CO2 Sequestration and Cave Pressure Problem

[cwill53]

Homework Statement

The carbon dioxide gas (CO2) can be separated from products of combustion (i.e., sequestration) and stored in an underground cavern which is initially at $p_{cavern,1}$= 100 kPa and $T_{cavern,1} = 320 K$ with a volume of 100,000 $m^3$. An adiabatic process compressor of $pV^k=C$ is used with an inlet of CO2 at T = 300 K and p = 100 kPa with gas delivered at the same pressure as cave pressure. In order to maintain the inlet temperature of 320 K the heat exchanger is installed to cool the gas. Pressure losses in the pipe can be ignored. Heat transfer from the ground to the cavern upholds constant temperature conditions within the cave. The process of CO2 charging proceeds until the pressure in the cave is 10,000 kPa. Assume ideal gas conditions, potential and kinetic energies are negligible.
Determine:
(a) the charge of mass to the cave, m,
(b) the total heat transfer to the cave, $Q_{cave}$,
(c) the total work energy required by the compressor, $W_{compressor}$, during this process,
(d) the heat transfer in the heat exchanger needed to keep a constant temperature of the CO2 within the cave during this process.

Relevant Equations

$$\frac{dE_{cv}}{dt}=\dot{Q}_{cv}-\dot{W}_{cv}+\sum_{i}\dot{m}_i(h_i+\frac{V_i^2}{2}+gz_i)-\sum_{e}\dot{m}_e(h_e+\frac{V_e^2}{2}+gz_e)$$
$$pV=mRT$$
$$dH=dU+pdV+Vdp$$
$$dU=\delta Q-\delta W$$
$$dh=c_p(T)dT$$

I've made it through the first 2 parts, it's just that part C has me stumped. I don't know how to manipulate the information I already know to figure out the total work energy required by the compressor, $W_{compressor}$, during the process.

 

[Chestermiller]

Show us what you did in parts 1 and 2 please.

 

[cwill53]

Show us what you did in parts 1 and 2 please.

This is going to be incredibly lazy, but I put the portion of the work pertaining to this problem inside of a PDF document accessible by this link:



To put the equations in Google Docs, which was the required format I had to use for this assignment, I had to copy LaTeX generated images and I no longer have the code typed up. Ignore everything from part C downwards.

 

[Chestermiller]

So, I assume we are working on part C. I feel much more comfortable working in terms of moles than in terms of mass, but, if you are confident in working in terms of mass, I can accept that.

For the adiabatic reversible operation of the compressor, what is the relationship between differential change of specific enthalpy, specific volume, and differential pressure change along the compressor? In terms of the inlet and outlet pressures, what is the overall change in specific enthalpy between the inlet and outlet of the compressor?

 

[cwill53]

So, I assume we are working on part C. I feel much more comfortable working in terms of moles than in terms of mass, but, if you are confident in working in terms of mass, I can accept that.

For the adiabatic reversible operation of the compressor, what is the relationship between differential change of specific enthalpy, specific volume, and differential pressure change along the compressor? In terms of the inlet and outlet pressures, what is the overall change in specific enthalpy between the inlet and outlet of the compressor?

Yes, that is correct. The reason I did it in mass amounts is because I'm just used to that. If you prefer molar that's fine. I just use mass basis because most of the data in Moran's book is given on a mass basis.
For the differential change in specific enthalpy in an adiabatic process, I wrote:
$$dh=du+pdv+vdp$$
$$dh=\delta q-\delta w+pdv+vdp$$
$$\delta q=0;\delta w=pdv$$
$$dh=vdp=c_pdT$$

 

[Chestermiller]

Yes, that is correct. The reason I did it in mass amounts is because I'm just used to that. If you prefer molar that's fine. I just use mass basis because most of the data in Moran's book is given on a mass basis.
For the differential change in specific enthalpy in an adiabatic process, I wrote:
$$dh=du+pdv+vdp$$
$$dh=\delta q-\delta w+pdv+vdp$$
$$\delta q=0;\delta w=pdv$$
$$dh=vdp=c_pdT$$

So what do you get when you integrate vdP over the compressor to get $\Delta h$?

 

[cwill53]

So what do you get when you integrate vdP over the compressor to get $\Delta h$?

Should you get $\Delta h= v(p_2-p_1)$?

 

[Chestermiller]

Should you get $\Delta h= v(p_2-p_1)$?

No. You need to integrate based on $Pv^k=C$.

 

[cwill53]

No. You need to integrate based on $Pv^k=C$.

I’m a tad confused, I don’t understand how the integration of vdp with respect to p wouldn’t be $h_2-h_1=v(p_2-p_1)$.

 

[Chestermiller]

I’m a tad confused, I don’t understand how the integration of vdp with respect to p wouldn’t be $h_2-h_1=v(p_2-p_1)$.

Is the gas specific volume in a compressor constant?

 

[cwill53]

Is the gas specific volume in a compressor constant?

No it can’t be because $pV^k=C$.

 

[cwill53]

Is the gas specific volume in a compressor constant?

Knowing the relationship $pV^k=C$, how should I carry out the integration?

 

[Chestermiller]

Knowing the relationship $pV^k=C$, how should I carry out the integration?

$$Pv^k=C=P_0v_0^k$$where the zero subscripts apply at the entrance to the compressor. So $$v=v_0\left(\frac{P}{P_0}\right)^{-1/k}$$So $$vdP=v_0\left(\frac{P}{P_0}\right)^{-1/k}dP$$