A slab of uniform cross section is such that , its thermal conductivityK varieswith distancex from one ends as K = K_(1) x + k_(2). Determine the effectivethermalconductivityof the slab .

A slab of uniform cross section is such that , its thermal conductivit

1.	A slab of uniform cross section is such that , its thermal conductivityK varieswith distancex from one ends as K = K_(1) x + k_(2). Determine the effectivethermalconductivityof the slab .
Answer» Solution :The entire slab of LENGTH l can besubdivided into innumerableelemental SECTION , each of a particularthermalconductivity . CONSIDER one suchsectionM atdistancex fromone of the ends X (say) .Let N be anothersectionat a distance x + dxfrom the endX . Further , letthe temperature of the sectionM and Nbe T and ` T + dT` Now , thermal currentacross the section M will be (in steady state). `(dQ)/(dt) = i - KA (dT)/(dx)= - (Kx+ K) = (-A)/(i) dT (or) int_(0)^(1) (dx)/(k_(1)x+ k_(2)) = -(A)/(i) int_(T_(1))^(T_(2)) dT` `rArr (1)/(K_(1)) [In(K_(1)x + K_(2)]_(0)^(t) = (-A)/(i)[T]_(T_(i))^(T_(2)) rArr (1)/(k_(1)) In((k_(1)l + k_(2))/(K_(2))) = (-A)/(i) (T_(2)-T_(1)) ` `i.e.,i = (-K_(1)A(T_(2) -T_(1)))/(In[(K_(1)l + k_(2))//K_(2)])"".......(1)` Now , if `K_(eff)` be theeffective thermalconductivityof the slab , then . `i = - k_(eff) (A (T_(2) - T_(1)))/(l)""......(2)` Comparing Eqs. (iv) `(sum_(i=1)^(n) l_(i)) //k_("eff") = sum_(i=l)^(n)(l_(i) //k_(i)) i.e., (l)/(k_("eff")) = int_(0)^(l)(dx)/(K)` [Replacing the summationsign byinergral sign] `= int_(0)^(i) = (dx)/(k_(1)x + K_(2))= (1)/(K_(1)) In((k_(1)l+ K_(2))/(k_(2))) i.e., K_("eff") = (lK_(1))/(In((K_(1)l + k_(2))/(K_(2))))`