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# Nodal Temperature Distribution of a Heated Pool

• A small pool of water, with dimensions of 3 meters in width, 2 meters in depth, and indefinite length, is subjected to energy transfer from several sources produced by energy input and solar radiation. The aim of this project is to determine the time required for this pool of water to reach a prescribed average temperature of 30 degrees C.

The pool is subject to the following conditions:
1. The sides of the pool are lined with heating elements which produce a constant heat flux of 250 W/m 2into the pool.
2. The surface of the pool is exposed to the air, the temperature of which varies sinusoidally throughout a 24hr cycle from 0 to 30 degrees C, the air can be considered to have a constant convection coefficient of 150 W/m 2K.
3. Solar radiation incoming on the pool also varies sinusoidally but from sunrise, at 6AM to sunset at 6PM, the flux as a result of solar radiation varies from 0 to 1000 W/m 2.
4. The heat flux at a particular depth z of the pool is given by: I(z)=I o exp (-z/0.3)
5. nitially at the start of the process, the entire pool is filled with water at a constant temperature of 10 degrees C.

1. The walls and floor of the pool are perfectly insulated and thus no heat loss occurs through them.
2. No water evaporates from the pool.

The following are to be determined:
1. The variation of the average temperature in the pool as a function of time.
2. The net energy gain of the pool as a function of time.
3. The time required at the specified conditions to reach an average temperature of 30 degrees C throughout the volume of the pool.
4. Several cross sections illustrating local temperature distribution.
This is to be accomplished using a finite element analysis method.

• Using finite element nodal array according to the figure above, equations 1-9 shown below are used. These equations follow a basic energy balance of heat flux in-heat flux out+heat generated=heat stored.

• Using Matlab to calculate the average temperature over these nodes using the equations above and incorporating each fluctuating element described in the conditions, the following results were determined:
• The first graph shows the average temperature of the pool as a function of time. The pool takes 2 day, 8 hours and 15 minutes to heat the pool from 10 degrees Celcius to 30 degrees Celcius.
The second graph shows the net energy gain or loss trend as a function of time.
• Finally, a figure of the temperature distribution was calculated when the pool reaches an average temperature of 30 degree Celcius.