Pumping equation

The specific speed for centrifugal pumps (equation 5.2) usually lies between 400 and 10,000, depending on the type of impeller. Generally, pump impellers are classified as radial for specific speeds between 400 and 1000, mixed flow between 1500 and 7000, and... 

For an inviscid fluid, ie frictionless flow, and no pump, equation (1.10) becomes... 

For a liquid of density p flowing with a constant mean velocity u through a pipeline of circular cross section and constant diameter between points 1 and 2 separated by a pump, equation 1.14 can be written as... 

In some cases filtration runs are made under conditions of constant rate rather than constant pressure. This occurs if the slurry is fed to the filter by a positive displacement pump. Equation (14.2-8) can be rearranged to give the following for a constant rate (dVldt)in /s. 

By solving the differential equation above and using the single-pump case (P2 = 0) to detennine a, it can be shown that the photo-current is... 

Depending on the method of pumping, the population of may be achieved by — Sq or S2 — Sq absorption processes, labelled 1 and 2 in Figure 9.18, or both. Following either process collisional relaxation to the lower vibrational levels of is rapid by process 3 or 4 for example the vibrational-rotational relaxation of process 3 takes of the order of 10 ps. Following relaxation the distribution among the levels of is that corresponding to thermal equilibrium, that is, there is a Boltzmann population (Equation 2.11). 

Figure 9.22 illustrates how a CARS experiment might be carried out. In order to vary (vj — V2) in Equation (9.18) one laser wavenumber, Vj, is fixed and V2 is varied. Here, Vj is frequency-doubled Nd YAG laser radiation at 532 nm, and the V2 radiation is that of a dye laser which is pumped by the same Nd YAG laser. The two laser beams are focused with a lens L into the sample cell C making a small angle 2a with each other. The collimated CARS radiation emerges at an angle 3 a to the optic axis, is spatially filtered from Vj and V2... 

Head. The tme meaning of the total developed pump head, H, is the amount of energy received by the unit of mass per unit of time (14). This concept is traceable to compressors and fans, where engineers operate with enthalpy, a close relation to the concept of total energy. However, because of the almost incompressible nature of Hquids, a simplification is possible to reduce enthalpy to a simpler form, a Bernoulli equation, as shown in equations 1—3, where g is the gravitational constant, SG is specific gravity, y is the density equivalent, is suction head, is discharge head, and H is the pump head, ie, the difference between H, and H. 

Specific Speed. A review of the dimensionless analysis (qv) as related to pumps can be found in Reference 14. One of these nondimensional quantities is called the specific speed. The universal dimensionless specific speed, Q, is defined as in equation 9 ... 

The simplest osmotic dosage form, ALZA Corporation s OROS elementary osmotic pump (Fig. 7), combines the dmg and sometimes an osmotic agent in a monolithic core and deflvers the dmg in solution (102). The mass dehvery rate with time dm df) of the dmg solution is described by equation 4, where is the hydrauHc permeabiUty of the membrane, a is the membrane reflection coefficient, Atz is the osmotic pressure gradient, APis the hydrostatic back pressure, A is the area of the membrane, C is the dissolved concentration of the dmg, and b is the membrane thickness. 

This equation describes the steady-state, or zero-order, release of the dmg. When the dmg completely dissolves, its concentration within the system begins to dilute, and the release rate foUows a parabohc decline with time (102). Acutrim (ALZA Corp.), dehvering phenylpropanolamine hydrochloride [154-41 -6] for appetite suppression, is an example of an elementary osmotic pump. 

Example 2 Simplified Ejector Figure 6-6 shows a very simplified sketch of an ejector, a device that uses a high velocity primary fluid to pump another (secondary) fluid. The continuity and momentum equations may he... 

Cavitation Loosely regarded as related to water hammer and hydrauhc transients because it may cause similar vibration and equipment damage, cavitation is the phenomenon of collapse of vapor bubbles in flowing liquid. These bubbles may be formed anywhere the local liquid pressure drops below the vapor pressure, or they may be injected into the hquid, as when steam is sparged into water. Local low-pressure zones may be produced by local velocity increases (in accordance with the Bernouhi equation see the preceding Conservation Equations subsection) as in eddies or vortices, or near bound-aiy contours by rapid vibration of a boundaiy by separation of liquid during water hammer or by an overaU reduction in static pressure, as due to pressure drop in the suction line of a pump. 

