Fluid flow mass balance

Since both residence times are related by the flow rate ratio (i.e., Tcooi = trx/ f), one rewrites the cooling fluid s mass balance in terms of the residence time for... 

Integration of the plug-flow mass balance is not possible until one equates the rate of mass transfer of reactant A from the bulk fluid phase toward the external surface of the catalyst and the volumetrically averaged rate of reactant consumption within the porous pellet via equation (30-48) ... 

The reference time, i f and concentration, Cp. f, are chosen for a specific application (e.g., in a flow reactor, the mean residence time and feed concentration, respectively). Equation 5.2.C-6 now permits a solution for the amount of poison, /Cpia, to be obtained as a function of the bulk concentration, Cp, and the physicochemical parameters. In a packed bed tubular reactor, Cp varies along the longitudinal direction, and so Eq. 5.2.C-6 would then be a partial differential equation coupled to the flowing fluid phase mass balance equation—these applications will be considered in Part Two—Chapter 11. 

As in Section 8.1 we consider an element of the bed through which a stream containing concentration c,(z,r) of adsorbabie species / is flowing. Assuming that the flow pattern can be described by the axially dispersed plug flow model, the differential fluid phase mass balance equation for each component is... 

A key concept in the equilibrium theory of multicomponent adsorption is the concept of coherence. Coherent behavior was assumed by most of the early workers, including Glueckauf, but the nature of this assumption appears to have been recognized only more recently by Helfferich. For a dilute equilibrium plug flow system the differential fluid phase mass balance [Eq. (9.1) may be written for each component in the form... 

With the development of improved numerical methods for solution of differential equations and faster computers it has recently become possible to extend the numerical simulation to more complex systems involving more than one adsorbable species. Such a solution for two adsorbable species in an inert carrier was presented by Harwell et al. The mathematical model, which is based on the assumptions of plug flow, constant fluid velocity, a linear solid film rate expression, and Langmuir equilibrium is identical with the model of Cooney (Table 9.6) except that the mass transfer rate and fluid phase mass balance equations are written for both adsorbable components, and the multicomponent extension of the Langmuir equation is used to represent the equilibrium. The solution was obtained by the method of orthogonal collocation. 

In Sections 2.6,2.7, and 2.8 overall mass, energy, and momentum balances allowed us to solve many elementary problems on fluid flow. These balances were done on an arbitrary finite volume sometimes called a control volume. In these total energy, mechanical energy, and momentum balances, we only needed to know the state of the inlet and outlet streams and the exchanges with the surroundings. 

In practice, the loss term AF is usually not deterrnined by detailed examination of the flow field. Instead, the momentum and mass balances are employed to determine the pressure and velocity changes these are substituted into the mechanical energy equation and AFis deterrnined by difference. Eor the sudden expansion of a turbulent fluid depicted in Eigure 21b, which deflvers no work to the surroundings, appHcation of equations 49, 60, and 68 yields... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

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. 

The modeling of mass transport from the bulk fluid to the interface in capillary flow typically applies an empirical mass transfer coefficient approach. The mass transfer coefficient is defined in terms of the flux and driving force J = kc(cbuik-c). For non-reactive steady state laminar flow in a square conduit with constant molecular diffusion D, the mass balance in the fluid takes the form... 

When fluid velocity is constant, a component mass balance for a chemical species A in plug flow can be written as... 

The Navier-Stokes equations have a complex form due to the necessity of treating many of the terms as vector quantities. To understand these equations, however, one need only recognize that they are not mass balances but an elaboration of Newton s second law of motion for a flowing fluid. Recall that Newton s second law states that the vector sum of all the forces acting on an object ( F) will be equal to the product of the object s mass (m) and its acceleration (a), or XF = ma. Now consider the first of the three Navier-Stokes equations listed above, Eq. (10). The object in this case is a differential fluid element, that is, a small cube of fluid with volume dx dy dz and mass p(dx dy dz). The left-hand side of the equation is essentially the product of mass and acceleration for this fluid element (ma), while the right-hand side represents the sum of the forces... 

Finite Volume Methods Finite volume methods are utilized extensively in computational fluid dynamics. In this method, a mass balance is made over a cell, accounting for the change in what is in the cell, and the flow in and out. Figure 3-52 illustrates the geometry of the ith cell. A mass balance made on this cell (with area A perpendicular to the paper) is... 

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,... 

