Unit area normalization

An ultrasonic horn has a small tip from which high intensity ultrasound is radiated. The acoustic intensity is defined as the energy passing through a unit area normal to the direction of sound propagation per unit time. Its units are watts per square meter (W/m2). It is related to the acoustic pressure amplitude (P) as follows for a plane traveling wave [1]. 

This expression applies to the transport of any conserved quantity Q, e.g., mass, energy, momentum, or charge. The rate of transport of Q per unit area normal to the direction of transport is called the flux of Q. This transport equation can be applied on a microscopic or molecular scale to a stationary medium or a fluid in laminar flow, in which the mechanism for the transport of Q is the intermolecular forces of attraction between molecules or groups of molecules. It also applies to fluids in turbulent flow, on a turbulent convective scale, in which the mechanism for transport is the result of the motion of turbulent eddies in the fluid that move in three directions and carry Q with them. 

Heat Flux - The rate of heat transfer per unit area normal to the direction of heat flow. It is the total heat transmitted by radiation, conduction, and convection. 

II. With respect to sound, symbolized by /, the rate of energy transfer per unit area normal to the direction of propagation. Thus, I = ll2)poCa p where po is the mean density of the medium, a is the amplitude, / is the frequency, and c is the velocity of the sound. 

The total intensity of sunlight outside the earth s atmosphere is characterized by the solar constant, defined as the total amount of light received per unit area normal to the direction of propagation of the light the mean value is 1368 W m-2, although variations from this mean are common (Lean, 1991). 

The incoming solar intensity 1/4Tt, where FT is the average total incoming solar light intensity per unit area normal to the direction of propagation outside the earth s atmosphere, 1368 W m-2, and the 1/4 factor takes into account that the solar energy is spread over the entire surface of the earth (see Fig. 14.2a). 

Consider an enclosure of dimensions large compared with any wavelengths under consideration, which is opaque but otherwise arbitrary in shape and composition (Fig. 4.11). If the enclosure is maintained at a constant absolute temperature T, the equilibrium radiation field will be isotropic, homogeneous, and unpolarized (see Reif, 1965, p. 373 et seq. for a good discussion of equilibrium radiation in an enclosure). At any point the amount of radiant energy per unit frequency interval, confined to a unit solid angle about any direction, which crosses a unit area normal to this direction in unit time is given by the Planck function... 

It is proportional to the flux jn i(Pi) of generation n — 1 incident at P times the reflection probability r = 1 — / . Furthermore, because we assume a cosine angular distribution of reflected particles, it is proportional to cos8. As the flux is defined as the number of particles per unit time and unit area normal to the surface also cos 82 enters. Finally, in the two dimensional case the distance d 2 between P and P2 enters with the power ( — 1). We obtain... 

The quantity that fully characterizes radiation in a medium is the spectral radiative intensity I r, s) (Wm pm sr ). This quantity is the formalization of the intuitive concept of a ray of light. It characterizes the local amount of power traveling along a given direction, per unit wavelength, per unit area normal to this direction, and per imit solid angle. It depends on five variables three spatial variables for the position vector r and two angular variables for the direction unit vector s. 

It is the aim now to show how the concept of drift velocity can be used to obtain an expression for the ionic current density flowing through an electrolyte in response to an externally applied electric field. Consider a transit plane of unit area normal to the direction of drift (Fig. 4.61). Both the positive and the negative ions will drift across this plane. Consider the positive ions first, and let their drift velocity be or simply v. Then, in 1 s, all positive ions within a distance cm of the transit plane will cross it. The flux 7+ of positive ions (i.e., the number of moles of these ions arriving in 1 second at the plane ofunit area) is equal to the number ofmoles ofpositive ions in a volume of 1 cm in area and v cm in length (with i = 1 s). Hence, 7 is equal to the volume in cubic centimeters times the concentration c. expressed in moles per cubic centimeter... 

In a collision experiment we have an incident beam of electrons of kinetic energy Eq. If one spin projection predominates then the beam is polarised, the polarisation P being given in terms of the intensities h of electrons with projection v. The intensity is the number of particles per second incident on unit area normal to the beam direction. 

Consider the emission of radiation by a differential area element dA of a surface, as shown in Fig. 12-18. Radiation is emitted in all directions into the hemispherical space, and ihe radiation streaming though the surface area dS is proporiional to the solid angle d(o subtended by dS. It is also proportional to ihe radiating area dA as seen by an observer on dS, wluch varies from a maximum of dA when dS is at the top directly above dA (d = 0 ) to a minimum of zero when dS is at the bottom (0 = 90 ). Therefore, the effective ar ea of dA for emission in the direction of B is the projection of dA on a plane normal to 9, which is dA cos 0. Radiation intensity in a given direction is based on a unit area normal to that direction to provide a comnioii basis for the comparison of radiation emitted in different directions. 

