I'm trying to make a cooling device to used in hot areas for cooling down newborns. I'm planning to cool down the mattress to a temperature of 37C, how effective will such a system be to keep the body core temp to 37C? I understand that the baby can not be analysed as a lumped body. Thanks!
I've had a waterbed since 1981. It is very effective at both cooling and heating one's body. At 37C, a mattress would make you feel like you're being cooked. At 27C, you'll feel like you're freezing.
I keep mine at around 30C, a little more in winter, a little less in summer, but either way, not much different. The thermostat might not be accurate, so don't rely on those numbers. The important point is that there is a very small Goldilocks zone.
It also helps to have a mattress pad, to provide some insulation to slow down the rate of heat transfer. As I learned when I first got the bed, without it, the temperature has to be very precisely set, and even a slight variation will be noticeable, while anything more will be uncomfortable.
You might start with a comfortable swimming pool temperature and make it a few degrees warmer since heat generation from muscular activity won't be happening.
Answering just the engineering side of the question, not the medical.
Skin temperature is usually 3 to 6 degrees lower than the body's core, more like 32-33 centigrade, and not uniform over all areas of the body.
A cooled mattress has to be frequently replaced after it warms up, or be connected to a cooling system, like a water bed with cold water circulating in it as CableStay's comment mentions, and has to have a thermometer to stabilise its temperature.
Questions of safety and hygiene and maintenance make the design more involved.