Fluid flow unit mass volume

One sign of progress is the extent to which sophisticated research on transport phenomena, particularly mass transfer, has penetrated several other fields, including those described in later papers of this volume. Examples include fundamental work on the mechanics of trickle beds [17] within reactor engineering studies of dispersion in laminar flows [18] in the context of separations important to biotechnology coupling between fluid flows and mass transfer in chemical vapor deposition processes for fabrication of semiconductor devices [19] and optical fiber preforms [20] and the simulation of flows in mixers, extruders, and other unit operations for processing polymers. 

Scale-Up Principles. Key factors affecting scale-up of reactor performance are nature of reaction zones, specific reaction rates, and mass- and heat-transport rates to and from reaction sites. Where considerable uncertainties exist or large quantities of products are needed for market evaluations, intermediate-sized demonstration units between pilot and industrial plants are usehil. Matching overall fluid flow characteristics within the reactor might determine the operative criteria. Ideally, the smaller reactor acts as a volume segment of the larger one. Elow distributions are not markedly influenced by... 

Here g is the gravity vector and tu is the force per unit area exerted by the surroundings on the fluid in the control volume. The integrand of the area integr on the left-hand side of Eq. (6-10) is nonzero only on the entrance and exit portions of the control volume boundary. For the special case of steady flow at a mass flow rate m through a control volume fixed in space with one inlet and one outlet, (Fig. 6-4) with the inlet and outlet velocity vectors perpendicular to planar inlet and outlet surfaces, giving average velocity vectors Vi and V9, the momentum equation becomes... 

Equation 10-5 is the unsteady, three-dimensional mass eonservation or eontinuity equation at a point in a eompressible fluid. The first term on the left side is the rate of ehange in time of the density (mass per unit volume). The seeond term deseribes the net flow of mass leaving the element aeross its boundaries and is ealled the eonveetive term. 

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... 

If the pilot reactor is turbulent and closely approximates piston flow, the larger unit will as well. In isothermal piston flow, reactor performance is determined by the feed composition, feed temperature, and the mean residence time in the reactor. Even when piston flow is a poor approximation, these parameters are rarely, if ever, varied in the scaleup of a tubular reactor. The scaleup factor for throughput is S. To keep t constant, the inventory of mass in the system must also scale as S. When the fluid is incompressible, the volume scales with S. The general case allows the number of tubes, the tube radius, and the tube length to be changed upon scaleup ... 

In model equations, Uf denotes the linear velocity in the positive direction of z, z is the distance in flow direction with total length zr, C is concentration of fuel, s represents the void volume per unit volume of canister, and t is time. In addition to that, A, is the overall mass transfer coefficient, a, denotes the interfacial area for mass transfer ifom the fluid to the solid phase, ah denotes the interfacial area for heat transfer, p is density of each phase, Cp is heat capacity for a unit mass, hs is heat transfer coefficient, T is temperature, P is pressure, and AHi represents heat of adsorption. The subscript d refers bulk phase, s is solid phase of adsorbent, i is the component index. The superscript represents the equilibrium concentration. 

Here Jv is the volumetric flow rate of fluid per unit surface area (the volume flux), and Js is the mass flux for a dissolved solute of interest. The driving forces for mass transfer are expressed in terms of the pressure gradient (AP) and the osmotic pressure gradient (All). The osmotic pressure (n) is related to the concentration of dissolved solutes (c) for dilute ideal solutions, this relationship is given by... 

This regime is characterized by the presence of one continuous fluid phase and one discrete fluid phase in tubular systems. The existence of the discrete phase generates a large interfacial area per unit tube volume for all flow configurations included in this regime. For that reason, Regime IV is of pragmatic interest when interphase heat and mass transfer are of key importance. 

It is found convenient to base compressible flow calculations on an energy balance per unit mass of fluid and to work in terms of the fluid s specific volume V rather than the density p. The specific volume is the volume per unit mass of fluid and is simply the reciprocal of the density ... 

Regardless of what other conservation equations may be appropriate, a bulk-fluid mass-conservation equation is invariably required in any fluid-flow situation. When N is the mass m, the associated intensive variable (extensive variable per unit mass) is r) = 1. That is, r) is the mass per unit mass is unity. For the circumstances considered here, there is no mass created or destroyed within a control volume. Chemical reaction, for example, may produce or consume individual species, but overall no mass is created or destroyed. Furthermore the only way that net mass can be transported across the control surfaces is by convection. While individual species may diffuse across the control surfaces by molecular actions, there can be no net transport by such processes. This fact will be developed in much depth in subsequent sections where mass transport is discussed. 

Coulson, J.M., and Richardson, J.F., Chemical Engineering, Volume I Fluid Flow, Heat Transfer and Mass Transfer, 3rd Edn (1977) Volume 2 Unit Operations, 3rd Edn... 

It may be noticed that the quantity Gdp/NPe does not have the dimensions of a diffusivity, but of a diffusivity multiplied by a density. This situation arises from the fact that the amount per unit mass, rather than the amount per unit volume, is used for a concentration. It is necessary to use the mass of fluid as a basis for concentration, since the mass but not the volume is conserved in the flow. 

The power or work rate consists of two parts. The first is the shaft-work rate shown in Fig. 7.1. Less obvious is the work associated with moving the flowing streams into and out of the control volume at entrances and exits. The fluid at any entrance or exit has a set of average properties, P, V, t/, H, etc. We imagine that a unit mass of fluid with these properties exists in a conduit adjacent to the entrance or exit, as shown in Fig. 7.1 at the entrance. This unit mass of fluid is pushed into the control volume by additional fluid, here replaced by a piston which exerts the constant pressure P. The work done by this piston in pushing the unit mass into the control volume is PV, and the work rate is (PV)m. The net work done at all entrance and exit sections is then A[(PV)fn]fg. Thus... 

The amount of mass flowing through a cross section of a flow device per unit time is called the mass flow rate, and is denoted by rii. A fluid may flow in and out of a control volume through pipes or ducts. The mass flow rate of a fluid flowing in a pipe or duct is proportional to the cross-sectional area of... 

Summary of Equations of Balance for Open Systems Only the most general equations of mass, energy, and entropy balance appear in the preceding sections. In each case important applications require less general versions. The most common restrictedTcase is for steady flow processes, wherein the mass and thermodynamic properties of the fluid within the control volume are not time-dependent. A further simplification results when there is but one entrance and one exit to the control volume. In this event, m is the same for both streams, and the equations may be divided through by this rate to put them on the basis of a unit amount of fluid flowing through the control volume. Summarized in Table 4-3 are the basic equations of balance and their important restricted forms. 

MACROSCOPIC MOMENTUM BALANCE. A momentum balance, similar to the overall mass balance, can be written for the control volume shown in Fig. 4.3, assuming that flow is steady and unidirectional in the x direction. The sum of all forces acting on the fluid in the x direction, by the momentum principle, equals the increase in the time rate of momentum of the flowing fluid. That is to say, the sum of forces acting in the x direction equals the difference between the momentum leaving with the fluid per unit time and that brought in per unit time by the fluid, or... 

No assumptions have been invoked to obtain this result. As illustrated below, the mass flux term with respect to a stationary reference frame, V p, v, contains contributions from bulk fluid flow (i.e., convection) and molecular mass transfer via diffusion. In fact, whenever the divergence of a flux appears in a microscopic balance expression, its origin was a dot product of that flux with the outward-directed unit normal vector on the surface of the control volume, accounting for input and output due to flux across the surface that bounds V(t). The divergence of a flux actually represents a surface-related phenomenon that has been transformed into a volume integral via Gauss s law. 

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