138.gif "width \u003d" 253 "height \u003d" 46 src \u003d "\u003e (4.2)
where https://pandia.ru/text/80/151/images/image005_105.gif "width \u003d" 39 "height \u003d" 41 "\u003e determines the temperature change in the Z direction.
After mathematical transformations, equation (4.2) will be recorded:
https://pandia.ru/text/80/151/images/image007_78.gif "width \u003d" 583 "height \u003d" 51 src \u003d "\u003e, after a reduction:
https://pandia.ru/text/80/151/images/image009_65.gif "width \u003d" 203 "height \u003d" 51 src \u003d "\u003e (4.4)
https://pandia.ru/text/80/151/images/image011_58.gif "width \u003d" 412 "height \u003d" 51 src \u003d "\u003e (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 \u003d" 68 "height \u003d" 22 src \u003d "\u003e. gif" width \u003d "203" height \u003d "51 src \u003d"\u003e. (4.8)
The value https://pandia.ru/text/80/151/images/image017_41.gif "width \u003d" 85 "height \u003d" 41 src \u003d "\u003e (4.9)
Equation (4.9) is referred to as the differential equation of thermal conductivity or the Fourier equation for a three-dimensional non-stationary temperature field in the absence of internal heat sources. It is the main equation when studying the heat conduction processes and sets the relationship between the temporary and spatial temperature change at any point of 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 \u003d" 181 "height \u003d" 50 "\u003e
It follows that the temperature change in time for any point of the body is proportional to the magnitude but.
The value https://pandia.ru/text/80/151/images/image021_29.gif "width \u003d" 26 "height \u003d" 44 "\u003e. Under the same conditions, the temperature in that body increases faster, which has a greater temperature coefficient. So Gas are small, and the metals are a large coefficient of thermal conductivity.
In nonstationary thermal processes but characterizes the rate of temperature change.
4.3 Uniqueness Conditions - Edge Conditions
Differential equation of thermal conductivity (or system of differential equations of convective heat exchange) describe these processes in the general. To study a specific phenomenon or group of fluid transfer phenomena with thermal conductivity or convection, you need to know: temperature distribution in the initial moment, temperature ambient, geometric shape and body sizes, physical parameters of the medium and body, boundary conditions characterizing the distribution of temperatures on the body surface or the conditions of thermal interaction of the body with the environment.
All these private features are combined into the so-called terms of unambiguity or boundary conditions which include:
1) Initial conditions . Set the conditions for the distribution of temperatures in the body and the ambient temperature at the initial moment of time τ \u003d 0.
2) Geometrical conditions . Set the shape, geometric size of the body and its position in space.
3) Physical conditions . Set the physical parameters of the medium and body.
4) Border conditions can be set three ways.
Boundary condition I kind : The temperature distribution on the body surface is set for any time;
Boundary condition II : Sets the density of the heat flux at each point of the body surface for any time.
Boundary condition III kind : The temperature of the medium surrounding the body, and the law of heat transfer between the body and the environment.
The laws of convective heat exchange between the surface solid And the environment is characterized by great complexity. The theory of convective heat exchange is based on the Newton-Richmann equation, which establishes the relationship between the heat flux density on the surface of the body q and the temperature pressure (TCT - TG), under the influence of which the heat transfer on the body surface occurs:
q \u003d α · (TCT - TG), W / m2 (4.11)
In this equation, α is the proportionality coefficient called the heat transfer coefficient, W / m2 · degrees.
The heat transfer coefficient characterizes the heat exchange intensity between the body and the environment. It is numerically equal to the amount of heat than a given (or perceived) unit of the body surface per unit of time with the temperature difference between the body surface and the environment in 1 degrees. The heat transfer ratio depends on very many factors and its definition is very difficult. When solving the problems of thermal conductivity, its value is usually taken permanent.
According to the law of energy conservation, the amount of heat released by the unit surface of the body of the environment per unit of time due to heat transfer must be equal to the heat, which by thermal conductivity is supplied to the unit of the surface per unit of times from the inner parts of the body:
https://pandia.ru/text/80/151/images/image023_31.gif "width \u003d" 55 "height \u003d" 47 src \u003d "\u003e - projection of the temperature gradient towards the normal direction to the DF site.
The reduced equality is mathematical formulation of the boundary condition III of the genus.
The solution of the differential equation of thermal conductivity (or system of equations for the processes of convective heat exchange) as given conditions of uniqueness allows you to determine the temperature field throughout the body for any point in time, i.e., find the function of the form: T \u003d F (x, y, z, τ).
Among the processes of complex heat exchange, radiation-but-convective and radiation-conductive heat exchange are distinguished.
it is divided by their sum. Radiation-conductive heat exchange in a flat layer for other source conditions is considered in [L. 5, 117, 163]; for a cylindrical layer - in [L. 116].
So why in the region classified as boiling layers of large particles, with increasing diameter increase and maximum heat exchange coefficients? It's all about gas convectionable heat exchange. In the layers of fine particles, the gas filtration rate is too small so that the convective component of heat exchange could "show". But with an increase in the diameter of the grains, it increases. Despite the low conductive heat exchange, in a boiling layer of large particles, the growth of the convective component compensates for this disadvantage.
