State educational institution

Higher professional education

SAMARA STATE TECHNICAL

UNIVERSITY

Laboratory work

Temperature distribution solution.

Completed by: students of group 1-ET-4

Kodina O. N. Lastochkin N. M. Afanasyev M. A.

Samara 2012

Brief theory.

Heat transfer is the physical process of transferring thermal energy from a hotter body to a colder one, either directly (by contact) or through a separating (body or medium) partition made of any material. When physical bodies One system is at different temperatures, then thermal energy is transferred, or heat is transferred from one body to another until thermodynamic equilibrium occurs. Spontaneous transfer of heat always occurs from a hotter body to a colder one, which is a consequence of the second law of thermodynamics

Thermal conductivity is the transfer of thermal energy by structural particles of a substance (molecules, atoms, ions) during their thermal movement. Such heat exchange can occur in any body with a non-uniform temperature distribution, but the mechanism of heat transfer will depend on the aggregate state of the substance. The phenomenon of thermal conductivity is that the kinetic energy of atom molecules, which determines the temperature of a body, is transferred to another body during their interaction or is transferred from more heated areas of the body to less heated areas. Sometimes thermal conductivity is also called a quantitative assessment of the ability of a particular substance to conduct heat.

The numerical characteristic of the thermal conductivity of a material is equal to the amount of heat passing through a material with an area of ​​1 square m per unit of time (second) with a unit temperature gradient. This numerical characteristic is used to calculate thermal conductivity for calibration and cooling of profile products.

Fourier's law of thermal conductivity.

In steady state, the energy flux density transmitted through thermal conductivity is proportional to the temperature gradient:

where - the heat flux density vector - the amount of energy passing per unit time through a unit area perpendicular to each axis, - the thermal conductivity coefficient (sometimes called simply thermal conductivity), - temperature. The minus on the right side shows that the heat flow is directed opposite to the vector grad T (that is, in the direction of a rapid decrease in temperature). This expression is known as Fourier's law of thermal conductivity.

In integral form, the same expression will be written as follows (if we are talking about a stationary heat flow from one facet to another):

where is the total heat loss power, is the cross-sectional area of ​​the parallelepiped, is the temperature difference between the faces, is the length of the parallelepiped, that is, the distance between the faces.

The thermal conductivity coefficient is measured in W/(m K).

Vacuum thermal conductivity coefficient

The thermal conductivity coefficient of vacuum is almost zero (the deeper the vacuum, the closer to zero). This is due to the low concentration in vacuum of material particles capable of transferring heat. However, heat in a vacuum is transferred by radiation. Therefore, for example, to reduce heat loss, the thermal walls are made double, silvered (such a surface reflects radiation better), and the air between them is pumped out.

Currently, there are many analytical and numerical methods for solving thermal problems for cylindrical and rectangular bodies. In the case of heating bodies of more complex shapes, only numerical methods are suitable for solution. Nevertheless, the use of analytical methods for bodies of regular cylindrical or rectangular shape (parallelepiped) is fully justified based on both the costs of creating the model and the convenience in solving control problems.

Basic provisions.

The temperature gradient is a vector directed normal to the isothermal surface in the direction of increasing temperature, i.e.

, (1)

Where - unit vector directed normally in the direction of increasing temperature.

The gradient is also indicated by the symbol (nabla). The components of the gradient along the axes of Cartesian coordinates are equal to the corresponding partial derivatives so that

. (2)

The expression in square brackets in the formula can be written as
.

Fourier's basic law of thermal conductivity.

Heat transfer by thermal conduction normal to an isothermal surface from places with higher temperatures to places with lower temperatures.

The amount of heat passing per unit time and per unit area of ​​an isothermal surface is called the heat flux density

, (3)

Where – the amount of heat passing per unit time or the rate of heat flow; S – surface area.

Law: Heat flux density is directly proportional to the temperature gradient

, (4)

where λ is the thermal conductivity coefficient.

The thermal conductivity coefficient is equal to the amount of heat flowing per unit time through a unit surface with a temperature difference per unit normal length equal to one degree.

[W/(mdeg)]

The thermal conductivity coefficient depends on temperature for metals; it decreases linearly; increases for gases; for liquids other than water and glycerin, it decreases.

Materials with
[W/(mgrad)] are called thermal insulation.

In addition to λ, the thermal diffusivity coefficient a is used

Thermal diffusivity coefficient a is equal to the amount of heat flowing per unit time through a unit surface, with a difference in the volumetric concentration of internal energy of 1 J/m³ per unit normal length.

C spruce work: Learn to use the Elcut program to solve problems of heat distribution over a solid body.

Completing of the work.

In the Elcut program we create a Thermal problem and draw a Solid body (brick) and set its characteristics and faces.

Thermal conductivity of the body 1 W/(cell*m)

After which we solve and launch the solution to the thermal problem. From which we can see that the heat flow decreases as it passes through the body. And the temperature of body parts decreases with distance from the wall.

Answers on questions.

