Heat transfer is conductive. Conductive heat transfer Heat transfer in a round pipe

It is carried out due to the collision of molecules, electrons and aggregates of elementary particles with each other. (Heat moves from a more heated body to a less heated one). Or in metals: gradual transfer of vibrations of the crystal lattice from one particle to another (elastic vibrations of lattice particles - phonon thermal conductivity).

Convective transport;

This transfer is associated with the movement of fluid particles and is caused by the movement of microscopic elements of substances; it is carried out by the free or forced movement of the coolant.

Under the influence of a temperature gradient in the earth's crust, convective flows of not only heat, but also matter arise. A thermohydrodynamic pressure gradient arises.

One can also observe the phenomenon that when a hydrodynamic pressure gradient occurs, oil is retained in the formation without a seal.

3. Heat transfer due to radiation.

A radioactive unit releases heat as it decays, and this heat is released through radiation.

33. Thermal properties of oil and gas formation, characteristics and area of ​​use.

Thermal properties are:

1) Heat capacity coefficient c

2) Thermal conductivity coefficient l

3) Thermal diffusivity coefficient a

1. Heat capacity:

c – the amount of heat required to increase the temperature of a substance by one degree under given conditions (V, P=const).

с=dQ/dТ

Average heat capacity of a substance: c=DQ/DT.

Because Rock samples can have different masses and volumes; for a more differentiated assessment, special types of heat capacity are introduced: mass, volumetric and molar.

· Specific mass heat capacity [J/(kg×deg)]:

С m =dQ/dТ=С/m

This is the amount of heat required to change a unit mass of the sample by one degree.

· Specific volumetric heat capacity [J/(m 3 ×K)]:

С v =dQ/(V×dТ)=r×С m,

where r is density

The amount of heat that must be imparted to a unit to increase it by one degree, in the case of P, V=const.

· Specific molar heat capacity [J/(mol×K)]:

С n =dQ/(n×dТ)=М×С m,

where M – relative molecular weight [kg/kmol]

The amount of heat that must be imparted to a mole of a substance to change its temperature by one degree.

Heat capacity is an additive property of the formation:

С i = j=1 N SC j ×К i , where SC i =1, К – number of phases.

The heat capacity depends on the porosity of the formation: the greater the porosity, the lower the heat capacity.

(s×r)=s sq ×r sq ×(1-k p)+s ×r s ×k p,

where с з – pore filling coefficient;

k p – porosity coefficient.

Thermal conductivity.

l [W/(m×K)] characterizes the property of a rock to transfer kinetic (or thermal) energy from one element to another.

Coefficient of thermal conductivity – the amount of heat passing per unit time through a cubic volume of a substance with a face of unit size, while on other faces a temperature difference of one degree is maintained (DT = 1°).

The thermal conductivity coefficient depends on:

ü mineral composition of the skeleton. The spread of coefficient values ​​can reach ten thousand times.

For example, the largest l for a diamond is 200 W/(m×K), because its crystal has virtually no structural defects. For comparison, l of air is 0.023 W/(m×K), water - 0.58 W/(m×K).

ü degree of fullness of the skeleton.

ü Thermal conductivity of fluids.

There is such a parameter as contact thermal conductivity coefficient .

Quartz has the highest contact coefficient - 7-12 W/(m×K). Next come hydrochemical sediments, rock salt, sylvite, and anhydrite.

Coal and asbestos have a reduced contact coefficient.

Additivity for the thermal conductivity coefficient is not observed, the dependence does not obey the additivity rule.

For example, the thermal conductivity of minerals can be written as follows:

1gl=Sv i ×1gl i ,

where 1gl i is the logarithm of the l i-th phase with the volumetric content v i .

An important property is the reciprocal of thermal conductivity, called thermal resistance.

Due to thermal resistance, we have a complex distribution of thermal fields. This leads to thermal convection, due to which special types of deposits can form - not an ordinary seal, but a thermodynamic one.

Thermodynamic resistance decreases with a decrease in density, permeability, humidity, and also (in northern regions) the degree of ice content.

It increases when water is replaced by oil, gas or air in the process of thermal pressure changes, with an increase in layered heterogeneity, the phenomenon of anisotropy.

Coals, dry and gas-saturated rocks have the greatest thermal resistance.

When moving from terrigenous rocks to carbonate rocks, the thermal resistance decreases.

Hydrochemical sediments such as halite, sylvite, mirabelite, anhydrite, i.e. have minimal thermal resistance. rocks with a lamellar salt structure.

Clay layers, among all layers, stand out for their maximum thermal resistance.

From all this we can conclude that thermal resistance determines the degree of thermal inertia, thermal conductivity.

Thermal diffusivity.

In practice, a coefficient such as thermal diffusivity, which characterizes the rate of temperature change during an unsteady heat transfer process.

а=l/(с×r), when l=const.

