Fluid continuity equation

The fomulation of Equation (8.68) gives the fully developed velocity profile, Fz(r), which corresponds to the local values of ix(r) and p(r) without regard to upstream or downstream conditions. Changes in Fz(r) must be gradual enough that the adjustment from one axial velocity profile to another requires only small velocities in the radial direction. We have assumed Vy to be small enough that it does not affect the equation of motion for V. This does not mean that Vr is zero. Instead, it can be calculated from the fluid continuity equation,... 

From the fluid continuity equation (8.3.13), the first term on the left-hand side cancels out with the last term on the right-hand side, giving the general conservation equation... 

The average multifluid model equations are outlined in the following together with the conventional interfacial closures that are frequently adopted in gas-liquid bubbly flow analyzes. The average multi-fluid continuity equation for phase k reads ... 

Analysis of time-dependent consolidation requires the solution of Biot s consolidation equations coupled to the equations describing flow. The transient hydro-mechanical coupling between pore pressure and volumetric strain for a linear elastic, mechanically isotropic porous medium and fully saturated with a single fluid phase (i.e. water), is given by the fluid continuity equation ... 

The fluid continuity equation when integrated across the cross section can be expressed as... 

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)... 

As already explained the necessity to satisfy the BB stability condition restricts the types of available elements in the modelling of incompressible flow problems by the U-V P method. To eliminate this restriction the continuity equation representing the incompressible flow is replaced by an equation corresponding to slightly compressible fluids, given as... 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Wc now obtain the integral of the continuity equation for incompressible fluids with respect to the local gap height hr this flow domain... 

Simplified forms of Eq. (6-8) apply to special cases frequently found in prac tice. For a control volume fixed in space with one inlet of area Ai through which an incompressible fluid enters the control volume at an average velocity Vi, and one outlet of area Ao through which fluid leaves at an average velocity V9, as shown in Fig. 6-4, the continuity equation becomes... 

The continuity equation is a mathematical formulation of the law of conservation of mass of a gas that is a continuum. The law of conservation of mass states that the mass of a volume moving with the fluid remains unchanged... 

Computational fluid dynamics (CFD) is the analysis of systems involving fluid flow, energy transfer, and associated phenomena such as combustion and chemical reactions by means of computer-based simulation. CFD codes numerically solve the mass-continuity equation over a specific domain set by the user. The technique is very powerful and covers a wide range of industrial applications. Examples in the field of chemical engineering are ... 

A drawback of the Lagrangean artificial-viscosity method is that, if sufficient artificial viscosity is added to produce an oscillation-free distribution, the solution becomes fairly inaccurate because wave amplitudes are damped, and sharp discontinuities are smeared over an increasing number of grid points during computation. To overcome these deficiencies a variety of new methods have been developed since 1970. Flux-corrected transport (FCT) is a popular exponent in this area of development in computational fluid dynamics. FCT is generally applicable to finite difference schemes to solve continuity equations, and, according to Boris and Book (1976), its principles may be represented as follows. 

We first derive the so-called continuity equation, which is a direct consequence of the conservation of mass. If p is the density, or mass per unit volume, then the total mass of a fluid contained in F is equal to M = fj p dF. Letting dS — fi dS be an element of the surface, with n a unit vector perpendicular to the surface, the mass flow per unit time through the surface element is pv dS. The total fluid flow out of the volume F is then given by... 

The creeping flow of a single fluid phase through a rigid permeable medium is modeled with the continuity equation and Darcy s Law ... 

The mass balance of soil air may be described by the classic continuity equation for compressible fluids ... 

The steady-state continuity equations which describe mass balance over a fluid volume element for the species in the stagnant film which are subject to uniaxial diffusion and reaction in the z direction are... 

As an alternative to film models, McNamara and Amidon [6] included convection, or mass transfer via fluid flow, into the general solid dissolution and reaction modeling scheme. The idea was to recognize that diffusion was not the only process by which mass could be transferred from the solid surface through the boundary layer [7], McNamara and Amidon constructed a set of steady-state convective diffusion continuity equations such as... 

The conservation of mass can be applied to an arbitrarily small fluid element to derive the microscopic continuity equation, which must be satisfied at all points within any continuous fluid. This can be done by considering an arbitrary (cubical) differential element of dimensions dx, dy, dz, with mass... 

This is the microscopic (local) continuity equation and must be satisfied at all points within any flowing fluid. If the fluid is incompressible (i.e., constant p), Eq. (5-7) reduces to... 

Example 5-3 Diffuser. A diffuser is a section in a conduit over which the flow area increases gradually from upstream to downstream, as illustrated in Fig. 5-3. If the inlet and outlet areas (Ai and A2) are known, and the upstream pressure and velocity (Pi and V ) are given, we would like to find the downstream pressure and velocity (P2 and If ). If the fluid is incompressible, the continuity equation gives V2 ... 

Example 5-5 The Toricelli Problem. Consider an open vessel with diameter Di containing a fluid at a depth h that is draining out of a hole of diameter D2 in the bottom of the tank. We would like to determine the velocity of the fluid flowing out of the hole in the bottom. As a first approximation, we neglect the friction loss in the tank and through the hole. Point 1 is taken at the surface of the fluid in the tank, and point 2 is taken at the exit from the hole, because the pressure is known to be atmospheric at both points. The velocity in the tank is related to that through the hole by the continuity equation... 

Example 5-6 Friction Loss in a Sudden Expansion. Figure 5-7 shows the flow in a sudden expansion from a small conduit to a larger one. We assume that the conditions upstream of the expansion (point 1) are known, as well as the areas A and A2. We desire to find the velocity and pressure downstream of the expansion (V2 and P2) and the loss coefficient, Kt. As before, V2 is determined from the mass balance (continuity equation) applied to the system (the fluid in the shaded area). Assuming constant density,... 

Note that there are 11 dependent variables, or unknowns in these equations (three u s, six r,y s, P, and p), all of which may depend on space and time. (For an incompressible fluid, p is constant so there are only 10 unknowns. ) There are four conservation equations involving these unknowns (the three momentum equations plus the conservation of mass or continuity equation), which means that we still need six more equations (seven, if the fluid is compressible). These additional equations are the con-... 

For an incompressible fluid, the term in parentheses is zero as a result of the conservation of mass (e.g., the microscopic continuity equation). Equation (13-25) can be generalized to three dimensions as... 

Some of the simplifications that may be possible are illustrated by the case of steady, fully-developed, laminar, incompressible flow of a Newtonian fluid in a horizontal pipe. The flow is assumed to be axisymmetric with no swirl component of velocity so that derivatives wrt 6 vanish and vg = 0. For fully-developed flow, derivatives wrt z are zero. With these simplifications and noting that the flow is incompressible, the continuity equation (equation A. 11) reduces to... 

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