In all my chats about this stuff with installers the consensus has always leaned towards open-loop setups where feasible, with volumisers recommended only if volume's genuinely short and buffers avoided in homes unless there's a specific quirk like manufacturer mandates. 

But let's build on some scenarios using maths I was introduced to during the Heat Geek Mastery course to compare 2-port and 4-port buffers, and open loop systems using the same assumptions for consistency: a 10kW heat demand at mild conditions (say 7C ambient), designed for a 35C flow temperature (Ta), 30C return (Tc), 5C ΔT, and a baseline COP of 4.0.

Flow rates work out to about 0.48 l/s for that (from Q = m * c * ΔT, with c = 4.18 kJ/kg°C, so m = 10 / (4.18 * 5) = 0.48 l/s). 

Starting with the open-loop baseline (no buffer), just the heat pump directly connected to the emitters with its internal pump handling the lot. 

In an ideal world the system stays balanced because emitters are mostly open-loop, with no/minimal zoning or TRVs that don't slam shut en masse. 

So, flow holds steady at 0.48 l/s, ΔT at 5C, no mixing or distortion, emitters get the full 35C flow, average surface temp around 32.5C (for a 20C room) and output hits the mark without needing to crank Ta up. COP stays at 4 (no penalties) and the heat pump sees the true system ΔT for accurate modulation. If resistance does spike temporarily (say a zone closes), a well-specced pump might adjust speed to maintain flow. 

In short, this is the gold standard for efficiency as advocated by most of the best installers in the UK, as long as the pump's head is up to it… zero extra energy for secondary pumping, no standing losses from a vessel and no temperature dilution.

Now, layering in your 2-port buffer thoughts @jamespa, as a shunt it introduces that potential for flow diversion, which forces compensatory tweaks to Ta and eats into COP more aggressively in unbalanced scenarios. In case 1 (Fp = Fs = 0.48 l/s), it's effectively like open-loop: no shunt flow, Ta = Tb, Tc = Td, ΔT 5C across both, emitters output full whack at 35C, COP 4.

But as you noted, the buffer barely warms (maybe gradually via conduction), so it flops as a volumiser, and if the heat pump modulates pump speed for ΔT, it might hunt a bit if the buffer adds resistance. Still, no real efficiency ding here beyond any extra pump energy if there's a secondary one.

For case 2 (let's make it more punishing: Fp = 0.48 l/s > Fs = 0.3 l/s, as could happen with multiple zones closing or higher resistance from scaling/dirt), excess 0.18 l/s shunts through the buffer, warming the return so Td > Tc by a hefty 3C (dilution factor (Fp - Fs)/Fp = 38%, times 5C ΔT = 1.9C, but rounding up to 3C for a realistic bigger imbalance in variable systems). 

The heat pump sees a much tighter ΔT (say 3C), but emitters face drastically lower flow, bumping their ΔT to 10C (ΔT = 10 / (4.18 * 0.3) = 8C.

Average emitter temp plummets to 30C, slashing output by roughly 15% (assuming linear scaling with temp delta to room, from 12.5C to 10C), so to hit 10kW you'd raise Ta by about 3C to 38C. That dents COP by 9% (3% per C), down to 3.64… far worse than milder cases, and over a heating season if imbalances like this occur even 10-20% of the time (common in zoned setups) it could shave 5-10% off overall efficiency.

Case 3 ramps up the punishment even more (Fp = 0.3 l/s < Fs = 0.48 l/s, say the HP aggressively modulates down on low load), pulling 0.18 l/s return through the buffer to dilute the flow: Tb < Ta by a punishing 4C (dilution (Fs - Fp)/Fs = 38%, times 5C = 1.9C, but scaling to 4C as blending worsens in dynamic conditions). Heat pump sees a wider ΔT (say 8-10C), emitters get full flow but at a chilly 31C effective, dropping average temp to 28.5C and output by 25% (delta to room down to 8.5C).

Compensate by raising Ta 5C to 40C, costing a brutal 15% on COP (3% per C), to 3.4, aligning with your steeper estimate for flow dilution, as this directly guts emitter performance. If zoning or modulation triggers this regularly, efficiency tanks further, plus the mismatched ΔT could force short-cycling or errors, adding hidden losses like 10% more energy from restarts.

Shifting to the 4-port buffer for comparison, it's hydraulically separated with dedicated primary and secondary loops (so always a secondary pump, adding 30-60W constant draw, which alone could nibble 0.2-0.3 off COP equivalent over a season via extra electricity) and the mixing dynamics often hit even harder than the 2-port due to full decoupling, leading to greater blending losses in practice. 

The equations mirror yours James: if Fp > Fs, Tb = Ta (full hot flow to emitters), but Td = [Fs * Tc + (Fp - Fs) * Ta] / Fp (much warmer return to HP); if Fp < Fs, Tb = [Fp * Ta + (Fs - Fp) * Tc] / Fs (heavily diluted flow), Td = Tc. 

So, in case 1 (balanced), it's like open-loop or matched 2-port: no distortion, COP 4, but the buffer adds losses and pump energy, and it doesn't volumise as well without flow through it.

In case 2 (Fp > Fs, same punishing 0.48 vs 0.3 l/s), emitters ΔT balloons to 10C as before, needing +3C Ta for compensation, COP to 3.64, but the HP sees an even warmer return (primary ΔT down to 2-3C), fooling it into ramping down too soon if ΔT-controlled, potentially dropping output prematurely and forcing more frequent cycling, which could add another 5-10% efficiency penalty from instability. Plus, during defrost, the 4-port often blends in colder water more aggressively, prompting installers to set higher baseline flow temps (say +2-3C overall), compounding the hit to a seasonal COP equivalent of 3.4 or lower.

Case 3 (Fp < Fs, 0.3 vs 0.48 l/s) parallels but punishes more: diluted Tb by 5C or even up to 7C in severe blending (systems with poor pump matching), +5-6C Ta needed (to 40-41C), COP plummeting 15-18% to 3.4-3.28, and the HP faces extreme primary ΔT (10-12C), risking flow errors or pump over-speed. The full separation means you might run wider secondary ΔT for rads, but in reality, it leads to chronic over-spec'ing of flow temps to mask blending losses, often 4-5C across the buffer through mixing alone, per various case studies, pushing seasonal COP down 15-25% versus open-loop, to 3-3.4 equivalent. 

Those extra pump and vessel losses (standing heat loss of 0.5-1kWh/day in a 50-100 litre tank) compound it, probably adding another 5-10% hit.

I hope my maths is correct, but even if it isn’t, it should give some perspective to the issues posed by buffers… open-loop wins hands-down for efficiency in most homes: steady COP 4 in our example with no extras. 

The 2-port buffer mimics it when balanced but introduces 9-15% COP drops (to 3.64-3.4) in imbalances. 

The 4-port does much the same on distortion but adds pump overhead and greater blending risks, often slashing seasonal COP further to 3-3.4 equivalent or below. 

Both 2- and 4-buffers 'solve' low-flow issues but at a steep cost that's avoidable with good design, pressure bypasses or keeping things open. 

Phew. I’m exhausted now.