I'll assume that each component in the speaker is running in parallel to the other. If that is right, then (to keep things simple) can I say that each component has a resistance of 16 ohms so that when I run the components in parallel I have a speaker that presents an 8 ohm load?No, the low and high frequency sections of the speaker are not in parallel, due to the presence of the crossover network in the speaker.
If all of this is right, then by bi-amping, I've effectively made it so that each channel of my amp is driving a 16 ohm load. Again, if this is true, then I've taken my 80 wpc amp and turned it into something around a 50 or 60 wpc amp, due to the greater load.
The amplifier section that is driving the speaker's woofer will NOMINALLY see an 8 ohm load for the very low frequency components of the signal, and a very high impedance for the very high frequency components of the signal. Therefore that amplifier section will have to supply little or no current and power at very high frequencies.
At frequencies in the crossover region, which for the 685 is specified as being centered at 4 kHz but is probably fairly broad, the impedance seen by that amplifier section will gradually transition between the nominal 8 ohm value and being very high.
The opposite situation will exist for the amplifier section driving the tweeter, i.e., it will have to supply little or no current and power at low frequencies, and it will see a nominally 8 ohm load at high frequencies, supplying whatever amount of current and power the signal calls for at high frequencies.
The actual values of the 8 ohm nominal loads will vary depending on how the speaker's impedance varies as a function of frequency. I note that the impedance of the 685 is specified as 8 ohms nominal, 3.7 ohms minimum. I would suspect that the 3.7 ohms occurs at frequencies that are handled by the woofer, which are also the frequencies that almost always require the most power.
I don't know why you might be finding that you have to turn the volume control to higher settings in the biamp configuration, unless the gain of the amplifier somehow changes when it is set up for four channel operation.