Plate momentum transfers

When the inlet length is expressed in terms of number of gap widths , the difference between the flow in a tube and the flow in an annulus of narrow gap differs only by 25% [(0.05 - 0.04)/0.05]. This situation is an indication that the growth of the laminar boundary layers from the wall to the center of the channel is similar in both cases. Because duct friction coefficients, a measure of momentum transfer, do not vary by more than a factor of 2 for ducts of regular cross sections when expressed in terms of hydraulic diameters, the use of the inlet length for tubes or parallel plates can be expected to be a reasonable approximation for the inlet lengths of other cross sections under laminar flow conditions. In the annular denuder, the dimensionless inlet length for laminar flow development, L, can be expressed as... 

As discussed in the previous chapter, most early efforts at trying to theoretically predict heat transfer rates in turbulent flow concentrated on trying to relate the wall heat transfer rate to the wall shear stress [1],[2],[3],[41. The reason for this is that a considerable body of experimental and semi-theoretical knowledge concerning the shear stress in various flow situations is available and that the mechanism of heat transfer in turbulent flow is obviously similar to the mechanism of momentum transfer. In the present section an attempt will be made to outline some of the simpler such analogy solutions for boundary layer flows, attention mainly being restricted to flow over a flat plate. 

FIGURE 8.2 Structure factors S(QZ) obtained from neutron-scattering patterns of butylam-monium vermiculite gels (upper panels) and from a 0.1 M protonated butylammonium salt solution with no clay (lowest panel). The upper panels show S(QZ) for gels prepared in a 0.1 M deuterated salt solution and in 0.1 and 0.01 M protonated salt solutions. The momentum transfer Q was perpendicular to the clay plates, and the structure factor S(QZ) has been normalized after correction for background scattering and absorption. 

The second condition is just the no-slip condition at the plate surface, whereas the third condition arises as a consequence of the assumption that the fluid is unbounded. At any finite t > 0, a region will always exist sufficiently far from the plate that no significant momentum transfer will yet have occurred, and in this region the fluid velocity will remain arbitrarily small. 

H. S. Mickley, R. C. Ross, A. L. Squyers, and W. E. Stewart, Heat, Mass, and Momentum Transfer for Flow Over a Flat Plate With Blowing or Suction, NACA TN 3208,1954. 

In a few limited situations mass-transfer coefficients can be deduced from theoretical principles. One very important case in which an analytical solution of the equations of momentum transfer, heat transfer, and mass transfer has been achieved is that for the laminar boundary layer on a flat plate in steady flow. 

Fig 9 Two Stages Se-t-up for the Study of Granular Explosive F.P. = flyer plate T.P, = momentum transfer plate L.I. = line initiator... 

Since this quantity is independent of z, it must also be equal to the net x momentum transferred in one second to one square metre of the lower plate. Since the momentum transfer in... 

The constant of proportionality p is the viscosity of the fluid and equation 7.4.1 is known as Newton s law of viscosity. The viscosity of a fluid is a measure of how easily momentum is transferred through the fluid. A high viscosity indicates rapid momentum transfer from the moving plate to the stationary plate and hence a high resistance to shear. The units of shear stress are the same as those of pressure, i.e. Nm or Pa, therefore viscosity has units of Pa s. The viscosity of water at room temperature is approximately 10" Pas whereas that of air is about 2 x 10" Pas. 

Equation (7.3-13) has been shown to be quite useful in correlating momentum, heat, and mass transfer data. It permits the prediction of an unknown transfer coefficient when one of the other coefficients is known. In momentum transfer the friction factor is obtained for the total drag or friction loss, which includes form drag or momentum losses due to blunt objects and also skin friction. For flow past a flat plate or in a pipe where no form drag is present, //2 = J = Jp- When form drag is present, such as in flow in packed beds or past other blunt objects,772 is greater thanJ, otJ andJ s Jg. 

Several important dimensionless numbers in combined heat and momentum transfer in fluids can be derived by considering the simple flow of a Newtonian fluid between two flat plates, one stationary and one moving at velocity, v see Fig. 5.1. 

Flow over a flat plate is another case where the analogy also carries over to momentum transfer. Here for gases (3)... 