In fact, “a” is not constant, because l is a function of coordinates and temperature, and c is a function of porosity coefficient, mass, etc.

During development, we can use processes in which an internal heat source may occur (for example, acid injection), in which case the equation will look like this:

dТ/dt=а×Ñ 2 Т+Q/(с×r),

where Q is the heat of the internal heat source, r is the density of the rock.

Heat transfer.

The next important parameter is heat transfer.

DQ=k t ×DТ×DS×Dt,

where k t is the heat transfer coefficient.

Its physical meaning: the amount of heat lost to neighboring layers, through a unit of surface, per unit of time when the temperature changes by one degree.

Typically, heat transfer is associated with displacement into the layers above and below.

34. The influence of temperature on changes in the physical properties of an oil and gas reservoir.

The heat that is absorbed by the rock is spent not only on kinetic thermal processes, but also on mechanical work, which is associated with the thermal expansion of the formation. This thermal expansion is associated with the dependence of the bonding forces of atoms in the lattice of individual phases on temperature, in particular appearing in the direction of bonds. If atoms are more easily displaced when moving away from each other than when approaching each other, a displacement of the centers of the fissile atoms occurs, i.e. deformation.

The relationship between temperature increase and linear deformation can be written:

dL=a×L×dT,

where L is the original length [m], a is the coefficient of linear thermal expansion.

Similarly for volumetric expansion:

dV/V=g t ×dT,

where g t is the coefficient of volumetric thermal deformation.

Since the coefficients of volumetric expansion vary greatly for different grains, uneven deformations will occur during the impact, which will lead to destruction of the formation.

At the points of contact there is a strong concentration of stress, which results in the removal of sand and, accordingly, the destruction of the rock.

The phenomenon of oil and gas displacement is also associated with volumetric expansion. This is the so-called Joule-Thompson process. During operation, a sharp change in volume occurs, and a throttling effect occurs (thermal expansion with a change in temperature). Thermodynamic debitometry is based on the study of this effect.

Let's introduce one more parameter - adiabatic coefficient : h s =dТ/dр.

The differential adiabatic coefficient determines the change in temperature depending on the change in pressure.

The value of h S >0 under adiabatic compression. In this case, the substance heats up. The exception is water, because... in the range from 0¼4° it cools down.

h S =V/(C p ×g)×a×T,

where V is volume, T is temperature, a is linear expansion coefficient, g is gravitational acceleration.

The Joule-Thompson coefficient determines the temperature change during throttling.

e=dТ/dр=V/(Ср ×g)×(1 - a×Т)=V/(Ср ×g) - h S ,

where V/(Ср×g) determines heating due to the work of friction forces

h S – cooling of the substance due to adiabatic expansion.

For liquid V/Ср×g>>hS Þ Liquids heat up.

For gases e<0 Þ Газы охлаждаются.

In practice they use noise metry wells - a method based on the phenomenon when gas, when the temperature changes, releases vibrational energy, causing noise.

35. Changes in the properties of an oil and gas reservoir during the development of a deposit.

1. In their natural state, the layers are located at great depths, and, judging by the geothermal steps, the temperature in these conditions is close to 150°, so it can be argued that the rocks change their properties, because when we penetrate into the layer We disrupt the thermal balance.

2. When we pumping water into the reservoir, this water has a surface temperature. Once water enters the formation, it begins to cool the formation, which will inevitably lead to various unfavorable phenomena, such as waxing of oil. Those. If there is a paraffin component in the oil, then as a result of cooling, paraffin will fall out and clog the formation. For example, at the Uzen field, the temperature of oil saturation with paraffin is Tn = 35° (40°), and during its development these conditions were violated, as a result the formation temperature decreased, the paraffin fell out, blockage occurred and the developers had to pump in hot water for a long time and warm up the formation until all the paraffin has dissolved in the oil.

3. High-viscosity oils.

To liquefy them, a coolant is used: hot water, superheated steam, as well as internal heat sources. So, a combustion front is used as a source: oil is ignited and an oxidizer is supplied.

The following projects are also being implemented in Switzerland, France, Austria, and Italy:

A method for reducing the viscosity of oils using radioactive waste. They are stored for 10 6 years, but at the same time heat highly viscous oil, making it easier to extract.

36. Physical state of hydrocarbon systems in oil and gas formations and characteristics of these states.

Let's take a simple substance and consider the state diagram:

R

Point C is the critical point at which the difference between properties disappears.

Pressure (P) and temperature (T), which characterize the formation, can be measured in a very wide range: from tenths of MPa to tens of MPa and from 20-40° to more than 150°C. Depending on this, our deposits containing hydrocarbons can be divided into gas, oil, etc.

Because at different depths, pressures vary from normal geostatic to abnormally high, then hydrocarbon compounds can be in gaseous, liquid or in the form of gas-liquid mixtures in the deposit.

