Spherical shell, conduction

Figure 9.8. Conduction through thick-walled tube or spherical shell The heat flow at any radius r is given by ...

Consider the heat conduction through a spherical shell of an inner diameter d and an outer diameter It can be shown that... 

This may be clear from Fig. 56. Consider a conducting, solid, spherical particle of radius a, carrying a positive charge q, immersed in a liquid of dielectric constant D. The potential of the sphere is q/Da. Next consider the contribution to the potential difference between the sphere and the liquid made by a spherical shell in the liquid, of radius r and thickness dr the charge dq on this will be opposite in sign to that on the sphere, and the contribution to the difference in potential between the surface of the solid and the liquid will be dqjDr. The total difference in potential between the surface of the solid and the liquid beyond the outer limit of the double layer will be the sum of the contributions from the sphere and all the shells, i.e. 

Derive an expression for the thermal resistance through a hollow spherical shell of inside radius r, and outside radius r having a thermal conductivity k. 

This model is very crude for small values of r, since it is based on the picture of the polarizable molecule as a conducting spherical shell. A further difficulty also arises from the fact that the polarizability a, which is observed for small field strengths, may not he applicable to the enormous fields in the neighborhoods of ions. 

Now consider a sphere, with density p, specific heat c, and outer radius R. The area of the sphere normal to the direclion of heat transfer at any location is A — 4vrr where r is the value of the radius at that location. Note that the heat transfer area A depends on r in this case also, and thus it varies with location. By considering a thin spherical shell element of thickness Ar and repeating tile approach described above for the cylinder by using A = 4 rrr instead of A = InrrL, the one-dimensional transient heat conduction equation for a sphere is determined to be (Fig. 2-17)... 

EXAMPLE 2-16 Heat Conduction through a Spherical Shell... 

Starting with an energy balance on a spherical shell volume clement, derive the one-dimensional transient heat conduction equation for a sphere with constant thermal conductivity and no heal generation. 

Consider a spherical shell of inner radius r outer radius Tj, thermal conductivity k, and emisslvity e. The outer surface of llie shell is subjected to radiation to surrouuding surfaces at but the direction of heat transfer is not known. Express the radiation boundary condition on the outer surface of the shell. 

Consider a spherical shell of inner radius r, and outer radius whose ihennal conductivity varies linearly in a specified temperature range as k(X) = (1 + fiT) where tfeo and /3 are two specified constants. The inner surface of the shell is maintained at a constant temperature of while the outer surface is mainlaincd al Tj. Assuming steady one-dimensional heat transfer, obtain a relation for (n) the heal transfer rate through the shell and (fe) the temprerature distribution 7(r) in the shell. 

Consider steady one-dimensional heat conduction through a plane wall, a cylindrical shell, and a spherical shell of unifonn thickness with constant thcimophysical propenies and no thermal energy generation. The geometry in which the variation of temperature in the direction of heal transfer will be linear is... 

Adding insulation to a cylindrical pipe or a spherical shell, however, is a different matter. The additional insulation increases the conduction re.sistance of... 

The structures within the homologous series Gdj + 2 2n + 3 C have been described in sect. 2 as a ccp arrangement of X and C atoms with Gd atoms in of the octahedral holes, i.e. as ordered defect derivatives of rock salt. That closed-shell situations do not always maximize stability is excellently demonstrated for the w = 1 and n = 2 members of the series because removal of one Gd atom per formula unit would produce closed-shell compounds in each case, Gd3X5C and Gd5X7C2 . Both metal-metal bonding interactions introduced by the three conduction electrons as well as the requirements of the highly charged interstitial atom to have a spherical shell of... 

Here we indicate how previous effectiveness factor analyses may be extended to situations where the pellet is not isothermal. Consider the case of a spherical pellet within which a catalytic reaction is taking place. If we examine an infinitesimally thin spherical shell with internal radius r similar to that shown in Figure 12.4 and write a steady-state energy balance over the interior core of the pellet, it is obvious that the heat flow outward by conduction across the sphere of radius r must be equal to the energy transformed by reaction within the central core. The latter quantity is just... 

Consider the nonisothermal porous spherical catalyst particle of radius R in which a single, irreversible, first-order reaction takes place at steady state (Figure 2.11). Taking the same spherical shell of thickness Ar at a radius r from the center, the steady-state energy balance over a differential shell of volume 4nt Ar includes conduction into and out of the control volume in the radial direction as well as heat release by reaction within the control volume ... 

Charging step - a cell was considered as a spherical shell with a dielectric membrane and with external and internal (cytoplasmic) conducting buffers. As a spherical dielectric, a position-dependent transmembrane potential was induced when the cell is submitted to an external field. This was a fast process. 

For heat conduction in a spherical shell (internal and external diameter dint and d.J we have ... 

