Sectional radiator: flow extender (pseudo-diagonal connection)

The radiator is solved as a network of pipes.

• Nodes: the top header at each section, the bottom header at each section; with a flow extender also the tube at each section it passes, the tube's open end and the outlet.
• Pieces between them: the vertical channel of each section; a header piece of one section width with its nipple; with an extender, the ring between the header (and nipple) and the tube, the tube pieces and the holes.

Each piece loses $\Delta p = \max(\Delta p_{lam}, \Delta p_{turb}) + K\,\rho v^2/2$: laminar friction $C\mu L v/(2 d_h^2)$ with $C = 64$ for a round bore and 96 for the ring around the tube, turbulent friction by Blasius $f = 0.316\,Re^{-1/4}$, and local losses (nipple 0.5 at its own velocity, vertical channel entry and exit 1.5, hole 2.7, tube open end 0.5).

A vertical channel also gets the gravity head of its water, $g H (\bar\rho - \rho_s)$: water that cools in a section is heavier than the supply and pulls flow down that section.

Water enters every section at the supply temperature (the top header carries supply water). A section gives off heat by the EN 442 law scaled to one section, $\Phi = \Phi_n (\Delta T_{ln}/\Delta T_n)^n$ with $\Delta T_{ln} = (\theta_{in} - \theta_{out})/\ln(\theta_{in}/\theta_{out})$, together with $\dot m c (\theta_{in} - \theta_{out}) = \Phi$; its mean water temperature is $T_a + \Delta T_{ln}$.

The node pressures are found by Newton–Picard iterations (secant resistances, the gravity head linearised with its slope) until no flow changes by more than 10⁻⁷ of the total. The same radiator connected on one side without an extender and diagonally is solved alongside for comparison, and the extender again for holes of 0–8 mm.

• The network equations, Blasius friction and the per-section EN 442 law are combined here; the calc is not taken from one publication. Loss coefficients are textbook values for sharp-edged contractions, entries and orifices, not measured on radiators.
• Headers give no heat of their own: their share is counted in the sections, and every section's water enters at the supply temperature. Heat exchange between the water in the tube and in the ring around it is neglected.
• The water in a vertical channel is taken as mixed across it; layering inside headers and channels is not modelled. A section whose flow turns upward is counted as cold.
• Water properties at the mean of supply and room temperature; density fitted to IAPWS values within 0.2 kg/m³ (20–100 °C), viscosity by the Vogel equation.
• Manufacturers rarely publish the bores of headers, nipples and channels: they are inputs. The water volume of one section computed from them can be compared with the datasheet.

• Supply at the top, return at the bottom (same side, diagonal, or same side with an extender in the bottom header). A supply connected at the bottom (supply and return swapped) is not modelled: there the gravity head works against the flow, the first one or two sections take the hot water and the rest stay cool (Mar'ina et al. 2022 measured 50.8, 41.5 and 39.7 °C).
• With the gravity head counted, the model evens out a same-side, top-to-bottom radiator at any flow — even a tenth of that head is enough. A measured 10-section radiator at 127–220 kg/h agrees (2–3 °C between sections). Yet radiators connected this way are reported to stay cold at the far end at low flow (Valtec names low flow as the usual cause), also with the air bled and the supply at the top. The likely cause is layering: hot water runs along the top header and cools, cold water stays in the lower part of the far sections. A network of pipes with mixed water in each passage cannot show this. Read the two settings of the gravity switch as bounds: on is the best case, off is the forced-flow case without any help from gravity.
• For the same reason the model can understate what an extender gains. Owners report a plain tube making such a radiator clearly warmer, while at the same flow neither bound here gives more than a diagonal connection (a few per cent above same side). To compare a real radiator, measure the supply and return pipe temperatures and set the flow so that the return temperature here matches.
• Steady operation, no air in the headers, clean channels. If an extender does not help, the channels are likely blocked (Valtec).
• 3–30 sections, 5–1500 kg/h. Differences under about 1 K between sections are within the model's uncertainty.
• The flow through the radiator is an input. On a one-pipe riser with a bypass, a higher pressure loss in the radiator (an extender, small holes) sends less water through it.
• The drawing is a scheme: headers and tube are drawn thicker than to scale, the water temperature over each section's height follows its heat balance, the dots move at the water's speed in each passage times the speed-up.