Characteristic Velocity C

Characteristic velocity (C ) is defined as the ratio of the product of the chamber pressure and throat area to the mass flow rate, that is, (Equation 4.11)  

It is denoted by C and depends on the flame temperature, mean molecular mass of the combustion products and propellant formulation. It is a fundamental parameter which gives the energy available on combustion and can be used to compare the efficiency of different chemical reactions independently of the Pc. For propellants, the value of C ranges between 1200 and 1600 ms-1. It is determined by firing a propellant grain in a motor and evaluating the area under the P-t profile and using Equation 4.11. 

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In forced convection, the velocity of the liquid must be characterized by a suitable characteristic value Vih e.g. the mean velocity of the liquid flow through a tube or the velocity of the edge of a disk rotating in the liquid, etc. For natural convection, this characteristic velocity can be set equal to zero. The dimension of the system in which liquid flow occurs has a certain characteristic value /, e.g. the length of a tube or the longitudinal dimension of the plate along which the liquid flows or the radius of a disk rotating in the liquid, etc. Solution of the differential equations (2.7.5), (2.7.7) and (2.7.8) should yield the value of the material flux at the phase boundary of the liquid with another phase, where the concentration equals c. ... 

This equation relates the hold-up to the flowrates of the phases and column diameter through the characteristic velocity, u0. It therefore gives a method of calculating the holdup for a given set of flowrates if u0 is known. Conversely, equation 13.33 may be used to calculate uq from experimental hold-up measurements made at different flowrates. Thus, if hold-up data are plotted with L d + (J/( 1 — j))L c as the ordinate against j( 1 — j) as the abscissa, a linear plot is obtained which passes through the origin and which has a slope equal to fir, Wl. 

The natural way of analyzing this problem is to introduce the appropriate scales. They would come from the characteristic concentration c, the characteristic length Lr, the characteristic velocity Qr, the characteristic diffusivity Dr and the characteristic time Tr. The characteristic length Lr coincides in fact with the "observation distance". Setting... 

Isp is also related to the characteristic exhaust velocity (C ), calculated from the P-t... 

The test chamber barometric pressure must not exceed absolute ambient air pressure by more than 1% at any rime. At the conclusion of the test, record the following a) chamber pressure b) muzzle velocity c) target accuracy d) sabot discarding characteristics e) fuze functioning time and f) observation of loading firing difficulties... 

If we consider the opposite extreme, where quadratic autocatalysis dominates, we should also change our characteristic timescale. So far we have based the wave velocity c on the cubic chemical time fch = /klal ... 

Fig 17 Computed Gurney velocity, [2(E0—Ej)], as a function of the characteristic velocity, [1.44po0] (mm/jusec), for C—H—N—0 explosives at various initial loading densities. Least squares fit to calculated values is shown. Also shown are experimental values of Gurney velocity (Ref 10), /2Eg, for maximum loading density for six pure explosives and three mixtures... 

If the effect of mixture ratio upon the characteristic velocity and thrust coefficient is examined, additional information is gained over that obtained from consideration of the specific impulse alone. Since the characteristic velocity bears the same dependence upon combustion temperature and molecular weight as does the specific impulse, the optimum characteristic velocity and optimum specific impulse would be expected and are observed to occur at approximately the same mixture ratio, see figure V. A. 6. The failure of these two performance parameters, c andcF. to have maximum values at the same propellant mixture ratio is traceable to their differing dependencies upon the specific heat ratio. The same effect is reflected in the dependency of the thrust coefficient upon the mixture ratio. For optimum expansion and a fixed pressure ratio across the nozzle, the thrust coefficient depends only upon the specific heat ratio. The dependence of the specific heat upon the mixture ratio in turn results in a dependence of the thrust coefficient upon the mixture ratio. Since thrust coefficient increases with decreasing specific heat ratio for a fixed pressure ratio, the maximum thrust coefficient should occur at the mixture ratio of maximum specific heat. The specific heat of the products increases both with temperature and with complexity of the product species. The maximum specific heat is found near the stoichiometric mixture ratio. For equilibrium expansion the specific heat also includes the effect of exothermic recombinations. This later effect is a maximum at the condition of maximum dissociation in the chamber which similarly occurs at the maximum combustion temperature or near the stoichiometric mixture ratio. 

Figure 3.19 Boundary layer flow over a droplet EPR (A) longitudinal velocities of air and droplets, (B) transversal air velocities, (C) shear stresses in the carrying flow, (D) integrated characteristics of the boundary layer.

On strongly adhesive substrates (Fig. 14b, c), friction increases with the substrate hydrophobicity in the low-velocity region, showing a weak velocity-strengthening, and a dramatic friction transition at around v = 10 m/s. This value is one order lower than the characteristic velocity of the polymer chain Vf = /Tf. The friction behavior is satisfactorily described by the repulsion-adsorption model below the transition region The friction transition is explained in terms of the elastic... 

C ODE the theoretical characteristic velocity for one-dimensional equilibrium flow. The precision of measured Fy can be estimated to be within 0.12 %. The precision of the calibration factors for the turbine-flow meters were generally within 0.25 % at a nominal flow rate, and the long term variation of K-factors were within 0.2 %. The overall precision of the measured Igpy, is calculated to be within 0.4 %. 

Here, u represents a characteristic velocity of the flow and usotmd is the speed of sound in the fluid at the same temperature and pressure. It may be noted that usound for air at room temperature and atmospheric pressure is approximately 300 m/s, whereas the same quantity for liquids such as water at 20°C is approximately 1500 m/s. Thus the motion of liquids will, in practice, rarely ever be influenced by compressibility effects. For nonisothermal systems, the density will vary with the temperature, and this can be quite important because it is the source of buoyancy-driven motions, which are known as natural convection flows. Even in this case, however, it is frequently possible to neglect the variations of density in the continuity equation. We will return to this issue of how to treat the density in nonisothermal flows later in the book. 

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).

To see how the thin-gap approximation e <governing equations and boundary conditions. The characteristic velocity for the polar velocity component is clearly... 

The minus sign implies that if c increases with n, the induced motion will be toward the surface (this is often associated with the name suction ), whereas the opposite will be true if c decreases with n (termed blowing ). If we introduce 0 = (c — Coo)/(co — c XJ), and the characteristic velocity and length scales used to derive (9 7), we can write this in dimensionless form as... 

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