A tank is charged initially with = 100 L of a solution of concentration Cio = 2 g moL/L. Another solution is then pumped in at V = 5 LVmin with concentration C q = 0.8 until a stoichiometric amount has been added. The rate equation is... 

Initially a reactor contains 2 m of a solvent. A solution containing 2 kg moPm of reactant A is pumped in at the rate of 0.06 m /min nntrl the volume becomes 4 m . The rate equation is / = 0.25C , 1/min. Compare the time-composition profile of this operation with charging all of the feed instantaneously. 

Equation (9-244) shows that, to have a low payback period (PBP), Cpc W and y should be small and y, (COP), and Cj large. Clearly as the unit cost of input energy Cj increases, the economics of heat pumps becomes more favorable. 

Equation (9-245) shows that in this particular case the fixed-capital cost per unit of input energy CpJW) must not exceed 160,000 (GJh" )" or 576 per kilowatt, to have a 1-year payback period if the heat pump is operational for 8000 h/year. For this case the corresponding value of y is about 0.12 for a heat pump with an operating life of 10 years purchased with money borrowed at a 10 percent rate of interest. 

Equation (18-31) contains no information about the ciystalhzer s influence on the nucleation rate. If the ciystaUizer is of a mixed-suspension, mixed-product-removal (MSMPR) type, satisfying the criteria for Eq. (18-31), and if the model of Clontz and McCabe is vahd, the contribution to the nucleation rate by the circulating pump can be calculated [Bennett, Fiedelman, and Randolph, Chem. E/ig, Prog., 69(7), 86(1973)] ... 

Equation (18-36) is the general expression for impeller-induced nucleation. In a fixed-geometry system in which only the speed of the circulating pump is changed and in which the flow is roughly proportional to the pump speed, Eq. (18-36) may be satisfactorily replaced with... 

Variable-Pressure, Variable-Rate Filtration The pattern of this categoiy comphcates the use of the basic rate equation. The method of Tiller and Crump (loc. cit.) can be used to integrate the equation when the characteristic curve of the feed pump is available. 

Constant Reynolds number is not used for fermentation scale-up it is only one factor in the aeration task. This is also true for considering the impeller as a pump and attempting scale-up by constant momentum. As mechanical mixing tends to predominate over bubble effects in improving aeration, scale-up equations including bubble effec ts have had httle use. 

For the pumped-discharge case, internal pressure and final fluid height are calculated by Eqs. (26-56) and (26-57). The final fluid level is the point at which the net positive suc tion head (NPSH) equation is satisfied. 

For these parameters, the equations predict a much higher vacuum (24.5 in Hg or 230 percent of the shortcut method) than the gravity-discharge case. Of course, different tank dimensions and pump characteristics coiild give different comparisons between cases. If conditions are such that the pump can completely empty the tank before backflow occurs, the vacuum is Rest calculated from Eq. (26-57). 

The equation for determining the Ns is similar to equation for the Nss, except that it substitutes the NPSHr in the denominator with the pump s discharge head ... 

Figure 3.4.3 illustrated that the pressure drop is independent of the catalyst quantity charged at any one RPM. This must be so, as will appear later on the modified Ergun equation. Since RPM is constant, so is AP on the RHS of the equation. Therefore, on the LHS, if bed depth (L/dp) is increasing, u must drop to maintain equality. Results over 5, 10, and 15 cm catalyst, and pumping air, all correlate well with the simple equation ... 

The torque eapaeity of the pump shaft must be greater than the torque eapaeity of the shear pin in all eases. We assume that failure of the pump shaft oeeurs at the interferenee of these two torque distributions. From equation 4.89, the torque eapaeity of the shear pin ean be determined by substituting the ultimate shear strength of the weak link material, for L, giving ... 

Solving equation 4.94 using Monte Carlo simulation for the variables involved, the shear yield strength required for the pump shaft material is found to have a Normal distribution with parameters ... 

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