The sun generates energy in the form of radiative heat throu

The sun generates energy in the form of radiative heat through nuclear fusion, the energy flux from the sun is 1.4 KW/m^2, but only 8.6 % of the heat actually reaches the Earth\'s surface due to absorption from the atmosphere. Furthermore, during the day, the water surface reflects (i.e. does not absorb) 90% of the incident energy. You want to configure your pool cover so that it uses the sun\'s energy to maintain a temperature of 22 degree C at a depth of 10 cm into the water during the day time. At your disposal, you have a pool cover that has a thermal conductivity of 25W/m-K and a thickness of 5 mm. The pool cover can either be directly placed on top of the pool water, such that the cover and the water are touching, or you can place the cover such that there is an air gap ¡n between the cover and the water The pool cover reflects 70% of the incident energy. You also have the option to not cover the pool. The thermal conductivity of air is 0 026 W/m-K. The thermal conductivity of water is 0.615 W/m-K. The eternal air temperature is an average of 38 degree C in the summer and in the 5 degree in the winter. The heat transfer coefficient of air is 30 W/m^3-K. Assume for each season that during the day the system is at steady state heat transfer and disregard differences in daytime hours, sunlight angle, etc. How would you configure the pool cover in the summer vs the winter? Provide drawings and a written explanation, as well as your mathematical solutions. What do your answers tell you about the usefulness of the air gap in maintaining the pool water temperature?

Solution

solution:

1)in summer without putting pool cover heat transfer is given by is

Q=Ts-T1/sum R

sum R=(1/ha)+(xw/5w)

xw=.1 m

kw=.615

ha=30

hence sum R=.1956

hence in summer Ts=38,T1=22

Q=81.79 W/m2

total heat supplied isQ=Qc+Qa=81.79

where Qa=.086*.1*Qs=.086*.1*1400=12.04 w

hence Qc=69.75 w/m2,hence in summer maximu heat transfer by convection is 81.79 but we require only 69.75 w/m2,hence it is possible to maintain temperature to 22 in summer without pool cover.

2)but in winter,heat transfer is

Q=Ts-T1/sumR

Ts=5,T1=22

Q=-86.91 W/m2

Qa=12.04 w/m2

Qr=-86.91+12.04=-74.87 w/m2

this much heat will move outward and temperature lower from 22 ,hence to achieved this we have increase resistance to heat transfer which can be done by putting pool cover,hence we need to put pool cover in winter

3)heat absorb by pool cover

Qa=.3*.086*Qs=36.12 w/m2

as in winter only radiation is positive source of heat energy,hence heat transfer in upward direction should be equal to heat transfer in downward direction by radiation

as in upward direction resistance to heat transfer without gap

sumR=(1/ha)+(t/Kp)+(Xw/Kw)

sum R=.2156

where heat transfer in upward direction is

Q=5-22/sum R=-78.84 w/m2

where Qa=36.12 w/m2

hence resultant

Qr=Q+Qa=-42.72 w/m2

hence again temperature would lower from 22 degree,hence it is neccesary to provide air gap such that

Qa+Q=0

Q=-Qa=-36.12=5-22/(sum R1)

sum R1=.4706

sumR1=sumR+xa/ka=(1/ha)+(t/Kp)+(Xw/Kw)+xa/ka

xa=6.63*10^-3 m

hence thickness of gap is 6.63 mm

5)in thisway in winter temperature can be maintain to 22 degree if air gap provided along with pool cover and without gap it is not possible to maintain temperature

6)hence in summer temperaure can maintain without pool cover but in winter pool cover need to provide with air gap of 6.63 mm.