Fluid mass density

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

Temperature correction factor Absolute viscosity of fluid Mass density of the gas phase (air)... 

The numerical application will be carried out with water as the fluid (mass density p = 10 kg m, specific heat C = 4.2 10 J kg K ). 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

For a one-component fluid, the vapour-liquid transition is characterized by density fluctuations here the order parameter, mass density p, is also conserved. The equilibrium structure factor S(k) of a one component fluid is... 

The quantum internal energy (fi /2m )(VY ) /p depends also on the derivative of the density, unlike in the fluid case, in which internal energy is a function of the mass density only. However, in both cases the internal energy is a positive quantity. 

Pxampk 2. A smooth spherical body of projected area Al moves through a fluid of density p and viscosity p with speed O. The total drag 8 encountered by the sphere is to be determined. Clearly, the total drag 8 is a function of O, Al, p, and p. As before, mass length /, and time t are chosen as the reference dimensions. From Table 1 the dimensional matrix is (eq. 23) ... 

Equations (4.1) or (4.2) are a set of N simultaneous equations in iV+1 unknowns, the unknowns being the N outlet concentrations aout,bout, , and the one volumetric flow rate Qout- Note that Qom is evaluated at the conditions within the reactor. If the mass density of the fluid is constant, as is approximately true for liquid systems, then Qout=Qm- This allows Equations (4.1) to be solved for the outlet compositions. If Qout is unknown, then the component balances must be supplemented by an equation of state for the system. Perhaps surprisingly, the algebraic equations governing the steady-state performance of a CSTR are usually more difficult to solve than the sets of simultaneous, first-order ODEs encountered in Chapters 2 and 3. We start with an example that is easy but important. 

In accordance with Equation (2.338) the determination of the figure of fiuid equilibrium is reduced to the following problem we have to find such a surface of the fluid, S(x,y,z), that its partial derivatives should be proportional to the corresponding components of the acting force. As we pointed out, when a fluid rotates uniformly around the same axis the total force can be represented as a sum of the attraction and centrifugal forces, and the former depends on the shape of the fluid mass in a rather complicated way. Besides, in the case of an inhomogeneous fluid the potential of the attraction field depends on the distribution of a density of a fiuid and for this reason this problem becomes even more complicated. 

Kinematic viscosity is the ratio of dynamic viscosity and density, and can be obtained by dividing the dynamic viscosity of a fluid with its mass density, as shown by Equation 18.2 ... 

Many engineering operations involve the separation of solid particles from fluids, in which the motion of the particles is a result of a gravitational (or other potential) force. To illustrate this, consider a spherical solid particle with diameter d and density ps, surrounded by a fluid of density p and viscosity /z, which is released and begins to fall (in the x = — z direction) under the influence of gravity. A momentum balance on the particle is simply T,FX = max, where the forces include gravity acting on the solid (T g), the buoyant force due to the fluid (Fb), and the drag exerted by the fluid (FD). The inertial term involves the product of the acceleration (ax = dVx/dt) and the mass (m). The mass that is accelerated includes that of the solid (ms) as well as the virtual mass (m() of the fluid that is displaced by the body as it accelerates. It can be shown that the latter is equal to one-half of the total mass of the displaced fluid, i.e., mf = jms(p/ps). Thus the momentum balance becomes... 

Jaberi, F. A., P. J. Colucci, S. James, P. Givi, and S. B. Pope (1999). Filtered mass density function for large-eddy simulation of turbulent reacting flows. Journal of Fluid Mechanics 401, 85-121. 

Here, v is the velocity vector field, p is the mass density of the fluid, D/Dt = S/Sf + V V is the material derivative, Vp is the gradient of the pressure, r[j is the shear viscosity, and F is the external force acting on the fluid volume. The right-hand side of Eq. (1) is a momentum balance between the internal pressure and viscous stress and the external forces on the fluid body. Any excess momentum contributes to the material acceleration of the fluid volume, on the left-hand side. 

A fluid composed of a single species is described by five fields the three components of the velocity, the mass density, and the temperature. This is a drastic reduction of the full description in terms of all the degrees of freedom of the particles. This reduction is possible by assuming the local thermodynamic equilibrium according to which the particles of each fluid element have a Maxwell-Boltzmann velocity distribution with local temperature, velocity, and density. This local equilibrium is reached on time scales longer than the intercollisional time. On shorter time scales, the degrees of freedom other than the five fields manifest themselves and the reduction is no longer possible. 

The effect of hills is interesting, in that no credit can be taken for the downhill side of the pipeline. The sum of all the uphill elevations appears as a pressure loss in actual operating practice. Baker includes an elevation correction factor which attempts to allow for the fact that the fluid-mixture density in the inclined uphill portion of the line is not accurately known. The gas mass-velocity seems to be the major variable affecting this correction factor, although liquid mass-velocity, phase properties. 

In the derivation of this equation we assume that the density is constant (we use molar density Nj/V rather than mass density pj), and that the difiusivity Dj is independent of composition, and we simplify the convection term somewhat. The gradient (V) and Laplacian (V ) operators should be familiar to students from fluid mechanics and heat and mass transfer courses. 

The combination of low viscosity and high density creates a risk situation, which might be described as Tanker effect . Assuming a tanker without separators and with a 50% filling of the tank accelerates, the inertia of the fluid mass creates pressure on the inner back side of the tank. If the driver pushes the break firmly, all liquid will move to the front and will create pressure on the inner front side of the tank. 

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