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Ebook Chemical engineering: Part 2

(BQ) Part 2 book "Chemical engineering" has contents: Heat transfer, mass transfer, the boundary layer, momentum, heat and mass transfer, humidification and water cooling. | SECTION 9 Heat Transfer PROBLEM 9.1 Calculate the time taken for the distant face of a brick wall, of thermal diffusivity, DH D 0.0042 cm2 /s and thickness l D 0.45 m, initially at 290 K, to rise to 470 K if the near face is suddenly raised to a temperature of  0 D 870 K and maintained at that temperature. Assume that all the heat flow is perpendicular to the faces of the wall and that the distant face is perfectly insulated. Solution The temperature at any distance x from the near face at time t is given by: ND1 ÂD p 1 N  0 ferfc[ 2lN C x / 2 DH t ] C erfc[2 N C 1 l p x/ 2 DH t ]g ND0 (equation 9.37) and the temperature at the distant face is: ND1 ÂD p 1 N  0 f2 erfc[ 2N C 1 l]/ 2 DH t g ND0 Choosing the temperature scale such that the initial temperature is everywhere zero, Â/2 0 D 470 290 /2 870 290 D 0.155 p DH D 0.0042 cm2 /s or 4.2 ð 10 7 m2 /s, DH D 6.481 ð 104 and l D 0.45 m ND1 1 erfc 347 2N C 1 /t0.5 Thus: 0.155 D ND0 D erfc 347t 0.5 erfc 1042t 0.5 C erfc 1736t 0.5 Considering the first term only, 347t 0.5 D 1.0 and t D 1.204 ð 105 s The second and higher terms are negligible compared with the first term at this value of t and hence: t D 0.120 Ms (33.5 h) 125 126 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS PROBLEM 9.2 Calculate the time for the distant face to reach 470 K under the same conditions as Problem 9.1, except that the distant face is not perfectly lagged but a very large thickness of material of the same thermal properties as the brickwork is stacked against it. Solution This problem involves the conduction of heat in an infinite medium where it is required to determine the time at which a point 0.45 m from the heated face reaches 470 K. The boundary conditions are therefore:  D 0,  D Â0 , t > 0 t D 0;  D 870 290 D 580 deg K,  D 0, x D 1,  D 0, x D 0, 2 x D 0, t > 0 t>0 tD0 ∂  ∂2  ∂  C 2C 2 ∂x 2 ∂y ∂z ∂ D DH ∂t D DH for all values of x ∂2  ∂x 2 2 (for unidirectional heat transfer) (equation .