Chapter Fourteenth Radiation Conductive Heat Exchange
14-2. Radiation-conductive heat exchange in a flat layer of gray absorbing medium without heat sources
14-3. Radiation-conductive heat exchange in a flat layer of selective and anisotropically scattering medium with heat sources
Thus, on the basis of listed and some other, more private works it becomes obvious that radiation-conductive heat exchange in systems containing volumetric sources of tapla is clearly not sufficiently studied. In particular, the effect of the selectivity of the medium and the boundary surfaces is not clear, the effect of surround and surface scattering anisotropy. In connection with this author, an approximate analytical solution was undertaken by the problem of radiation-keeper heat exchange in a flat layer
total and convective heat transfer. In particular cases of this GAID heat exchange. The radiation heat exchange in a moving medium (in the absence of a co-ductive transfer), radiation-conductive heat exchange in a fixed medium (in the absence of convective (transfer) and purely "convective heat exchange in a moving medium, when there is no radiation transfer. The complete system of equations describing the processes of radiation-convective heat exchange, was considered and analyzed by IB ch. 12.
In equation (15-1), the total heat transfer coefficient from the flow to the wall of the channel can be found on the base (14-14) and (14-15). For this purpose, we consider in the framework of the adopted scheme the process of heat transfer of the current medium with a boundary surface as a radiation-no-conductive heat exchange of the core of the flow and the wall of the channel through the border layer thick b. We equate the temperature of the nucleus of the middle calorimetric temperature of the medium in this section, which can be done, given the small thickness of the "boundary layer compared to the channel diameter. Considering as one of the boundary surfaces of the core of the flow [with a temperature in this section of the channel T (x) and the absorption The ability to AG], and as another - "the wall of the channel (with the temperature TW and the absorption capacity AW), consider the process of radiation-conductive heat exchange through the boundary layer. Using (14-14), we obtain an expression for the local heat transfer coefficient A in this section: the problems of radiation-convective heat exchange, even for simple cases are usually more difficult than the problem of radiation-conductive heat exchange. The following is an approximate solution [L. 205] One common task of radiation-convective heat exchange. Significant simplifications allow you to bring the decision to the end.
As shown in [L. 88, 350], the tensor approximation under certain conditions is a more accurate method that opens up new possibilities in the study of heat transfer processes. In (L. 351] The proposed tensor approximation (L. 88, 350] was used to solve the combined problem of radia-conductive heat exchange and gave good results. In the future, the author the tensor approximation was generalized "and the case of spectral and complete radiation at arbitrary indicatricries volumetric and surface scattering in radiating systems [L. 29, 89].
Applying an iterative method for solving problems of complex heat exchange, it should be first set to the values \u200b\u200bof Qpea.i in all zones and determine the resulting distribution of Qpea.i (I \u003d L 2, ..., P) the temperature field on the basis of which is calculated on the electric sensor. The second approximation of all values
Radiation-conductive heat exchange is considered in relation to the flat layer of weaking medium. Two tasks have been solved. The first is the analytical consideration of the radiation-conductive heat exchange in a flat layer of the medium without any restrictions in the "carrying temperature of the surface surfaces. At the same time, the medium and boundary surfaces were assumed to be gray, and the internal sources of heat in the medium were absent. The second solution belongs to the symmetric problem of radiation -conductive heat exchange in a flat layer of selective and anisotropically scattering medium with heat sources inside the layer. Results of the decision of the first task
As special cases of the system of equations of complex heat exchange, all individual equations considered in the hydrodynamics and the theory of heat exchange are measured: the equations of motion and continuity of the medium, the equations of pure conductive, convective and radiation heat exchange, the equations of radiation-conductive heat exchange in the fixed medium and, finally, the radiation heat exchange equations in a moving, but intimate-pro-army.
Radiation-conductive heat exchange, which is one of the types of complex heat exchange, takes place in various fields of science and datronics (astro- and geophysics, metallurgical and glass industry, electrovacuum technology ,.The production of new materials, etc.). To the need to study the processes of radiation-conductive heat exchange, there are also problems of energy transfer in the boundary layers of fluxes of liquid and gaseous media and the problems of studying the thermal conductivity of various translucent materials.
but to calculate the process of radihodio-"conductive heat exchange Ib those conditions for which the solutions obtained are valid. Numerical solutions of the problem give a visual. Cartin of the process under study for (specific cases, without requiring the introduction of many restrictions inherent in approximate analytical research. both analytical and numerical solutionsundoubtedly, they are known (progress in the study of the processes of radiation-tanguctive heat exchange, despite its limited and private character.
This chapter discusses the two analytic solutions to the problem of radiation-no-conductive heat exchange in a flat layer of the medium. The first solution considers the problem in the absence of restrictions on the temperature, the absorption capacity of the boundary surfaces and the optical thicknesses of the medium layer [L. 89, 203]. This solution is carried out by iterations, and the environment I.Border surfaces are assumed to be gray, and there are no heat in the medium.
Fig. 14-1. The scheme to solve the problem of the Ra-diagonal-conductive heat exchange in a flat layer of the absorbing and heat-conducting medium in the absence of internal heat sources in the medium.
