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user12277
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First a disclaimer: This is just an idea I had last night, so if my reasoning here is way off, please let me now.

Let's say that we have an initial thermodynamic system which is in complete balance. We assume, in order to have an easy number to work with, that the incoming radiation is 100 $W/m^2$ and this is perfectly balanced by an outgoing radiation of 100 $W/m^2$.

Now let us assume that a "lid" is placed over this system so that 50 % of the outgoing radiation is reflected back again through a greenhouse effect. The initial incoming radiation is, however, unaffacted. So in total, as 50 % is now reflected back, the total incoming radiation will now be 150 $W/m^2$.

As the system is now in imbalance, the temperature will rise, so that a new balance may be achieved. Eventually, the outgoing radiation will then also become 150 $W/m^2$.However, as 50 % will still be reflected back, the amount of radiation reflected back will now be 50 % of this. So now 75 $W/m^2$. will be reflected. If we add this to the original incoming radiation of 100 $W/m^2$, the total incoming radiation is now 175 $W/m^2$. And as the temperature rise again, and the outgoing radiation rises to this level, we now find that 50 % of this, that is 87.5 $W/m^2$. will be reflected back, and so forth.

The scenario may be easily summarized as a geometric series, where the total incoming and outgoing radiation, in $W/m^2$, becomes:

$100 + 50 + 25 + 12.5 + ...$

This we recognize as a convergent, infinite geometric series, and the sum may be easily calculated via the sum formula:

$s = \frac{a_1}{1 -k} = \frac{100}{1-0.5} = 200$

Through this reasoing, the system will therefore eventually stabilize at an incoming and outgoing radiation value of 200 $W/m^2$.

Now, obviously, this is a very simple example, and in a real, thermodynamic system, there would be a multitude of other factors which would influence the radiation balance. Furthermore, in reality, the process described here would be continuous, and not discrete. However, I still wondered if the general reasoing here may be applied if we have such a simple thermodynamic system, and if we assume that there are no extraother factors influencing the radiation balance.

Thanks in advance!

First a disclaimer: This is just an idea I had last night, so if my reasoning here is way off, please let me now.

Let's say that we have an initial thermodynamic system which is in complete balance. We assume, in order to have an easy number to work with, that the incoming radiation is 100 $W/m^2$ and this is perfectly balanced by an outgoing radiation of 100 $W/m^2$.

Now let us assume that a "lid" is placed over this system so that 50 % of the outgoing radiation is reflected back again through a greenhouse effect. The initial incoming radiation is, however, unaffacted. So in total, as 50 % is now reflected back, the total incoming radiation will now be 150 $W/m^2$.

As the system is now in imbalance, the temperature will rise, so that a new balance may be achieved. Eventually, the outgoing radiation will then also become 150 $W/m^2$.However, as 50 % will still be reflected back, the amount of radiation reflected back will now be 50 % of this. So now 75 $W/m^2$. will be reflected. If we add this to the original incoming radiation of 100 $W/m^2$, the total incoming radiation is now 175 $W/m^2$. And as the temperature rise again, and the outgoing radiation rises to this level, we now find that 50 % of this, that is 87.5 $W/m^2$. will be reflected back, and so forth.

The scenario may be easily summarized as a geometric series, where the total incoming and outgoing radiation, in $W/m^2$, becomes:

$100 + 50 + 25 + 12.5 + ...$

This we recognize as a convergent, infinite geometric series, and the sum may be easily calculated via the sum formula:

$s = \frac{a_1}{1 -k} = \frac{100}{1-0.5} = 200$

Through this reasoing, the system will therefore eventually stabilize at an incoming and outgoing radiation value of 200 $W/m^2$.

Now, obviously, this is a very simple example, and in a real, thermodynamic system, there would be a multitude of other factors which would influence the radiation balance. Furthermore, in reality, the process described here would be continuous, and not discrete. However, I still wondered if the general reasoing here may be applied if we have such a simple thermodynamic system, and if we assume that there are no extra factors influencing the radiation balance.

Thanks in advance!

First a disclaimer: This is just an idea I had last night, so if my reasoning here is way off, please let me now.

