Pressure-particle velocity curves

Hugoniot curves, such as those depicted in Figs. 4.3 and 4.4, may be transformed from the pressure-volume plane to the pressure-particle velocity plane (Fig. 4.5) using 

We assume that in (4.38) and (4.39), all velocities are measured with respect to the same coordinate system (at rest in the laboratory) and the particle velocity is normal to the shock front. When a plane shock wave propagates from one material into another the pressure (stress) and particle velocity across the interface are continuous. Therefore, the pressure-particle velocity plane representation proves a convenient framework from which to describe the plane Impact of a gun- or explosive-accelerated flyer plate with a sample target. Also of importance (and discussed below) is the interaction of plane shock waves with a free surface or higher- or lower-impedance media. 

The physical state of the sample before and after impact is sketched in Fig. 4.6(a). Positive velocity, indicating mass motion to the right (in the laboratory), is plotted toward the positive, u, axis. Hence, in the initial state 0, the target B is at Up = 0 and P = 0, whereas the initial state in the flyer plate O is Up = Ufp and P = 0. Upon interaction of flyer plate A with target B, a shock wave propagates forward in the sample and rearward in the flyer plate. Because the pressure and particle velocity are continuous at the flyer- 

If the Hugoniot of the flyer plate (.4) and the target (B) are known and expressed in the form of (4.7), the particle velocity Ui and pressure Pi of the shock state produced upon impact of a flyer plate at velocity, Ufp, may be calculated from the solution of the equation equating the shock pressures in the flyer and driver plate  

In general, when a solid returns to its initial phase upon unloading from high pressure, or when the postshock temperature is sufficiently low that 

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An important application of the impedance match method is demonstrated by the pressure-particle velocity curves of Fig. 4.9 for various explosives. Using the above method, the pressure in shock waves in various explosives is inferred from the intersection of the explosive Hugoniot with the explosive product release isentropes and reflected shock-compression Hugoniots (Zel dovich and Kompaneets, 1960). The amplitudes of explosively induced shock waves which can be propagated into nonreacting materials are calculable using results such as those of Fig. 4.9. 

By plotting Hugoniot curves in the pressure-particle velocity plane (P-u diagrams), a number of interactions between surfaces, shocks, and rarefactions were solved graphically. Also, the equation for entropy on the Hugoniot was expanded in terms of specific volume to show that the Hugoniot and isentrope for a material is the same in the limit of small strains. Finally, the Riemann function was derived and used to define the Riemann Invarient. 

Phase Diagram (Zenz and Othmer) Zenz and Othmer (op. cit.) have graphically represented (Fig. 17-2) all gas-solid svstems in which the gas is flowing counter to gravity as a function o pressure drop per unit of height versus velocity. Note that line OAB in Fig. 17-2 is the pressure-drop versus gas-velocity curve for a packed bed and BD the cui ve for a fluid bed. Zenz indicates an instability between D and H because with no sohds flow all the particles will be entrained from the bed however, if sohds are added to replace those entrained, system JJ prevails. The area DHJJ will be discussed further. 

Hugoniot curve A curve representing all possible final states that can be attained by a single shock wave passing into a given initial state. It may be expressed in terms of any two of the five variables shock velocity, particle velocity, density (or specific volume), normal stress (or pressure), and specific internal energy. This curve it not the loading path in thermodynamic space. 

Figure 4.9. Shock pressure versus particle velocity for engineering materials, geological material, and explosive detonation products. Intersection of detonation product curves with nonreactive media predicts shock pressure and particle velocity at an explosive sample interface. (After Jones (1972).)...

Fig. 3.3. Stress-particle velocity characterizations of many materials have been documented. The explosive cross curves superposed on the materials responses provide approximate loading stress levels to be determined from the intersection of the explosive and material curves. For example, the detonation of TNT produces a pressure of 25 GPa in 2024 aluminum alloy.

