as a plane wall. Plot the temperature distribution withand without the TBC. Are there any limits to the thick-ness of the TBC?3.31A commercial grade cubical freezer, 3 m on a side,has a composite wall consisting of an exterior sheet of 6.35-mm-thick plain carbon steel, an intermediatelayer of 100-mm-thick cork insulation, and an innersheet of 6.35-mm-thick aluminum alloy (2024). Adhesive interfaces between the insulation and themetallic strips are each characterized by a thermal con-tact resistance of . What isthe steady-state cooling load that must be maintainedby the refrigerator under conditions for which the outerand inner surface temperatures are 22 C and 6 C,respectively?3.32Physicists have determined the theoretical value of thethermal conductivity of a carbon nanotube to be kcn,T5000 W/m K.(a) Assuming the actual thermal conductivity of thecarbon nanotube is the same as its theoretical value,find the thermal contact resistance, Rt,c, that existsbetween the carbon nanotube and the top surfacesof the heated and sensing islands in Example 3.4 .(b) Using the value of the thermal contact resistancecalculated in part (a), plot the fraction of the totalresistance between the heated and sensing islandsthat is due to the thermal contact resistances forisland separation distances of 5 ms20 m.3.33Consider a power transistor encapsulated in an alu-minum case that is attached at its base to a squarealuminum plate of thermal conductivity k240 W/m K,thickness L6 mm, and width W20 mm. The caseis joined to the plate by screws that maintain a contactpressure of 1 bar, and the back surface of the platetransfers heat by natural convection and radiation toambient air and large surroundings at TTsur25 C. The surface has an emissivity of 0.9, and theconvection coefficient is h4 W/m2K. The case iscompletely enclosed such that heat transfer may beassumed to occur exclusively through the base plate.Rt,c2.5104m2K/W(a) If the air-filled aluminum-to-aluminum interfaceis characterized by an area of Ac2104m2anda roughness of 10m, what is the maximum allow-able power dissipation if the surface temperature ofthe case, Ts,c, is not to exceed 85 C?(b) The convection coefficient may be increased bysubjecting the plate surface to a forced ﬂow of air.Explore the effect of increasing the coefficient overthe range 4h200 W/m2K.Porous Media3.34Ring-porouswoods, such as oak, are characterized bygrains. The dark grains consist of very low-densitymaterial that forms early in the springtime. The sur-rounding lighter-colored wood is composed of high-density material that forms slowly throughout most ofthe growing season. Assuming the low-density material is highly porous andthe oak is dry, determine the fraction of the oak cross-section that appears as being grained. Hint: Assume thethermal conductivity parallel to the grains is the sameas the radial conductivity of Table A.3.