Question

Controlling carbon emissions has become a global challenge. Considering the earth’s atmosphere as the “control volume” or the process, construct a set of logical expressions for defining a material balance on carbon. Use any reasonable assumptions. You should not neglect the accumulation term – express it as a rate of change of carbon in the process. What are the resulting equations? How can they be solved?

Answer 1

Hi
Let us assume the environment as a control volume of Volume V m^3.

Let we have emissions source of carbon in the atmosphere and also oceans are the sinks.

We know carbon is neither getting created nor destroyed in the process thus we can write.

Rate of carbon coming in atmosphere +rate of generation of carbon - rate of the adsorption of the carbon = rate of accumulation of the carbon in the atmosphere.

Carbon is not getting generated in the carbon automatically so we can assume the rate of carbon coming in atmosphere =0 , we are taking care of the inlet of carbon by the emission in the next term.
Let us assume that rate of carbon that is getting generated is K.

Also let us assume that the rate at which it is getting absorbed in the oceans is Y.

Let us assume that Concentration of carbon in the atmosphere is given by C .
Then accumulation =d(VC)/dt

Writing all in the conservation equation.

K-Y=VdC/dt

This can be solved between the two time limits if we know the concentration of carbon at the initial point.
Let us assume it is Co.
Then
Solving it between time t=0 and time t
((K-Y)/V)*t)=CF-Co

Here Cf is the final concentration.
Unit of concentration can be taken as per the choice but we should make sure to equalise the units in all the terms.

Hope this helps

Thanks