The tubing goes in the slab.
Any thermal resistant material between you and the tubing decreases system efficiency (hotter water, more pumping). As concrete has virtually no thermal resistance it is nearly the perfect radiator.
The carpet and pad should be considered by your radiant floor heating designer, who will also be concerned about floor coverings over Warmboard® or any radiant panel. Carpet is used over radiant floor all the time but will decrease operating efficiency and is never recommended in basements. Special care must be taken when carpeting a basement floor, including insulation and an adequate vapor barrier. |