A tank containing 20 kg of water at 20 degrees C is fittedwith a stirrer that de

Question

A tank containing 20 kg of water at 20 degrees C is fittedwith a stirrer that delivers work to the water at the rate of 0.25kW. For water, Cp = 4.184 kJ/kg-K a. How long does it take for the temperature of the water torise to 30 degrees C is no heat is lost from the water? b. how long does it take for the temperature of the water torise to 30 degrees C is heat is lost from the tank at a rate( = 0.01 kW/K) that is proportional to the temperaturedifference between that for the tank (Ttank) and thesurrounding ambient (Tambient = 20 degrees C)? Specifically, the rate of heat loss from the tank equals(Ttank - Tambient). Hink: beginby writing your expression for the 1st law in differential form andthen as a time derivative. The textbook states the following about the first law:although energy assumes many forms, the total quantity of energy isconstant, and when energy disappears in one form it appearssimultaneously in other forms. For any process, the first lawrequires: (Energy of the system) + (Energy ofsurroundings) = 0. Also, Ut = Q +W. If someone could solve the problem and try to explain it tome, that would be excellent! A tank containing 20 kg of water at 20 degrees C is fittedwith a stirrer that delivers work to the water at the rate of 0.25kW. For water, Cp = 4.184 kJ/kg-K a. How long does it take for the temperature of the water torise to 30 degrees C is no heat is lost from the water? b. how long does it take for the temperature of the water torise to 30 degrees C is heat is lost from the tank at a rate( = 0.01 kW/K) that is proportional to the temperaturedifference between that for the tank (Ttank) and thesurrounding ambient (Tambient = 20 degrees C)? Specifically, the rate of heat loss from the tank equals(Ttank - Tambient). Hink: beginby writing your expression for the 1st law in differential form andthen as a time derivative. The textbook states the following about the first law:although energy assumes many forms, the total quantity of energy isconstant, and when energy disappears in one form it appearssimultaneously in other forms. For any process, the first lawrequires: (Energy of the system) + (Energy ofsurroundings) = 0. Also, Ut = Q +W. If someone could solve the problem and try to explain it tome, that would be excellent!

Explanation / Answer

a) energy required to raise the temperature to 300C =mCpdT = 20*4.184*10 = 836.8 kJ so, time required = 836.8 kJ / (0.25kJ/s) = 3347.2 s = 55min 47.2 sec ......................................ans(a) b)let us say Ttank = T and Tambient =T0 , T0=200C applying 1st law of thermodynamics, q = u + w or, q' = u' +w'         where, ' represents time derivative or, - (T-T0) = mCp(dT/dt) -0.25 ............................{w = -0.25 kJ/s as itis work done on the system} or, 0.25 - 0.01*(T-20) = 20*4.184*(dT/dt) or, dT/(25-(T-20)) = [1/(100*83.68)]dt or, dT/(45-T) = (1/8368)dt integrating within the limits T = 200C to300C and t = 0 to t = t -[ln(45-30) - ln(45-20)] = (1/8368)t or, t = (8368)*ln(25/15) = 4274.58 s = 71 min 14.48 sec............................................................ans(b)

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