Let us consider heat transfer between two plates at temperatures under vacuum conditions. To a first approximation, we will assume that the gas molecules colliding with the first plate acquire energy, and the temperature corresponding to the temperature will characterize the energy of the molecules colliding with the second plate. It's easy to see what's between the plates

energy transfer occurs between practically non-colliding molecules. In this case, there will be no temperature gradient inside the gas.

Let us write the expression for the internal energy flux density in the direction from plate 1 to plate 2 (Fig. 5.6):

Here is the heat capacity at constant volume per molecule. The corresponding energy flux density in the reverse direction is:

where and c are the average values ​​of the concentration of molecules and the speed of their thermal movement. The difference will obviously determine the heat transfer density (heat transfer through a unit area per unit time):

Using the relation, we rewrite (52.3) in the form

Here specific heat gas at constant volume. The obtained result shows that heat transfer under vacuum conditions is proportional to the gas density.

In fact, under vacuum conditions, the degree of contact of falling molecules with the walls is insufficient to transfer to them upon reflection the average energy corresponding to the temperature solid; in this case, temperature jumps occur at the boundary of the gas with the walls. Taking the latter circumstance into account, formula (52.5) ​​takes the form

where is the accommodation coefficient, which takes into account the above temperature jumps and depends on the properties of the gas and solid surfaces.

The dependence of the thermal conductivity of gases on pressure under vacuum conditions can be observed using the device shown in Figure 5.7. Through two tubes 1 and 2, connected by a rubber stopper A, a wire is stretched, heated electric shock until it glows red. If air is pumped out from tube 2 through branch B using a forevacuum pump, then the glow of the wire in this

the tube changes from red to higher temperature (white) due to a decrease in heat removal by the gaseous medium.

According to (52.6), by lowering the pressure, thermal conductivity in a vacuum can be made extremely small. This circumstance is used in Dewar vessels (Fig. 5.8) intended for storing liquefied gases and implementation of adiabaticity in a number of devices. Dewar vessels have double walls, between which a high vacuum is created, due to which the thermal conductivity of the vessels is extremely low. The transfer of heat from the outside into vessels of this kind is carried out mainly by radiation, to reduce which the walls of the vessels are coated with a thin layer of silver.

The transfer of energy in the form of heat that occurs between bodies at different temperatures is called heat exchange. Driving force of any heat exchange process is the temperature difference between the more heated and less heated bodies, in the presence of which spontaneous heat transfer takes place.

According to the second law of thermodynamics, the spontaneous process of heat transfer in space occurs under the influence of a temperature difference and is directed towards decreasing temperature.

Heat transfer is the exchange of energy between molecules, atoms and free electrons. As a result of heat exchange, the intensity of movement of particles of a more heated body decreases, and that of a less heated body increases.

Heat transfer– the science of heat propagation processes. The laws of heat transfer underlie thermal processes - heating, cooling, vapor condensation, boiling of liquids, evaporation - and have great importance for carrying out many mass transfer processes (distillation, drying, etc.), as well as reaction processes of chemical technology that occur with the supply or removal of heat.

Bodies participating in heat exchange are called coolants. Heat can spread in any substance and even in a vacuum. There are no ideal heat insulators.

Heat is transferred in all substances thermal conductivity due to energy transfer by microparticles. Molecules, atoms, electrons and other microparticles that make up matter move at speeds proportional to temperature. Due to the interaction of particles with each other, faster ones give energy to slower particles, thus transferring heat from the zone with faster high temperature to an area with lower temperatures.

In liquids and gases, heat transfer can also occur due to the mixing of moving particles. In this case, not individual molecules, but large macroscopic volumes of a more heated liquid (gas) move to zones with lower temperatures, and less heated ones move to zones with a higher temperature. The transfer of heat along with macroscopic volumes of matter is called convection.

At the same time, thermal conductivity occurs along with convection. Such complex look heat transfer is called convective. Convection is the determining process of heat transfer in liquids and gases, since it is much more intense than thermal conductivity.

Heat exchange between a liquid (gas) and the surface of a solid (or vice versa) has become widespread. This process is called convective heat transfer or simply heat transfer.

Radiation is the third method of heat transfer . Heat is transmitted by radiation through all transparent media, including in vacuum (in space). Energy carriers during radiation are photons, emitted and absorbed by bodies participating in heat exchange.

In most cases, heat transfer is carried out in several ways simultaneously. The process of heat transfer involves all methods of heat transfer - thermal conductivity, convection and radiation. More complex is the process of heat transfer from a more heated coolant to a less heated one through the wall separating them, called heat transfer. In the process of heat transfer, heat transfer by convection is accompanied by thermal conductivity and heat transfer by radiation. However, when considering complex heat transfer processes, one or two of the three methods of heat propagation are predominant under certain conditions.

In continuously operating temperature devices various points do not change over time and the ongoing heat exchange processes are considered established(stationary). In periodically operating devices, where temperatures change over time, unsteady(non-stationary) heat transfer processes.