Unknown driving force

For this problem, we want to know the net driving force (DF) that is required to move a given fluid (/a, p) at a specified rate (Q) through a specified pipe (D, L, e). The Bernoulli equation in the form DF = ef applies. 

All the relevant variables and parameters are uniquely related through the three dimensionless variables / /VRe, and e/D by the Moody diagram or the Churchill equation. Furthermore, the unknown (DF = ef) appears in only one of these groups (/). The procedure is thus straightforward  

From the resulting value of DF, the required pump head (—w/g) can be determined, for example, from a knowledge of the upstream and downstream pressures and elevations using Eq. (6-69). 

Calculate the Reynolds number (jVRc pi), using Eq. (6-45) and the volumetric flow rate instead of the velocity, i.e., 

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All criteria proposed here are constructed such that if absolutely no gradient of a particular type exists, then the value of the corresponding criterion is zero. For fast catalytic processes this is not reasonable to expect and therefore a value judgment must be made for how much deviation from zero can be ignored. For the dimensionless expressions the Damkdhler numbers are used as these are applied to each particular condition. The approach is that the Damkdhler numbers can be calculated from known system values, which are related to the unknown driving forces for the transport processes. 

In this case, the flow rate is to be determined when a given fluid is transported in a given pipe with a known net driving force (e.g., pump head, pressure head, and/or hydrostatic head). The same total variables are involved, and hence the dimensionless variables are the same and are related in the same way as for the unknown driving force problems. The main difference is that now the unknown (Q) appears in two of the dimensionless variables (/ and 7VRe), which requires a different solution strategy. 

The relationship between flow rate, pressure drop, and pipe diameter for water flowing at 60°F in Schedule 40 horizontal pipe is tabulated in Appendix G over a range of pipe velocities that cover the most likely conditions. For this special case, no iteration or other calculation procedures are required for any of the unknown driving force, unknown flow rate, or unknown diameter problems (although interpolation in the table is usually necessary). Note that the friction loss is tabulated in this table as pressure drop (in psi) per 100 ft of pipe, which is equivalent to 100pef/144L in Bernoulli s equation, where p is in lbm/ft3, ef is in ft lbf/lbm, and L is in ft. 

The inclusion of significant fitting friction loss in piping systems requires a somewhat different procedure for the solution of flow problems than that which was used in the absence of fitting losses in Chapter 6. We will consider the same classes of problems as before, i.e. unknown driving force, unknown flow rate, and unknown diameter for Newtonian, power law, and Bingham plastics. The governing equation, as before, is the Bernoulli equation, written in the form... 

These network equations can be solved for the unknown driving force (across each branch) or the unknown flow rate (in each branch of the net-... 

We will illustrate the procedure for solving the three types of pipe flow problems for high-speed gas flows unknown driving force, unknown flow rate, and unknown diameter. 

The unknown driving force could be either the upstream pressure, Px, or the downstream pressure, P2. However, one of these must be known, and the other can be determined as follows. 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

First, the variation in the intrinsic barriers, AG, for related electrochemical reactions can be expected to be closely similar to those for the same series of homogeneous reactions using a fixed coreactant. If the comparison is made at a fixed electrode potential, E, the (often unknown) driving-force terms cancel provided that the free-energy profiles are symmetrical (the symmetry factor a. = 0.5) so that ... 

The differenee in reaction rates of the amino alcohols to isobutyraldehyde and the secondary amine in strong acidic solutions is determined by the reactivity as well as the concentration of the intermediate zwitterions [Fig. 2, Eq. (10)]. Since several of the equilibrium constants of the foregoing reactions are unknown, an estimate of the relative concentrations of these dipolar species is difficult. As far as the reactivity is concerned, the rate of decomposition is expected to be higher, according as the basicity of the secondary amines is lower, since the necessary driving force to expel the amine will increase with increasing basicity of the secondary amine. The kinetics and mechanism of the hydrolysis of enamines demonstrate that not only resonance in the starting material is an important factor [e.g., if... 

There are numerous reports of allelopathy in the literature, but often the identity of the allelochemical(s) is unknown. There are, however, many cases where specific compounds or groups of compounds have been implicated as allelopathic agents. Table 1 summarizes some examples of sources and identities of allelochemicals that directly inhibit plant growth. These secondary compounds have been implicated as a driving force in ecological succession ( 1 ). 

In this problem, it is desired to determine the size of the pipe (D) that will transport a given fluid (Newtonian or non-Newtonian) at a given flow rate (Q) over a given distance (L) with a given driving force (DF). Because the unknown (D) appears in each of the dimensionless variables, it is appropriate to regroup these variables in a more convenient form for this problem. 

Because the integral in Figure 10.5 reflects more than just a transient stress on the system (it also includes the cumulative contribution of unknown hand-written checks), this search for a useful indicator of system stress that could be used as a driving force for the estimation of future clearings was abandoned. 

Consideration of the thermodynamics of a representative reaction coordinate reveals a number of interesting aspects of the equilibrium (Fig. 5). Because the complex is in spin equilibrium, AG° x 0. Only complexes which fulfill this condition can be studied by the Raman laser temperature-jump or ultrasonic relaxation methods, because these methods require perturbation of an equilibrium with appreciable concentrations of both species present. The photoperturbation technique does not suffer from this limitation and can be used to examine complexes with a larger driving force, i.e., AG° 0. In such cases, however, AG° is difficult to measure and will generally be unknown. 

Figure 6. Schematic of driving forces for flows in Czochralski crystal growth system, which shows the regions where the driving forces will produce the strongest motions. The shape functions describing the unknown interface shapes are listed also.

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