Plane walls

Strictly speaking, this expression is correct for a semi-infinite region bounded by a plane wall and containing a gas at rest. Here it is applied to a bounded region surrounded by a curved wall, and the molecules have a drift velocity parallel to che wall. Knudsen was concerned that this drift velocity might invalidate the treatment, but Pollard and Present [8] showed Chat this is not che case. 

The Plane Wall. To calculate the heat-transfer rate through a plane wall, Fourier s law can be appHed directly. 

Fig. 1. Sketch of heat flow through (a) a plane wall where the arrow indicates the direction of heat flow and (b) a series of composite walls, (c) The...

When the plane wall is made of mote than one material (Fig. lb), the heat-transfer rate is given as... 

HEAT TRANSFER BY CONDUCTION 9.3.1. Conduction through a plane wall... 

This important mechanism of heat transfer is now considered in more detail for the flow of heat through a plane wall of thickness x as shown in Figure 9.5. 

Planck s constant, 23 33 Planck spectrum, 23 2 Plan-Do-Check-Act (PDCA) model, 21 174 Plane-polarized light (PPL), 16 470, 476 Planetary rotation reactor, 22 154, 155 Planetary type mixers, 16 720 Plane wall, heat-transfer rate through,... 

For conduction through a plane wall of area A and thickness x, eqn. (82) may be integrated to give... 

Parallel plane walls Midplane Parallel to walls 1.004... 

A. Grib, Prikladnaya Mat.ematika i Mekhanika 8, 169 (1944) (On the propagation of a plane wave for an ordinary explosion near a plane wall) 12) S.J, Jacobs R.S. Grabenstetter, "Shaped Detonation Waves, OSRD 5603 (1946) 12a) K.O. Friedrichs,... 

Thermal Conductivity of Laminar Composites. In the case of laminar composites or layered materials (cf. Figure 1.74), the thermal conductance can be modeled as heat flow through plane walls in a series, as shown in Figure 4.36. At steady state, the heat flux through each wall in the x direction must be the same, qx, resulting in a different temperature gradient across each wall. Equation (4.2) then becomes... 

The exact treatment yields expressions which have the same form as the expressions given above only the numerical factors are different. The more detailed theory for the diffusion-convection problem between plane walls was developed by Furry, Jones, and Onsager (F10) and that for the column constructed from two concentric cylinders by Furry and Jones (Fll). Recently more attention has been given to the r61e of the temperature dependence of the transport coefficients in column operation (B9, S15). 

Surprisingly enough, it is possible for the steady-state assumption in a moving-boundary system to be essentially exact. This occurs in the deposition of a filter cake under the influence of a constant pressure difference. An interesting example is given by Brenner (B13), in a study of the unconfined growth of a filter cake on a circular cloth-covered aperture in a plane wall. It is assumed that the velocity vector of the filtrate within the cake is everywhere and at all times proportional to the pressure gradient (Darcy s law). 

The governing equation is therefore identical with that for the irrotational flow of an ideal fluid through a circular aperture in a plane wall. The stream lines and equipotential surfaces in this rotationally symmetric flow turn out to be given by oblate spheroidal coordinates. Since, from Eq. (157), the rate of deposition of filter cake depends upon the pressure gradient at the surface, the governing equation and boundary conditions are of precisely the same form as in the quasi-steady-state approximation... 

Motion of a Sphere Parallel to a Plane Wall—1 Motion... 

Figure 1.1 Conduction in a plane wall. In one dimension, Fourier s law becomes... 

Fourier s law is applied to conduction in a plane wall, as shown in Fig. 1.1. The heat flow Q x is the heat energy transfer in the x direction. 

A control volume drawn around a plane wall with three layers is shown in Fig. 1.2. Three different materials, M, N and P, of different thicknesses, AxM, AxN and AxP, make up the three layers. The thermal conductivities of the three substances are kM, kN and kp respectively. By the conservation of energy, the heat conducted through each of the three layers have to be equal. Fourier s law for this control volume gives... 

Problem The thermal conductivity of a plane wall varies with... 

A plane wall, 0.15 m thick, internally generates heat at a rate of 6 x 104 W/m3. One side of the wall is insulated and the other side is exposed to an environment at 25°C. The heat transfer coefficient between the wall and the environment is 750 W/(m2.K). The thermal conductivity of the wall is 20 W/(m.K). Calculate the maximum temperature in the wall. 

In order to illustrate how fully developed flov)/ through a duct filled with a porous medium can be analyzed, consider flow through a wide duct with plane walls, i.e., flow between parallel plates, with a uniform heat flux at the wall. The flow situation is thus as shown in Fig. 10.24. 

Let X be the ratio of the cylinder radius a to the distance between the long axis of the cylinder and the plane wall d. For the cylinder migrating perpendicular to the plane, the electrophoretic velocity is expressed by... 

FIG. 5. Variation of the dimensionless electrophoretic velocity U/Uq of a cylinder with distance parameter X in the vicinity of a plane wall (1) parallel to a dielectric plane, (2) perpendicular to a conducting wall. 

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