Fluid constant density

A similar linear logarithmic relationship, known as a van t Hoff plot, usually exists between adjusted retention data and the reciprocal of column temperature in gas, liquid (constant composition) and supercritical fluid (constant density) chromatography. The effect of temperature on retention is based on the Gibbs-Helmholtz equation and has a sound thermodynamic basis, Eq. (1.9)... 

The foregoing forms of the Energy Equation can be rewritten for Newtonian fluids (constant density and thermal conductivity) together with the appropriate... 

Example 5-5 Find the temperature profile for the laminar flow of a Newtonian fluid (constant density and thermal conductivity) if there is a constant energy flux at the wall. 

The term static bead generally denotes the pressure in a fluid due to the head of fluid above the point in question. Its magnitude is given by the apphcation of Newton s law (force = mass X acceleration). In the case of bquids (constant density), the static headp/, Pa (lbf/ft ) is given by... 

Steady-state operation (i.e., accumulation in the reactor is zero) Constant fluid mixture density Stirrer input energy is neglected Wj,... 

A theoretical ideal fluid situation, a perfect fluid having a constant density and no viscosity, is often used in a theoretical analysis. 

An incompressible fluid is a fluid whose density remains constant during flow. Liquids are normally treated as being incompressible, as a gas can be when only slight pressure variation occurs. 

Here in Chapter 1 we make the additional assumptions that the fluid has constant density, that the cross-sectional area of the tube is constant, and that the walls of the tube are impenetrable (i.e., no transpiration through the walls), but these assumptions are not required in the general definition of piston flow. In the general case, it is possible for u, temperature, and pressure to vary as a function of z. The axis of the tube need not be straight. Helically coiled tubes sometimes approximate piston flow more closely than straight tubes. Reactors with square or triangular cross sections are occasionally used. However, in most of this book, we will assume that PFRs are circular tubes of length L and constant radius R. 

Chapter 2 developed a methodology for treating multiple and complex reactions in batch reactors. The methodology is now applied to piston flow reactors. Chapter 3 also generalizes the design equations for piston flow beyond the simple case of constant density and constant velocity. The key assumption of piston flow remains intact there must be complete mixing in the direction perpendicular to flow and no mixing in the direction of flow. The fluid density and reactor cross section are allowed to vary. The pressure drop in the reactor is calculated. Transpiration is briefly considered. Scaleup and scaledown techniques for tubular reactors are developed in some detail. 

Example 14.6 Explore the consequences of the following shutdown strategy for an isothermal, constant-density CSTR that has been operating at steady state. At time zero, the discharge flow rate is increased by a factor of 1 -b 5. Simultaneously, the inlet flow rate is made proportional to the fluid volume in the vessel. When does the vessel empty and what happens to the composition of the discharge stream during the shutdown interval ... 

In order to derive specific numbers for the temperature rise, a first-order reaction was considered and Eqs. (10) and (11) were solved numerically for a constant-density fluid. In Figure 1.17 the results are presented in dimensionless form as a function of k/tjjg. The y-axis represents the temperature rise normalized by the adiabatic temperature rise, which is the increase in temperature that would have been observed without any heat transfer to the channel walls. The curves are differentiated by the activation temperature, defined as = EJR. As expected, the temperature rise approaches the adiabatic one for very small reaction time-scales. In the opposite case, the temperature rise approaches zero. For a non-zero activation temperature, the actual reaction time-scale is shorter than the one defined in Eq. (13), due to the temperature dependence of the exponential factor in Eq. (12). For this reason, a larger temperature rise is foimd when the activation temperature increases. 

It is particularly convenient to choose the reference conditions at which the volumetric flow rate is measured as the temperature and pressure prevailing at the reactor inlet, because this choice leads to a convenient physical interpretation of the parameters and CA0 and, in many cases, one finds that the latter quantity cancels a similar term appearing in the reaction rate expression. Unless otherwise specified, this choice of reference conditions is used throughout the remainder of this text. For constant density systems and this choice of reference conditions, the space time t then becomes numerically equal to the average residence time of the fluid in the reactor. 

For semibatch operation, the term fraction conversion is somewhat ambiguous for many of the cases of interest. If reactant is present initially in the reactor and is added or removed in feed and effluent streams, the question arises as to the proper basis for the definition of /. In such cases it is best to work either in terms of the weight fraction of a particular component present in the fluid of interest or in terms of concentrations when constant density systems are under consideration. In terms of the symbols shown in Figure 8.20 the fundamental material balance relation becomes ... 