At high pressures, the density of gases approaches the density of light hydrocarbon liquids. Under these conditions, heavy oil fractions can dissolve in compressed gas. As a result, the oil will be partially dissolved in the gas. If the amount of gas is insignificant, then with increasing pressure the gas dissolves in oil. Therefore, depending on the amount of gas and its condition, deposits are distinguished:

1. pure gas;

2. gas condensate;

3. gas and oil;

4. petroleum containing dissolved gas.

The boundary between gas-oil and oil-and-gas deposits is arbitrary. It developed historically, in connection with the existence of two ministries: the oil and gas industries.

In the USA, hydrocarbon deposits are divided according to the value of the gas-condensate factor, density and color of liquid hydrocarbons into:

1) gas;

2) gas condensate;

3) gas and oil.

The gas condensate factor is the amount of gas in cubic meters per cubic meter of liquid product.

According to the American standard, gas condensates include deposits from which lightly colored or colorless hydrocarbon liquids with a density of 740-780 kg/m 3 and a gas condensate factor of 900-1100 m 3 /m 3 are extracted.

Gas deposits may contain adsorbed bound oil, consisting of heavy hydrocarbon fractions, constituting up to 30% of the pore volume.

In addition, at certain pressures and temperatures, the existence of gas hydrate deposits, where gas is in a solid state, is possible. The presence of such deposits is a large reserve for increasing gas production.

During the development process, initial pressures and temperatures change and technogenic transformations of hydrocarbons into deposits occur.

Somehow, gas may be released from oil during a continuous development system, as a result of which there will be a decrease in phase permeability, an increase in viscosity, a sharp decrease in pressure occurs in the bottomhole zone, followed by the loss of condensate, which will lead to the formation of condensate plugs.

In addition, during gas transportation, gas phase transformations can occur.

38. Phase diagrams of single-component and multicomponent systems.

Gypsum phase rule (shows the variability of the system - the number of degrees of freedom)

N - number of system components

m is the number of its phases.

Example: H 2 O (1 set) N=1 m=2 Þ r=1

When jammed R only one T

One-component system.

Compress from A to B - the first drop of liquid (dew point or condensation point P = P us)

At point D the last bubble of steam remains, the point of vaporization or boiling

Each isotherm has its own boiling and vaporization points.

Two-component system

Changes R And T, i.e., the condensation onset pressure is always less than the vaporization pressure.

Related information.

The process of heat transfer by thermal conductivity is explained by the exchange of kinetic energy between the molecules of a substance and the diffusion of electrons. These phenomena occur when the temperature of a substance is different at different points or when two bodies with different degrees of heating come into contact.

The basic law of thermal conductivity (Fourier's law) states that the amount of heat passing through a homogeneous (homogeneous) body per unit time is 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 flow transmitted by thermal conductivity, W;

l - thermal conductivity coefficient, ;

d - wall thickness, m;

t 1, t 2 - temperature of the heated and cold surface, K;

S - surface area, m2.

From this expression we can conclude that when developing the design of RES, the heat-conducting walls should be made thin, thermal contact should be ensured over the entire area in the connections of parts, and materials with a high thermal conductivity coefficient should be selected.

Let us consider the case of heat transfer through a flat wall of thickness d.

Figure 7.2 – Heat transfer through the wall

The amount of heat transferred per unit time through a section of the wall with area S will be determined by the already known formula

This formula is compared to Ohm's law equation for electrical circuits. It is not difficult to see their complete analogy. So the amount of heat per unit time P T corresponds to the current value I, the temperature gradient (t 1 - t 2) corresponds to the potential difference U.

The attitude is called T e r m i c h e s k i m resistance and denoted by R T,

The considered analogy between the flow of heat flow and electric current not only allows us to note the commonality of physical processes, but also facilitates the calculation of thermal conductivity in complex structures.

If in the considered case the element that needs to be cooled is located on a plane having a temperature t CT1, then

t ST1 = P T d/(lS) + t ST2.

Therefore, to reduce t CT1, it is necessary to increase the area of ​​the heat-removing surface, reduce the thickness of the heat-transmitting wall, and select materials with a high thermal conductivity coefficient.

To improve thermal contact, it is necessary to reduce the roughness of the contacting surfaces, cover them with heat-conducting materials and create contact pressure between them.

The quality of thermal contact between structural elements also depends on electrical resistance. The lower the electrical resistance of the contact surface, the lower its thermal resistance, the better the heat dissipation.

The lower the heat removal capacity of the environment, the longer it will take to establish a stationary heat transfer regime.

Typically the cooling part of the design is the chassis, housing or casing. Therefore, when choosing a design layout option, you need to look at whether the cooling part of the structure chosen for mounting has the conditions for good heat exchange with the environment or is heat-resistant.

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.

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

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 considered 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 coolants). 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.