Thermal conductivity defining equation

To determine the asymptotic behavior of the thermal conductivity A., equations (6.50)-(6.52) are inserted into equation (6.47) and the background is isolated by defining... 

Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theoiy. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950 2138, 1952 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundaiy. According to the Dukler theoiy, three fixed factors must be known to estabhsh the value of the average film coefficient the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group defined as follows ... 

The laser we use in these experiments is an exclmer laser with a pulse width of approximately 20 nsec. In this time regime the laser heating can be treated using the differential equation for heat flow with a well defined value for the thermal diffusivity (k) and the thermal conductivity (K) (4). 

Equations (8) are based on the assumption of plug flow in each phase but one may take account of any axial mixing in each liquid phase by replacing the molecular thermal conductivities fc, and ku with the effective thermal conductivities /c, eff and kn eff in the definition of the Peclet numbers. The evaluation of these conductivity terms is discussed in Section II,B,1. The wall heat-transfer terms may be defined as... 

In this equation S includes heat of chemical reaction, any interphase exchange of heat, and any other user-defined volumetric heat sources. At is defined as the thermal conductivity due to turbulent transport, and is obtained from the turbulent Prandtl number... 

ORR rate constant as defined by eq 61, 1/s ORR rate constant in Figure 11, cm/s thermal conductivity of phase k, J/cm K relative hydraulic permeability saturated hydraulic permeability, cm electrokinetic permeability, cm catalyst layer thickness, cm parameter in the polarization equation (eq 20) loading of platinum, g/cm molecular weight of species i, g/mol symbol for the chemical formula of species i in phase k having charge Zi... 

If the temperature gradient across the laminar sublayer and the value of thermal conductivity were known, it would be possible to calculate the rate of heat transfer by Equation 2.1. This is usually impossible, however, because the thickness ofthe laminar sublayer and the temperature distribution, such as shown in Figure 2.5, are usually immeasurable and vary with fluid velocity and other factors. Thus, a common engineering practice is the use of the film (or individual) coefficient of heat transfer, h, which is defined by Equation 2.16 and based on the difference between the temperature at the interface, and the temperature of the bulk of fluid, f], ... 

Chapman-Enskog theory provides the basis for the multicomponent transport properties laid out by Hirschfelder, Curtiss, and Bird [178] and by Dixon-Lewis [103]. The multi-component diffusion coefficients, thermal conductivities, and thermal diffusion coefficients are computed from the solution of a system of equations defined by the L matrix [103], seen below. It is convenient to refer to the L matrix in terms of its nine block submatrices, and in this form the system is given by... 

As a final example, consider line D of Table 9.1. We represent this problem as a body of density p, and heat capacity cp and whose surface is in contact with another medium of temperature Ts. Assume the initial body temperature is the same as the temperature of the other medium at I m = Ts. From the fundamental equation we can write, pcpLdTb/dt = X(TS — Tb)/L, where L is the characteristic conductive length, and X is the thermal conductivity. We now scale this problem over the entire time of the thermal transient. Once the entire time of the transient passes fe - t ), the body will have reached the new temperature of 7, 2. For the overall transient, the temperature rate of change is (Tb2 - Tb )/Gi -t ). and the average driving potential for the thermal conduction will be TS2 - T, )/2 = ( 7),2 - Tj, )/2. We now define the first-order relationship between the parameter as... 

The thermal conductivity has to be carefully defined. The fluid and the particulate matter will, in general, have different thermal conductivities. The conductivity, ka, in the above equation is an area averaged or apparent thermal conductivity of the porous material. 

Thermal conductivity is the most difficult quantity to understand in terms of the electronic structure. Thermal energy can be stored in vibrational normal modes of the crystal, and one can transport thermal energy through the lattice of ions. These concepts seem to be macroscopic. Therefore, one can set up suitable wave packets to treat thermal conductivity as quantized matter. In particular, electron plus induced lattice polarization can be defined as polarons. For conduction electrons, the electrical conductivity and the thermal conductivity were first observed by Wiedemann and Franz as indicated in the following equation ... 

Equation (1-1) is the defining equation for thermal conductivity. On the basis of this definition, experimental measurements may be made to determine the thermal conductivity of different materials. For gases at moderately low temperatures, analytical treatments in the kinetic theory of gases may be used to predict accurately the experimentally observed values. In some cases, theories are available for the prediction of thermal conductivities in liquids and solids, but in general, many open questions and concepts still need clarification where liquids and solids are concerned. 

These equations obey the Onsager reciprocal relations, which state that the phenomenological coefficient matrix is symmetric. The coefficients Lqq and Lu arc associated with the thermal conductivity k and the mutual diffusivity >, respectively. In contrast, the cross coefficients Llq and Lql define the coupling phenomena, namely the thermal diffusion (Soret effect) and the heat flow due to the diffusion of substance / (Dufour effect). 

By comparison with q = —/T((/77cA), the defining equation for the coefficient of thermal conductivity K, we find that... 

In order to treat thermal conduction in one-component systems, we may let Q = = U, the total energy of the molecule defined in equation... 

Equation 1-21 for the rate of conduction heat transfer under.steady conditions can also be viewed as the defining equation for thermal conductivity. Thus the thermal conductivity of a material can be defined as the-rate of... 

A two-dimensional bar has the geometry shown in Fig. P5-114 with. specified temperature T,iOn the upper surface and Tg on the lower surfaces, and insulation on the sides. Tiie thermal conductivity of tlie upper part of the bar is while that of the lower part is kg. For a grid defined by A.r = Ay = /, write the simplest form of the matrix equation, AT = C, used to find the steady-state temperature field in the cross section of the bar. Identify on the figure the grid nodes where you write tlie energy balance. 

In order to calculate the overall heat transfer coefficient in the equation above, the heat transfer properties of the heating medium and the liquor are described in terms of individual heat transfer coefficients, and the heat transfer properties of the separating solid in terms of its thermal conductivity. Deposits at the interface, which one might expect to be described by a thermal conductivity, are usually described instead in terms of either a fouling (heat transfer) coefficient or a fouling factor (which is usually defined to have a value of 1000/[fouling coefficient]). 

The mixture theories were originally developed for dielectric properties, but can be applied to other properties that are governed on a macroscopic level by Laplace s equation ( 10). Consequently, the generalized conductivity in the above equations can be the ionic conductivity o, thermal conductivity K, or complex electrical conductivity o defined by ... 

Equation (2.185) enables an explicit calculation for the heating or cooling time to be carried out. This time tk is defined such that once the time has passed a given temperature k is reached in the centre of the thermally conductive body (r+ = 0). This temperature corresponds to a particular value... 

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