Approximate integral-heat-balance methods

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. 

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. 

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