Integrals direct

Diffusion problems in one dimension lead to boundaiy value problems. The boundaiy conditions are applied at two different spatial locations at one side the concentration may be fixed and at the other side the flux may be fixed. Because the conditions are specified at two different locations, the problems are not initial value in character. It is not possible to begin at one position and integrate directly because at least one of the conditions is specified somewhere else and there are not enough conditions to begin the calculation. Thus, methods have been developed especially for boundary value problems. 

After eliminating /if or cf using the adsorption isotherm, Eq. (16-140) can be integrated directly to obtain the constant pattern. 

ARR. 4 SWSI For direct drive Impeller overhung on prime mover shaft. No bearings on fan Prime mover base mounted or integrally directly connected. 

This equation cannot be integrated directly because the velocity u increases as the pressure falls and is, therefore, a function of l (Figure 4.7). It is, therefore, convenient to work in terms of the mass flow G which remains constant throughout the length of pipe. 

This equation cannot be integrated directly since the temperature 9 is expressed as a function of two independent variables, distance jc and time t. The method of solution involves transforming the equation so that the Laplace transform of 6 with respect to time is used in place of 9. The equation then involves only the Laplace transform 0 and the distance jc. The Laplace transform of 9 is defined by the relation ... 

Equation 11.72 cannot be integrated directly, however, because the temperature driving force (0j — 0O) is not known as a function of location x on the plate. The solution of equation 11.72 involves a quite complex procedure which is given by KAYS and CRAWFORD 45 and takes the following form ... 

These equations cannot be integrated directly since the conditions at the interface are not necessarily constant nor can they be expressed directly in terms of the corresponding property in the bulk of the gas or liquid. 

For the case where t8 = rp — tp, Equation 13 can be integrated directly to give ITto1al, the total amount of tertiary ion formed. For the other two cases, integration cannot be performed directly, and values of ITtotal were evaluated numerically on a KDF 9 computer, using a procedure for Simpson s rule. (Numerical evaluation of the directly integrable case provided a check on this procedure.) Ip and I8 are then given by... 

For design purposes the two equations can be integrated directly starting for the known temperature conditions at one end of the exchanger and integrating towards the known conditions at the other end, hence enabling the required heat exchange surface to be determined. This procedure is very similar to that... 

In liquid chromatography, equation (1.14) can be integrated directly, neglecting the variation of viscosity with pressure and the compressibility of the mobile phase... 

The CC2 method [74] is an approximation to coupled cluster with singles and doubles (CCSD), and the excited state energies calculated have MP2 quality. An implementation that employs the resolution of identity (RI) approximation for two-electron integrals to reduce the CPU time is also available, RI-CC2 [75], which is suitable for large scale integral-direct calculations. This method has been implemented in TURBOMOLE [76],... 

No heat integration Direct heat integration Heat storage... 

Figure 7.4. Sketch for the deduction of the intensity, It, transmitted into the detector for symmetrical-reflection geometry. The photon is scattered in a depth of x. Integration direction is indicated by a straight dashed arrow... 

However, very economical measurement of acceleration is possible when the washing machine is equipped with a high-resolution analog pressure sensor with microprocessor control. These pressure sensors come as a separate part or are integrated directly on to the PCB of the washing machine control. They work as differential pressure sensors and measure the static pressure via a hose in the lower part of the suds container, thus measuring the water level in the suds container (Fig. 5.57). 

Once nuclear hormone receptors are bound to their hormone, they are capable of being integrated directly into the machinery that regulates the transcription of specific genes. This action is more direct, and apparently more... 

The equilibrium data given are represented by a straight line of slope m = 1.20. As shown in Problem 12.12, the equation for Nog may be integrated directly when the equilibrium line is given by ye = mx to give ... 

Gilmore, C.J. (1984) MITHRIL - an integrated direct-methods computer program... 

Where there is an analytical expression for the equilibrium such as Eq. (8.3), then Eq. (8.15) may be integrated directly. Otherwise it is necessary to perform the integration numerically or graphically. 

Further, if the concentration difference across the laminar sublayer does not vary appreciably with time, Eqs. (2) to (4) may be integrated directly to give linear relations between q and fciAc, between k and D, and between q and Dt( ic/ x). 

This expression must be solved numerically, but it is a single integral expression, which can be integrated directly rather than two simultaneous differential equations. 

An integral-direct procedure—integrals are evaluated as required. 

Direct use of geothermal heat has developed rapidly in China during the last decade. In the year 2000, integrated direct use of geothermal... 

Equation (10) defines the selectivity kinetic problem since K is the only parameter to be determined. Equation (10) can be integrated directly to give... 

The next step is to evaluate the coulomb and resonance integrals. Direct calculation of the latter requires specification of the Hamiltonian and hence explicit account of electron-repulsion terms, which is very difficult for these complex molecules. Accordingly, it is usual to make the earlier assumptions that is, either the resonance integral is directly proportional to the overlap integral or is related to it by a Wolfsberg-Helmholz formula... 

Equation (2.1) is of a particularly simple form and can be integrated directly to give the concentration of the reactant P at any time ... 

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