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By A. J. Ede

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This may be regarded as accumulating in a layer PQ of thickness a; the resulting change of temperature in a time At is therefore k(0A + ec-2eB)At . k Ai _ ^apc = 2- (ö^ + ö c - 2 ö B ) . a pea If the time interval is chosen so that At = pea2/2k, the temperature rise will be (θΑ + θε)/2 — θΒ. Since the original temperature was ΘΒ the new temperature becomes θΒ>=(θΑ + θ€)/2, so that the corres­ ponding point B' is found simply by joining A and C with a straight line, as shown. In the same way, by joining B and D, a new point C" is found, and so on; a complete new temperature distribution, corresponding to the situation after a time interval At = cpa2/2k, is thus built up.

NUMERICAL METHODS The most powerful method is that known as the "finite difference" or "numerical" method. It is assumed that the body in which the flow of heat is taking place is subdivided into small but finite cells by some regular series of intersecting planes, and that each of these cells is at a uniform temperature. The heat conduction law is applied to each cell in turn, together with the fact that, for the steady state, the net gain or loss of heat must be zero. Instead of 50 INTRODUCTION TO HEAT TRANSFER the differential equation, therefore, a number of very simple simul­ taneous algebraic equations are produced and these may, in prin­ ciple, be solved by straightforward methods.

Since surface tem­ peratures have been specified, the heat transfer coefficients need not be considered. For silica aerogel k =0-025 W/m degC. Then for case (a), CONDUCTION q= H 47 2<150-30)xl2 {1ο§β(0·035/0·025)}/0·07 + {1ο&(0·045/0·035)}/0·025 and for case (b), a=z q 2π(150-30)χ12 {loge(0-035/0-025)}/0-025 + {loge(0-045/0-035)}/0-07 The order in which the materials are applied affects the heat transfer because the flow lines are not parallel but radial; more advantage is extracted from the better of the two insulators when it is placed in the region of the greater heat flow per unit area.

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