Thermal conductivity, expression

These two expressions differ only by the leading constant terms. The simple thermal conductivity expression derived here is roughly 40% the size of the rigorous result. It captures the functional dependence on temperature, molecular mass, heat capacity, and pressure (independent of pressure) of the exact result. Experimentally the thermal conductivity is generally found to be independent of pressure, except at very low pressures. The thermal conductivity is predicted to increase as the square root of temperature, which somewhat underestimates the actual temperature dependence. Consideration of interactions between molecules, as in the next section, brings the temperature dependence into better accord with observation. 

K-factor K-factor is a term sometimes used for the thermal insulation value or coefBcient of thermal conductivity. It is numerically equivalent to the thermal conductivity expressed in British units (BTU in.)/(ft h°FA and its inverse (1/K) is known as the R-factor. K-factors and R-factors are commonly used for thermal insulating materials such as plastic foam. 

The numeric value of the thermal conductivity expressed in British units is sometimes referred to as the K-factor, and its inverse (1/K) as the R-factor. 

In treating the thermal conductivity due to electrons, the solid is considered to contain an electron gas. The gas thermal conductivity expression may again be applied, that is. 

The reactor shell and the inner pipes are made of austenitic stainless steel with thermal conductivity expressed as follows ... 

Thermal conductivity is expressed in W/(m K) and measures the ease in which heat is transmitted through a thin layer of material. Conductivity of liquids, written as A, decreases in an essentially linear manner between the triple point and the boiling point temperatures. Beyond a reduced temperature of 0.8, the relationship is not at all linear. For estimation of conductivity we will distinguish two cases < )... 

As a good first approximation (187), the heat conduction of low density foams through the soHd and gas phases can be expressed as the product of the thermal conductivity of each phase times its volume fraction. Most rigid polymers have thermal conductivities of 0.07-0.28 W/(m-K) and the corresponding conduction through the soHd phase of a 32 kg/m (2 lbs/fT) foam (3 vol %) ranges 0.003-0.009 W/(m-K). In most cellular polymers this value is deterrnined primarily by the density of the foam and the polymer-phase composition. Smaller variations can result from changes in cell stmcture. 

The noise is expressed as noise density in units of V/(Hz), or integrated over a frequency range and given as volts rms. Typically, photoconductors are characterized by a g-r noise plateau from 10 to 10 Hz. Photovoltaic detectors exhibit similar behavior, but the 1/f knee may be less than 100 Hz and the high frequency noise roU off is deterrnined by the p—n junction impedance—capacitance product or the amplifier (AMP) circuit when operated in a transimpedance mode. Bolometers exhibit an additional noise, associated with thermal conductance. 

A sohdus indicates the quotient of two unit symbols and the word per the division of two unit names m/s for meter per second. The horizontal line or negative powers are also permissible. The sohdus or the word per is not repeated in the same expression, eg, acceleration as m/s for meter per second squared and thermal conductivity as W/(m-K) for watt per meter kelvin. 

Electrical—Thermal Conductivities. Electrical conductivities of alloys (Table 5) are often expressed as a percentage relative to an International Annealed Copper Standard (lACS), ie, units of % lACS, where the value of 100 % lACS is assigned to pure copper having a measured resistivity value of 0.017241 Q mm /m. The measurement of resistivity and its conversion to % lACS is covered under ASTM B193 (8). 

Since each ratio is dimensionless, any consistent units may be employed in any ratio. The significance of the symbols is as follows t = temperature of the surroundings tb = initial uniform temperature of the body t = temperature at a given point in the body at the time 0 measured from the start of the heating or coohng operations k = uniform thermal conductivity of the body p = uniform density of the boc c = specific heat of the body hf = coefficient of total heat transfer between the surroundings and the surface of the body expressed as heat transferred per unit time per unit area of the surface per unit difference in temperature between surroundings and surface r = distance, in the direction of heat conduction, from the midpoint or midplane of the body to the point under consideration / = radius of... 

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 ... 

Expressions for the a. derivatives are of the same form r = rate of reaction, a fi motion of s and T G = mass flow rate, mass/(time)(siiperficial cross section) u = linear velocity D = diffiisivity k = thermal conductivity... 

The ability of a material to retard the flow of heat is expressed by its thermal conductivity (for unit thickness) or conductance (for a specific thickness). Low values for thermal conduc tivity or conductance (or high thermal resistivity or resistance value) are characteristics of thermal insulation. 

FIG. 27-4 Thermal conductivity of petroleum liquids. The solid lines refer to density expressed as degrees API the broken lines refer to relative density at 288 K (15 C). (K = [ F + 459.7]/1.8)... 

Dinwiddle et al. [14] proposed a model for the behavior of the CBCF in which the temperature dependence of thermal conductivity (Aj-) may be expressed as... 

Thermal conductivities for gaseous compounds are important in unit operations involving heat transfer coefficients. Thermal conductivities can be readily computed from an empirical polynomial expression that has the following form ... 

Table 1 provides states the temperature range over which the correlation constants are reported from the literature. Estimates from this expression using the constants in Table 1 are generally accurate and typically provide agreement to well within + 1% compared to experimentally measured thermal conductivities. 

Liquified gases are sometimes stored in well-insulated spherical containers that are vented to the atmosphere. Examples in the industry are the storage of liquid oxygen and liquid ammonia in spheres. If the radii of the inner and outer walls are r, and r, and the temperatures at these sections are T, and T, an expression for the steady-state heat loss from the walls of the container may be obtained. A key assumption is that the thermal conductivity of the insulation varies linearly with the temperature according to the relation ... 