The most detailed analytical study was obtained above the problem of radiation-conductive heat exchange through a layer of gray, a pure absorbing medium when specifying the temperatures of gray boundary surfaces of the layer and in the absence of heat sources in the environment itself. The problem of radiation-conductive heat exchange layer of the radiating and heat-conducting medium with boundary surfaces in the existence of heat sources was considered in a very limited number of works with the adoption of certain assumptions.
For the first time, an attempt to take into account internal heat sources in the processes "radiation-conductive heat exchange was taken in [L. 208], where the problem of heat transfer by radiation and thermal conductivity through a layer of gray, a non-scattering medium with a uniform distribution of sources by volume. However, a mathematical error made in the work has reduced the results obtained.
This type of heat exchange occurs between the inhibited particles of the body located in the temperature field.
T. = f. ( x. , y, z. , t. ), characterized by grad temperature gradient T.The temperature gradient is a vector directed by normal N 0 to an isothermal surface in the direction of increasing temperature:
grad.T. = p o. DT / DN. = p o. T.
Distinguish thermal fields: one-dimensional, two-dimensional and three-dimensional; stationary and nonstationary; Isotropic and anisotropic.
Analytical description of the process of conductive heat exchange is based on the Fundamental Fourier law, tied the characteristics of a stationary heat flux, propagating in a one-dimensional isotropic medium, geometric and thermophysical parameters of the medium:
Q. \u003d λ (t 1 -T. 2 ) S / L T or p \u003d Q. / T \u003d λ (T. 1 -T. 2 ) S / L
where: - Q. - the amount of heat transferred through the sample during the time t. , Cal;
λ - coefficient of thermal conductivity of the material of the sample, W / (M- region);
T. 1 , T. 2 - accordingly, the temperature of the "hot" and "cold" sections of the sample, hail;
SS. - sample cross section, m 2;
l. - the length of the sample, m;
R - heat flow, W.
Based on the concept of an electrothermal analogy, according to which thermal values R andT. put an electric current I. and electric potential U. , imagine the Fourier law in the form of the "Ohm Law" for the heat circuit site:
P \u003d ( T. 1 -T. 2 ) / L / λs. = (T. 1 -T. 2 ) / R. T. (4.2)
Here in physical meaning parameter R. T. there is thermal resistance to the heat chain area, and 1 / λ - Specific thermal resistance. Such a presentation of the process of conductive heat exchange makes it possible to calculate the parameters of heat circuits represented by topological models known to the methods of calculating electrical circuits. Then, just as for the electrical circuit, the expression for current density in vector form has the form
j. = – σ grad.U. ,
for the thermal chain, Fourier law in vector form will look
p. = - λ Grad. T. ,
where r - the density of the heat flux, and the minus sign indicates that the heat flux is distributed from the body heated to a colder cross section.
Comparing expressions (4.1) and (4.2), we will see that for the conductive heat exchange
a.= a. KD \u003d λ / l.
Thus, to increase the efficiency of the heat transfer process, it is necessary to reduce the length. l. thermal chain and increase its thermal conductivity λ
The generalized form of describing the process of conductive heat exchange is the differential equation of thermal conductivity, which is a mathematical expression of energy conservation laws and Fourier:
cf. dt. / dt. = λ x. d. 2 T. / dX. 2 + λ y. d. 2 T. / dY. 2 + λ z. d. 2 T. / dZ. 2 + W. v.
where from -specific heat capacity, J / (CG-);
p is the density of the medium, kg / m 3;
W. v. - bulk density of internal sources, W / m 3;
λ x. λ y. λ z. - specific thermal conductivity in the directions of the coordinate axes (for anisotropic medium).
4.2.2. Convective heat exchange
This type of heat exchange is a complex physical process at which the heat transfer from the surface of the heated body into the surrounding space due to the washing of its flow of coolant - liquid or gas - with a lower than that of the heated body, temperature. In this case, the parameters of the temperature field and the intensity of the convective heat exchange depend on the nature of the movement of the coolant, its thermal physical-schzki characteristics, as well as on the shape and size of the body.
So, the flow of the coolant flow can be free and forced, which corresponds to phenomena naturaland forcedconvection. Also distinguished laminar and turbulent j.the flow modes of the flow, as well as their intermediate states depending on the ratio of forces that determine these movements of the stream - forces of internal friction, viscosity and inertia.
Simultaneously with convective, conductive heat exchange occurs due to thermal conductivity of the coolant, but its effectiveness is low due to the relatively small values \u200b\u200bof the thermal conductivity of fluids and gases. In the general case, this heat exchange mechanism describes Newton Richmanas;
P \u003d. a. KB. S. ( T. 1 - T. 2 ), (4.3)
where: a. KB. - heat transfer coefficient convection, W / (m 2 -Grad.);
T. 1 - T. 2 2 - respectively, the temperature of the wall and the coolant, K;
S. - The surface of heat exchange, m 2.
With the external simplicity of the description of the Newton-Richman, the complexity of a quantitative assessment of the effectiveness of the convective heat exchange process is that the value of the coefficient a. KB. Depends on the set of factors, i.e. It is the function of many process parameters. Find explicitly addiction a. KB. = f.but 1 , a. 2 , ..., but j. , ..., but n. ) it is often impossible, since the process parameters also depend on temperature.
