5.37

PROBLEM 5.37 K NOWN: Thickness, properties and initial temperature of steel slab. Convection conditions. F IND: Heating time required to achieve a minimum temperature of 550°C in the slab. SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction, (2) Negligible radiation effects, (3) Constant roperties. p ANALYSIS: With a Biot number of hL/k = (250 W/m 2 ?K × 0.05m)/48 W/m?K = 0.260, a lumped capacitance analysis should not be performed. At any time during heating, the lowest temperature in the slab is at the midplane, and from the one-term approximation to the transient thermal response of a lane wall, Eq. (5.41), we obtain p () () ( ) 2o o1 1 i 550 800 C TT 0.417 C exp Fo T T 200 800 C ?? ? ? ? ?° ? == ==? ??° With and from Table 5.1 and 1 0.488 rad? ? 1 C 1.0396? 52 k / c 1.115 10 m / s,?? ? ==× ( ) () 22 1 t / L ln 0.401 0.914???==? () () 2 2 22 52 1 0.841 0.05m 0.914 L t 861s 0.488 1.115 10 m / s ?? ? == = × < COMMENTS: The surface temperature at t = 861s may be obtained from Eq. (5.40b), where Hence, () () o1 cos 0.417 cos 0.488 rad 0.368.x?? ? ?? ? == = ( )( si T L, 792s T T 0.368 T T ) ? ? ?=+ ? Assuming a surface emissivity of ? = 1 and surroundings that are at the radiation heat transfer coefficient corresponding to this surface temperature is Since this value is comparable to the convection coefficient, radiation is not negligible and the desired heating will occur well before t = 861s. 800 C 221 C 579 C.=°?°=° sur T T 800 C, ? ==° () () 22 2 r s sur s sur hTTTT205W/m??=+ += ?K. Frank P. Incropera PROBLEM