Bernoulli coefficients

Box 18.1 Deterministic and Random Processes Bernoulli Coefficients Normal Distributions... 

Figure 18.1 Random walk of an object through an infinite array of discrete boxes numbered by m = 0, 1, + 2,.... At time t = 0 the object is located in box m = 0 (probability 1) and then moves with equal probability to the two adjacent boxes m = 1 (probabilities 1/2). The time steps are numbered by n. The resulting occupation probabilities, p(n,m), of being in box m after time step n are the Bernoulli coefficients (Eq. 18-1). Curve A shows a typical individual path. Curve B represents the unlikely case in which the object jumps six times in the same direction.

The analytical expression, Eq. 18-1, is not easy to evaluate for large values of n. Fortunately, the French mathematicians, DeMoivre and Laplace, found that with increasing n, the Bernoulli coefficients converge to the function ... 

In this expression n and m are no longer restricted to integer values. As shown in Fig. 18.2 for n = 4 and n = 6, the continuous representation of the Bernoulli coefficients is surprisingly good even for small n values it becomes even better if n is large. 

Figure 18.2 Bernoulli coefficients p(n,m) for n - 4 and n = 6 (open circles with numbers) compared to the normal density approximation p (m) by DeMoivre and Laplace. Note that in most mathematical handbooks, the Bernoulli coefficients are listed as p(n,k) =...

Now, everything falls into place We set out to study the laws of random walk by using the simple model of Fig. 18 and found the Bernoulli coefficients. We then saw that for large n (which is equivalent to large times), the Bernoulli coefficients can be approximated by a normal distribution whose standard deviation, a, grows in proportion to the square root of time, tm (Eq. 18-3). And now it turns out that the solution of the Fick s second law for unbounded diffusion is also a normal distribution. In fact, the analogy between Eqs. 18-3b and 18-17 gave the basis for the law by Einstein and Smoluchowski (Eq. 18-17) that we used earlier (Eq. 18-8). The expression (2Dt)U2 will also show up in other solutions of the diffusion equation. 

Explain the difference between the Bernoulli coefficient, p(n,m), and the function / (w)ofEq. 18-2. 

By which mathematical or physical principle are the Bernoulli coefficients and Fickian diffusion linked ... 

Looking at the Bernoulli equation, we see that the friction loss (ef) can be made dimensionless by dividing it by the kinetic energy per unit mass of fluid. The result is the dimensionless loss coefficient, K ... 

Bernoulli s equation applied across the valve relates the pressure drop and flow rate in terms of the valve loss coefficient. This equation can be rearranged to give the flow rate as follows ... 

Equation (5-14) is combined with Bernoulli s equation. Assuming flow on a horizontal axis and using a coefficient of discharge to account for friction at the orifice, the mass flow rate of an ideal gas through a thin hole in the containment wall is ... 

Finally these coefficients may be reexpressed more compactly in terms of Bernoulli polynomials2211... 

The coefficient of velocity may be determined by a velocity traverse of the jet with a fine pitot tube in order to obtain the mean velocity. This is subject to some slight error, as it is impossible to measure the velocity at the outer edge of the jet. The velocity may also be computed approximately from the coordinates of the trajectory. The ideal velocity is computed by the Bernoulli theorem. 

In case the velocity of approach is not negligible, as was assumed in the preceding discussion, it is necessary to consider the velocity head at point 1. Thus, for a conical nozzle, writing the Bernoulli equation between a point inside the pipe immediately upstream of the nozzle (point 1) and a point immediately downstream of the nozzle (point 2), and then introducing the velocity coefficient, the actual jet velocity is... 

Venturi tubes, flow nozzles, and flow tubes, similar to all differential pressure producers, are based on Bernoulli s theorem. Meter coefficients for venturi tubes and flow nozzles are approximately 0.98-0.99, whereas for orifice plates it averages about 0.62. Therefore, almost 60% (98/62) more flow can be obtained through these elements for the same differential pressure (see Figure 3.82). 

Adding Cd, a discharge coefficient, to account for frictional losses (assumed zero in Bernoulli s Equation) and other non-idealities, we get Equation (36), the operating equation for the Venturi meter. 

Remember from fluid mechanics that the Bernoulli equation is an equation for frictionless flow along a streamline. The flow through the screen is similar to the flow through an orifice, and it is standard in the derivation of the flow through an orifice to assume that the flow is frictionless by applying the Bernoulli equation. To consider the friction that obviously is present, an orifice coefficient is simply prefix to the derived equation. 

Recognizing that the Bernoulli equation was the one applied, a coefficient of discharge must now be prefixed into Equation (5.3). Calling this coefficient C ... 

The reduced partition functions of isotopic molecules determine the isotope separation factors in all equilibrium and many non-equilibrium processes. Power series expansion of the function in terms of even powers of the molecular vibrations has given explicit relationships between the separation factor and molecular structure and molecular forces. A significant extension to the Bernoulli expansion, developed previously, which has the restriction u = hv/kT < 2n, is developed through truncated series, derived from the hyper-geometric function. The finite expansion can be written in the Bernoulli form with determinable modulating coefficients for each term. They are convergent for all values of u and yield better approximations to the reduced partition function than the Bernoulli expansion. The utility of the present method is illustrated through calcidations on numerous molecular systems. 

In Equation 56 the coefficients modulating the Bernoulli terms—i.e., W/z m), are all in the range between zero and unity, as can be seen from Equation 53. The range for Equation 56 is again defined (see Equation 48) as (w max/27r). Given an order of expansion n, the highest frequency, t/ max, and a value of L, the weighting function W(n,m,w n,ax,E) becomes a function of m only, so that each term (Sur " = u - — in Equation 56 has a frequency-independent coefficient. 

Table VIII. The Bernoulli-Modulating Coefficients for Various...

This is the first rule of the mean. It holds if the coeflScients c are independent of the isotopic substitutions, as they are for the Bernoulli series. The same condition is also satisfied for the Jacobi expansions, when a quantity common to a given molecular species, such as v max of the lightest isotopic molecule, is used for evaluating the modulating coefficients. For special combinations of isotopic pairs the rule holds to higher orders and... 

Using a quantity such as v, ax of HoO for evaluating the modulating coefficients for both isotopic pairs, the one-term Jacobi expansion predicts the quantum correction to be zero, thus satisfying the first rule of the mean. The first contribution to the quantum correction arises from n = 2 in the expansion. To describe the bending vibrations adequately, however, we need at least n = 3. In Table XVI, quantum corrections predicted by expansion formulae are compared with the exact quantum correction for the disproportionation among the isotopic water molecules. No entry is made for the Bernoulli series at 300°K. because the series does not exist at this temperature. 

A considerable amount of data are available relating to the effect of substituents upon the rate of decarboxylation. Unfortunately, rates or activation parameters are usually not given individually for the free acid and the anion. As indicated previously, caution must be used in the interpretation of these overall rate coefficients. A number of years ago, Bernoulli et reported an extensive study of the... 

A loss coefficient can be defined for any element in which energy is dissipated (pipe, fittings, valves, etc.), although the friction factor is defined only for pipe flow. All that is necessary to describe the pressure-flow relation for pipe flows is Bernoulli s equation and a knowledge of the friction factor, which depends upon flow conditions, pipe size, and fluid properties. 

For incompressible fluids, Bernoulli s equation relates the pressure drop across a valve and the flow rate through the valve in terms of the loss coefficient, Kf... 

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