Solve this task for each specific case helps similarity theorylearning properties of such phenomena and methods for establishing their similarity. In particular, it is proved that the flow of a complex physical process is determined not separate. It is physical and geometric parameters, but dimensionless power complexes composed of the parameters substantial to the flow of this process called criteria like . Then the mathematical description of the complex process is reduced to the preparation of these criteria, one of which contains the desired value of AV, criterial equation , the type of which is valid for any of the species of this process. If the similarity criteria cannot be made, this means that either some important parameter of the process is missed from consideration, or some parameter of this process can be removed from consideration without much damage.
The heat transfer process with thermal conductivity is explained by the exchange kinetic energy Between the molecules of the substance and the diffusion of electrons. These phenomena take place when the temperature of the substance at different points is different or when two bodies with varying degrees of heating are contacted.
The main law of thermal conductivity (Fourier law) states that the amount of heat passing through a homogeneous (uniform) body per unit of time, directly proportional to the cross-sectional area, normal to the heat flow, and the temperature gradient along the flow
where R T is the power of the heat flux transmitted by thermal conductivity, W;
l is the coefficient of thermal conductivity;
d - wall thickness, m;
t 1, T 2 - the temperature of the heated and cold surface, K;
S - surface area, m 2.
From this expression, it can be concluded that when developing the design of the RES, the heat-conducting walls should be done fine, in parts of parts to provide thermal contact over the entire area, choose materials with a large thermal conductivity coefficient.
Consider the case of heat transfer through a flat wall thickness d.
Figure 7.2 - heat transfer through the wall
The amount of heat transmitted per unit of time through the wall section of the square S is determined by the already known formula
This formula is compared with the equation of the Ohm law for electrical circuits. It is not difficult to make sure of their complete analogy. So the amount of heat per unit of time p t corresponds to the value of current I, the temperature gradient (T 1 - T 2) corresponds to the difference in the potentials U.
The relationship is called t e r m and ch e with to and m resistance and denote by R T,
The considered analogy between the flow of thermal flux and the electric current not only allows to note the generality of physical processes, but also facilitates the calculation of thermal conductivity in complex structures.
If in the considered case of the element to be cooled, is located on the plane having a temperature T ST1, then
t ST1 \u003d P T D / (LS) + T ST2.
Therefore, to reduce T St1, it is necessary to increase the area of \u200b\u200bthe heat sink surface, reduce the thickness of the heat transmitting wall and choose materials with a large thermal conductivity coefficient.
To improve thermal contact, it is necessary to reduce the roughness of contacting surfaces, covering them with heat-conducting materials and create contact pressure between them.
The quality of thermal contact between the elements of the structure depends also on electrical resistance. The smaller the electrical resistance of the contact surface, the less its thermal resistance, the better the heat sink.
The smaller the heat sink of the environment, the longer it is necessary to establish a stationary heat exchange mode.
Typically, the cooling part of the design is the chassis, housing or casing. Therefore, when choosing a design version of the design, you need to look, whether the condition chosen for the attachment is the coolant condition for a good heat exchange with the environment or heat-resistant.
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 basis technological processes;
Typical processes and industrial technology devices.
The second part includes three main sections:
Hydrodynamics and hydrodynamic processes;
Thermal processes and devices;
Mass exchange processes and devices.
In the first part of the discipline, abstracts of N.Kh lectures were published. Zinnatullina, A.I. Guryanova, V.K. Ilyina (Hydraulics
and heat engineering, 2005); On the first partition of the second part of the discipline - the Tutorial N.Kh. Zinnatullina, A.I. Guryanova, V.K. Ilina, D.A. Eldasheva (hydrodynamics and hydrodynamic processes, 2010).
This manual sets out the second section of the second part. In this section, the most common cases of conductive and convective heat exchange, industrial methods of heat transfer, evaporation, as well as the principle of operation and design of heat exchanger equipment will be considered.
The textbook consists of three chapters, each of them ends with questions that students can use for self-control.
The main task is presented tutorial - Teach students to carry out engineering calculations of thermal processes and selection of the necessary equipment for their conduct.
PART. 1. Heat exchange
Industrial technological processes flow in a given direction only at certain temperatures, which are created by supplying or removing thermal energy (heat). Processes, the rate of leakage of which depends on the speed of the supply or removal of heat, are called thermal. The driving force of thermal processes is the difference in temperatures between the phases. The devices in which heat processes are carried out are called heat exchangers, heat transferred heat to them.
The calculation of heat exchange processes is usually reduced to the determination of the interfacial surface of the heat exchange. This surface is located
From the heat transfer equation in integrated form. The heat transfer coefficient, as is well known, depends on the phase heat transfer coefficients,
as well as on thermal resistance of the wall. Below will be considered methods for their definition, finding the temperature field and heat fluxes. Where possible, the desired values \u200b\u200bare made from solving the equations of conservation laws, and in other cases simplified mathematical models or physical modeling method are used.
Convective heat exchange
When convection, the heat transfer occurs with macrobal particles of the coolant stream. Convection is always accompanied by thermal conductivity. As is known, thermal conductivity - phenomenon molecular, convection - the phenomenon of macroscopic, in which
In the transfer of heat, all layers of coolant with different temperatures are involved. The heat convection is transferred much faster than thermal conductivity. Convection at the surface of the wall of the apparatus fades.
