Basement bathroom remodel; tub insulation, heating/cooling?

We have a full bathroom in our basement which we are planning to remodel. New tub, new tile (floor and tub surround), new faucets, etc.

The room is mostly below grade, but there is a small window just above grade near the top of the tub wall. The long side of the tub is directly agains the foundation wall. Because of the layout, the new tub will have to go in the same spot. There is a bathroom vent fan, but no ducting to heat or cool the room. Adding ductwork is not an option, according to a contractor friend. We would have to cut through several floor joists. I'm not sure that's an issue though (not having the ducting). It rarely gets hot since it is in the corner of the basement, but it gets ridiculously cold in the winter. We've been using an electric space heater, but I would like to install a radiant floor under tile. Would that be enough to heat the whole room?

Also, since one side of the tub is against the foundation wall, and the bottom of the tub is on the concrete slab, I'm wondering if/how we should insulate it? Should we put something under the tub to insulate? How would we do that and make sure it stays dry? Or should we put the tub over tile with radiant heat?

What about the tub wall? There is standard wood framing that could be filled with insulation. What's the best type of insulation? What type of drywall should be installed? We want to tile the walls surrounding the tub.

Obviously I'm a complete novice, and literally need advice on the project from the ground up!

If your slab wasn't installed with a moisture barrier and insulation, a lot of the heat you apply in the floor will go towards heating the earth and any moisture down there. You can specify various amounts of heat/sq ft you want in a mat (within limits), or if using cables, you can wind them closer together which will increse the amont of heat available. Usually, the tile is just warmed to feel more comfortable - there is a limit how much heat you can source before the tile ends up being too warm and becomes dangerous both for you and for the floor. So, you'd need some analysis of the actual heat load of the room, compare that with the max density of radiant heat allowed, to then see whether that would be enough.

Don't know whether this would work for you or not, but maybe a high velocity branch duct might allow you to go through the joists with a small enough hole. These use a high velocity fan, small, insulated ducts, and small diffuser heads in the room.

They also make radiant ceiling panels which wouldn't have as much of a heat loss to the earth. You can get a heated towel rack, too, but that doesn't produce all that much heat (unless it is hydronic, then it can source a lot more).

On the tub, are you considering a tiled enclosure, or a 3-4 piece manufacturered wall enclosure? If you decided to tile it, you could use something like KerdiBoard or Wediboard, which are tileable foam panels. They come in various thicknesses (up to 2" thick). If you can build the wall out a little to allow for more insulation on that outside wall, things will be nicer. I don't think I'd want infloor radiant underneath the tub (well, I would, but check with the manufacturer to see if it would be an allowed installation). Depending on the tub, some come with an applied layer of foam on the underside, designed to set directly on the floor. Those would help it from conducting heat to the slab. Stuffing insulation around the sides can help.

A common max on radiant floor heating is in the order of 12W/sqFt, or about 41BTU/hour. Some will allow more, some less, but that gives you an idea of the max you might be able to put into the room from below. There is a limit on how hot you can make the floor, or, you could provide lots more.

What is the ceiling height in your proposed bathroom?

Raising the floor a few inches is what will enable you to insulate effectively.

If you do not mind the operating costs, there are small wall mounted individual room heating/cooling units.