Three different principles govern the design of bench-scale calorimetric units heat flow, heat balance, and power consumption. The RC1 [184], for example, is based on the heat-flow principle, by measuring the temperature difference between the reaction mixture and the heat transfer fluid in the reactor jacket. In order to determine the heat release rate, the heat transfer coefficient and area must be known. The Contalab [185], as originally marketed by Contraves, is based on the heat balance principle, by measuring the difference between the temperature of the heat transfer fluid at the jacket inlet and the outlet. Knowledge of the characteristics of the heat transfer fluid, such as mass flow rates and the specific heat, is required. ThermoMetric instruments, such as the CPA [188], are designed on the power compensation principle (i.e., the supply or removal of heat to or from the reactor vessel to maintain reactor contents at a prescribed temperature is measured). 

Consider flow through the pipe-work shown in Figure 1.3, in which the fluid occupies the whole cross section of the pipe. A mass balance can be written for the fixed section between planes 1 and 2, which are normal to the axis of the pipe. The mass flow rate across plane 1 into the section is equal to p Q and the mass flow rate across plane 2 out of the section is equal to P2Q.2, where p denotes the density of the fluid and Q the volumetric flow rate. 

A mass balance of adsorbate in the fluid flowing through an increment dz of bed as shown in Figure 17.17 gives ... 

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... 

The fluid in a packed bed reactor flows from one end of the reactor to the other so our first approximation will be to assume no mixing, and therefore the mass balance on the fluid win be a PFTR,... 

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 a multicomponent fluid (the only situation of interest with chemical reactions) we next have to solve mass balances for the individual chemical species. This has been implicitly the subject of this book until now. The species balance is written as flow in minus flow... 

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... 

In the ideal CSTR, the fluid concentration is uniform and the fluid flows in and out of the reactor. Under the steady state condition, the accumulation term in the general material balance, eq. (3.70), is zero. Furthermore, the exit concentration is equal to the concentration in the reactor. For a volume element of fluid (F,), the mass balance for the limiting reactant becomes (Levenspiel, 1972)... 

In most adsorption processes the adsorbent is contacted with fluid in a packed bed. An understanding of the dynamic behavior of such systems is therefore needed for rational process design and optimization. What is required is a mathematical model which allows the effluent concentration to be predicted for any defined change in the feed concentration or flow rate to the bed. The flow pattern can generally be represented adequately by the axial dispersed plug-flow model, according to which a mass balance for an element of the column yields, for the basic differential equation governing llie dynamic behavior,... 

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 ... 

In a tubular reactor, the reactants are fed in at one end and the products withdrawn from the other. If we consider the reactor operated at steady state, the composition of the fluid varies inside the reactor volume along the flow path. Therefore, the mass balance must be established for a differential element of volume dV. We assume the flow as ideal plug flow, that is, that there is no back mixing along the reactor axis. Hence, this type of reactor is often referred to as Plug Flow Reactor (PFR). 

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... 

Now, we specialize the theory to an isothermal flow of a fluid component through the channels of a solid skeleton, namely a part of soil. It serves as carrier for an adsorbate whose mass balance contains a source term (> ,) = a, so that we admit mass exchanges between the solid and the adsorbate phase due to adsorption/desorption processes only. 

The flow coefficient is constant for the system based mainly on the construction characteristics of the pipe and type of fluid flowing through the pipe. The flow coefficient in each equation contains the appropriate units to balance the equation and provide the proper units for the resulting mass flow rate. The area of the pipe and differential pressure are used to calculate volumetric flow rate. As stated above, this volumetric flow rate is converted to mass flow rate by compensating for system temperature or pressure. 

The characteristics of electrokinetically controlled fluid flow in microchannel manifolds has been studied in a systematic way by Harrison and coworkers [28, 30]. An illustrative demonstration of the potential of this approach is shown in Fig. 2 for the controlled dilution of a fluorescein solution under voltage control. In parallel with a stepwise decrease of the potential applied to the fluorescein reservoir, a decrease of fluorescence signal downstream after the junction is visible in Fig. 2. As long as the ionic strength and pH in each supply channel is the same (same jieo), mass balance is automatically fulfilled, and the incoming flows at the intersection will be exactly balanced by the outgoing flow of the mixed components (otherwise, mass balance would be enforced by additional hydrodynamic or secondary internal flows). This way of mixing fluids was also... 

Macroscopic Mass Balance in a Steady Continuous System In the flow system shown in the accompanying figure, fluid at velocity V and density p 1 enters the system over the inlet surface Si, and it leaves at density p2 with velocity V2 over surface S2- The flow is steady state. Derive a mass balance using Eq. 2.4.1. 

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