2.52 A chermically neacting mixture is stond in a thin-wallod sherical conkiner of rafius F4=200mm, and the ewhermic neaction peracrates beat at a uniform, but ismperaturedepesaknt volansinic nate of q=4, exp (Te where 4. =5000W/m,A=75K, and T,is, the msture inmperature in kehins. The vessel is enclened by an insulating material of outer ralias r s thermal eedictivity k, and emissivity _. The euter surface of the insularion experienees convection heat tramofer and net raliative ecchuenge with the afpoining air and lagse survounalings, respectively. (a) Wrime the sacady-state form of the heat diffusion equative for the inculation. Verify that this equation is satialied by the emperature distribution T(r)=Ta,1(Ta,1Ts)[1(r/r2)1(r/p)] Skewb the iempersare diveribulion. Tirl. labeling key features. (b) Applying Feuricr's law, show that the rate of heat transfer by cenduction through the inoulation may be expressed as qr=(1/r12)(1/r22)4mk(T21Ts2) Applying an encrey balance to a conerol surface abosi the conainer, ebtain an alicmative expression for qe expresing yoer tesult in lerms of q and rp. (c) Applying an enengy balance to a control surface flaced around the euter surface of the invulation. ebeaie an expecsion fruen which Ts1 may be determined as a function of q.Fn,h,T,F, and Tm (d) The process engineer wishes to maineais a reactor kemperanure of Tv=T(r1)=95C under conditions for which k=0.05W/mK,r2=208mm,h=5 What is the actual reacter temperature and the euter surface iemperature T2 of the invulation