Convective warmth transfer is described by the Fourier-Kirchhoff equation. The patterns of the medium flow are described by the Navier-Stokes equations (laminar mode) and Reynolds (turbulent mode), as well as the equation of continuity. The study of the patterns of convective heat exchange can be carried out in isothermal and non-erotic formulation.
In isothermal formulation, the Navier-Stokes equations first solve and continuously, then the obtained speed values \u200b\u200bare used to solve the Fourier-Kirchhoff equation. The values \u200b\u200bof heat transfer coefficients obtained in this way are subsequently updated, corrected.
In the non-erotic formulation of the Navier-Stokes equation, continuity and Fourier-Kirchhoff are solved together, taking into account the dependence of the thermophysical properties of the medium from temperature.
As shown experimental data, dependencies with R.(T.), L ( T.)
and R ( T.) weak, and M ( T.) - very strong. Therefore, only M dependence M is usually taken into account ( T.). It, this dependence, can be represented as a dependence of Arrhenius or, simpler, in the form of an algebraic equation. Thus, the so-called conjugate tasks arise.
IN lately Methods have been developed to solve many heat transfer problems in laminar fluid flows, taking into account the dependence of the viscosity of fluid from temperature. For turbulent flows, everything is more complicated. However, you can use approximate numerical solutions using computer technologies.
To solve these equations, it is necessary to put the conditions for unambiguity, which include the initial and boundary conditions.
Boundary heat exchange conditions can be given in a different way:
The boundary conditions of the first kind - are given by the distribution of the temperature of the wall:
; (19)
the simplest case when T. C T \u003d const;
The boundary conditions of the second kind - the heat flux on the wall is set.
; (20)
The boundary conditions of the third kind - the distribution of the temperature of the medium surrounding the channel and the heat transfer coefficient is set.
on the medium to the wall or vice versa
. (21)
The choice of the type of boundary condition depends on the conditions of the heat exchange equipment.
On a flat plate
Consider a stream with unchanged thermophysical characteristics (R, M, L, c P. \u003d const), making a forced movement along a flat semi-infinite thin plate and hauling heat. Suppose the unlimited flow at speeds
and temperatures T.° flies on a semi-infinite plate coinciding
With plane h. – z. and having temperature T. Art \u003d const.
Highlight hydrodynamic and thermal border layers
With a thickness of D g and D T, respectively (region 99% Speed \u200b\u200bChanging w X.
and temperature T.). In the core of the flow and T.° constant.
Analyze the equations of continuity and Navier-Stokes. The task is two-dimensional, since w Z., 
. According to experimental data it is known that in the hydrodynamic borderline layer 
. In the core of the flow 
const, therefore, according to the Bernoulli equation 
, in the border layer the same
.
As known " h.»D g, therefore 
.
Therefore, have
; (22)
. (23)
Record similar equations for the axis w. does not make sense because w y. It can be found from the equation of continuity (22). Using similar procedures, you can simplify the Fourier-Kirchhoff equation.
. (24)
The system of differential equations (22) - (24) is an isothermal mathematical model of a flat stationary thermal laminar boundary layer. We formulate the boundary conditions
On the border with the plate, i.e. for w. \u003d 0: at any h. speed w X.\u003d 0 (adhesion condition). On the border and outside the hydrodynamic border
those. for w.≥ D g ( h.), as well as h.\u003d 0 for any w.: w X.\u003d. For temperature field, similar arguments.
So, the boundary conditions:
w. x ( x., 0) = 0, x. > 0; w X. (x., ∞) = ; w X. (0, y) \u003d; (25)
T. (x., 0) = T. st x. > 0; T. (x., ∞) = T. ° ; T. (0, y.) = T. °. (26)
The exact solution of this task in the form of infinite series was obtained by Blazius. There are simpler approximate solutions: the method of integral relations (Yudaev) and the pulse theorem (schlichting). A.I. Raspical task was solved by the method of conjugate physical
and mathematical modeling. Speed \u200b\u200bprofiles were obtained
w X. (x., y.), w. y ( x., y.) and temperatures T., as well as the thickness of the border layers
D g ( x.) and d t ( h.)
; (27)
, Pr. ≥ 1; (28)
Pr. \u003d ν / a.
Coefficient BUT In the formula (27) at Realov - 5.83; Yudaev - 4.64; Blauzius - 4; Schlichting - 5.0. The approximate type of dependency found is shown in Fig. 1.3.
As you know, for gases Pr. ≈ 1, drip liquids Pr. > 1.
The results obtained allow the impulse and heat transfer coefficients to determine the coefficients. Local values \u200b\u200bΓ ( x.) I. Nu. g x.
, 
. (29)
| y. |
 |
| w X. |
| T. Art |
| (T-T. ST) |
| D g ( x.) |
| D T ( x.) |
| x. |
Fig. 1.3. Hydrodynamic and thermal laminar border layers
on a flat plate
Averaged values \u200b\u200bI. 
in length l.
