Balance equations fluid mass

Three types of theoretical approaches can be used for modeling the gas-particles flows in the pneumatic dryers, namely Two-Fluid Theory [1], Eulerian-Granular [2] and the Discrete Element Method [3]. Traditionally the Two-Fluid Theory was used to model dilute phase flow. In this theory, the solid phase is being considering as a pseudo-fluid. It is assumed that both phases are occupying every point of the computational domain with its own volume fraction. Thus, macroscopic balance equations of mass, momentum and energy for both the gas and the solid... 

The balance equations for mass, momentum and energy describe the entire flow situation. The continuity assumption of smooth fluid properties and no-shp flow conditions at the wall hold for most cases in microprocess engineering, hence the change in density p with time is correlated with the velocity vector w as... 

The two-fluid model allows the phases to have thermal nonequilibrium as well as unequal velocities. In this model, each phase or component is treated as a separate fluid with its own velocity, temperature, and pressure. Thus, each phase has three independent set of governing balance equations for mass momentum and energy. The velocity difference as in the separated flow is induced by density differences and the temperature differences between the phases is fundamentally induced by the time lag of energy transfer between the phases at the interface as thermal equilibrium is reached. The two-fluid model... 

Based on continuum mechanics, the motion of fluid particles is modeled as a two-phase flow with phase boundary represented by a sharp interface. More precisely, we consider isothermal flows of two fluids of constant material properties each, which are immiscible on the molecular scale. Within each phase, the balance equations of mass and momentum read... 

Instead of assigning different shear rates, he employed different breakage rate expressions for the two zones. The problem of coupling population balance models with fluid flow models has received some attention recently and coupled PB-CFD models have been developed for a wide variety of processes such as fluidization [70], gas-liquid reactions in bubble columns [71] and nanoparticle synthesis in flame aerosol reactors [72]. Complete description of aggregation in turbulent environments requires simultaneous solution of basic balance equations for mass, momentum, energy and concentration of species present along with population balances for particles/aggregates of different size classes. 

In the GAIM process, the gas and the polymer are considered immiscible. Thus, the gas penetration can be modeled as a sequential co-injection process, where a fluid of much lower viscosity displaces a fluid of higher viscosity in the mold cavity. To avoid numerical instabilities due to the large polymer/gas viscosity ratio, the gas is replaced with a fictitious fluid with a viscosity being less than 10 of the polymer viscosity. Furthermore, the curvature of the spiral is not considered. Thus, the governing equations for this process are derived by considering the balance equations of mass, momentum, and energy for each domain coupled with the kinematic interface condition [5]. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

It may be noted that the energy and mass balance equations assume that the fluid is continuous. This is so in the case of a liquid, provided that the pressure does not fall to such a low value that boiling, or the evolution of dissolved gases, takes place. For water... 

Since in the energy balance equation, the kinetic energy per unit mass is expressed as a2/2a, hence a = 0.5 for the streamline flow of a fluid in a round pipe. 

In considering the flow in a pipe, the differential form of the general energy balance equation 2.54 are used, and the friction term 8F will be written in terms of the energy dissipated per unit mass of fluid for flow through a length d/ of pipe. In the first instance, isothermal flow of an ideal gas is considered and the flowrate is expressed as a function of upstream and downstream pressures. Non-isothermal and adiabatic flow are discussed later. 

The work done by the pump is found by setting up an energy balance equation. If W, is the shaft work done by unit mass of fluid on the surroundings, then —Ws is the shaft work done on the fluid by the pump. 

If a concentration gradient exists within a fluid flowing over a surface, mass transfer will take place, and the whole of the resistance to transfer can be regarded as lying within a diffusion boundary layer in the vicinity of the surface. If the concentration gradients, and hence the mass transfer rates, are small, variations in physical properties may be neglected and it can be shown that the velocity and thermal boundary layers are unaffected 55. For low concentrations of the diffusing component, the effects of bulk flow will be small and the mass balance equation for component A is ... 

Kinetic Rate Lam y/Vfateriat Balance reaction/deactivation/ reactor design equation, heat/mass transpat/ fluid-flow model,... 

As will be shown later the equation above is identical to the mass balance equation for a continuous stirred-tank reactor. The recycle can be provided either by an external pump as shown in Fig. 5.4-18 or by an impeller installed within the reaction chamber. The latter design was proposed by Weychert and Trela (1968). A commercial and advantageously modified version of such a reactor has been developed by Berty (1974, 1979), see Fig. 5.4-19. In these reactors, the relative velocity between the catalyst particles and the fluid phases is incretised without increasing the overall feed and outlet flow rates. 

Momentum balance equations are of importance in problems involving the flow of fluids. Momentum is defined as the product of mass and velocity and as stated by Newton s second law of motion, force which is defined as mass times acceleration is also equal to the rate of change of momentum. The general balance equation for momentum transfer is expressed by... 

Mass balance equations can be developed to describe mass transfer in the stirred tank. The system is defined as the fluid within the tank, excluding any headspace above the fluid and the solid tank itself. The total mass balance on this sytem, given above in verbal form as Eq. (1), takes the following form when fluid density (p) is assumed to be constant ... 

Bowers and Taylor (1985) were the first to incorporate isotope fractionation into a reaction model. They used a modified version of EQ3/EQ6 (Wolery, 1979) to study the convection of hydrothermal fluids through the oceanic crust, along midocean ridges. Their calculation method is based on evaluating mass balance equations, as described in this chapter. 

For liquids stored at their saturation vapor pressure, P = Ps, and Equation 4-91 is no longer valid. A much more detailed approach is required. Consider a fluid that is initially quiescent and is accelerated through the leak. Assume that kinetic energy is dominant and that potential energy effects are negligible. Then, from a mechanical energy balance (Equation 4-1), and realizing that the specific volume (with units of volume/mass) v = 1/p, we can write... 

