Mean integral value

The vane anemometer s physical dimensions are often quite large (compared with other local velocity measurement instruments). It does not strictly measure a local velocity at all, but rather provides a spatially integrated mean value. This is an advantage in many cases where the air volume flow rate has to be predicted using local velocities and an integration principle. 

X u>h, so that Uh = VhU Hh, where Vh is a linear operator from Hq into Hh- It is possible to establish this correspondence in a number of different ways by approval of different operators Vh-If u(x) is a continuous function, we might accept Uf (x) = u x) for X ojh- Sometimes Uh x ) is determined at a node x G as the integral mean value of (j ) over some neighborhood of this node (for instance, of diameter 0(h)). In the sequel we will always assume that u(x) is a continuous function and keep Uh(x ) = u(xj for all Xf u>h unless otherwise is explicitly stated. 

In the definition of the shape coefficient in (2.105) and its calculation according to (2.107) and (2.108), constant thermal conductivity A was presumed. The temperature dependence of A = A( ) is accounted for by the transformed temperature from (2.26), which was introduced in section 2.1.4. It is found that a shape coefficient S calculated for constant A can be used unaltered, for cases in which A = A( ), thereby allowing the heat flow between two isothermal surfaces to be calculated. Equation (2.105) can be used for this, provided that A is replaced by the integral mean value... 

The mean heat transfer coefficient am is the integral mean value over the plate length L... 

We will introduce so-called boundary layer coordinates, Fig. 3.13, in which the coordinate x1 = x is chosen to be along the surface of the body and x.2 = y as perpendicular to it. We will presume an initial velocity wa(y) its integral mean value will be wm. 

The Nusselt number is equal to the dimensionless temperature gradient at the wall. It is a universal function of x+, Re and Pr for every fluid with a body of a given shape. The mean Nusselt number is independent of x+, as it is the integral mean value over the heat transfer surface... 

The mole fraction yG of the more volatile components here is, like in (4.67), an integral mean value over a cross section of the vapour space. From the last relationship with diV = — dNG the so-called Rayleigh equation is obtained... 

Itan/1 = Z with and = 2j--I. Finally one obtains for the integrated mean value the limiting relations... 

The solution of the system (Equation 14.30) can be obtained numerically by time discretization. Because of the nonlinearities, the integrated mean value of the conduction matrix K = K T(t)) for a given period of time can be created and a numerical scheme of one or two time levels can be applied [96]. Favorable results have been obtained by the linear Galerkin scheme [97] ... 

The calculated Hq and Hl values can vary from location to location in each section. When this occurs, an integrated mean value should be used. The overall HTU values can be obtained from Eqs. (16-27T Even if Hq and Hl are constant, Hqq and Hql will vary owing to the curvature of the equilibrium curve. 

Generally, if g is a function defined on P, let g(X) be the (random) variable associating the value g(x) with any value x of X in vector space Then the integral mean value of g, with density /x, equals by definition... 

The solution of the system (Equation 13.30) can be obtained numerically by time discretization. Because of the nonlinearities, the integrated mean value of the conduction matrix K = for a given period of... 

Using the mean value theorem for definite integrals... 

To integrate Eq. (11-3), and AT must be known as functions of Q. For some problems, varies strongly and nonlinearly throughout the exchanger. In these cases, it is necessary to evaluate and AT at several intermediate values and numerically or graphically integrate. For many practical cases, it is possible to calculate a constant mean overall coefficient from Eq. (11-2) and define a corresponding mean value of AT,n, such that... 

For physical absorption, values of the mass-transfer coefficients may not vary greatly, so a mean value could be adequate and coiild be taken outside the integral sign, but for reactive absorption the variation usually is too great. 

The function QG occurring in the integrand is represented by two straight line segments below the axis. Xf the mean value theorem is applied to the separate integrals, the result is... 

The power required for pumping will be given by the product of the volumetric flowrate and the pressure difference between the pump outlet and the discharge end of the pipeline. Taking note of the fluctuating nature of the flow, it is necessary to consider the energy transferred to the fluid over a small time interval and to integrate over the cycle to obtain the mean value of the power. 

This expression will give the point value of the Stanton number and hence of the heat transfer coefficient. The mean value over the whole surface is obtained by integration. No general expression for the mean coefficient can be obtained and a graphical or numerical integration must be carried out after the insertion of the appropriate values of the constants. 

Mean values may be obtained by graphical or numerical integration. 

Mean value Interval Value of integral in interval over interval... 

Then, applying the mean value theorem, the surface integral around the observation point p can be represented as... 

Note that since the mean value theorem involves an integration over all allowed values of the independent variable, the integration relevant to average speed is from zero to infinity. In terms of the mean value theorem stated in Eq. (25), the probability of a negative speed is zero, so the integral in Eq. (37) is only over positive values of v. The integral in Eq. (37) may be evaluated by making the... 

The average value of drift velocity, um, is given by integration of Eq. (79) over the radius of the tube according to the relevant mean value theorem,... 

The mean value of the interaction potential energy should provide some guidance on the value of the first of the terms on the right it helps that those interaction energies will have a lower bound. The second term then primarily addresses entropic contributions to jLt x that integral accumulates the weight of the favorable configurations, well-bound to the solute, that the solvent host offers the solute without coercion. 

In order to assess the accuracy of the present method, we compared it with two other methods. One was the Track Etch detector manufactured by the Terradex Corp. (type SF). Simultaneous measurements with our detectors and the Terradex detectors in 207 locations were made over 10 months. The correlation coefficient between radon concentrations derived from these methods was 0.875, but the mean value by the Terradex method was about twice that by our detectors. The other method used was the passive integrated detector using activated charcoal which is in a canister (Iwata, 1986). After 24 hour exposure, the amount of radon absorbed in the charcoal was measured with Nal (Tl) scintillation counter. The method was calibrated with the grab sampling method using activated charcoal in the coolant and cross-calibrated with other methods. Measurements for comparison with the bare track detector were made in 57 indoor locations. The correlation coefficient between the results by the two methods was 0.323. In the case of comparisons in five locations where frequent measurements with the charcoal method were made or where the radon concentration was approximately constant, the correlation coefficient was 0.996 and mean value by the charcoal method was higher by only 12% than that by the present method. 

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