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Two liquidsof thermalcapacitance C_(1) and C_(2)are connected by a metallic rod of length l , unifrom cross sectionalarea A and thermal conductivityK. The entire system is thermally insulated , to check any heat loss to the surrouding . Initially the temperaturedifference betweenthe twobodies as a function of time t. |
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Answer» Solution :In timerelated processes , one of the variables in usually time t . The other variable here is the temperaturedifference `theta ` betweenthe two BODIES . Letus firstof all establish a relation ( in the form of differentialequation) betweenthesetwo variables. Forthis , let us consider the process takingplaceduringa small time interval DT . IfdQ be the amountof heatlost from one BODYTO the pther (i.e., heat FLOW through the rod) , then `dQ = (KA theta dt)/(L).........(i)` (Here `theta` is takenpositive). Now, if T and `T^(1)`be theinstantaneous temperatures of the hotand cold bodyat time t. then `T - T^(1) = theta........(ii)` Moreover , as a result of theheat flowdQ, we have `dQ, = C_(1) (-dT) and dQ = C_(2) (dT^(1))` wheredT and `dT^(1)` are therespectivetemperaturechanges occurringin the hot and cold body . `therefore "" - dT + dT^(1) = dQ ((1)/(C_(1)) + (1)/(C_(2))) rArr - d(T-T^(1)) = dQ((1)/(C_(1)) + (1)/(C_(2))).........(iii)` Differentaining Eq.(ii) with respect to time t , we get`(d(T - T^(1)))/(dt) = (d theta)/(dt) rArr d(T - T^(1)) = d theta` Therefore , Eq.(iii) canbe rewritten as ` - d theta = dQ ((C_(1) + C_(2))/(C_(1)C_(2))) = (KA theta)/(l) dt ((C_(1) + C_(2))/(C_(1)C_(2)))` i..e,`int_(theta_(0))^(theta) - (d theta)/(theta) int (KA)/(l) ((C_(1) + C_(2))/(C_(1)C_(2))) dt rArr [ - In theta ]_(theta_(0))^(theta) = (KA)/(l) ((C_(1) + C_(2))/(C_(1)C_(2))) [t]_(0)^(t)` (or) `In((theta)/(theta_(0))) = (-KA)/(l) ((C_(1) + C_(2))/(C_(1)C_(2)))t ` Adopting `(KA)/(l)((C_(1) + C_(2))/(C_(1)C_(2))) = mu` we haveIn `((theta)/(theta_(0))) = mu t "" or "" theta = theta_(0e^(-mut))` |
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