I'm working on a setting with merfolk that I'd prefer to be fully mammalian. However, they have underwater cities and I find that hard to believe for a species that needs to surface every few hours at most to breathe. So, I was wondering if pharyngeal and cutaneous respiration similar to what softshell turtles are capable of would be plausible for a mammalian species. Would this method of breathing capable of providing the amount of oxygen necessary to maintain a warm-blooded metabolism? Are there any other concerns that would make this unrealistic for a mammal?
Sadly, this is not possible. Warm-blooded animals such as mammals simply use too much oxygen to maintain their internal body temperature for cutaneous breathing to be effective.
Let's get some numbers for that. The human body uses about 8mg of oxygen per second. In the ocean, oxygen concentrations vary between 0 and 10mg oxygen per liter. So in the best of circumstances, we're flowing 1 liter of oxygen over the merperson body per second, which doesn't actually sound that implausible. However, the merperson skin isn't actually going to be able to absorb all that oxygen- it's limited by the rate of diffusion.
According to this paper, the human body can take up about 1mL of oxygen per minute, based on both their measured assessment and the theoretical model that required an algebraic static solution of Fick's second law. I've grabbed the relevant section since the article may be behind a paywall for some users:
Esentially, we're facing a difference of two orders of magnitude between what we need (8mg/sec) and what is possible to obtain (0.021mg/sec).
However, there are some other ideas out there that might help you get around this limit- I'll link to my answer here, a different question here, and one more here, just in case you haven't seen them already.
Sadly, it's physically impossible to build an endotherm that breaths water nomatter how hard you try.
In best case scenario, oxygen saturation of water reaches about 10 milligrams per liter. Consider the oxidation of beryllium metal in oxygen, one of the most efficient heat generation method possible with a standard enthalpy of formation of -599 kJ/mol.
As the product beryllium oxide has a formula BeO, with a molecular mass of 25, and contains a ratio of Be to O of 9 to 16, one liter of water can only form about 15.625 milligrams of beryllium oxide if the oxygen saturation is fully consumed. 0.015625/25*599000= 374.375 J per litre of water consumed.
At a heat capacity of 4.2kJ/kg*C, 374.375 J of energy is only enough to raise the temperatire of that litre of water for 0.09 degrees C. As a matter of fact, such a low temperature difference is impossible to capture even with a recuperating heat exchanger, which maxes out at 95 percent efficiency, or a retained temperature difference of 1.78 degrees C above the highest ocean temperature recorded of 31 degrees C= 32.78 degrees C, or a core temperature of about 33 degrees C.
Below 35 degrees C, hypothermia sets in, below 33degrees C, the heart stops. Below 30 degrees C, a mammal will die of hypothermia. Mean ocean temp near the tropics is about 28 degrees C. So, nomatter how large your merfolk is, the very water they breath will prevent their body temperature from reaching high enough to keep their hearts beating!
This is why ALL marine endotherms breaths air-- not because they had lungs instead off gills, but because the oxygen content of seawater alone is not enough to maintain even the bare minimum of core body temperature required for an endotherm to function!
Fish and ectotherms uses gills because their physiology does not require heat generation nor require a high temperature. The energy from dissolved oxygen is enough for most biological activity IF maintaining body heat is not necessary.
In DC comics, the atlanteans get away with this by using magic to supply most of their physical energy requirement, which may not even be endothermic at all. Without magic or another power source that does not rely on chemical reaction with oxygen gas, water breathing endotherm are physically impossible unless you count stable body temperatures atmost half a degree above the average water temperature as warm-blooded.
One cheating, one based on reactions that does not require Di-Oxygen, that is, an An-aerobic reaction.
The Cheating method: have your merfolk live in extra hot water that approaches the average human body temperature, or at least above which a mammalian heart can beat with minimal extra heating. I.e. have your merfolk live in geothermal areas like the saltwater kingdom in DC aquaman. The extra heat requirement for a mammalian physiology is supplied by an external source, or like in the movie, literally bathing in molten lava.
The Non-Cheating method: use an alternative oxidizer for your merfolk's respiratory process, preferably something that can be
(a. Stored compactly in physiological conditions, preferably at least as dense as glycogen in terms of oxidative energy density.
(b. Can be resupplied with relative ease, preferably by ingestion, or is readily available at concentrations high enough to support endothermic metabolism within the environment your merfolk lives in.