Physically, the solid and the fluid are linked by the mass transfer between them. The equilibrium concentrations in the solution are continually changing as the analytical concentrations change the adjustments are constrained to be such that the mass action expressions and balance equations are always... 

Two additional points about Equation (8) need to be discussed here. Equation (8) contains mj in the denominator. Thus the solution concentrations must be known before the first increment dE, is taken and none of them can be zero. In practice this means that the set of nonlinear equations (mass action and balance equations) describing the fluid phase in its initial unperturbed equilibrium state must be solved once. Further, Equation (8) does not completely describe a heterogeneous system at partial equilibrium. 

Another key point of differentiation is the fact that nearly all PSA separations are bulk separations and any investigator interested in a high fidelity description of the problem of adsorption must solve a mass balance equation such as Eq. (9.9), the bulk separation equation, together with the uptake rate model and a set of thermal balance equations of similar form. In addition to the more complicated pde and its attendant boundary and initial conditions the investigator must also solve some approximate form of a momentum balance on the fluid flow as a whole. 

If the density of the fluid is constant, then the volumetric flow rates in and out of the reactor are equal, v = Vq. The mass-balance equation then simplifies to become... 

This equation is not appropriate if all five of these conditions are not met. We can relax the third and fourth restrictions for the PFTR by considering the differential element of volume dV = At dz rather than the differential element of length dz. The mass-balance equation at a position where the fluid has moved from volume V to volume V + d V then becomes... 

We always use Cj in moles per liter (or in moles per cubic decimeter or 1 kilomole/m for the SI purist) as the only unit of concentration. The subscript j always signifies species, while the subscript i always signifies reaction. We use j as the species designation and species A as the key reactant. For gases the natural concentration unit is partial pressure Pj, but we always convert this to concentration, Cj = Pj RT, before writing the mass-balance equations. Conversion X means the fiaction of this reactant that is consumed in the reactor, Ca = Cao( 1 — X), but we prefer to use C i rather than X and find the conversion after we have solved the equation in terms of G. We cannot use this unit of density of a species when the density of the fluid varies with conversion, but we prefer to do so whenever possible because the equations are simpler to write and solve. 

Any fluid flow situation is described completely by momentum, mass, and energy balances. We have thus far looked at only simplified forms of the relevant balance equations for our simple models, as is done implicitly in aU engineering courses. It is interesting to go back to the basic equations and see how these simple approximations arise. We need to examine the full equations to determine the errors we are making in describing real reactors with... 

For any more complex flow pattern we must solve the fluid mechanics to describe the fluid flow in each phase, along with the mass balances. The cases where we can still attempt to find descriptions are the nonideal reactor models considered previously in Chapter 8, where laminar flow, a series of CSTRs, a recycle TR, and dispersion in a TR allow us to modify the ideal mass-balance equations. 

This is the differential form of the mass balance equation in three dimensions. Since J can be written as pu, where u is the flow velocity of the fluid, the above equation can also be written as... 

The interrelationships are embodied in variations of the Navier-Stokes equations, which describe mass and momentum balances in fluid systems (23). 

The particle and bulk densities are commonly used in mass balance equations, since the mass and the external volume of the particles are involved. On the other hand, the hydraulic density should be preferably used in hydrodynamic calculations, because buoyancy forces are involved, and so the total mass of the particle should be taken into account, including the fluid in the open pores. It is obvious that the particle density is equal to the skeletal and hydrodynamic density in the case of nonporous particles. Moreover, in the case of a porous solid in a gas-solid system, the gas density is much lower than the particle density, and tlius... 

If flow is cocurrent the lower sign is used if countercurrent the upper sign is used. Since the mass flowrate of the cooling fluid is based upon the cross-sectional area of the reactor tube the ratio G Ip Gq SpC(= H is a measure of the capacities of the two streams to exchange heat. In terms of the limitations imposed by the onedimensional model, the system is fully described by equations 3.9S and 3.96 together with the mass balance equation ... 

The mass balance equations of the traditional multicomponent rate-based model (see, e.g., Refs. 57 and 58) are written separately for each phase. In order to give a common description to all three considered RSPs (where it is possible, of course) we will use the notion of two contacting fluid phases. The first one is always the liquid phase, whereas the second fluid phase represents the gas phase for RA, the vapor phase for RD and the liquid phase for RE. Considering homogeneous chemical reactions taking place in the fluid phases, the steady-state balance equations should include the reaction source terms ... 

The boundary conditions are defined in the same way as with the flow analysis network. The nodes whose control volumes are empty or partially filled are assigned a zero pressure, and the gate nodes are either assigned an injection pressure or an injection volume flow rate. Just as is the case with flow analysis network, a mass balance about each nodal control volume will lead to a linear set of algebraic equations, identical to the set finite element formulation of Poisson s or Laplace s equation. The mass balance (volume balance for incompressible fluids) is given by... 

The Mean Velocity of Laminar Pipe Flow Use the macroscopic mass-balance equation (Eq. 2.4.1) to calculate the mean velocity in laminar pipe flow of a Newtonian fluid. The velocity profile is the celebrated Poisseuille equation ... 

It should be understood that this rate expression may in fact represent a set of diffusion and mass transfer equations with their associated boundary conditions, rather than a simple explicit expression. In addition one may write a differential heat balance for a column element, which has the same general form as Eq. (17), and a heat balance for heat transfer between particle and fluid. In a nonisothermal system the heat and mass balance equations are therefore coupled through the temperature dependence of the rate of adsorption and the adsorption equilibrium, as expressed in Eq. (18). 

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