Approximate integral-heat-balance

The mathematical model for the borehole grouting in the regions of permafrost is proposed and solved analytically by the approximate integral-heat-balance method. 

Pleshanov (P4) extends the integral heat balance method to bodies symmetric in one, two, or three dimensions, using a quadratic polynomial for the approximate temperature function. Solutions are obtained in terms of modified Bessel functions which agree well with numerical finite-difference calculations. 

In the original problem, the solution was also required to satisfy the initial condition 0 = 0o (or 0 = 0) at z (or z) = 0. Of course, the present approximate formulation is valid only for z aPe, and so we cannot apply these initial conditions to the solution of (3-202). However we can see from (3 200a) that an overall integral heat balance from z = 0 to a large value of z (where (3 202) is vahd) requires... 

For the time being, low viscous systems shall be considered only. With this restriction the power input by the agitator may be neglected as a first approximation. In addition, the absence of any further heat sources or sinks, such as evaporation cooling, is assumed. In a next step the special theoretical case that all components fed into the reactor do not react with each other shall be discussed. If the two terms defined in Equ. (4-48) and (4-49) are inserted into the general integral heat balance and if the steady state is evaluated, a mixing temperature Tij will be observed ... 

Unlike the radiant loss from an optically thin flame, conductive or convective losses never can be consistent exactly with the plane-flame assumption that has been employed in our development. Loss analyses must consider non-one-dimensional heat transfer and should also take flame shapes into account if high accuracy is to be achieved. This is difficult to accomplish by methods other than numerical integration of partial differential equations. Therefore, extinction formulas that in principle can be used with an accuracy as great as that of equation (21) for radiant loss are unavailable for convective or conductive loss. The most convenient approach in accounting for convective or conductive losses appears to be to employ equation (24) with L(7 ) estimated from an approximate analysis. The accuracy of the extinction prediction then depends mainly on the accuracy of the heat-loss estimate. Rough heat-loss estimates are readily obtained from overall balances. 

The simulator will calculate material and energy balances. These would be usefol in looking for heat integration opportunities. Approximate equ ment specifications would also be established. 

The applicability of the component balance equations with reaction terms is limited. It requires the knowledge of the reaction kinetics and then, the equations are rather part of a more complex mathematical model involving heat and mass transfer equations and the like. In balancing proper, the integral reaction rates W (n) in (4.2.11) can be known only approximately or rather, they are unknown. We will now show how they can be eliminated from the set of balance equations. 

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