The radialian intensity for emitted radiation l, 9, ( >) is defined as the rate at which mdiqtion energy dQe emitted in the (9, ) direction per unit area normal to this direction and per unit solid angle about this direction. That is. 

This value of intensity is the same in ail directions since a blackbody i.s a diffuse emitter. Intensity represents the rale of radiation emission per unit area normal to the direction of emission per unit solid angle. Therefore, the rale of radiation energy emitted by A, in the direction of di through the solid angle is determined by multiplying /] by the area of Aj normal to 0i and the solid angle oij.,. That is,... 

Here is the (diffusive) mass flux of species A (mass transfer by diffusion per unit time and per unit area normal to the direction of mass transfer, in kg/s m ) and is the (diffusive) molar flux (in kmol/s m ). The mass flux of a species at a location is propoitional to the density of the mixture at that location. Note that p = Px + Pb density and C = Q + is the molar concentration of the binary mixture, and in general, they may vary throughout the mixture. Therefore, pd(pjp) dp or Cd(C /C) + dC - But in the special case of constant mixture density p or constant molar concentration C, the relations above simplify to... 

Evidence has been presented of values of Da or Da which increase or decrease with concentration. It is of particular interest to consider the theoretical basis of this behavior in sorbed fluids (9). If the driving force per molecule is proportional to —d/xx/dx for flow in the x-direction, then the flux, /, through unit area normal to x, is for species A... 

By definition it is a symmetric second-rank tensor. The stress tensor ffy, i,j= 1,3), is also a symmetric second-rank tensor defined as follows (Landau and Lifchitz ) the element Oy is the i component of the force acting on the unit area normal to the axis x. The symmetry of the stress tensor is imposed by the condition of mechanical equilibrium. 

The spectral intensity is defined as the energy per unit area normal to the direction of propagation, per unit solid angle about the direction, per metre of wavelength. 

In three dimensions has the dimensionality [length] / nd the dimension of flux is [Z/ ] . When multiplied by the total number of particles N, the flux vector gives the number of particles that cross a unit area normal to its direction per unit time. 

If there are dN particles in the size range dj, to dp + d dp) per unit volume of air, this corresponds to a total particle cross-sectional area of Kd /4)dNdz over the light path length, dz, per unit area normal to the beam. The attenuation of light over this length is given by the relation... 

The quantity pu is the mass flux through a unit area normal to u. A molar average velocity u may be correspondingly defined by... 

Consider the idealized picture of a mass-transfer process based on the two-film theory as shown in Figure 7.27. The partial pressure profile in the gas film is linear, as called for by steady-state diffusion, but the concentration profile in the liquid film falls below a linear measure as a result of a first-order chemical reaction removing the absorbed gas. A normal mass balance over the differential segment dz (we may assume unit area normal to the direction of diffusion) produces a familiar result ... 

P is the electric dipole moment per unit volume, P = P being the polarization charge per unit area normal to the vector P. D = 1 D 1 also expresses the charge density of a condenser required to maintain the field E at the interior of the dielectric (Fig. 4.11). 

Equation (9.1) shows that P is the dipole moment per unit volume, a vector quantity that has its direction parallel to Ei for an isotropie medium. Imagine a small cylindrical volume of length d/ parallel to the polarisation and of cross-sectional area dA. Let the apparent surface charges at the two ends of the cylinder be dq. It then follows that P = dq dl/dv, where do is the volume of the small cylinder. However, do = dl dA, so that P = dq/dA = a, where a is the apparent surface charge per unit area normal to the polarisation. (Note a stands for conductivity in section 9.3.)... 

The reason for the lower experimental value lies, of course, in the facts that not all chain segments are parallel to the direction of alignment and that the average number of chains crossing unit area normal to the alignment direction is less than the theoretical number. In the next section models for highly oriented polyethylene are described. A model that has been found particularly useful for understanding the moduli of oriented LCPs is considered in section 12.4.7. 

The cathode emits electrons that are accelerated towards the anode with a defined voltage, typically 50-30,000 V. There are basically two types of electrodes thermionic cathodes (tungsten or LaBs (lanthanum hexaboride)) and field emission cathodes. The Wehnelt cylinder controls the current density and brightness of the electron beam. Brightness is defined as current per unit area normal to the given direction, per unit solid angle, and a criterion for beam quality. 

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