Momentum radial

The simplest and most common device for measuring flow rate in a pipe is the orifice meter, illustrated in Fig. 10-7. This is an obstruction meter that consists of a plate with a hole in it that is inserted into the pipe, and the pressure drop across the plate is measured. The major difference between this device and the venturi and nozzle meters is the fact that the fluid stream leaving the orifice hole contracts to an area considerably smaller than that of the orifice hole itself. This is called the vena contracta, and it occurs because the fluid has considerable inward radial momentum as it converges into the orifice hole, which causes it to continue to flow inward for a distance downstream of the orifice before it starts to expand to fill the pipe. If the pipe diameter is D, the orifice diameter is d, and the diameter of the vena contracta is d2, the contraction ratio for the vena contracta is defined as Cc = A2/A0 = (d2/d)2. For highly turbulent flow, Cc 0.6. 

From the momentum density one can find the radial momentum density by integration over the angular variables Of in momentum space ... 

Figure 5.4. The radial electron density D(r) (left) and radial momentum density I p) (right) for I S Be li contribution (dotted), 2s contribution (dashed), and total (soUd).

The circumferential-momentum equation is a parabolic partial differential that requires solution for the radial dependence of the circumferential velocity. With the velocity profiles in hand, the radial-momentum equation can be used to determine the resulting radial pressure dependence. 

Assuming that the radial momentum equation reduces to... 

Substituting the functional form of the velocity (Eq. 5.54) into the radial momentum equation produces a second-order nonlinear equation as... 

W(z) — w/r must be a function of z alone. Notice that the radial momentum equation has been divided by r, so the quantity... 

Because the right-hand sides of the momentum equations are functions of z alone, it must be the case that both dp/dz and 1 /r(dp/dr) are functions of z alone. By differentiating the radial-momentum equation with respect to z, it could be concluded that... 

In the inviscid stagnation flow, the single parameter a establishes both velocity components as well as the pressure field. In the inviscid regions, the radial momentum equation, Eq. 6.19, loses two terms, leaving only... 

This difference formula propagates axial-velocity u information from the lower boundary (e.g., stagnation surface) up toward the inlet-manifold boundary. At the stagnation surface, a boundary value of the axial velocity is known, u = 0. A dilemma occurs at the upper boundary, however. At the upper boundary, Eq. 6.106 can be evaluated to determine a value for the inlet velocity u j. However, in the finite-gap problem, the inlet velocity is specified as a boundary condition. In general, the velocity evaluated from the discrete continuity equation is not equal to the known boundary condition, which is a temporary contradiction. The dilemma must be resolved through the eigenvalue, which is coupled to the continuity equation through the V velocity and the radial-momentum equation. 

Care must be taken with the convective terms in the transport equations to account for the axial flow direction. In the stagnation-flow problems for flow against a surface, the axial velocity is always negative (i.e., flowing toward the surface). The convective term in the radial-momentum equation uses the following upwind difference approximation ... 

The governing equations for the rotating disk must include a circumferential momentum equation, and the circumferential velocity becomes a dependent variable. Also the circumferential velocity contributes to the radial-momentum equation. As simplified by the general equations of Section 6.2, the nonreacting, constant-property equations are summarized... 

The radial momentum equation is decoupled from the rest of the system, but can be used to determine the radial pressure gradient if needed. Again, substituting the scaled axial velocity and other definitions yields... 

Derive the relationship between the vorticity-transport equation (Eq. 6.47) and the radial-momentum equation for semi-infinite stagnation flow. In other words, carefully fill in all the missing pieces in the derivation leading to Eq. 6.55. In the course of this exercise, review how to apply integration by parts. 

For channels that are narrow compared to their length (rs zs) and for Rer > 1, it is apparent from Eq. 7.15 that the only order-one term is the pressure gradient. Therefore we conclude that in the boundary-layer approximation the entire radial-momentum equation reduces to... 

The continuity equation at the inlet boundary can be viewed as a constraint equation. Referring to the difference stencil (Fig. 17.14), it is seen that this first-order equation itself is evaluated at the boundary and no explicit boundary condition is needed. Moreover, since the inlet temperature, pressure, and composition are specified, the density is fixed and thus dp/dt = 0. Therefore, at the boundary, the continuity equation (Eq. 17.15) has no time derivative it is an algebraic constraint. There is no explicit boundary condition for A. At the inlet boundary, the value of A must be determined in such a way that all the other boundary conditions are satisfied. Being an eigenvalue, A s effect is felt through its influence on the V velocity in the radial momentum equation, and subsequently by V s influence on u through the continuity equation. 

It is possible and beneficial to reduce the system to index-one by replacing A with a new dependent variable , where A = 3/31 [13], The initial condition for is arbitrary, since itself never appears in the equations—a suitable choice is 4> = 0. Anywhere A appears, it is simply replaced with 3/31, which is conveniently done in the DAE software interface. The index reduction can be seen from the following procedure The continuity at the inlet boundary (an algebraic constraint) can be differentiated once with respect to t to yield an equation for dV/dt. Then dV/dt is replaced by substitution of the radial-momentum equation. This substitution introduces A = 34>/31, which makes the continuity equation (at the inlet boundary) an independent differential equation for 4>. Thus the modified system is index-one. This set of substitutions is not actually done in practice—it simply must be possible to do them to achieve the index reduction. 

The columns of cells below row 16 contain the values of the dependent variables at the node points. They will all be iterated until a final solution is achieved. The formula in each cell represents an appropriate form of the difference equations. Each column represents an equation. Column B represents the continuity equation, column C represents the radial momentum equation, column D represents the circumferential momentum equation, and column E represents the thermal energy equation. Column F represents the perfect-gas equation of state, from which the nondimensional density is evaluated. The difference equations involve interactions within a column and between columns. Within a column the finite-difference formulas involve the relationships with nearest-neighbor cells. For example, the temperature in some cell j depends on the temperatures in cells j — 1 and j + 1, that is, the cells one row above and one row below the target cell. Also, because the system is coupled, there is interaction with other columns. For example, the density, column F, appears in all other equations. The axial velocity, column B, also appears in all other equations. 

C47 In cell CI9, enter the difference formula for the radial momentum equation... 

Gas enters at the center of the electrode along the axis of rotation from the gas panel. It was found that if the gas were allowed to flow in directly and impinge normally to the upper electrode, that excessive deposits built up rapidly at the center of this upper electrode, and poor deposition uniformities were observed. Therefore, a gas injection shield was placed at the center of the lower electrode to provide some radial momentum to the incoming gas flow. This shield is illustrated in Figure 19. 

Fig. 8.3). Vertical gas flow is in turn used to fluidized the milled material out of the grinding chamber. Typically fluidized bed jet mills are fitted with a classification wheel. Based on the rotational speed of the classifier wheel, particles of too large a size have a radial momentum imparted upon them returning these particles back to the grinding zone of the fluidized bed. This type of arrangement will typically lead to much narrower particle size distributions than other types of jet milling equipment. 

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