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,... 

Several age-distribution functions may be used (Danckwerts, 1953), but they are all interrelated. Some are residence-time distributions and some are not. In the discussion to follow in this section and in Section 13.4, we assume steady-flow of a Newtonian, single-phase fluid of constant density through a vessel without chemical reaction. Ultimately, we are interested in the effect of a spread of residence times on the performance of a chemical reactor, but we concentrate on the characterization of flow here. 

Furthermore, for the steady flow of a constant-density fluid, f is constant, and, from equation (13.3-5), dd = dt/t. Thus, on combining equations 13.3-5 and -6, we obtain... 

Consider the steady flow of fluid at a volumetric rate q through a stirred tank as a closed vessel, containing a volume V of fluid, as illustrated in Figure 13.4. We assume the flow is ideal in the form of BMF at constant density, and that no chemical reaction occurs. We wish to derive an expression for E(t) describing the residence-time distribution (RTD) for this situation. 

Consider an element of fluid (as tracer ) entering the vessel at t = 0. Visualizing what happens to the element of fluid is relatively simple, but describing it quantitatively as Eft) requires an unusual mathematical expression. The element of fluid moves through the vessel without mixing with fluid ahead of or behind it, and leaves the vessel all at once at a time equal to the mean residence time ft = Vlq for constant density). Thus, Eft) = 0 for 0 < t < f, but what is Eft) at t = F ... 

With these results, the general equations of Section 15.2.1 can be transformed into equations analogous to those for a constant-density BR. The analogy follows if we consider an element of fluid (of arbitrary size) flowing through a PFR as a closed system, that is, as a batch of fluid. Elapsed time (t) in a BR is equivalent to residence time (t) or space time (r) in a PFR for a constant-density system. For example, substituting into equation 15.2-1 [dWd/A - FAJ(-rA) = 0] for dV from equation 15.2-15 and for d/A from 15.2-13, we obtain, since FAo = cAoq0,... 

Consider the steady flow of a constant-density fluid at qg m3 s 1 through the N stirred tanks (Figure 19.11). At t = 0, a quantity of na moles of a nonreacting tracer A is introduced into the first tank as a pulse or Dirac input, S(t — 0) s S(t). At any subsequent time t. a material balance for tracer around the i th tank is ... 

If the fluid has a constant density or behaves as an ideal gas, then the internal energy remains constant if the temperature is constant. If no heat transfer to the fluid takes place, <7=0. For these conditions, equations 1.8 and 1.9 may be combined and written as... 

For the constant-density flows considered in this work,27 the fundamental governing equations are the Navier-Stokes equation for the fluid velocity U (Bird et al. 2002) ... 

For constant-density flow, the fundamental relationship (Pope 2000) between the fluid-particle PDF and the Eulerian PDF of the flow is... 

Thus, in summary, the two necessary conditions for correspondence between the notional-particle system and the fluid-particle system in constant-density flows are... 

In a Lagrangian PDF simulation, each notional particle represents a fluid element with mass u)wAm. In a constant-density system, the unit mass is defined by... 

A fluid of constant density p is pumped into a cone-shaped tank of total volume HnR I3. The flow out of the bottom of the tank is proportional to the square root of the height h of hquid in the tank. Eterive the equations describing the system. 

We will restrict this treatment to steady-state flow with one entering stream and one leaving stream of a single fluid of constant density. Before proceeding further let us define a number of terms used in connection with the nonideal flow of fluids. 

Further, we want to be able to work problems with numerical solutions. This will require simplifying assumptions wherever possible so that the equations we need to solve are not too messy. This wiU require that fluids are at constant density so that we can use concentrations in molesA olume. This is a good approximation for hquid solutions but not for gases, where a reaction produces a changing number of moles, and temperatures and... 

The molar flow rate of a species in a flow reactor is Fj = vCj. The batch reactor is a closed system in which v = 0. The volumetric flow rate is ti, while thelinear velocity in a tubular reactor is u, We usually assume that the density of the fluid in the reactor does not change with conversion or position in the reactor (the constant-density reactor) because the equations for a constant-density reactor are easier to solve. 

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