Kirchhoff integral equation

To calculate P at any other temperature we integrate Kirchhoff s equation for the heat of dilution ... 

A further equation (Frame 11) involving an integration is Kirchhoff s equation which relates the enthalpy changes, AH% and AHj for a chemical reaction taking place at temperatures, T and T2, to an integral involving the heat capacity difference, ACp, between the products and reactants in the reaction ... 

Generalized Kirchhoff integral formulae for the Lame equation and the vector wave equation... 

In particular, assuming that the domain D with the external forces is located outside the volume V, we arrive at the generalized Kirchhoff integral formula for the Lame equation (Zhdanov, 1988) ... 

Assuming that all external forces are located outside the homogeneous domain V c = const), we arrive at the Kirchhoff integral formula for the vector wave equation... 

The generalized Kirchhoff integral formulae enable the values of the elastic displacement field (equation (13.121)) or vector wavefield (equation (13.125)) to be reconstructed everywhere inside the domain V from the known values of these fields and their normal derivatives at the domain boundary S. 

Equating the right-hand sides of formulae (13.195) and (13.198), we find that integral, u ) and the Kirchhoff integral over surface Or are both equal to zero for a sufficiently large but finite value of r (for which S = S) ... 

One can assume that the Kirchhoff integral over the surface Or characterizes the superposition of waves traveling across that surface from the surrounding outer space inside domain K-- Therefore, equations (13.199) and (13.200) show that in the given model, only outward waves travel across the surface Or, providing the radius r is large enough. 

To integrate this equation accurately we must take account of the variation of Afh with temperature, and this may be done by using Kirchhoff s equation (2.32), which in this case gives us... 

The proof of Theorem 5.1.6 also rests upon the separatrix splitting phenomenon. To this end, one should represent the Kirchhoff equations as perturbations of integrable equations. Introduce a small parameter e replacing in the Kirchhoff equations e by ee. Then on the fixed four-dimensional level surface of two integrals... 

Summary Non-stationary random vibrations of polvcfonally shaped slightly damped Kirchhoff-plates are presented. The frequency response function of the undamped structure is calculated by an advanced bound-ary-integral equation method with Green s functions of finite domains. Subsequently, light hysteretic damping is built in by applying the quadrature type of elastic-viscoelastic correspondence. ... 

However, a more accurate integration of Eq.(4.6) can be carried out if the variation of heat of vaporization with temperature is taken into account. Using Kirchhoff s equation at constant pressure. 

Even though in classical lamination theory by virtue of the Kirchhoff hypothesis we assume the stresses and are zero, we can still obtain these stresses approximately by integration of the stress equilibrium equations... 

This equation is known as Kirchhoff s law in integral form. It enables us to calculate the heat of reaction at different temperatures by knowing die heat of reaction at one temperature, say, 298K, and heat capacities of reactants and products. 

Equation (8-8) is called Kirchhoffs identity. At this point we note that the emissivities and absorptivities which have been discussed are the total properties of the particular material i.e., they represent the integrated behavior of the material over all wavelengths. Real substances emit less radiation than ideal black surfaces as measured by the emissivity of the material. In reality, the emissivity of a material varies with temperature and the wavelength of the radiation. 

The equality resulting from Kirchhoff s law between the directional spectral absorptivity and the emissivity, aA = eA, suggests that investigation of whether the other three (integrated) absorptivities aA, a and a can be calculated from the corresponding emissivities sx, s and e should be carried out. This will be impossible without additional assumptions, as the absorptivities ax, a and a are not alone material properties of the absorbing body, they also depend on the incident spectral intensity Kx of the incident radiation, see Table 5.1. The emissivities sx, s and s are, in contrast, purely material properties. An accurate test is therefore required to see whether, and under what conditions, the equations analogous to (5.69), ax = sx, a = s and a = e are valid. 

This relation is known as Kirchhoff s law. Equation 7.27 may be substituted into the various relationships for the integrated emissivity or absorptivity. However, it does not follow that such quantities as directional total, hemispherical-spectral, or hemispherical total emissivity and absorptivity are necessarily equal. In fact, the integrated properties are only equal if certain restrictions are met. These are given in Table 7.1. 

Equation (3-66) is the mathematical formulation of Kirchhoff s law. Exact integration of this equation requires a knowledge of ACp as a function of temperature. An expansion of ACp as a power series in T is usually employed. 

Equations El. 4 and El. 6 form Kirchhoff circuit laws for a parallel mounting, which are the classical starting point in electrodynamics for modeling circuits. By integration of the charge variations, one obtains the multipole charge as equal to the total charge ... 

Equation 7.49 is known as Kirchhoff s law, after the German physicist Gustav-Robert Kirchhoff. According to Kirchhoff s law, the difference between the enthalpies of a reaction at two different temperatures is just the difference in the enthalpies of heating the products and reactants from Ti to T2. Note that in deriving Equation 7.48 we have assumed that the constant-pressure heat capacities are independent of temperature. Otherwise, they must be expressed as functions of T, and the integral in Equation 7.48 must be done explicitly. 

The change in activity (and also in the activity coefficient, if the composition is expressed in a way which docs not depend on the temperature) over a range of temperature can be obtained by integrating this equation. For a wide range of temperatures it may be necessary to give Li as a function of temperature as in the Kirchhoff equation. 

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