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I am planning a solar collector system for domestic hot water and heating. I am attempting to calculate the heat loss in a storage tank. If we assume:

Tank Volume = 1m^3

Tank Surface Area = 6m^2

Tank Contents = 1000L water

Initial Water Temp = 90C (194F)

The Temp Outside the Tank = 10C (50F)

Tank Insulation Thickness = 5.08cm (2in)

Insulation Material = Closed Cell Urethane Foam

Insulation k = 0.02 W/mC

Our initial energy in the tank =

1000L * (90 + 273)K * 4184J = 1,518,792,000J

Calculating the heat loss we have:

k * A * dT * d^-1 ==> .02 * 6 * 80 / .0508 = 189W = 189J/s

So, 1,518,792,000J - 189J = 1,518,791,811J is the energy remaining after 1 second. The new temperature is then:

T = (1,518,791,811 /(4184 * 1000)) - 273 = 89.9999548C

If we then repeat the process 60 times our water should be

89.997245C with total energy being 1,518,780,662J.

After 3600 Iterations (1 hour) our water should be:

89.837566C with the total energy being 1,518,112,376J

After 86,400 Iterations (24 Hours) our water should be:

86.191268C with the total energy being 1,502,856,267J

After 2,592,000 Iterations (30 Days) our water should be:

28.515970C with the total energy being 1,261,542,820J

But I just do not believe these numbers. Even if I change k from .02 to .2 After 86,400 Iterations (24 Hours) our water should be:

69.412071C with the total energy being 1,432,652,104J

(I do understand that the "useful" energy is considerably less than the total energy, with 1000L of water @ 90C having roughly 61.6kWh of "useful energy" if we define "useful energy" is any water temp > 37C. That is Useful Energy = Total Energy - Energy @ 37C).

Anyway, I have had a fair amount of life experience with attempting to keep things cold and hot, and I simply do not believe 1000L of 90C water will only "cool down" to 86C in 24 hours with a mere 2in of closed cell foam surrounding it. What am I doing wrong? Obviously I have something wrong, I just cannot figure out what it is.

Thanks In Advance,

Fish

Tank Volume = 1m^3

Tank Surface Area = 6m^2

Tank Contents = 1000L water

Initial Water Temp = 90C (194F)

The Temp Outside the Tank = 10C (50F)

Tank Insulation Thickness = 5.08cm (2in)

Insulation Material = Closed Cell Urethane Foam

Insulation k = 0.02 W/mC

Our initial energy in the tank =

1000L * (90 + 273)K * 4184J = 1,518,792,000J

Calculating the heat loss we have:

k * A * dT * d^-1 ==> .02 * 6 * 80 / .0508 = 189W = 189J/s

So, 1,518,792,000J - 189J = 1,518,791,811J is the energy remaining after 1 second. The new temperature is then:

T = (1,518,791,811 /(4184 * 1000)) - 273 = 89.9999548C

If we then repeat the process 60 times our water should be

89.997245C with total energy being 1,518,780,662J.

After 3600 Iterations (1 hour) our water should be:

89.837566C with the total energy being 1,518,112,376J

After 86,400 Iterations (24 Hours) our water should be:

86.191268C with the total energy being 1,502,856,267J

After 2,592,000 Iterations (30 Days) our water should be:

28.515970C with the total energy being 1,261,542,820J

But I just do not believe these numbers. Even if I change k from .02 to .2 After 86,400 Iterations (24 Hours) our water should be:

69.412071C with the total energy being 1,432,652,104J

(I do understand that the "useful" energy is considerably less than the total energy, with 1000L of water @ 90C having roughly 61.6kWh of "useful energy" if we define "useful energy" is any water temp > 37C. That is Useful Energy = Total Energy - Energy @ 37C).

Anyway, I have had a fair amount of life experience with attempting to keep things cold and hot, and I simply do not believe 1000L of 90C water will only "cool down" to 86C in 24 hours with a mere 2in of closed cell foam surrounding it. What am I doing wrong? Obviously I have something wrong, I just cannot figure out what it is.

Thanks In Advance,

Fish

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