Doughty, J.R., 1971. Heat and momentum transfer between parallel porous plates. PhD Thesis, University of Arizona. 

Because momentum transfer is vectorial, there can only be an analogy between all three transport phenomena if momentum transfer can be considered unidimensional (e.g. for momentum transfer in cylindrical tubes or along a flat plate). 

Two-dimensional compressible momentum and energy equations were solved by Asako and Toriyama (2005) to obtain the heat transfer characteristics of gaseous flows in parallel-plate micro-channels. The problem is modeled as a parallel-plate channel, as shown in Fig. 4.19, with a chamber at the stagnation temperature Tstg and the stagnation pressure T stg attached to its upstream section. The flow is assumed to be steady, two-dimensional, and laminar. The fluid is assumed to be an ideal gas. The computations were performed to obtain the adiabatic wall temperature and also to obtain the total temperature of channels with the isothermal walls. The governing equations can be expressed as... 

In practice, as shown in the accompanying Figure, attenuators do reduce the impulse (Ref 8) delivered. What the above simple-minded picture does not consider is that the attenuator traps some of the momentum delivered by the explosive. Stated differently, a thin attenuator does not transfer to the massive plate all the momentum, delivered to it by the explosive, even after making allowance for the mass of the attenuator. This does not mean that conservation of momentum is violated. It simply means that the attenuator retains (traps) more momentum than would be expected on the basis of its weight relative to the weight of the plate. Conversely, Altshuler et al (JETP 34, 614 (1958)) have shown that a thin metal plate, propelled across a void by an HE charge, delivers its momentum to a massive... 

In the discussion of the use of the Reynolds analogy for the prediction of the heat transfer rate from a flat plate it was assumed that when there was transition on the plate, the x-coordinate in the turbulent portion of the flow could be measured from the leading edge. Develop an alternative expression based on the assumption that the momentum thickness before and after transition is the same. This assumption allows an effective origin for the x-coordinate in die turbulent portion of the flow to be obtained. 

Because, for flow over a heated surface. r>ulc>x is positive and ST/ y is negative. S will normally be a negative. Hence, in assisting flow, the buoyancy forces will tend to decrease e and e, i.e., to damp the turbulence, and thus to decrease the heat transfer rate below the purely forced convective flow value. However, the buoyancy force in the momentum equation tends to increase thle mean velocity and, therefore, to increase the heat transfer rate. In turbulent assisting flow over a flat plate, this can lead to a Nusselt number variation with Reynolds number that resembles that shown in Fig. 9.22. 

Qfis may ask the reason for the functional form of Eq. (6-4). Physical reasoning, based on the experience gained with the analyses of Chap. 5, would certainly indicate a dependence of the heat-transfer process on the flow field, and hence on the Reynolds number. The relative rates of diffusion of heat and momentum are related by the Prandtl number, so that the Prandtl nunfber is expected to be a significant parameter in the final solution. We can be rather confident of the dependence of the heat transfer on the Reynolds and Prandtl numbers. But the question arises as to the correct functional form of the relation i.e., would one necessarily expect a product of two exponential functions of the Reynolds and Prandtl numbers The answer is that one might expect this functional form since it appears in the flat-plate analytical solutions of Chap. 5, as well as the Reynolds analogy for turbulent flow. In addition, this type of functional relation is convenient to use in correlating experimental data, as described below. 

To analyze the heat-transfer problem, we must first obtain the differential equation of motion for the boundary layer. For this purpose we choose the jc coordinate along the plate and the y coordinate perpendicular to the plate as in the analyses of Chap. 5. The only new force which must be considered in the derivation is the weight of the element of fluid. As before, we equate the sum of the external forces in the x direction to the change in momentum flux through the control volume dx dy. There results... 

Three general flow regimes may be anticipated for the flow over a flat plate shown in Fig. 12 12. First, the continuum flow region is encountered when the mean free path A is very small in comparison with a characteristic body dimension. This is the convection heat-transfer situation analyzed in preceding chapters. At lo wer gas pressures, when A L, the flow seems to slip along the surface and u 4= 0 at y = 0. This situation is appropriately called slip flow. At still lower densities, all momentum and energy exchange is the result of... 

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