Physics of the Fine wire heat exchanger

We will treat the physics of heat transfer to a fine wire, find the optimal wire diameter (100 μ) at which wire cost/performance is minimal, and come at a heat transfer from air to wire cloth as a function of air velocity. Then we will treat the pressure drop through wire cloth, and find the optimal air speed (0.4 m/s) at wich the sum of the exergy loss of pumping and heat transfer is minimal. With these values we find the optimal temperature drop (2.4°C) over the heat exchanger by minimizing the yearly cost of the exchanger due to investment, and the cost of exergy loss due to heat transfer

Theory of heat transfer

Heat transfer around a fine wire

A widely used correlation for heat transfer from a cylinder in a perpendicular flow is: Nu = 0.57 · Re^0.5 · Pr^0.33 and α = Nu · l/d. When we fill in η = 1.85e-5 Pa ·s; ρ = 1.3 kg/m³, Cp=1010 J/kgK and λ = 0.025 W/mK for the material constants of air at room temperature, this correlation becomes α = 3.2 ·(v/d)^0.5. We see that the heat transfer coefficient is inversely proportional to the square root of the wire diameter, which is the reason for the development of fine wire heat exchangers after all. With an air velocity v of 0.5 m/s and a wire of 100 m, we have α =226 W/m²K, which is around ten times the typical value of flat plate heat exchangers to air.

Heat transfer through a wire cloth

Fine wires can only be efficiently incorporated into a device using textile technology, such as weaving, and in the case of heat transfer from water to air, we have to weave copper capillaries into copper fine wires. This leads naturally to a cloth where at the capillaries, the wires have a spacing equal to their diameter, and at the wire crossing in the mid point between the capillaries, a zero spacing between the capillaries. This cloth, transferring heat to the air streaming through it, can be represented as two rows of cylinders φd, spaced 2 ·d, in series, perpendicular to the flow. Per square meter there are then 1/d wires, each with surface π ·d m², so that the heat transfer coefficient related to the cloth surface is α = π ·3.2 ·(v/d)0.5 = 10 ·(v/d)^0.5 W/m²K

Optimal wire diameter

The kilogram price of copper fine wire increases with decreasing wire diameter, because of the wire drawing cost. If Pc is the copper wire price per kg, and the density of copper is 8900 kg/m³, then per square meter cloth there is 1/d · π/4 ·d² ·8900 ·Pc or 7e3 ·d ·Pc nlg copper. Per unit of heat transfer coefficient, 10 ·(v/d)^0.5 W/m²K, there is 700 ·d^1.5 ·v^-0.5 ·Pe nlg copper. So we have the economically optimal wire diameter, when the product Pe ·d^1.5 is minimal. This is the case with a wire diameter of 100μm. With this optimal wire diameter, the calculated heat transfer coefficient related to the cloth surface is 1000 · v^0.5 W/m²K.

Theory of pressure drop

Pressure drop through wire bundles

We find the following correlation for pressure drop for flow perpendicular to pipe bundles for Re < 25 · x/(x-1), where x is the ratio of pipe spacing to pipe diameter, v means the   mean fluid velocity over the bundle front surface, l the pipe bundle length in the flow direction and δ the pipe diameter. In our case, we have two bundles in series with x=2 and with l/ δ = 1 at a distance of more than 2 · δ, and δ = d = 1e-4 m, so we can write here for the pressure drop: Δp = 2 · 28.4/ π · η /ρ/v/d ·1 · ½ · ρv² or Δp= 6.6 · v, for Re<25 or v<3.8 m/s.

Optimal air velocity

By blowing air through the cloth we have to use exergy, or electric energy. When the ceiling ventilator motor has a constant efficiency of 6W_mech / 45 W_el, and the total pressure drop, inclusive the acceleration term, through the fan Δp=6.6 ·v+½ρv², the surface of the heat exchanger is A, then we need 45/6 ·Δp ·A electric power to move the air. By moving the air faster, we increase the heat transfer, and so save exergy by lowering the temperature drop through which the heat flows. We can express this exergy by estimating the electric energy we need to pump heat from outside air at a mean temperature over the Dutch heating season of 4.8°C, to heat inside air at a mean temperature of 17.3°C. For each increase of the ΔT of one of the heat exchangers by 1°C, we need an extra Φ_w /(17.3-4.8)/ 8 amount of electric power when the COP of the heat pump is 8. When we use the heat transfer and pressure drop equations in these functions for a heat exchanger of Φ_w =1kW, the sum of the exergy losses is: 45/6 ·A ·(0.5 ·v²+6.6 ·v) + (1kW)² /8/12.5/ 1000 ·v^0.5 ·A or  A ·(3.75 ·v² + 49.5 ·v + 10 ·v^-0.5). The value of  v  at which this function has its minimum value is 0.21 m/s. In practice, the efficiency of a standard one-phase "außenlaufer" fan motor  increases with its output, and this increases the optimum air velocity to about v = 0.4 m/s in a typical case. This low optimal air speed is the reason that the heat exchanger surface should be plied in a zig-zag fashion to increase the frontal air speed to a more acceptable value of about 2 m/s, in order to keep the apparatus in a compact form.

Optimal temperature drop

When we take v to be 0.4 m/s, our cloth surface at 1 kW becomes A = 1e3 / 1000.0.4^0.5/ΔT = 1.6 / ΔT m²(nlg), or when we write the cloth cost of  Pa off in 7 years, 1.6/7 ·Pa/ΔT  nlg/y. The  exergy loss is 1kW ·ΔT/12.5/COP = 10 ·ΔT (W), this power during the heating season of 5080 hours, 10 ·ΔT ·5080 = ΔT ·50.8 kWh/y, or with a kWh price at 0.2 nlg, 10.2 ·ΔT nlg/y. The sum of capital cost and exergy loss cost is  0.23 ·Pa/ΔT + 10.2 ·ΔT. With Pa = 300 nlg/m² cloth, mounted in a heat exchanger, this function is minimal with Pa = 300 nlg/m², at T = 2.6°C, and with Pa = 200 nlg/m², at ΔT = 2.2°C. The cost price is one of the last parameters that becomes known, but a value for the optimal ΔT = 2.5 °C can be taken. This in sharp contrast with usual tube-plate-fin heat exchangers, that have optimal temperature drops of 15..20 °C.

Design considerations

Home heating systems are mass products, and shuld be preferably constructed from mass produced components. One such component is the ceiling fan motor. A ceiling fan is a noiseless, slow (4 turns/second) turning axial fan with a fan span of about 1 meter, a 240 V, one-phase, 50 Hz, 45 W input motor with central 10 pole pair windings and a peripheral short circuit iron core. Its maximum mechanical output is about 14 Watt at 280 rpm.

Measurements on fine wire heat exchangers

Marian Vlot made her masters thesis TUDelft, 2003, by measuring the friction and the heat transfer from water to air on very carefully dimensioned stacks of Fine Wire cloths, woven from 1.1mm/1.7 mm copper capillaries and 0.1 mm copper wires in a small air tunnel.