Was the Apollo 13 CO2 problem a matter of capacity, or of rate?

Question

During Apollo 13, an explosion in the service module caused the command module to lose electrical power. This required using the environmental control system of the lunar module to remove carbon dioxide from the air, instead of the one in the command module. However, CO$_2$ levels continued to rise.

The movie Apollo 13 explains the problem as such:

FRED HAISE: (expletive) I know why my numbers are wrong. I only figured it for two people.

JACK SWIGERT: Maybe I should just hold my breath.

and

NEWS ANCHOR (on TV): Well, the astronauts appear to have enough oxygen to keep them alive. One thing they have too much of is carbon dioxide. With each breath that three men exhale more of the poisonous gas into the lunar module cockpit and the scrubbers intended to keep the atmosphere breathable are quickly becoming saturated.

The movie suggests that the CO$_2$ problem was a limited rate. In brief, they "can't keep up." That is, the LiOH canisters can only filter so many moles of CO$_2$ per minute. A normal LM crew of 2 produces CO$_2$ below this rate, so the filter keeps up. Apollo 13's crew of 3 produced CO$_2$ above this rate, the canisters "saturated", and the excess CO$_2$ accumulated in the cabin.

However, another possibility is that the LiOH canisters have a limited capacity. In brief, they "run out." They can only filter so many total moles of CO$_2$. More astronauts means more total moles of CO$_2$ to be handled by the LM. Once a canister's limit is reached, no more CO$_2$ can be filtered, and the CO$_2$ level rises until a new canister is installed. The CO$_2$ level could even deliberately be allowed to rise in order to ration the canister supply.

Which of these factors was more important in causing the CO$_2$ levels to rise during Apollo 13? (It's possible that both play a role, but I am looking for which one was more relevant to the Apollo 13 situation.)

Please support your answer with official NASA sources or published scientific research. Answers based solely on speculation, the movie, or other popular accounts are not acceptable.

Answer 1

From this NASA page about Apollo 13:

There were four cartridges from the LM and four from the backpacks, counting backups. However, the LM was designed to support two men for two days and was being asked to care for three men for about four days. After a day and a half in the LM, a warning light showed that the carbon dioxide had built up to a dangerous level. Mission control devised a way to attach the CM canisters to the LM system by using plastic bags, cardboard and to tape all materials carried on board.

The four cartridges of the LM were designed for two man for two days, that is 4 man-days. When used by 3 man, they should last for 4/3 = 1.33 days. The dangerous level was noticed after 1.5 days or 4.5 man-days. The needed capacity for 3 man for 4 days or 12 man-days was not available by far using LM cartridges only. 12 cartridges were needed but only 4 were available.

A rate problem should have arised much earlier.

From the Apollo 13 flight journal:

085:22:14 Lousma: Roger. Earth at 24. And it looks like you're getting up to about 15 on the CO2, so we want you to select Secondary and swap out the primary cartridge. Over.

085:22:30 Haise: Okay. I'll select Secondary and swap out the primary cartridge. [Long pause.]

085:22:53 Lousma: Okay, Fred. And when you select - When you swap out the primary cartridge, don't reselect Primary. Stay on Secondary until we use the secondary up. Over.

085:23:10 Haise: Okay. I'm changing out primary and - stay in Secondary until we use it up. Very long comm break.

This is Apollo Control at 85 hours 26 minutes. In Mission Control our displays were very quick to show the effects of the change over from the primary lithium hydroxide canister to the fresh backup canister. The partial pressure of carbon dioxide which had been reading 14.9 millimeters of mercury has dropped to 4.6 at the present time and is continuing to go down from there.

085:29:43 Haise: And the change out is complete, Jack.

085:29:50 Lousma: Okay. Copy the changeout complete, and we're reading 4.5 on the CO2 here.

So after a change of the LM cartridges to a fresh one the CO2 level dropped from 14.9 to 4.5 mm Hg. This drop would not have been possible when a rate problem existed.

4.5 mm Hg partial pressure of CO2 is equivalent to 5900 ppm (part per million) of CO2 in air at sea level. 14.9 is 19,600 ppm. The mean CO2 content of fresh outdoor air is about 400 ppm, low quality room air may have more than 1400 ppm.

The Apollo 13, Mission Report has some different information:

The LM primary cartridges (41 man-hours) and the secondary cartridges (17.9 man-hours) capacities are different, but the sum of 2 primary and 2 secondary is 117.8 man-hours or 4.9 man-days.

Air quality was improved when using four CM cartridges simultaneously to lower the CO2 content.

The CO2 partial pressure in the CM was 1.1 mm Hg one minute after the oxygen tank explosion.