, 
, 
. (30)
Similar to heat transfer
, 
; (31)
, . (32)
IN this case The analogy of heat and impulsotuds persists (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 a Pr. \u003d 1, then p T-g, x. \u003d 1, therefore, a complete analogy of the processes of impulso and heat transfer.
From the obtained equations should
γ ~, m; A ~, L. (34)
As a rule, such a qualitative dependence is performed
Not only for flat border, but also for more complex cases.
The task is considered in isothermal formulation, thermal boundary conditions of the first kind T. Art \u003d const.
As removal from the edge of the plate (zoom in the coordinate h.) Growing D G ( h.). In this case, the heterogeneity of the speed field w X. distributed in the region more and more distant from the border of the phase partition,
What is the prerequisite for the emergence of turbulence. Finally, for RE x Kp begins the transition of a laminar mode to turbulent. The transition zone corresponds to the values h.calculated by Re X. From 3.5 × 10 5 ÷ 5 × 10 5.
At distances Re X. \u003e 5 × 10 5 The entire border layer is turbulized,
With the exception of a viscous or laminar sublayer thick D 1g. In the core of the stream, the speed does not change. If a Pr. \u003e 1 That inside the viscous sublayer can be isolated by a thermal sublayer D 1T, in which the molecular transfer of heat prevails over turbulent.
The thickness of the entire turbulent thermal boundary layer is usually determined from the condition ν T \u003d a t, therefore d r \u003d d t.
First, consider the turbulent hydrodynamic border layer (Fig. 1.4). Let's leave all the approximations made for the laminar layer. The only difference is the presence of ν t ( w.), so
. (35)
Save and boundary conditions. By solving the system of equations (35)
and (22) with boundary conditions (25), using a semi-empirical model of the Prantl-bearing turbulence, one can obtain the characteristics of the turbulent boundary layer. In a viscous sublets, where the linear law of the speed distribution is implemented, you can neglect turbulent pulse transfer, and outside it molecular. In a wall area
(less than a viscous sublayer) is usually the logarithmic speed profile, and in the external region - a power law with an indicator 1/7 (Fig. 1.4).
Fig. 1.4. Hydrodynamic and thermal turbulent border layers
on a flat plate
As in the case of a laminar borderline layer, it is possible to use the length averaged l. Pullery coefficients
. (36)
Consider a thermal turbulent borderline layer. Energy equation has a view
. (37)
If a Pr. \u003e 1, then inside the viscous sublayer you can highlight the heat sublayer, where the molecular transfer of heat can be
. (38)
For local heat transfer coefficient mathematical model Has appearance
Middle Plate Length Value 
determined by that
Below are the formation of a turbulent border layer (A) and the distribution of the local heat transfer coefficient (b) with a longitudinal flow of a flat semi-infinite plate (Fig. 1.5).
Fig. 1.5. Border Layers D g and D T and local heat transfer coefficient a
on a flat plate
In a laminar layer ( h. ≤ l. CR) thermal flow only due to thermal conductivity, for a qualitative estimate, a ratio A ~ can be used.
In the transition zone, the overall thickness of the boundary layer increases. However, the value of a increases, because the thickness of the laminar sublayer decreases, and in the resulting turbulent layer, the heat is transferred not only to thermal conductivity, but also convection together
with a moving mass of fluid, i.e. more intense. As a result, the total thermal resistance of heat transfer decreases. In the zone of the developed turbulent mode, the heat transfer coefficient again begins to decrease due to the increase in the overall thickness of the boundary layer A ~.
So, the hydrodynamic and thermal border layers on a flat plate are considered. The qualitative nature of the dependencies of the dependences is also for the border layers, which are formed when streamlining more complex surfaces.
Heat exchange in a round tube
Consider stationary heat exchange between the walls of the horizontal direct pipe of the circular cross section and the stream with unchanged thermophysical characteristics and moving due to the forced convection within it. We will take the thermal boundary conditions of the first kind, i.e. T. Art \u003d const.
I.Sections of hydrodynamic and thermal stabilization.
At the entrance of the fluid into the pipe due to the braking caused by the walls, the hydrodynamic boundary layer is formed on them.
As you remove from the entrance, the thickness of the border layer increases,
While the border layers adjacent to the opposite walls,
Do not close. This plot is called the initial or hydrodynamic stabilization section - l. ng.
Like a change in the speed profile along the pipe length changes
and temperature profile.
II.Consider the laminar movement of the fluid.
Earlier, in the section of the disciplines "Hydrodynamics and hydrodynamic processes", we considered the hydrodynamic initial portion. To determine the length of the initial site, the following dependence was proposed
.
For fluid Pr. \u003e 1, therefore, the thermal border layer will be inside the hydrodynamic boundary layer.
This circumstance suggests that the thermal border layer develops in a stabilized hydrodynamic area and the speed profile is known - parabolic.
The temperature of the fluid in the input section of the heat exchange section is constant in cross section and equal T.° and in the core of the stream it does not change. Under these conditions, the equation of the thermal boundary layer has the form
. (41)
The solution of this equation under the above conditions gives:
· For the length of the thermal entry plot
; (42)
· For local heat transfer coefficient
; (43)
· For average heat transfer coefficient length
; (44)
· For a local number of Nusselt
; (45)
· For the average number of Nusselt
. (46)
Consider equation (42). If a 
T. 
