Find the solution of the heat equation ut = uxx with u(x, 0) = x 2 (answer should be explicit and contain no "R "). Hint: One approach just trial and error. Alternative approach take v = uxxx and write the problem for v

Athin walled concentric tube exchanger is used to cool engine oil from 160°c to 60°c with water that is available at 25°c acting as a coolant. the oil and water flow rates are each at 2 kg/s, and the diameter of the inner tube is 0.5 m and the corresponding value of the overall heat transfer coefficient is 250 w/m2. oc. how long must the heat exchanger be to accomplish the desired cooling? cpwater=4.187 kj/kg-candcpengine el=2.035 kj/kg·°c, oil . 120]

Asimple speed reducer is composed of 2 spur gears. the pinion gear has a pitch diameter of 0.75" and 36 teeth while the driven gear has a pitch diameter of 4.0" and 192 teeth. a)-what is the diametral pitch of each gear? b)-if an electric motor rotating ccw at 3000 rpm is coupled to the pinion, what is the rotational speed of the driven gear? c)-if the torque delivered to the pinion is 1 n-m, what is the torque on the driven gear? d)-what is the power transmitted by the gear train?

Air is flows through an adiabatic nozzle with an inlet pressure of 400 psia at a temperature of 300°f and a velocity of 20 ft/s. the outlet pressure is 40 psia and the velocity is 800 ftl/s. determine the outlet temperature of the air using: a) property tables b) specific heat (using the specific heat value for air at 300°f)