Particle velocity is an important parameter in the so-called impedance-mismatch method of determining whether the shock from one material enters as a shock or rarefaction into another material in contact with the first material (see Vol 7, HI79-83 and Vol 9, S60). Two of the three commonly used Hugoniot curves (see Vol 7, H179—83) are in the form of P vs u or U vs u plots, and the third form, P vs v, depends on v, usually obtained via Eq 2 (see Vol 7, HI 80) The writer has suggested that input shock particle velocity is a better criterion for the shock sensitivity threshold of expls than input shock pressure (see Vol 9, S76)... 

Each velocity is assigned to a midpoint of the interval over which the measurement is made, and the initial velocity is found by extrapolating the velocity vs thickness curve to zero thickness. The initial pressure and particle velocity are found from a graphical solution as in Technique 1. 

The transport velocity can also be evaluated from the variations of the local pressure drop per unit length (Ap/Az) with respect to the gas velocity and the solids circulation rate, Jp. An example of such a relationship is shown in Fig. 10.4. It is seen in the figure that, along the curve AB, the solids circulation rates are lower than the saturation carrying capacity of the flow. Particles with low particle terminal velocities are carried over from the riser, while others remain at the bottom of the riser. With increasing solids circulation rate, more particles accumulate at the bottom. At point B in the curve, the solids fed into the riser are balanced by the saturated carrying capacity. A slight increase in the solids circulation rate yields a sharp increase in the pressure drop (see curve BC in Fig. 10.4). This behavior reflects the collapse of the solid particles into a dense-phase fluidized bed. When the gas... 

This data collected for determination of the Knox parameters can also be used to establish the linearity of the pressure versus linear velocity curve to evaluate compression of the bed. Lastly, these data can be used to estimate the effective particle size from the pressure drop. The pressure drop data are aseful to assess the effective particle size with the vendors nominal particle size and particle size distribution data. Calculation of the effective particle size is given by Eq. (7.9), where dp is the particle size in cm, u is the linear velocity in cm/s, p is the viscosity in cP, L is the bed length in cm, k[) is the column permeability (e.g. 1 x 10 for irregular particles and 1.2 x 10 for spherical), and AP is the pressure in psi. 

Figure 9.10. Shock pressure versus particle velocity for water ice, and snow with different densities. Curves for serpentine with the impact velocity of Earth s escape velocity and ice with Ganymede s escape velocity are plotted for the estimation of shock pressure by means of impedance matching method. Escape velocities for satellites are indicated on the particle velocity axis. (Figure from /Vhrens and O Keefe [31].)...

At any interface between two media, shock transfer occurs such that the pressure and particle velocity across the interface are equal. The strengths of the transmitted and reflected shock waves depend on the relative impedance which can be calculated from the Hugoniot curves (Figure 18). 

On the other hand, we have seen in Chapter 3 that for small molecules a reduction of the particle diameter below about 1pm stops making sense because of the pressure requirements and the high linear velocities associated with the optimum of the plate height-velocity curve. However, for larger molecules with lower diffusion coeflBcients, the optimal linear velocity moves to smaller values in direct proportion to the diffusion co dent. This means that the pressure is also reduced by the same factor, if we keep column size and particle size constant Since the pressure constraints are lowered, we can reduce the particle size further, beyond that which made sense for small molecules. So while a good choice of a particle size for a small molecule was around 3pm, it may be around 0.3pm (300nm) for a macromolecule. 

If the particles have a wide size distribution, the pressure drop vs. gas velocity curve shows a rather vague transition from a fluidized condition to the fixed bed condition, as already shown in Fig. 44b. From this figure one can obtain gas velocities corresponding to the beginning of fiuidization M),f and to the completion of fiuidization w f. 

As shown in Fig. 18.63, the critical penetration velocities derived for a 6.1 pm sized particle with different solid-liquid contact angles (see Fig. 18.61 (left)) are related to the atomizer distance for Sn droplets of different sizes in different gas flow fields (by changing the atomization gas pressure) via the dependency of the solidification degree on the droplet-flight distance in Fig. 18.29. The value of the droplet size represents the mass median diameter ( iso.s) of metal powder particles produced at a specified atomization pressure The dashed curve denotes the critical penetration velocity. Because a smaller droplet in a stronger gas flow has a... 

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