.
For liquids Pr. \u003e 1, so in most cases, especially
for liquids with large Pr.The heat exchange during the laminar mode of motion is carried out mainly on the heat stabilization site. As can be seen from the ratio (43) a for the pipe on the heat stabilization section, it is reduced by removal from the entrance (the thickness of the thermal boundary layer D T increase increases) (Fig. 1.6).
Fig. 1.6. Temperature profile on the initial and stable plot
with a laminar flow of fluid in a cylindrical pipe
For turbulent flow The flow in the pipe, as well as on a flat plate, firstly, the thickness of the hydrodynamic and thermal boundary layers coincide; And secondly, they grow much faster than for laminar. This leads to a decrease in the length of the part of the thermal
and hydrodynamic stabilization, which allows in most cases to neglect them when calculating heat transfer
. (47)
III.Stabilized heat exchange during laminar movement medium.
Consider stationary heat exchange in a round tube when the thermophysical properties of the liquid are constant (isothermal case), the speed profile does not change along the length, the temperature of the pipe wall is constant and equal T. Art, there are no internal heat sources in the stream,
And the amount of heat released due to dissipation of energy is negligible. Under these conditions, the heat exchange equation is the same as for the boundary layer. Consequently, the initial equation for the study of heat exchange is equation (41).
Border conditions:
(48)
The decision of this problem was first obtained by Gretz, then Nusselt, as the sum of the infinite series. A slightly different solution was obtained by Shumilov and Yablonsky. The solution obtained is fair
and for the thermal stabilization site under the condition of preliminary hydrodynamic stabilization of the flow.
For the region of stabilized heat exchange, the local heat transfer coefficient is equal to the limit
or 
(49)
As can be seen from the figure (Fig. 1.7), with increasing 
number Nu. decreases asymptotically approaching the second section of the curve
to a permanent value Nu. \u003d 3.66. This happens because for the stabilized heat exchange temperature profile on the length of the pipe
does not change. The first area is the formation of a temperature profile. The first plot corresponds to the thermal initial area.
| 10 –5 10 –4 10 –3 10 –2 10 –1 10 0 |
| 1 |
| 3,66 |
| Nu. |
| Nu. |
|
Fig. 1.7. Change local and medium Nu. along the length of the round tube T. Art \u003d Const.
IV.Stabilized heat exchange with turbulent motion environment.
Source equation
. (50)
Border conditions:
(51)
When solving the problem, the problem of selecting the speed profile occurs w X.. Some for w X. Logarithmic law (A.I. Razinov) is used, others - Law 1/7 (VB Kogan). There is a conservatism of turbulent currents, which lies in the weak effect of boundary conditions and the speed field w X. on heat transfer coefficients.
For the number of Nusselt, the following formula is proposed
. (52)
As with a laminar movement in the field of stabilized heat exchange during turbulent medium Nu. does not depend on the coordinate h..
We have considered above the special cases of heat exchange, namely: with isothermal formulation of the problem and thermal boundary conditions of the first kind of heat exchange in smooth cylindrical pipes and flat horizontal plates.
The literature has solutions of thermal tasks and for other cases. Note that the surface roughness of the pipe and plate leads
To an increase in the heat transfer coefficient.
Solid heat
To solve this problem, various coolants are used.
TN are classified by:
1. By appointment:
Heating TN;
Cooling TN, cheerful;
Intermediate tn;
Drying agent.
2. P. aggregate state:
· Single-phase:
Low-temperature plasma;
Non-confidential couples;
Do not boiling and non-evaporated with this pressure of the fluid;
Solutions;
Grainy materials.
· Multi-, two-phase:
Boiling, evaporating and sprayed fluid;
Condensing pairs;
Melting, hardening materials;
Foams, gas-covers;
Aerosols;
Emulsions, suspensions, etc.
3. By temperature range and pressure:
High-temperature TN (smoke, flue gases, salting melts, liquid metals);
Medium-temperature TN (water vapor, water, air);
Low-temperature TN (atmospheric pressure T. kip ≤ 0 ° C);
cryogenic (liquefied gases - oxygen, hydrogen, nitrogen, air, etc.).
With increasing pressure, the boiling point of liquids is growing.
As direct sources of thermal energy at industrial enterprises, flue (smoke) gases and electricity are used. Substances transmitting heat from these sources, called intermediate TN. The most common intermediate TN:
Water steam saturated;
Overheated water;
Organic fluids and their pairs;
Mineral oils, liquid metals.
Requirements for TN:
Big r, with R.;
The high value of the heat of vaporization;
Low viscosity;
Non-combustibility, non-toxicity, heat resistance;
Cheapness.
Warm removal
Many industrial technology processes proceed under conditions when there is a need to remove heat, for example, when cooling gases, liquids, or when condensing vapors.
Consider 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 - 8-12 ° C, and a turnover (summer) is about 30 ° C.
Cooling water consumption 
Determine from the heat balance equation
. (83)
Here - the consumption of cooled coolant; N. N I. N. K - initial
and finite enthalpy of cooled coolant; N. NV I. N. KV - initial
and finite enthalpy of cooling water; - Environmental loss.
