Where with p, J/(kg×K) – isobaric heat capacity; r, kg/m 3 – density; l, W/(m×K) – thermal conductivity coefficient; w x, w y, w z– projections of the fluid velocity vector; q v, W/m 3 – volumetric density of internal heat release of the liquid.
Equation (1.12) is written for the case l=const.
Differential for solids is called the differential heat equation and can be obtained from (1.12) under the condition w x = w y = w z = 0, with p=with v=With:
,
where is the thermal diffusivity coefficient, characterizing the rate of temperature change in the body. Values a = f(t) for various bodies are given in reference books.
Differential heat equation
 | (1.13) |
describes the non-stationary temperature field of solids with internal heat release (with internal heat sources). Such heat sources can be: Joule heat released when electric current passes through conductors; heat released by fuel rods of nuclear reactors, etc.
The differential heat equation (1.13), written in Cartesian coordinates, can be represented in cylindrical (r,z, φ) and spherical (r, φ , ψ).
In particular, in cylindrical coordinates ( r – radius; φ – polar angle; z- applicate) the differential equation of thermal conductivity has the form
 | (1.14) |
Uniqueness conditions
The differential equation describes many heat conduction processes. To select a specific process from this set, it is necessary to formulate the features of this process, which are called conditions of unambiguity and include:
· geometric conditions , characterizing the shape and size of the body;
· physical conditions , characterizing the properties of bodies participating in heat exchange;
· border conditions , characterizing the conditions of the process at the boundary of the body;
· initial conditions , characterizing the initial state of the system at non-stationary processes.
When solving thermal conductivity problems, the following are distinguished:
· boundary conditions of the first kind, when the temperature distribution on the body surface is specified:
t c = f (x, y, z, τ) or t c =const;
· boundary conditions of the second kind, when the heat flux density on the body surface is specified:
q c = f (x, y, z, τ) or q c =const;
· boundary conditions of the third kind, when the ambient temperature is set t and the heat transfer coefficient between the surface and the environment.
In accordance with the Newton-Richmann law, the heat flow transferred from 1 m 2 surface to a medium with a temperature t,
At the same time, this heat flow is supplied to 1 m 2 surface from the deep layers of the body by thermal conductivity
Then the heat balance equation for the body surface will be written in the form
 | (1.15) |
Equation (1.15) is a mathematical formulation of boundary conditions of the third kind.
The system of differential equations, together with the conditions of uniqueness, represents a mathematical formulation of the problem. Solutions of differential equations contain integration constants, which are determined using uniqueness conditions.
Test questions and assignments
1. Analyze in what ways heat is transferred from hot water to air through the wall of a heating radiator: from water to the inner surface, through the wall, from the outer surface to the air.
2. Why is there a minus on the right side of equation (1.3)?
3. Analyze the relationship using reference literature λ(t) for metals, alloys, thermal insulation materials, gases, liquids and answer the question: how does the thermal conductivity coefficient change with temperature for these materials?
4. How is heat flow determined? (Q, W ) with convective heat transfer, thermal conductivity, thermal radiation?
5. Write down the differential equation of thermal conductivity in Cartesian coordinates, describing a three-dimensional stationary temperature field without internal heat sources.
6. Write down the differential equation for the temperature field of a wire that is energized for a long time under a constant electrical load.
2. THERMAL CONDUCTIVITY AND HEAT TRANSFER
IN STATIONARY MODE
2.1. Thermal conductivity of a flat wall
Given: flat uniform wall thickness δ (Fig. 2.1) with a constant thermal conductivity coefficient λ and constant temperatures t 1 And t 2 on surfaces.
Define:
temperature field equation t=f(x) and heat flux density q, W/m2.
The temperature field of the wall is described by the differential equation of thermal conductivity (1.3) under the following conditions:
· because the mode is stationary;
· because there are no internal heat sources;
· 
because temperature t 1 And t 2 on surfaces the walls are constant.
Wall temperature is a function of only one coordinate X and equation (1.13) takes the form
Expressions (2.1), (2.2), (2.3) are a mathematical formulation of the problem, the solution of which will allow us to obtain the desired temperature field equation t=f(x).
Integrating equation (2.1) gives
Upon repeated integration, we obtain a solution to the differential equation in the form
Addiction t=f(x), according to (2.5) – a straight line (Fig. 2.1), which is true when λ=const.
To determine the heat flux density passing through the wall, we use Fourier's law
Taking into account 
we obtain a calculation formula for the heat flux density transmitted through a flat wall,
Formula (2.6) can be written in the form
Where 
The quantity is called thermal resistance of thermal conductivity flat wall.
