Pool boiling experiment

For the first diag below, it's unclear how accurate the clamp meter is for DC. It's going to have a large error (possibly - I'm just guessing). You could try measuring a known constant value for a while, see if it drifts. 

For the second diagram below, perhaps the resistance of the wires between the alligator clamp, and the heating element, is too large. Can you measure the voltage directly across the heating element?From the photo too, it looks like you're not directly connecting the multimeter wires on the copper block, but on a length of wire (looks like maybe 25-30cm of wire at a guess on each end, which is a total of maybe 0.5 milliohm).

I can actually make the connections direct to the heating element for now (while using this open tank), but we need to be able to get those measurements through some copper wires in the future because we want to implement this under higher pressures. In summary, we will need to get the measurements using this setup in the future anyway, due to restrictions in our pressure vessel penetrations.

I am not sure what resistance x temperature is, although you can compute a temperature coefficient of resistance (R1 - R2 / T1- T2).

Stainless steel has a positive temperature coefficient of resistance - when it gets hotter its resistance increases.

I assume you are calculating the resistance by measuring voltage and current (R = V / I) and also the power (P = V x I)

To calculate resistance you need to measure voltage only across the material whose resistance you want calculate - there should be no other current carrying wire or shunt in this voltage measurement and the meter should not be grounded. To calculate current you measure voltage across a shunt that is carrying all the current. To calculate power, measure voltage across the load and current though the load.

There would seem to be some little mistake in how / where you are measuring voltage and calculating current and resistance.

If you want to reduce the effect of the copper wire, use heavier gage wire.

I tried the last suggestion to measure the voltage across the heating element (30 AWG 316L SS wire) only. At the same time, we took voltage measurements on three other points in the system: (1) between the two terminals of the power supply (positive and negative), (2) in the positive side only (between the positive terminal and the positive side of the heating element), and (3) in the negative side only (between the negative terminal and the negative side of the heating element.

Then, the total voltage across the power supply (measurement 1) should be equal to the sum of the other three measurements. Based on that we observed the following:

(1) The voltage measurements across the terminals (total) and across the positive side showed a stable behavior (measurements do not vary much).

(2) The voltage measurements across the negative side and across the heating element showed an unstable behavior (measurements vary considerably).

(3) The negative side copper block in the test section was generating a big amount of bubbles compared with the positive side copper block. That means the negative side of our system is getting hotter than our positive side.

Note 1: Based on (2) and (3), we believe the unstable behavior happens because both, the negative copper block (part of the negative side section) and the 316L SS wire, are transferring more heat to the water than the positive side. That make sense for me if we think the electrons move from the negative to the positive terminals (electrons move in the opposite direction of the current flow). That means the power supply is providing power to the system from the negative to the positive terminals, across our system. Hence, the negative side gets hotter and most of the heat is transferred to the tank  through the copper block (due to the temperature difference between this copper block and the water) and the SS wire (since its temperature increases much faster than the copper blocks). Then, just a small portion of the power actually reaches and is released in the positive side.

Note 2: Based on (2), we believe the instability observed in the measurements means those two section's voltages are balancing each other. That happens because the bubbles coming out of the copper block and the heating element help to cool down those parts, according to the pool boiling curve and bubble dynamics. According to bubble dynamics, when a single bubble (hot bubble) departs from a surface, it takes that heat out of the surface and the water comes in contact with the surface at this spot, increasing the heat transfer mechanisms there.

Please give me some comments/feedbacks on the above discussion.

If you are using bare metal, some of the current will flow through the water and not the wire.

If one end of the wire is getting hotter, it may be due to a poor connection. With this much current, a little extra impedance at the connection will cause significant heat. Alligator clips don't provide much contact area - for a good connection, you need more total contact area than the cross-sectional area of the wire. Contact means direct metal-to-metal contact - no air or water gaps. This can be quite a problem since many connectors do not meet this criterion.