Infinite Parallel Planes

The value of J2 is determined from proportioning the resistances between J, and Jy, so that 

When two infinite parallel planes are considered, At and A2 are equal and the radiation shape factor is unity since all the radiation leaving one plane reaches the other. The network is the same as in Fig. 8-26, and the heat flow per unit area may be obtained from Eq. (8-40) by letting A, = A2 and Fn = 1.0. Thus 

When two long concentric cylinders as shown in Fig. 8-29 exchange heat we may again apply Eq. (8-40). Rewriting the equation and noting that Fi2 = 1.0, 

The area ratio AjA2 may be replaced by the diameter ratio dtld2 when cylindrical bodies are concerned. 

Equation (8-43) is particularly important when applied to the limiting case of a convex object completely enclosed by a very large concave surface. In this instance A A2 — 0 and the following simple relation results  

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Sphere—radiation to surface Infinite circular cylinder— radiation to curved surface Semi-infinite cylinder— radiation to base Cylinder of equal height and diameter — radiation to entire surface Infinite parallel planes — radiation to planes... 

Co = 0.8768 for the configuration described by Eqs. (3) and (4). We note that Cq = 1 for infinite parallel planes. Numerical solutions by Philip et al. (1983) and by Sloane and Elmoursi (1987) yielded Cq = 0.80. The latter investigators performed numerical computations of the balance constant for several electrode shapes. 

If the emissivities of all surfaces are equal, a rather simple relation may be derived for the heat transfer when the surfaces may be considered as infinite parallel planes. Let the number of shields be n. Considering the radiation network for the system, all the surface resistances would be the same since the emissivities are equal. There would be two of these resistances for each shield and one for each heat-transfer surface. There would be n + I space resistances, and these would all be unity since the radiation shape factors are unity for the infinite parallel planes. The total resistance in the network would thus be... 

Notice that the analyses above, dealing with infinite parallel planes, have been carried out on a per-unit-area basis because all areas are the same. 

Fig. 8-61 Radiation netwo- tor infinite parallel planes separated by a transmit ,ng specular-diffuse plane...

Repeat Prob. 8-56 for two infinite parallel planes with the same temperatures and emissivities. Calculate the heat-transfer rates per unit area of the parallel planes. 

We begin with the problem of steady flow between two infinite, parallel plane boundaries. We assume that the pressure gradient G is a nonzero constant and that the upper boundary moves in the same direction as the pressure gradient with a constant velocity U, while the lower boundary is stationary. A sketch of the flow configuration is given in Fig. 3-1. It is conventional to describe the resulting motion of the fluid in terms of Cartesian coordinates x, rather than z, in the direction of motion (parallel to the pressure gradient and the wall... 

Figure 3-1. A schematic of the steady, unidirectional flow between two infinite parallel plane boundaries, with pressure gradient G acting from left to right in the x direction and the upper plate translating with velocity u = U. The arrows represent the magnitude and direction of the fluid velocity at steady state.

Figure 3-2. Typical velocity profiles for unidirectional flow between infinite parallel plane boundaries (a) Gdr piU = 0 (simple shear flow), (b) Gd2 / pU = 1, and (c) Gd2 jpU = 7. The length of the arrows is proportional to the local dimensionless velocity, with u = 1 at J = 1 in all cases. The characteristic velocity scale in this case has been chosen as the velocity of the upper wall, uc = U. The profiles are calculated from Eq. (3-24).

Figure 3-3. Velocity profile for unidirectional flow driven by a pressure gradient between two infinite parallel plane boundaries that are stationary [2D Poiseuille flow, Eq. (3-30)]. The characteristic velocity in this case has been chosen as the centerline velocity divided by 8.

Figure 3-12. The velocity profiles at different times t for start-up of simple shear flow between two infinite, parallel plane walls.

Here we consider two generalizations of the problem of pressure-gradient driven unidirectional flow between two infinite parallel plane boundaries. The difference is that, in the present problems, we assume that the boundaries are corrugated rather than flat. In particular, instead of flat walls, we assume that the walls are located at... 

First, let us consider the flow between two infinite parallel planes at a distance 2h from each other. The coordinate X is measured from one of the planes along the normal. Since the fluid velocity is independent of the coordinate Y, we can rewrite (1.5.1) in the form... 

The primary current distribution equations were first derived by Kasper [11-13] for simple geometries of uniform distributions, such as infinite parallel plane plates, infinite height cylindrical surfaces, and concentric spheres (Figure 13.1). Further, some of these parameters were approached using the secondary current distribution. 

Infinite parallel planes 2 (distance between planes)... 

C, respectively, and the emissivities of A and B are 0,90 and 0.25, respectively. Both surfaces are gray, (a) Surfaces A and B are infinite parallel planes 3 m apart, (b) Surface is a spherical shell 3 m in diameter, and surface. S is a similar shell concentric with A and 0,3 m in diameter, (c) Surfaces A and B are fiat parallel squares 2 by 2 m, one exactly above the other, 2 m apart, (d) Surfaces A and B are concentric cylindrical tubes with diameters of 300 and 275 mm, respectively, (e) Surface A is an infinite plane, and surface B is an infinite row of lOO-mm-OD tubes set on 200-mm centers. (/) Same as (e) except that 200 mm above the centerlines of the tubes is another infinite plane having an emissivity of 0,90, which does not transmit any of the energy incident upon it. (g) Same as /) except that surface. 6 is a double row of lOO-mm-OD tubes set on equilateral 200-mm centers. 

Infinite parallel planes — radiation to planes Spacing between ... 

V/ Potential of molecular attraction force betv een tv o infinite parallel planes J-m ... 

EXAMPLE 4.11-4. Radiation Between Infinite Parallel Gray Planes Derive Eq. (4.11-22) by starting with the general equation for radiation between two gray bodies A, and A2 which are infinite parallel planes having emissivities s, and 2. respectively. 

A typical case for equation (9-15) is that of infinite parallel planes. 

Two gray surfaces (one at 482°C with an emissivity of 0.90 and the other at 204°C and an emissivity of 0.25) have a net transfer of radiant energy. Determine the W/m for the following cases infinite parallel planes 3 m apart flat squares 2.0 m on a side 1 m apart. 

Let us suppose we have steady isothermal flow (i.e., the temperature is constant throughout the flow field and all d/dt = 0) between two infinite parallel planes, as shown in Figure 3.1. The flow is in the x direction. We assume for generality that there is a finite pressure gradient (dp/dx / O) and that the surface at y = 0 moves relative to the surface aty = H with a constant velocity V. We shall see subsequently that the results obtained here will form the foundation for the modeling of singlescrew extrusion and the extrusion coating of flat sheets. 

Mention was already made of the variety of geometries and boundary conditions that can arise in these problems, and we proposed to limit ourselves to the semi-infinite, parallel plane, spherical, and cylindrical geometries subject to constant initial and surface concentrations. We start with the simplest of these geometries, the semi-infinite medium, and follow this with a discussion of the other three principal geometries. 

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