A 50-liter liquid nitrogen container is constructed of concentric copper spheres separated by a vacuum space. If the emissivities of both inner surfaces of the vacuum space are 0.018, what loss in liters per day of the contents and percent of rated contents per day would be expected just due to radiation heat transfer Assume specular reflection for the surfaces. The warm outer surface is at 300 K. What is the heat transfer by molecular conduction when the gas in the vacuum space is air at a pressure of 15 mPa measured at 300 K The area ratio of the inner to the outer spherical shells may be assumed to be 0.8. 

The residual gas pressure of air measured at 300 K in the vacuum space of a spherical liquid hydrogen storage container is 1.3 mPa. The temperature of the outer surface is 300 K while that for the inner surface is 20.3 K. The inner vessel has an outside diameter of 1.524 m and the outer vessel has an inside diameter of 1.676 m. Emissivity of the inner vessel is 0.04 while that of the outer vessel is 0.09. Determine the heat transfer rate by radiation, the heat-transfer rate by molecular conduction, and the heat transfer rate by radiation if a spherical floating shield is inserted equidistant between the warm and cold spherical shells. The emissivity of the floating shield on both sides is 0.05. 

Recently, a study[265] has been conducted on the formation of droplets by the capillary wave instability of a spherical liquid shell with pulsating cavity. 

Electrical bulk properties of ionic solids can be rather inhomogeneous (Sec. 3.1). In the following it is shown that microelectrodes are a very useful tool to gain spatially resolved information on the conductivity of such inhomogeneous solids. Let us first consider the case of a spherical microelectrode (radius rme) atop a sample with homogeneous bulk conductivity Ubuik- The bulk resistance R between the microelectrode and a hemispherical counter-electrode of radius rce (Fig. 12a) can be calculated by integrating the infinitesimal resistances of hemispherical shells according to... 

Consider a spherical container of inner radius = 8 cm, outer radius fa = 10 ern, and thermal conductivity A = 45 W/m C, as shown in Fig. 2-52. The inner and outer surfaces of the container are maintained at constant temperatures of Ti = 200 0 and T2 - 80°C, respectively, as a result of some chemical reactions occurring inside. Obtain a general relation for the temperature distribution inside the shell under steady conditions, and determine the rate of heat loss from the container. 

Consider a 4 m-diameter spherical lank that is initially filled with liquid nitrogen at 1 atm and - 196°C. The tank is exposed to 20°C ambient air and 40 km/h winds. The tempeiature of the thin-shelled spherical tank is observed to be almost the same as the temperature of the nitrogen inside. Disregarding any radiation heat exchange, determine the rate of evaporation of the liquid nitrogen in the tank as a result of heat transfer from the ambient air if the tank is (a) not insulated, (h) insulated with 5-cm-thick fiberglass insulation (/ = 0.035 W/m °C), and (c) insulaterl with 2-cm-tliick superinsulation that has an effective thermal conductivity of 0.00005 W/m °C. 

Wind ai 30°C flows over a 0,5-m-diamctcr spherical tank containing iced water at 0°C with a velocity of 25 kni/h. If the lank is thin-shelled wilh a high Iheimal conductivity material, the rate at which ice melts is (a) 4.78kgdi (6) 6.15 kg/h (c) 7.45 kgfli... 

Impedance is the ratio of the voltage across a system to the current passing through the system. It measures the dielectric properties (permittivity and conductivity) of the system. The dielectric behavior of colloidal particles in suspension is generally described by Maxwell s mixture theory [26]. This relates the complex permittivity of the suspension to the complex permittivity of the particle, the suspending medium and the volume fraction. Based-on Maxwell s mixture theory, shelled-models have been widely used to model the dielectric properties of particles in suspension [35-40]. A single shelled spherical model is shown in Fig. la. 

Figure 1. (a) Diagram of a single shelled spherical particle, representing a cell in suspension, (b) Plot showing the real and imaginary parts of the Clausius-Mossotti factor of the mixture, calculated for different conductivities of the medium. The following parameters for the medium and a cell were used = 8.854 x 10 Fm , = 3 x lo m, [Pg.509]

Fig. 4.6a considers a spherical core-shell particle in which the core is taken to be vacuum and the shell is silver. The particle radius is 50 nm, so when the shell thickness is 50 nm we recover the solid particle result. As the shell becomes thinner, the plasmon resonance red-shifts considerably, very much like we see for highly oblate spheroids. Fig. 4.6a assumes that the dielectric constant of silver is independent of shell thickness, so the resonance width does not change much when the shell becomes thin. However, the correct dielectric response needs to include for finite size effects (as noted above) when the shell thickness is smaller than the conduction electron mean free path. Fig. 4.6b shows what happens to the spectrum in Fig. 4.6a when the finite size effect is incorporated, and we see that it has a significant effect for shells below 10 nm thickness, leading to much broader plasmon lineshapes. 

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