Based on Eq.
q R=t 1 –t 2
we can conclude that the thermal resistance of the wall is directly proportional to the temperature difference across the wall thickness.
Take into account the dependence of the thermal conductivity coefficient on temperature, λ(t), it is possible if we substitute the values into equations (2.6) and (2.7) λ avg for temperature range t 1 –t 2.
Let's consider thermal conductivity multilayer flat wall, consisting, for example, of three layers
(Fig. 2.2).
Given:δ 1, δ2, δ 3, λ 1, λ 2, λ 3, t 1 =const, t 4 =const.
Define: q, W/m2; t 2, t 3.
Under stationary conditions and constant temperatures of the wall surfaces, the heat flow transmitted through a three-layer wall can be represented by a system of equations:
Temperatures at layer boundaries t 2 And t 3 can be calculated using equations (2.8) – (2.10) after the heat flux density ( q) by (2.12).
General form of equation (2.12) for a multilayer flat wall consisting of P homogeneous layers with constant temperatures on the outer surfaces and , has the form
2.2. Thermal conductivity of a cylindrical wall
under boundary conditions of the first kind
Given:
Homogeneous cylindrical wall (pipe wall) with inner radius r 1, external – r 2, length , with a constant thermal conductivity coefficient λ
, with constant temperatures on surfaces t 1 And t 2.
(Fig. 2.3).
Define: temperature field equation
t = f(r), heat flux transferred through the wall
Q, Tue.
Differential heat equation in cylindrical coordinates (1.14) for the conditions of this problem:
takes the form
The procedure for solving the system of equations (2.15) – (2.17) is the same as in the case of a flat wall: the general integral of the second-order differential equation (2.15) is found, which contains two integration constants
from 1 And from 2. The latter are determined using boundary conditions (2.16) and (2.17) and after substituting their values into the solution of the differential equation (general integral) we obtain equation of the temperature field of a cylindrical wall t = f (r) as
If we take the derivative of the right side of equation (2.18) and substitute it into (2.19), we obtain the calculation formula for heat flux of a cylindrical wall
 | (2.20) |
In technical calculations, heat flow is often calculated for 1 m pipe length:
and is called linear heat flux density.
Let us write equation (2.20) in the form
Where 
– thermal resistance to thermal conductivity of a cylindrical wall.
For three-layer cylindrical wall(a pipe covered with two layers of thermal insulation) with known constant surface temperatures ( t 1 And t 4), with known geometric dimensions ( r 1, r 2, r 3, r 4, ) and thermal conductivity coefficients of layers ( λ 1, λ 2, λ 3) (Fig. 2.4) we can write the following equations for heat flow Q:
Temperatures at the boundaries of layers (t 2,t 3) can be calculated using equations (2.21).
For multilayer cylindrical wall, consisting of P layers, formula (2.22) can be written in the general form
 | (2.23) |
Effective thermal conductivity coefficient for a multilayer cylindrical wall, as well as for a multilayer flat wall, is determined from the equality of the sum of the thermal resistances of the multilayer wall to the thermal resistance of a homogeneous wall of the same thickness as the multilayer wall. So, for two-layer thermal insulation of a pipe
(Fig. 2.4) effective thermal conductivity coefficient (λeff) will be determined from the equality
2.3. Thermal conductivity of flat and cylindrical walls
under boundary conditions of the third kind (heat transfer)
Boundary conditions of the third kind consist of setting the temperature of the liquid (t) and heat transfer coefficient () between the wall surface and the liquid.
The transfer of heat from one liquid to another through the wall separating them is called heat transfer.
Examples of heat transfer are the transfer of heat from flue gases to water through the pipe wall of a steam boiler, the transfer of heat from hot water to the surrounding air through the wall of a heating radiator, etc.
Heat exchange between the surface and the medium (coolant) can be convective, if the coolant is liquid (water, oil, etc.) or radiation-convective when heat is transferred by convective heat exchange and radiation, if the coolant is gas (flue gases, air, etc.).
Let us consider heat transfer through flat and cylindrical walls under the condition of only convective heat exchange on the surfaces. Heat transfer with radiation-convective heat transfer (complex heat transfer) on surfaces will be discussed later. W/m 2 heat transfer (Q
If a 1 And a 2 commensurate.
Heat transfer through a multi-layer cylindrical wall calculated by the formula
| (2.35) |
Where F 1 And F 2– area of the inner and outer surfaces of the multilayer cylindrical wall.
Solving problems of determining the temperature field is carried out on the basis of the differential equation of thermal conductivity, the conclusions of which are shown in the specialized literature. This manual provides options for differential equations without conclusions.