Three natural pathways, Four if you count organic materials as valible electron receptors, exists at power and energy densities high enough to potentially support endothermic metabolism:
Denitrification, or Dissimilatory nitrate reduction (to ammonia) This is the reaction employed by anaerobic soil bacteria to produce energy from nitrates, a product of ammonia oxidation or nitrification within aerobic top layers of the soil. This process uses dissolved nitrates as the electron acceptor, with a saturation point of near 300 grams per liter of water, the maximum energy density of this reaction is enough to boil water over for at least 3 times for its stored mass of oxidizer. Nitrates could be collected from the environment, preferably during non-endothermic "hibernation" periods, produced from nitrogenous compounds during bidaily to weekly (if up to 3 kilograms of nitrates is made during one surfacing) surfacing to collect atmospheric oxygen from air, or ingested as part of your merfolk's diet. The latter may well be because of anoxygenic photosynthesis of some of your worlds plants, like purple/green non-sulfur bacteria.
Hydrogenosome and hydrogenesis This is the prevalent way of disposing of unwanted electrons in archaea and certain cyanobacteria. In certain small (euxinophilic)animals, devolved, mitochondria derived hydrogenosomes uses protons themselves as a terminal electron acceptor, producing hydrogen gas as a byproduct. The reaction is inefficient, in terms that a one pair of electrons can at best pump six protons (usually just four), but as this reaction uses water as oxidizer, the energy density per kilogram of water consumed is almost the same as air breathing organisms. As the product of this reaction is hydrogen gas, this also causes your merfolk to exhale bubbles, but not inhale anything.
Methanogenesis Methanogenesis is the last option for anaerobic respiration that does not involve the exhaustion of organic compounds as byproducts. Methane is produced by obligate anaerobic archaea that uses carbon dioxide as the terminal electron acceptor. This reaction pumps 6 protons per pair of electrons in Methanobacilus, 4 to 5 in Methanosarcina. Methanogenesis produces marginally more energy than Hydrogenesis, and it uses the byproduct of respiration itself as the oxidizer. The energy density of methanogenesis is often high enough to keep a pile of compost at self limiting temperature for two to two and a half days.
Fermentative processes Contrast to popular beliefs, very Deep forms of fermentation are very efficient in terms of energy production per substrate molecule consumed. For example, the process of acetone butanol ethanol fermentation produces at maximum 5.25 ATP per glucose consumed, and the oxaloacetate/malate disproportionation reactions of roundworms makes 6.1 ATP per glucose consumed. these are one of the reasons why compost piles can spontaneously catch fire. Of course, your merfolk will not catch fire because they are underwater, but the heat generation is more than enough to keep your endothermic merfolk warm.
Drawbacks of anaerobic respiration: Although anaerobic respiration conserves body heat by nearly eliminating the necessary diffusive(and therefore thermally contacting) water flow and the resulting heat exchange with your merfolks's core organs, they are relatively inefficient when compared to oxidation by dioxygen gas in term of organic carbon consumed. However, as the limiting factor is heat exchange not food(as this is not earth), having a race of merfolk that are exceptionally voracious eaters would likely make good story details, or at least prevent obesity, as most marine mammals had exceptionally thick blubbers......
Let's start with what we consume in O2. The average human consumes about 25% of the oxygen in the air with every breath, and that means that the average human is going to consume about 550 litres of O2 in a given day.
A Merman is going to use more because water is cold  meaning that he or she needs more energy to keep internal body temps regulated. Let's say double, to be sure. That means 1.1 KL/day
First question is whether or not an ocean can support that kind of O2 saturation level. Assuming an earth like world, the answer is probably, but only to a certain depth. This is because the O2 saturation in the water is caused by photosynthesis in the sub-surface plant life (microscopic and macroscopic) and the sunlight that drives this process can only reach down so far. Bottom line is that regardless of breathing method, your merfolk are likely to suffocate at extreme depths. This isn't that big a deal, because conventional mammals can't regulate their internal pressure so you want to keep within a certain range of the surface for practicality, not to mention just seeing where you're going.
For now, let's assume that there's enough plant life in the ocean to support your O2 levels.
The thing about mammals is that they all have lungs. This is one of the primary limitations to our capacity to dive to great depths currently. That said, is it possible that sustained natural selection could evolve a mammal that uses (say) gills instead?
Well, I'd have to say yes, but I don't know what underwater breathing would actually look like in a mammal that has gone back to the oceans. All I can really say is that if fish like animals could adapt to land by evolving lungs and limbs, then it's entirely feasible that in time, mammals could evolve in a manner that allowed them to breathe underwater. The eventual removal of lungs (or their atrophy-cation like the appendix) would also remove some of the natural impediments from a species being able to take full advantage of underwater life by removing the primary impediment to pressure variations; the lungs.