Flow rate problems

to the other extreme, the discharge rate is insufficient for the downstream process and changing the speed of the feeder does not correct the problem, the solution is to enlarge the outlet or add an air permeation system. If the mismatch between actual and required flow rates is so great that these alternatives are not capable of solving the problem, a fluidised or air-assisted discharge should be considered. 

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Equations 4.1-4.3 describe an ideal constant flow rate problem. The explicit finite difference technique was applied for the numerical solution of Equations 4.1-4.3. An experimental study was carried out of the flow of carbon nanotube-PEDOT PSS solution (p 8.8 mPa-s and o = 68 mN/m) through a needle of 60 pm internal diameter at a constant flow rate to validate the computer simulations. Flow rates of 60 ml/h resulted in a continuous jet stream, and this was also predicted by the computer simulation, see Figure 4.1. On the other hand, as the flow rate was lowered below Wecriticai = 1 5 drop formation occurred. This was also predicted by the computer simulation as is illustrated in Figure 4.2. 

Since the integral in Equation 2-90 is known from inputs, the flow rate problem is solved. The net result on combining Equations 2-87 and 2-90 is +1... 

Volume flow rate problem without skin effects. In this formulation. Equations 18-18a,b,c apply, but the total volumetric production rate Q(t) of the ellipsoidal source is specified instead of the pressure in Equation 18-18d. This rate is not generally equal to the Darcy VFR(t) unless wellbore storage effects completely vanish. The physical volumetric balance equation requires us to consider the more general statement VFR(t) - VC dp/dt = Q(t), where V is the storage volume and C is the compressibility of the wellbore fluid. For convenience, we will take a dimensional production rate Q(t) in the form... 

Flow rate problem with skin. Equation 18-20d provides the boundary condition for the flow rate problem without skin effects, and an exact solution can be obtained in closed analytical form (Proett and Chin, 1998). However, it is possible to obtain an exact solution for the more difficult problem including skin effects (Proett and Chin, 2000). Since this more general solution is available, we will not discuss the skin-free model in this book. Perhaps the greatest difficulty in formulating the problem correctly lies in the form of the skin model used. Conventionally, the ad hoc skin model p, = p - SR 9p/ar is... 

General flow rate problem formulation. If we now return to Equation 18-20 and review the basic formulation, it is obvious that only Equation 18-20d needs to be changed to model skin. If we rewrite Equation 18-20d in the form (47ir Pj.gf kyl kjj /p) (9p /ar ) - VC 9p/5t = Q q F(t )= h is clear that 9p/9t must be replaced by 5p /9t, to differentiate between the pressure inside the sandface and that in the well. Doing so, we obtained the extended law... 

Thus loops, utility paths, and stream splits offer the degrees of freedom for manipulating the network cost. The problem is one of multivariable nonlinear optimization. The constraints are only those of feasible heat transfer positive temperature difference and nonnegative heat duty for each exchanger. Furthermore, if stream splits exist, then positive bremch flow rates are additional constraints. 

The framework for the solution of porous media flow problems was estabUshed by the experiments of Henri Darcy in the 1800s. The relationship between fluid volumetric flow rate, hydraulic gradient, and cross-sectional area, yi, of flow is given by the Darcy formula ... 

Flooding. When a stable rathole forms in a bin and fresh material is added, or when material falls into the channel from above, a flood can occur if the bulk sohd is a fine powder. As the powder falls into the channel, it becomes entrained in the air in the channel and becomes fluidized (aerated). When this fluidized material reaches the outlet, it is likely to flood from the bin, because most feeders are designed to handle sohds, not fluids (see Eluidization). Fimited Discharge Kate. Bulk sohds, especially fine powders, sometimes flow at a rate lower than required for a process. This flow rate limitation is often a function of the material s air or gas permeabihty. Simply increasing the speed of the feeder does not solve the problem. There is a limit to how fast material... 

A potential problem for rotary valve usage is that they tend to pull material preferentially from the upside of the valve, which can affect the mass flow pattern. Another problem is that once soHd drops from the vane, the air or gas that replaces it is often pumped back up into the bin. In addition, air can leak around the valve rotor. Such air flows can decrease the soflds flow rates and/or cause flooding problems. A vertical section shown in Figure 13 can alleviate the preferential flow problem because the flow channel expands in this area, usually opening up to the full outlet. To rectify the countercurrent air flow problem, a vent line helps to take the air away to a dust collector or at least back into the top of the bin. 

Adaptive Control. An adaptive control strategy is one in which the controller characteristics, ie, the algorithm or the control parameters within it, are automatically adjusted for changes in the dynamic characteristics of the process itself (34). The incentives for an adaptive control strategy generally arise from two factors common in many process plants (/) the process and portions thereof are really nonlinear and (2) the process state, environment, and equipment s performance all vary over time. Because of these factors, the process gain and process time constants vary with process conditions, eg, flow rates and temperatures, and over time. Often such variations do not cause an unacceptable problem. In some instances, however, these variations do cause deterioration in control performance, and the controllers need to be retuned for the different conditions. 

Example 3 Venturi Flowmeter An incompressible fluid flows through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can he solved using the mechanical energy balance. In a well-made venturi, viscous losses are neghgihle, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. 

Three examples of simple multivariable control problems are shown in Fig. 8-40. The in-line blending system blends pure components A and B to produce a product stream with flow rate w and mass fraction of A, x. Adjusting either inlet flow rate or Wg affects both of the controlled variables andi. For the pH neutrahzation process in Figure 8-40(Z ), liquid level h and the pH of the exit stream are to be controlled by adjusting the acid and base flow rates and w>b. Each of the manipulated variables affects both of the controlled variables. Thus, both the blending system and the pH neutralization process are said to exhibit strong process interacHons. In contrast, the process interactions for the gas-liquid separator in Fig. 8-40(c) are not as strong because one manipulated variable, liquid flow rate L, has only a small and indirec t effect on one controlled variable, pressure P. 

The problem presented to the designer of a gas-absorption unit usually specifies the following quantities (1) gas flow rate (2) gas composition, at least with respect to the component or components to be sorbed (3) operating pressure and allowable pressure drop across the absorber (4) minimum degree of recoverv of one or more solutes and, possibly, (5) the solvent to be employed. Items 3, 4, and 5 may be subject to economic considerations and therefore are sometimes left up to the designer. For determining the number of variables that must be specified in order to fix a unique solution for the design of an absorber one can use the same phase-rule approach described in Sec. 13 for distillation systems. 

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