When solving problems of thermal conductivity in moving fluids that characterize a nonstationary three-dimensional temperature field with internal heat sources, the equation is used
Equation (4.10) is a differential energy equation in a Cartesian coordinate system (Fourier equation Kirchhoff). In this form, it is used in studying the process of thermal conductivity in any body.
If x = y = z =0, i.e. a solid body is considered, and in the absence of internal heat sources q v =0, then the energy equation (4.10) turns into the heat conduction equation for solids (Fourier equation)
(4.11)
The value C=a, m 2 sec in equation (4.10) is called the thermal diffusivity coefficient, which is a physical parameter of a substance that characterizes the rate of temperature change in the body during unsteady processes.
If the thermal conductivity coefficient characterizes the ability of bodies to conduct heat, then the thermal diffusivity coefficient is a measure of the thermal inertial properties of the body. From equation (4.10) it follows that the change in temperature over time t for any point in space is proportional to the value “a”, i.e. the rate of temperature change at any point of the body will be greater, the greater the thermal conductivity coefficient. Therefore, other things being equal, temperature equalization at all points in space will occur faster in the body that has a large thermal diffusivity coefficient. The thermal diffusivity coefficient depends on the nature of the substance. For example, liquids and gases have high thermal inertia and, therefore, a low thermal diffusivity coefficient. Metals have low thermal inertia, since they have a high thermal diffusivity coefficient.
To denote the sum of second derivatives with respect to coordinates in equations (4.10) and (4.11), you can use the symbol 2, the so-called Laplace operator, and then in the Cartesian coordinate system
The expression 2 t in a cylindrical coordinate system has the form
For a solid body under stationary conditions with an internal heat source, equation (4.10) is transformed into the Poisson equation
(4.12)
Finally, for stationary thermal conductivity and in the absence of internal heat sources, equation (4.10) takes the form of the Laplace equation
(4.13)
Differential equation of thermal conductivity in cylindrical coordinates with an internal heat source
(4.14)
4.2.6. Uniqueness conditions for heat conduction processes
Since the differential equation of thermal conductivity is derived on the basis of the general laws of physics, it characterizes the phenomenon of thermal conductivity in the most general form. Therefore, we can say that the resulting differential equation characterizes a whole class of heat conduction phenomena. In order to single out the specifically considered process from the countless number and give its complete mathematical description, it is necessary to add a mathematical description of all the particular features of the process under consideration to the differential equation. These particular features, which together with the differential equation provide a complete mathematical description of a specific heat conduction process, are called uniqueness or boundary conditions, which include:
a) geometric conditions characterizing the shape and size of the body in which the process takes place;
b) physical conditions characterizing the physical properties of the environment and the body (, C z, , a, etc.);
c) temporary (initial) conditions characterizing the distribution of temperatures in the body under study at the initial moment of time;
d) boundary conditions characterizing the interaction of the body in question with the environment.
Initial conditions are necessary when considering non-stationary processes and consist in specifying the law of temperature distribution inside the body at the initial moment of time. In the general case, the initial condition can be written analytically as follows for =0:
t = 1 x, y, z. (4.15)
In the case of uniform temperature distribution in the body, the initial condition is simplified: at =0; t=t 0 =idem.
Boundary conditions can be specified in several ways.
A. Boundary conditions of the first kind, specifying the temperature distribution on the surface of the body t c for each moment of time:
t c = 2 x, y, z, . (4.16)
In the particular case when the temperature on the surface is constant throughout the entire duration of heat transfer processes, equation (4.16) is simplified and takes the form t c =idem.
B. Boundary conditions of the second kind, specifying the value of the heat flux density for each point on the surface and any moment in time. Analytically this can be represented as follows:
q n = x, y, z, , (4.17)
where q n heat flux density on the surface of the body.
In the simplest case, the heat flux density over the surface and over time remains constant q n =idem. This case of heat exchange occurs, for example, when heating various metal products in high-temperature furnaces.
B. Boundary conditions of the third kind, specifying the ambient temperature tf and the law of heat exchange between the surface of the body and the environment. Newton's law is used to describe the process of heat exchange between the surface of a body and the environment.
According to Newton's law, the amount of heat given off by a unit surface of a body per unit time is proportional to the difference in temperature of the body t c and the environment t f
q = t c t f . (4.18)
The heat transfer coefficient characterizes the intensity of heat exchange between the surface of the body and the environment. Numerically, it is equal to the amount of heat given off (or perceived) by a unit of surface per unit of time when the temperature difference between the surface of the body and the environment is equal to one degree.
