Transient Energy Balance for Heating a Tank of Water

[hinjab]

Homework Statement

In an industrial process, a tank containing 200 liters of water must be heated from a temperature of 293 K to 372 K at a constant pressure. There is a negligible change in the volume of the water. The water is stirred during this process to maintain a uniformly distributed temperature. During the process, the convective heat transfer coefficient between the water and the ambient air is 8.5 W/(m2K) and the surface area equals 2.0 m2. The ambient air remains at a constant temperature of 293 K. Neglect the work associated with stirring the tank.

I was first told to find the energy required to heat the water to its final temperature assuming the tank is perfectly insulated and does not convectively lose energy. I got 66107 kJ and it is correct.

I think had to calculate the rate of heat transfer at the final temperature from the water, which is its steady state temperature. I got Qdot = 1343 W which is correct.

I now have to calculate the time to heat the water to within 0.985 of its final temperature assuming that the heat transfer calculated in part (b) is the heat input to the device. For this problem you need to use the transient form of the energy balance. Plot the temperature time history of the water. While this question can be answered using a numerical method, you can investigate it using a differential equation and a closed form solution.

Homework Equations

I know that my transient energy balance is power in = Time derivative of the internal energy change + convective heat loss.

The Attempt at a Solution

power in (1364) = mcdT/dt + hA ((T - Tair), T = Twater which is what I need to solve for. I know that my solution will be something like T = Cexp (-t) but I'm having such a hard time getting to this conclusion.

 

[BvU]

Hello Hinjab,

Since the ambient air temperature is constant, you can add (or subtract) its time derivative to your equation:$$\dot Q = mc {d\over dt}(T-T_{air}) + hA(T-T_{air} )$$Does that make it easier for you ?

 

[hinjab]

i see

 

[Chestermiller]

Hi - kind of but not really, can you go over what my next step would be ?

Are you saying that you don't know how to solve a linear first order ordinary differential equation with constant coefficients?

 

[hinjab]

Are you saying that you don't know how to solve a linear first order ordinary differential equation with constant coefficients?

sorry read it wrong- got it thanks