John Siegenthaler often gets involved with discussions of the surface temperature of a radiant-heated floor. Issue: 1/06

I often get involved with discussions of the surface temperature of a heated floor. Perhaps it's in the context of not allowing the slab to exceed the recommended 85°F surface temperature in situations of prolonged foot contact with the floor. Other times, it's a discussion of "striping"

## Painting by Numbers

Not being much of an artist, I occasionally resort to mathematically generated scenes using a technique called finite element analysis (FEA). This is a technique that engineers who suffered through the complex theoretical mathematics of two-dimensional heat conduction can really appreciate. It approximates the solution of partial differential equations that govern such heat transfer. This is done by breaking the cross-section of the material assembly being analyzed into hundreds or even thousands of small adjacent pieces (e.g., finite elements), and calculating heat flows in and out of each one of them.

For steady state heat transfer, the heat flowing into and out of each of these elements must be equal. The temperature of each element will settle to a value that makes this possible. The balanced heat transfer is described by hundreds or thousands of simultaneous linear equations, which would be virtually impossible to solve without the number crunching power of a computer. Thirty years ago, such a situation required a mainframe computer and lots of computing time. Today, the cheapest department store computer can do these calculations in a couple of seconds. That solution is a temperature value for every element in the cross-section. Figure 1
Figure 1 shows the finite element mesh used to model a portion of a typical 6-inch thick concrete floor slab with 1-inch extruded polystyrene underside insulation and 5/8-inch PEX tubing embedded at mid-slab thickness.

Notice the adiabatic boundaries in the slab cross-section. These are boundaries across which no heat transfer is allowed due to symmetry of tubing placement and the assumption that water in the adjacent tubing is at the same temperature.

Arguably, the latter assumption is slightly flawed in that water temperature continuously decreased along the length of the circuit. However, in a system with serpentine tubing placement, it's still a reasonable approximation, and one that's necessary to develop an FEA model without resorting to 3D complexities. Figure 2
The FEA mesh is shown below the section of the slab it corresponds to. In this case, it exists from the centerline of the tubing to a line halfway to the adjacent tube. Thus, the left and right sides of the FEA mesh are adiabatic surfaces. Assuming such symmetry cuts the number of nodes required for the model in half.

The PEX tube is modeled as a multi-sided polygon that approximates the round shape in a manner that simplifies FEA mesh creation. The tube wall is shown within the blue outline in Figure 2. The number of nodes in the vicinity of the tube is higher to better approximate the higher heat flux in this region of the cross-section. Figure 3
Surface temperatures are calculated at 0.5-inch intervals. When plotted, they reveal the ability of the slab to laterally diffuse heat, as shown in Figure 3. Figure 4
The temperatures generated by the FEA model can also be displayed as isotherms (lines of constant temperature). This is shown in Figure 4, along with a composite graph of the surface temperature profile. Notice the correspondence between the peaks in the surface temperature profile and tube position. Figure 5

## The Big Picture

FEA models were created and run for a 6-inch thick concrete slab with tube spacing of 6, 12, 18, and 24 inches. For each tube spacing, the model was run for water temperatures of 80°F, 100°F, and 120°F. The surface temperature profiles based on the output of these models are shown in Figure 5. Figure 6
These graphs show how the temperature difference between the peaks and valleys increases with wider tube spacing. With tubing spaced 6 inches apart, there is very little variation in floor surface temperature, especially at low water temperatures where the variation between peak and valley temperatures is less than 1°F. At 24-inch tube spacing, the difference between peak and valley temperatures is approximately 12.5°F when the water temperature in the tubing is 120°F.

It's also evident that the area under the curves decreases as the water temperature is lowered. This area is proportional to the heat output from the surface of the slab.

Figure 6 is a composite showing the surface temperature of all four tube spacings and three water temperatures. In each case, the surface temperature profile exists from the center of the tube to a point halfway to the adjacent tube. Figure 7 