Achieving lower cooling temperatures can be provided
Using low-temperature liquid refrigerants.
Air cooling. The most widely air is used as a cooling agent in mixing heat exchangers - cooling towers, which are the main element of the water-co-airing cycle equipment (Fig. 2.5).
Fig. 2.5. Cooling towers with natural (a) and forced (b) traction
Hot water in the cooling towers is cooled both due to the contact with cold air and as a result of the so-called evaporative cooling,
In the process of evaporation of part of the water flow.
Mixing heat exchangers
In mixing heat exchangers (ST), heat transmission from one coolant to another occurs during their direct contact or mixing, therefore, the thermal resistance of the wall (separating coolants) is absent. The most common hundred is used to condensate vapors, heating and cooling water and vapor. On the principle of the device, the STR is divided into bubble, shelf, nozzle and hollow (with splashing of the liquid) (Fig. 2.18).
| par |
| water |
| in |
| air |
| water |
| water |
| water |
| par |
| G. |
| par |
| heated liquid |
| but |
| air |
| water |
| par |
| water + condensate |
| B. |
| liquid |
Fig. 2.18. Schemes one hundred: a) Barboratory mixing heat exchanger for water heating;
b) a capacitor's plating heat exchanger; c) a spherical barometric condenser; d) hollow
Part 3. Examination
The evaporation is the process of concentrating solutions of solid non-volatile substances by removing a volatile solvent in the form of vapors. Evaluation is usually carried out when boiling. Usually, only part of the solvent is removed from the solution, since the substance must remain
in fluctuate condition.
There are three evaporation methods:
Surface evaporation is carried out by heating the solution on the heat exchange surface due to heat supply to the solution through the wall from the heating steam;
Adiabatic evaporation, which occurs by instant evaporation of the solution in the chamber, where the pressure is lower than the saturated pair pressure;
Evaporation by contact evaporation - the heating of the solution is carried out with direct contact between the moving solution
and hot heat carrier (gas or liquid).
In industrial technology, the first evaporation method is mainly applied. Next about the first method. To carry out the evaporation process, it is necessary to transfer heat from the coolant to a boiling solution, which is possible only in the presence of temperature difference between them. The temperature difference between the coolant and the boiling solution is called the useful difference in temperature.
A saturated water vapor (heating or primary) is used as a coolant in evaporation devices. Evaluation is a typical heat exchange process - heat transfer by condensation of a saturated water vapor to a boiling solution.
In contrast to conventional heat exchangers, evaporators consist of two main nodes: a heating chamber or a boiler and separator. The separator is designed to capture the drops of a solution of steam, which is formed when boiling. This pair is called secondary or juice. The temperature of the secondary pair is always less than the boiling point of the solution. To maintain a permanent vacuum in the condenser, it is necessary to suck the vapor pump with a vacuum pump.
Depending on the pressure of the secondary pair distinguish the evaporation when r atm r ram r wak. In case of evaporation when r VAK decreases the boiling point of the solution, with p. Has - secondary steam is used for technological purposes. The boiling point of the solution is always above the boiling point of a clean solvent. For example, for saturated aquatic solution
NaCl (26%) T. kip \u003d 110 ° С, for water T. Kip \u003d 100 ° C. Secondary steam, selected from evaporator installation for other needs, is called extra ferry.
Temperature loss
Usually, pressure of heating and secondary vapors is known in single-populated evaporation plants, i.e. There are their temperatures. The difference between the temperatures of heating and secondary vapors is called the total difference in the temperatures of evaporators
. (96)
Total temperature difference 
related to the useful difference in temperatures by relation
Here d ¢ - concentration temperature depression; D ¢¢ - hydrostatic temperature depression; D ¢ Determine as the difference of boiling temperatures T. kip. P and Pure Solvent T. kip. Czech Republic p \u003d \u003d.const.
D ¢ \u003d. T. kip. R - T. kip. Czech T. kip. Czech, D ¢ \u003d T. kip. R - T. VP. (98)
The temperature of the subsequent vapor formed by boiling is lower than the boiling point of the solution itself, i.e. part of temperatures is lost useless; D ¢¢ characterizes the increase in the boiling point of the solution with an increase in hydrostatic pressure. Typically, the height of boiling pipes determine the average pressure, and for this pressure, the average boiling point of the solvent is determined. T. cf.
Here p. a - pressure in the device; R PZ - Pick-Sciential Mixture Density
in boiling pipes 
; H. - Height of boiling pipes.
D² \u003d T. cf - T. VP, (99)
where T. Wed - the boiling point of the solvent when p \u003d P. cf; T. VP - the temperature of the secondary pair at pressure p. but.
Multicpus evaporation
In a multi-circuable evaporator installation, secondary pairs (Fig. 3.2, 3.3) of the previous body is used as heating steam
In the subsequent case. Such an evaporation organization leads
To significant savings of heating steam. If taken 
In all buildings, the total consumption of heating steam on the process is reduced in proportion to the number of housings. Practically, in real conditions, this ratio is not maintained, it is usually higher. Next, we consider the equations of material and thermal balances for a multi-circuable evaporation unit (see Fig. 3.2), which are a system of equations recorded for each case separately.
