Thermal conductivity in a gas

Suppose that two large metal plates parallel to the xy-plane and separated by a distance Z are at temperatures and T2, the hotter plate (T2) being the upper one. After some time a steady state will be established in which there is a downward flow of heat at a constant rate. This flow of heat results from the fact that the molecules at the upper levels have a greater thermal energy than those at the lower levels the molecules moving downward carry more energy than do those moving upward. 

If the gas is monatomic with an average thermal energy e = f/cT, then the average energy of the molecules at the height z is 

To calculate the heat flow, we consider an area 1 m in a horizontal plane at the height z (Fig. 30.3). The energy carried by a molecule as it passes through the plane depends on the temperature of the layer of gas at which the molecule had its last opportunity to adjust its 

The factor appears since, on the average, only of the molecules are going down and only are going up. The net flow upward is denoted by and is 

Before writing out the equation in detail, we should note that if the gas is not to have net motion through the surface we require that the number of molecules going up in unit time must equal the number going down, so that 

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Thus, from Eqs. (4.16) and (4.17), Nup 10, indicating that the Nup of a single particle due to thermal conduction in a gas-solid suspension roughly varies from 2 to 10. A more detailed discussion on the variation of Nup with S/dp is given by Zabrodsky (1966). 

The mechanism of thermal conduction in a gas is a simple one. We identify the kinetic energy of a molecule with its temperature thus, in a high-temperature region, the molecules have higher velocities than in some lower-temper-... 

This equation is applicable to an isotropic medium only, so the heat is conducted with the same thermal conductivity k in all directions. The thermal conductivity A is a property of a conducting medium, and is mainly a function of temperature. High pressure affects the thermal conductivity in a gas medium. 

What physical properties of gas molecules influence their ability to conduct heat A quantitative dependence of the thermal conductivity of a gas can be expressed in terms of the gas molecules specific heat, mass, and cross-sectional area. Specif-... 

These measure the change in thermal conductivity of a gas due to variations in pressure—usually in the range 0.75 torr (100 N/m2) to 7.5 x 10"4 torr (0.1 N/m2). At low pressures the relation between pressure and thermal conductivity of a gas is linear and can be predicted from the kinetic theory of gases. A coiled wire filament is heated by a current and forms one arm of a Wheatstone bridge network (Fig. 6.21). Any increase in vacuum will reduce the conduction of heat away from the filament and thus the temperature of the filament will rise so altering its electrical resistance. Temperature variations in the filament are monitored by means of a thermocouple placed at the centre of the coil. A similar filament which is maintained at standard conditions is inserted in another arm of the bridge as a reference. This type of sensor is often termed a Pirani gauge. 

The coefficients of transport properties considered here include the viscosity, diffusivity, and thermal conductivity of a gas. The transport coefficients vary with gas properties if the flow is laminar. When the flow is turbulent, the transport coefficients become strongly dependent on the turbulence structure. Here we only deal with the laminar transport coefficients the discussion of the turbulent transport coefficients is given in 5.2.4. 

Consider a packet of emulsion phase being swept into contact with the heating surface for a certain period. During the contact, the heat is transferred by unsteady-state conduction at the surface until the packet is replaced by a fresh packet as a result of bed circulation, as shown in Fig. 12.6. The heat transfer rate depends on the rate of heating of the packets (or emulsion phase) and on the frequency of their replacement at the surface. To simplify the model, the packet of particles and interstitial gas can be regarded as having the uniform thermal properties of the quiescent bed. The simplest case is represented by the problem of one-dimensional unsteady thermal conduction in a semiinfinite medium. Thus, the governing equation with the boundary conditions and initial condition can be imposed as... 

The thermal conductivity of a gas is the quantity which is measured in the Pirani gauge (page 125) and in the detector of a gas-phase chromatography column (page 171). The thermal conductivity is related to the heat capacity of the gas, which measures the amount of energy that can be absorbed per molecule to the velocity of the molecules, which is a measure of the number of collisions with the heated surface per unit time and pressure and to the pressure of the gas. 