According to the law of conservation of energy, the amount of heat that is removed from a unit surface per unit time due to heat transfer (4.18) must be equal to the heat supplied to a unit surface per unit time due to thermal conductivity from the internal volumes of the body (4.7), i.e.
, (4.19)
where n normal to the body surface; the index “C” indicates that the temperature and gradient relate to the surface of the body (with n=0).
Finally, the boundary condition of the third kind can be written as
. (4.20)
Equation (4.20), in essence, is a particular expression of the law of conservation of energy for the surface of a body.
D. Boundary conditions of the fourth kind, characterizing the conditions of heat exchange between a system of bodies or a body with the environment according to the law of thermal conductivity. It is assumed that there is perfect contact between the bodies (the temperatures of the contacting surfaces are the same). Under the conditions under consideration, there is equality of heat flows passing through the contact surface:
. (4.21)
The study of any physical process is associated with the establishment of relationships between quantities characterizing this process. For complex processes, which include heat transfer by thermal conductivity, when establishing a relationship between quantities, it is convenient to use the methods of mathematical physics, which considers the course of the process not in the entire space under study, but in an elementary volume of matter during an infinitesimal period of time. The connection between the quantities involved in the transfer of heat by thermal conductivity is established in this case by the so-called differential equation of thermal conductivity. Within the limits of a selected elementary volume and an infinitely small period of time, it becomes possible to neglect the change in some quantities characterizing the process.
When deriving the differential equation of thermal conductivity, the following assumptions are made: physical quantities λ, with p And ρ permanent; there are no internal heat sources; the body is homogeneous and isotropic; the law of conservation of energy is used, which for this case is formulated as follows: the difference between the amount of heat entering due to thermal conductivity into an elementary parallelepiped during the time dτ and leaving it for the same time, is spent on changing the internal energy of the elementary volume under consideration. As a result, we arrive at the equation:
The quantity is called Laplace operator and is usually abbreviated as 2 t(the sign reads “nabla”); size λ /cρ called thermal diffusivity coefficient and denoted by the letter A. With the indicated notation, the differential heat equation takes the form
Equation (1-10) is called differential equation of thermal conductivity, or the Fourier equation, for a three-dimensional unsteady temperature field in the absence of internal heat sources. It is the main equation in the study of heating and cooling of bodies in the process of heat transfer by thermal conductivity and establishes a connection between temporal and spatial changes in temperature at any point in the field.
Thermal diffusivity coefficient A= λ/cρ is a physical parameter of a substance and has a unit of measurement m 2 / s. In non-stationary thermal processes the value A characterizes the rate of temperature change. If the thermal conductivity coefficient characterizes the ability of bodies to conduct heat, then the thermal diffusivity coefficient A is a measure of the thermal inertial properties of bodies. From equation (1-10) it follows that the change in temperature over time ∂t / ∂τ for any point of the body is proportional to the value A Therefore, under the same conditions, the temperature of the body that has a higher thermal diffusivity will increase faster. Gases have small, and metals have large, thermal diffusivity coefficients.
The differential equation of thermal conductivity with heat sources inside the body will have the form
Where q v- the amount of heat released per unit volume of a substance per unit time, With- mass heat capacity of the body, ρ - body density .
The differential equation of thermal conductivity in cylindrical coordinates with an internal heat source will have the form
Where r- radius vector in a cylindrical coordinate system; φ - corner.
Setting TMO objectives
We have a volume that is affected by thermal loads, it is necessary to determine the numerical value q V and its distribution by volume.
Fig. 2 - External and internal sources of friction
1. Determine the geometry of the volume under study in any selected coordinate system.
2. Determine the physical characteristics of the volume under study.
3. Determine the conditions that initiate the TMT process.
4. Clarify the laws that determine heat transfer in the volume under study.
5. Determine the initial thermal state in the volume under study.
Problems solved when analyzing solid waste:
1. “Direct” tasks of TMO
Given: 1,2,3,4,5
Determine: temperature distribution in space and time (further 6).
2. “Inverse” TMT problems (inverse):
a) inverse boundary tasks
Given: 1,2,4,5,6
Define: 3;
b) inverse odds tasks
Given: 1,3,4,5,6
Define: 2;
c) reverse retrospective task
Given: 1,2,3,4,6
Define: 5.
3. “Inductive” tasks of TMO
Given: 1,2,3,5,6
Define: 4.
FORMS OF HEAT TRANSFER AND THERMAL PROCESSES
There are 3 forms of heat transfer:
1) thermal conductivity in solids (determined by microparticles, and in metals by free electrons);
2) convection (determined by macroparticles of the moving medium);
3) thermal radiation (determined by electromagnetic waves).
