As shown in Figure P1-8, suppose that a composite solid consists of alternating layers of mate-...

  

Transcribed Image Text:

As shown in Figure P1-8, suppose that a composite solid consists of alternating layers of mate- rials A and B, with layer thicknesses LA and LB, respectively. Both materials are isotropic, with ther- mal conductivities kA and kg. It is desired to predict the average, steady-state heat flux in a sample that contains many layers. The flux in the composite is written as Q = -K G where Q and G are the average heat flux and temperature gradient, respectively, and K is the effec- live thermal conductivity. With the coordinate axes aligned as shown, the effective conductivity will be of the form K= Kxx 0 0 0 0 Kyy 0 0 Kzz and (a) Evaluate Kxx, Kyy, Kzz in terms of k₁, kB, and , where = LA/(LA + LB) is the volume fraction of material A. (Hint: The steady heat flux q through a slab of thickness L and thermal conductivity k is q=kAT/L, where AT is the temperature difference between the surfaces.) (b) Show that, in general, the vectors Q and -G will not be parallel to one another. That is, the directions of the heat flux and temperature gradient will tend to differ. A B X Z LA LB Figure P1-8. Composite solid consisting of alternating layers of materials A and B. y As shown in Figure P1-8, suppose that a composite solid consists of alternating layers of mate- rials A and B, with layer thicknesses LA and LB, respectively. Both materials are isotropic, with ther- mal conductivities kA and kg. It is desired to predict the average, steady-state heat flux in a sample that contains many layers. The flux in the composite is written as Q = -K G where Q and G are the average heat flux and temperature gradient, respectively, and K is the effec- live thermal conductivity. With the coordinate axes aligned as shown, the effective conductivity will be of the form K= Kxx 0 0 0 0 Kyy 0 0 Kzz and (a) Evaluate Kxx, Kyy, Kzz in terms of k₁, kB, and , where = LA/(LA + LB) is the volume fraction of material A. (Hint: The steady heat flux q through a slab of thickness L and thermal conductivity k is q=kAT/L, where AT is the temperature difference between the surfaces.) (b) Show that, in general, the vectors Q and -G will not be parallel to one another. That is, the directions of the heat flux and temperature gradient will tend to differ. A B X Z LA LB Figure P1-8. Composite solid consisting of alternating layers of materials A and B. y