Chapman S (1916) On the Law of Distribution of Molecular Velocities, and on the Theory of Viscosity and Thermal Conduction, in a Non-Uniform Simple Monoatomic Gas. Phil Trans Roy Soc London 216A 279-348... 

Heat Transfer in a Packed Bed (Effective Thermal Conductivity) In a bed of solid particles through which a reacting fluid is passing, heat can be transferred in the radial direction by a number of mechanisms. However, it is customary to consider that the bed of particles and the gas may be replaced by a hypothetical solid in which conduction is the only mechanism for heat transfer. The thermal conductivity of this solid has been termed the effective thermal conductivity k. With this scheme the temperature T of any point in the bed may be related to and the position parameters r and z by the differential equation... 

Chapman, S. 1916 On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas. Philosophical... 

This reflects the fact that the thermal conductivity of a gas obeying simple kinetic theory is independent of the pressure. The transition in thermal conductivity between asymptotes usually occurs between 0.1 and 100 mmHg, which includes the pressures characteristic of freeze drying processes. The pressure range over which the transition in thermal conductivity between asymptotes occurs is characteristic of the pore size distribution of the void spaces within the freeze-dried material [50]. A smaller pore dimension means that the gas must achieve a higher pressure in order for the mean free path of the gas to become comparable to the pore spacing and, hence, means that the transition between asymptotes will occur at higher pressures. 

In a stationary gas, all heat transfer is conductive. An example of this is the boundary layer immediately adjacent to a wall surface, such as a heat exchanger surface. The thermal conductivity of a gas is, in principle, independent of pressure but increases with rising tem-... 

Each of these thermal conductions mechanism is shown in Figure 13-3. Here, the thermal conduction through the air can be regarded as a transport phenomenon with kinetic energy driven by the collision of gas molecules in the air imder a temperature gradient. Therefore, the thermal conductivity of a gas depends on the mean free path of the gas, and the mean free path of a gas (If) enclosed in a narrow space can be given by equation (13-2), from the mean pore size (I,) and the mean free path of the gas in free space (Lg), and this can be transformed as in equation (13-3) (Takahama, 1995 Takita,... 

The thermal conductivity of a gas mixture which is measured directly is not the quantity X introduced in equation (4.79), because measurements are always performed in the absence of a net diffusive flux. In order to evaluate the measured thermal conductivity in the zero-density limit X, the multicomponent diffusion coefficients are employed (Ross et al. 1992) and then one obtains, in a consistent first-order approximation. 

The evaluation of the thermal conductivity of a gas mixture is rather more complicated and difficult than for the other two properties. The difficulty stems from equations (4.92)-(4.102), which make it clear that, in general, many more cross sections are involved for each binary interaction than for the viscosity or diffusion coefficient. This is a result of the presence of internal energy and its relaxation. Of course, if these cross sections could be evaluated from a pair potential, the additional difficulty would be rather minor, since the calculation of extra cross sections is a relatively small additional burden compared with the treatment of the collision dynamics. However, as has been pointed out before, the evaluation of the collision dynamics prevents such calculations from being performed routinely. As a result, the cross sections that enter the expressions for the thermal conductivity must be evaluated by other means for some of them this is extremely difficult, since there is little guidance from experiment or model calculations. 

Equations (4.92) to (4.102), which express the thermal conductivity of a gas mixture in terms of the cross sections of the Wang Chang-Uhlenbeck theory have been rewritten by Monchick et al. (1965) in an approximate form, in which so-called complex collisions have been neglected, as... 

This is the proof of the variational principle that Eq. 93 gives the solution of the Chapman-Enskog equation. The variational principle given in the form of Eq. 93 is much more convenient for practical purposes, because we need not consider restrictions other than simple ones such as the auxiliary conditions, Eqs. 70, 71, and 72. In the case of thermal conduction in a simple gas, Eq. 93 reduces to... 

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