The inputs required for the calculation are the radii, iimer and outer temperatures, and thermal conductivities at the two temperatures. This expression enables an estimate of the heat flow into a spherical storage tank containing liquified gas. 

W. The for this equation should be expressed m terms of Dj(N].., = hjDj/k) as should the Reynolds number (Np = D vp/p). k being thermal conductivity, v being velocity, and p being density. 

Thermal conductance defines a material s ability to transmit heat measured in watts per square meter of surface area for a temperature gradient of one Kelvin in terms of a specific thickness expressed in meters. Its dimensions are therefore W/m K. 

Thermal resistance is the reciprocal of thermal conductance. It is expressed as m KTW. Since the purpose of thermal insulation is to resist heat flow, it is convenient to measure a material s performance in terms of its thermal resistance, which is calculated by dividing the thickness expressed in meters by the thermal conductivity. Being additive, thermal resistances facilitate the computation of overall thermal transmittance values (t/-values). 

Thermal conductivity is a function of temperature and experimental data may often be expressed by a linear relationship of the form ... 

Bi very small, (say, <0.1). Here the main resistance to heat transfer lies within the fluid this occurs when the thermal conductivity of the particle in very high and/or when the particle is very small. Under these conditions, the temperature within the particle is uniform and a lumped capacity analysis may be performed. Thus, if a solid body of volume V and initial temperature Oo is suddenly immersed in a volume of fluid large enough for its temperature 0 to remain effectively constant, the rate of heat transfer from the body may be expressed as ... 

Explain the concepts of momentum thickness" and displacement thickness for the boundary layer formed during flow over a plane surface. Develop a similar concept to displacement thickness in relation to heat flux across the surface for laminar flow and heat transfer by thermal conduction, for the case where the surface has a constant temperature and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain an expression for this thermal thickness in terms of the thicknesses of the velocity and temperature boundary layers. 

In the buffer zone the value of d +/dy+ is twice this value. Obtain an expression for the eddy kinematic viscosity E in terms of the kinematic viscosity (pt/p) and y+. On the assumption that the eddy thermal diffusivity Eh and the eddy kinematic viscosity E are equal, calculate the value of the temperature gradient in a liquid flowing over the surface at y =15 (which lies within the buffer layer) for a surface heat flux of 1000 W/m The liquid has a Prandtl number of 7 and a thermal conductivity of 0.62 W/m K. 

In these equations the independent variable x is the distance normal to the disk surface. The dependent variables are the velocities, the temperature T, and the species mass fractions Tit. The axial velocity is u, and the radial and circumferential velocities are scaled by the radius as F = vjr and W = wjr. The viscosity and thermal conductivity are given by /x and A. The chemical production rate cOjt is presumed to result from a system of elementary chemical reactions that proceed according to the law of mass action, and Kg is the number of gas-phase species. Equation (10) is not solved for the carrier gas mass fraction, which is determined by ensuring that the mass fractions sum to one. An Arrhenius rate expression is presumed for each of the elementary reaction steps. 

The ordinary multicomponent diffusion coefficients D j and the viscosity and thermal conductivity are computed from appropriate kinetic theory expressions. First, pure species properties are computed from the standard kinetic theory expressions. For example, the binary diffusion coefficients are given in terms of pressure and temperature as... 

In the laser flash method, the heat is put in by laser flash instead of electric current in the stepwise heating method mentioned above. Thus this method may be classified as a stepwise heating method. A two-layered laser flash method was developed by Tada et al. " The experimental method and the data analysis, including a case involving radiative heat flow, are described in detail in the review article by Waseda and Ohta. A thin metal plate is placed at the surface of a melt. A laser pulse is irradiated onto a metal plate of thickness / having high thermal conductivity. The sample liquid under the metal plate and the inert gas above the plate are designated as the third and first layers, respectively. The temperature of the second layer becomes uniform in a short time" and the response thereafter is expressed by... 

Table 1.6 Characteristic quantities to be considered for micro-reactor dimensioning and layout. Steps 1, 2, and 3 correspond to the dimensioning of the channel diameter, channel length and channel walls, respectively. Symbols appearing in these expressions not previously defined are the effective axial diffusion coefficient D, the density thermal conductivity specific heat Cp and total cross-sectional area S, of the wall material, the total process gas mass flow m, and the reactant concentration Cg [114].

Effectively, Eqs. (86) and (87) describe two interpenetrating continua which are thermally coupled. The value of the heat transfer coefficient a depends on the specific shape of the channels considered suitable correlations have been determined for circular or for rectangular channels [100]. In general, the temperature fields obtained from Eqs. (86) and (87) for the solid and the fluid phases are different, in contrast to the assumptions made in most other models for heat transfer in porous media [117]. Kim et al. [118] have used a model similar to that described here to compute the temperature distribution in a micro channel heat sink. They considered various values of the channel width (expressed in dimensionless form as the Darcy number) and various ratios of the solid and fluid thermal conductivity and determined the regimes where major deviations of the fluid temperature from the solid temperature are found. 

Here, q is the flux of heat (W m ), X is the thermal conductivity (W K ), T is temperature (K), and r is the distance from the center of the spherical heat source. Under the steady state approximation, the heat generated in the small sphere, Qj , is equal to the heat flow, Qflow. from the surface of the small sphere to the surrounding medium, as expressed by Eq. (8.7). 

The constant of proportionality k is known as the thermal conductivity of the material and the above relationship is known as Fourier s law for conduction in one dimension. The thermal conductivity k is the heat flux which results from unit temperature gradient in unit distance. In s.i. units the thermal conductivity, k, is expressed in Wm"1 K. Integration of Fourier s law yields... 

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