Mean speed

Note Generally, the ratio of rated to mean wind speed may be quite high due to long lean periods, when the maehine may slay idle, redueing the value of the mean speed. 

Wind has a highly turbulent and gusting character. In addition, a time-mean speed varies with the height from the ground and the roughness of the terrain over which the wind passes. The time-mean wind speed profile can be determined using the following expression ... 

The root mean square speed rrrm of gas molecules was derived in Section 4.10. Using the Maxwell distribution of speeds, we can also calculate the mean speed and most probable (mp) speed of a collection of molecules. The equations used to calculate these two quantities are i/mean = (8RT/-nM),a and... 

When a probe is inserted into a plasma, it will experience electrons and ions colliding with its tip. Due to the high mean speed of electrons, the flow of electrons is higher than the flow of ions. Consequently, the tip will charge up negatively until the electrons are repelled, and the net current then is zero. The probe potential then is the floating potential, Vfl. The electron current density Je then balances the ion current density 7,. At potentials lower than Vfl the ion current cannot increase further—in fact, only ions are collected from the plasma—and the ion saturation current /,s is measured. The plasma potential Vpi is defined as the potential at which all electrons arriving near the probe are collected and the probe current equals the electron current. Note that the plasma assumes the plasma potential in the absence of a probe hence probe perturbation at Vpi is... 

Substituting for number density in terms of pressure and expressing mean speed in terms of absolute temperature and molecular mass m then gives the desired final result for total intensity, or number of molecules in an equilibrium gas striking a surface of unit surface area per unit time,... 

The reaction rate is then taken to be the product of the frequency at which activated complexes cross the energy barrier and their concentration at the top of the barrier. If (Cx) represents the concentration lying within a region of length dx at the top of the barrier (see Figure 4.2) and if vx is the mean speed at which molecules move from left to right across the barrier, the rate is given by... 

Rotation speed (rpm) Pressure Ratio Mass Flow (kg/s) Inlet Temperature (°C) Inlet Pressure (MPa) Rim Speed (m/s) Number of Stages Inlet Temperature Inlet Pressure (MPa) Mean Speed (m/s) Number of Stages... 

And each particle in the gaseous state can move at amazingly high speeds indeed, they are often supersonic. For example, an average atom of helium travels at a mean speed of 1204 ms-1 at 273.15 K. Table 1.4 lists the mean speeds of a few other gas molecules at 273.15 K. Notice how heavier molecules travel more slowly, so carbon dioxide has a mean speed of 363 ms-1 at the same temperature. This high speed of atomic and molecular gases as they move is a manifestation of their enormous kinetic energy. It would not be possible to travel so fast in a liquid or solid because they are so much denser - we call them condensed phases. 

These root mean speeds are very high. Flydrogen gas, H2, at 20.°C has a value of approximately 2000 m/s. 

Here m is the mass of a molecule with an effective diameter of first derived by Maxwell in 1860. We can immediately see the importance of the nature of the molecules through their mass and through the effective diameter, which is not just a geometric size but includes a contribution from the interaction with its neighbours. 

First Moments. For both of the dispersed plug-flow cases Mi = 0. This means that the center of gravity of the solute moves with the mean speed of the flowing fluid. For the uniform and the general dispersion models, however, this is not always true. If the solute concentration is initially uniform over a cross-sectional plane, it can be shown (A6) that... 

Distribution functions are usually first met in physical chemistiy when the crude treatment of molecular velocities in the kinetic theory of gases (all the molecules taken as having the same mean speed) is replaced by Maxwell s seminal equation showing that the number of molecules having velocities between narrow limits depends very much on what velocities are chosen. This is shown in Fig. 9.1. Thus, this first and basic distribution law of Maxwell, the distribution of velocities, gives an unexpected result (the nonsymmetrical nature of the distribution), which still causes us to think, more than a century after its publication. 

The distribution of the speeds about the most probable value.. We know, for example, what proportion of the molecules have speeds more than double the mean speed, less than half the mean speed, and so on. 

Shape factors of a different sort are involved in the Taylor dispersion problem. With parabolic flow at mean speed U through a cylindrical tube of radius R, Taylor found that the longitudinal dispersion of a solute from the interaction of the flow distribution and transverse diffusion was R2U2/48D. The number 48 depends on both the geometry of the cross-section and the flow profile. If, however, we insist that the flow should be laminar, then the geometry of the cross-section determines the flow and hence the numerical constant in the Taylor dispersion coefficient. 

It is convenient to take an origin moving with the mean speed of the stream and to reduce the variables to dimensionless form. Let a be a dimension characteristic of the cross-section S and let... 

Thus as t— a), drai/dr— 0 and the centre of gravity ultimately moves with the mean speed of the stream. Choosing the origin in the original plane of the centre of gravity, mio = 0 and... 

These are the relations which exist between the moments of the normal distribution and in this sense the mean concentration is ultimately distributed about a point moving with the mean speed of the stream according to the normal law of error, the variance being 2(1 + kh2)t. It should be noted that the approach to normality is as r 1, a very much slower process than the vanishing of terms in the expressions for the moments, which is as exp(-Air). 

Na is the Avogadro constant and c is the mean speed at which the molecules approach each other in a gas. When the temperature is T and the molar masses are MA and MB, this mean speed is... 

The mean speed is calculated by multiplying each possible speed by the fraction of molecules that have that speed and then adding all the products together. 

The mean speed of the molecules increases with temperature, so the collision rate increases too. However Eq. 20 shows that the mean speed increases only as the square root of the temperature, which is far too weak a dependence to account for observation. An increase in temperature by 10°C at about room temperature (from 273 K to 283 K) increases the rate of collision by a factor of only 1.02, whereas many reaction rates double over that range. Another factor must be affecting the rate. 

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