Towel Warmer Moved Away from the Riser: Will It Heat?

The loop

The warmer and its two lines form a loop in parallel with the riser between its two outlets. The calculation walks around the loop in the flow direction and balances pressures at a loop flow $G$:

Gravity head. $\Delta p_g = g \oint \rho \, (-dz)$ around the loop, element by element (each piece of line, the warmer, the riser between the outlets) with the density of the water in it. Equivalently, water cooled at a height $z$ drives the loop by $g\,\Delta\rho\,(z - z_{mix})$, where $z_{mix}$ is the outlet where the cooled water rejoins the riser: cooling above it drives the flow, cooling below it (in a dip that is not insulated) holds it back. The warmer cools its water at the middle of its heated height.

Push of the riser. The riser flow between the outlets loses pressure to friction and the run of the two tees ($\zeta = 1$). A downward riser flow raises the pressure at the upper outlet and helps the loop; an upward flow pushes the other way.

Losses. Each line piece: $\lambda = 64/Re$ (laminar) or Blasius; fittings: $(K_\infty + K_1/Re)\,\rho v^2/2$ with $K_\infty$ of PP-R fittings (90° bend 1.2, 45° 0.5, dividing tee 1.2, combining tee 0.8, adapter 0.25), ball valves 0.3, warmer entry and exit 1.5, and the low-flow term $K_1$ of the two-K method (bend 800, 45° 500, valve 300, tee branch 500).

Temperatures. A line cools as $T = T_a + (T_0 - T_a)\,e^{-U' L/(G c)}$. The warmer follows the EN 442 law on the log-mean difference, $Q = Q_{50}\,(\Delta T_{ln}/50)^n$. Where the riser carries the returned water back to the inlet (an upward riser), the inlet water is the mix, solved together with the loop.

Every steady state. $F(G)$ is scanned over both directions; each place where it falls through zero is a stable state. The hottest is shown; a second one (usually a standstill) is reported.

Reference. The same warmer hung at the riser: 0.3 m stubs plus the vertical runs, two bends per line, no dip.

• One-dimensional, steady flow; the equations are derived here from the energy and momentum balance, not taken from one source.
• The warmer cools its water at the middle of its heated height; its internal resistance is its entry and exit (ζ 1.5 in all).
• Fitting coefficients: PP-R values of a manufacturer's design guide (FD-Plast); low-flow terms from the two-K method (Hooper, 1981), meant for threaded fittings; ball valve ζ 0.3 from typical Kv of DN15 valves. At Re of a few hundred they are the least certain part.
• Heat loss of the lines per metre and kelvin is estimated here for each insulation choice (0.2, 0.3, 0.5, 0.8 W/(m·K)), not measured.
• Water density from Kell (1975); viscosity from a Vogel-type fit (within 1 % of IAPWS values at 20–80 °C).
• The riser water between the outlets keeps the riser temperature unless the riser carries returned water back to the inlet.

• No published measurement of a towel warmer on a riser was found to compare with; the result is an engineering estimate. Measure the riser and the warmer's inlet and outlet to calibrate it.
• Below about 2 kg/h the water in long horizontal lines may layer (hot on top, cold below) and creep both ways in one pipe, which a one-dimensional model cannot show.
• Air in the lines, sludge, half-closed valves and a riser that stops circulating at night are not modelled; any of them can leave the warmer cold.
• Which steady state the loop takes when there are two depends on its history (for example a start after the hot water was off).
• Installer rules of thumb (natural circulation up to 4–5 m from the riser, slope 3–30 mm/m, pipes DN 20 or more) are not norms; this tool calculates the actual layout instead.
• Moving a towel warmer fed from a shared riser may need approval (in Russia it is a re-arrangement of the flat under the Housing Code, art. 25–26).