Let's say that we have an initial thermodynamic system which is in complete balance. We assume, in order to have an easy number to work with, that the incoming radiation is 100 $W/m^2$ and this is perfectly balanced by an outgoing radiation of 100 $W/m^2$.

Now let us assume that a "lid" is placed over this system so that 50 % of the outgoing radiation is reflected back again through a greenhouse effect. The initial incoming radiation is, however, unaffacted. So in total, as 50 % is now reflected back, the total incoming radiation will now be 150 $W/m^2$.

As the system is now in imbalance, the temperature will rise, so that a new balance may be achieved. Eventually, the outgoing radiation will then also become 150 $W/m^2$.However, as 50 % will still be reflected back, the amount of radiation reflected back will now be 50 % of this. So now 75 $W/m^2$. will be reflected. If we add this to the original incoming radiation of 100 $W/m^2$, the total incoming radiation is now 175 $W/m^2$. And as the temperature rise again, and the outgoing radiation rises to this level, we now find that 50 % of this, that is 87.5 $W/m^2$. will be reflected back, and so forth.

The scenario may be easily summarized as a geometric series, where the total incoming and outgoing radiation, in $W/m^2$, becomes:

$100 + 50 + 25 + 12.5 + ...$

This we recognize as a convergent, infinite geometric series, and the sum may be easily calculated via the sum formula:

$s = \frac{a_1}{1 -k} = \frac{100}{1-0.5} = 200$

Through this reasoing, the system will therefore eventually stabilize at an incoming and outgoing radiation value of 200 $W/m^2$.

Now, obviously, this is a very simple example, and in a real, thermodynamic system, there would be a multitude of other factors which would influence the radiation balance. Furthermore, in reality, the process described here would be continuous, and not discrete. However, I still wondered if the general reasoing here may be applied if we have such a simple thermodynamic system, and if we assume that there are no other factors influencing the radiation balance.

Thanks in advance!

Source Link
user12277
user12277
  • 405
  • 2
  • 6
  • 14

Can the radiation balance value be estimated through an infinite geometric series?

First a disclaimer: This is just an idea I had last night, so if my reasoning here is way off, please let me now.

Let's say that we have an initial thermodynamic system which is in complete balance. We assume, in order to have an easy number to work with, that the incoming radiation is 100 $W/m^2$ and this is perfectly balanced by an outgoing radiation of 100 $W/m^2$.

Now let us assume that a "lid" is placed over this system so that 50 % of the outgoing radiation is reflected back again through a greenhouse effect. The initial incoming radiation is, however, unaffacted. So in total, as 50 % is now reflected back, the total incoming radiation will now be 150 $W/m^2$.

As the system is now in imbalance, the temperature will rise, so that a new balance may be achieved. Eventually, the outgoing radiation will then also become 150 $W/m^2$.However, as 50 % will still be reflected back, the amount of radiation reflected back will now be 50 % of this. So now 75 $W/m^2$. will be reflected. If we add this to the original incoming radiation of 100 $W/m^2$, the total incoming radiation is now 175 $W/m^2$. And as the temperature rise again, and the outgoing radiation rises to this level, we now find that 50 % of this, that is 87.5 $W/m^2$. will be reflected back, and so forth.

The scenario may be easily summarized as a geometric series, where the total incoming and outgoing radiation, in $W/m^2$, becomes:

$100 + 50 + 25 + 12.5 + ...$

This we recognize as a convergent, infinite geometric series, and the sum may be easily calculated via the sum formula:

$s = \frac{a_1}{1 -k} = \frac{100}{1-0.5} = 200$

Through this reasoing, the system will therefore eventually stabilize at an incoming and outgoing radiation value of 200 $W/m^2$.

Now, obviously, this is a very simple example, and in a real, thermodynamic system, there would be a multitude of other factors which would influence the radiation balance. Furthermore, in reality, the process described here would be continuous, and not discrete. However, I still wondered if the general reasoing here may be applied if we have such a simple thermodynamic system, and if we assume that there are no extra factors influencing the radiation balance.

Thanks in advance!