D • CONDUCTION OF HEAT 



that the problem involves only one essential parameter, namely, the 

 Biot number 



AT = y (4-9) 



In the solution (Eq. 4-7) the Biot number enters via the eigenvalues which 

 depend on A'', according to Eq. 4-6. 



Except for small values of r, corresponding to <<C 1, the series in 

 Eq. 4-7 converges rather rapidly and one or at the most two terms (in 

 case of large A'') are sufficient for practical purposes. When the conver- 

 gence of the series is slow, good approximations of the temperature dis- 

 tribution in the slab can be obtained by application of results for the 

 semi-infinite solid discussed in the next article. 



D,5. The Semi-Infinite Solid. The heat flow problem for this case 

 is formulated by the differential equation (Eq. 4-1) with its initial and 

 boundary conditions for d-^ oo . Its solution [6, Chap. 2] can be put in 

 the form, 



e{N., T*) = erfc -^ - e(^^+^*> erfc f^= + V^\ (5-1) 



where erfc z is the complementary error function, 



erfc z = I — —=. / e~'''dz 

 Vtt Jo 



while the dimensionless variables are 



= i ^- = T ^* = B (5-2) 



Curves of 6(Nx, t*) vs. vt* with Nx as parameter are shown in Fig. D,5. 



In application to slabs of thickness d, the significant feature of these 

 curves is that 6 at x ^ d in the semi-infinite solid is close to zero, until 

 some time t* when the temperature at the position represented by Nx=d 

 begins to rise rapidly. For shorter times t* < t*, the temperature dis- 

 tribution in the slab is approximately represented by that in the semi- 

 infinite solid with increasing accuracy as x = ^d approaches the flame 

 side (^^0). 



However, as ^ — > 1 the slab temperature given by Eq. 4-7 rises above 

 that for the semi-infinite solid at Nx^^d, given by Eq. 5-1, owing to the 

 influence of the boundary at x — d where heat flow is blocked in accord- 

 ance with the boundary condition dT/dx = for the slab. The effect of 

 this thermal barrier can be represented by supposing that, in addition to 

 the heat source of strength h{Tg — T) at x = 0, the region defined by 

 ^ a; ^ d in the semi-infinite solid receives heat from an image source 



( 262 > 