Thermal conductivity of solids
General concepts
Temperature field is a set of temperature values in the volume under study, taken at a certain point in time.
t(x, y, z, τ)- a function that determines the temperature field.
There are stationary and non-stationary temperature fields:
stationary - t(x,y,z);
non-stationary - t(x, y, z, τ).
The condition for stationarity is:
Let's take a certain body and connect points with equal temperatures
Fig. 3-Temperature gradient and heat flow
grad t- temperature gradient;
on the other side: 
.
Fourier's law - heat flow in solids is proportional to the temperature gradient, the surface through which it passes and the time interval under consideration.
The proportionality coefficient is called the thermal conductivity coefficient λ , W/m·K.
shows that heat spreads in the direction opposite to the temperature gradient vector.
; 
For an infinitesimal surface and time interval:
Heat equation (Fourier equation)
Consider an infinitesimal volume: dv =dx dy dz
Fig. 4 - Thermal state of an infinitesimal volume
We have a Taylor series:
Likewise:
; 
; 
.
In the general case we have in a cube q V. The conclusion is based on the generalized law of conservation of energy:
.
According to Fourier's law:
; 
; 
.
After transformations we have:
.
For a stationary process:
The spatial dimension of problems is determined by the number of directions in which heat transfer occurs.
One-dimensional problem: 
;
for a stationary process: 
;
For : 
For : 
;
a- thermal diffusivity coefficient, 
.Cartesian system;
k = 1, ξ = x - cylindrical system;
k = 2, ξ = x - spherical system.
Uniqueness conditions
Uniqueness condition – These are conditions that make it possible to select from the set of feasible solutions a single one that corresponds to the task at hand.
Question 23 What is the specific heat of fusion of ice?
The specific heat of fusion is found by the formula:
where Q is the amount of heat required to melt a body of mass m.
when solidifying, substances release the same amount of heat that was required to melt them. Molecules, losing energy, form crystals, being unable to resist the attraction of other molecules. And again, the body temperature will not decrease until the entire body hardens, and until all the energy that was expended on its melting is released. That is, the specific heat of fusion shows both how much energy must be expended to melt a body of mass m, and how much energy will be released when a given body solidifies.
For example, the specific heat of fusion of water in the solid state, that is, the specific heat of fusion of ice is 3.4*10^5 J/kg
The specific heat of fusion of ice is 3.4 times 10 to the 5th power joule/kg
The specific heat of fusion is denoted by the Greek letter λ (lambda), and the unit of measurement is 1 J/kg
Question 24 Let us denote L1 as the specific heat of vaporization and L2 as the specific heat of fusion. That more?
Since a body gains energy during vaporization, we can conclude that the internal energy of a body in a gaseous state is greater than the internal energy of a body of the same mass in a liquid state. Therefore, during condensation, steam releases the amount of energy that was required for its formation
Specific heat of vaporization– a physical quantity showing the amount of heat required to convert 1 kg of a substance into steam without changing its temperature. Odds " r
Specific heat of fusion– a physical quantity showing the amount of heat required to transform 1 kg of a substance into liquid without changing its temperature. Odds " λ » for different substances, as a rule, are different. They are measured empirically and entered into special tables
The specific heat of vaporization is greater
Question 25: differential heat equation for a two-dimensional unsteady temperature field in Cartesian coordinates?
x i = x, y, z – Cartesian coordinate system;
If the temperature remains constant along one of the coordinates, then mathematically this condition is written (for example, for the z coordinate) as follows: dT/dz=0.
In this case, the field is called two-dimensional and is written:
for non-stationary mode T=T(x, y, t);
for stationary mode T=T(x, y).
Equations of a two-dimensional temperature field for the mode
non-stationary:
Question 26: differential heat equation for a nonstationary temperature field in cylindrical coordinates?
x i = r, φ, z – cylindrical coordinate system;
Temperature field is a set of temperature values at all points of a given computational domain and over time.
The temperature field is measured in degrees Celsius and Kelvin and is designated in the same way as in TTD: , where x i are the coordinates of the point in space at which the temperature is found, in meters [m]; τ – time of the heat exchange process in seconds, [s]. That. the temperature field is characterized by the number of coordinates and its behavior over time.
The following coordinate systems are used in thermal calculations:
x i = r, φ, z – cylindrical coordinate system;
The temperature field, which changes over time, called non-stationary temperature field. And vice versa, the temperature field, which does not change over time, called stationary temperature field.
cylindrical coordinates (r – radius; φ – polar angle; z – applicate), the differential equation of thermal conductivity has the form
,
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