Awright. We're closer to sympatico on this. One thing at a time:
I believe that you are looking to model the entire response of the melting ice.
More specifically put: My experience with thermodynamics and heat transfer has me convinced that the "entire response" as you put it is a lot more dynamic and a lot more relevant to the discussion than:
I'm just trying to show that the amount of ice melted taking fluid from T1 down to T2 will be the same regardless of the shape of the ice.
And I get that. What I'm saying is that you can't "just" do that the way you want because the variables you're choosing to model are a lot less important than the variables you're choosing to ignore.
The time that it takes to get from T1 to T2 will be longer for the ball than the cube, so one could claim that "the ball of ice melts slower" - which is true, but the amount of ice that has melted by the time the temperature reaches T2 (the drinking temperature) will be the same.
T1 to T2 will be faster for the ball, not slower. It will be faster through sheer volume. That's the argument for "ice balls." Simply put, "more ice, less of it in your drink." The argument for an ice ball is similar to the argument for an ice luge. Note that these are not arguments I put much faith in - not to say they aren't going to do anything, just that they're not worth the hassle.
You wanted to eliminate the "ice ball advantage" to make for a simpler (but much less practical) experiment. Okay, but in doing that you're tilting the dominant effects of convection and specific heat change further into the fore, and they're chaotic. That cue 1st law of thermo - all things being equal - drops right out of the problem and gets replaced with something that's like, but isn't quite, heat sink equations. Note that in addition to all the heinous math here, you're also dealing with variable geometry, variable hf, variable k and the specific heat equations.
The discussion really comes down to this: You think in terms of T1 and T2. You're presuming the rest of that stuff doesn't matter. The part you're missing is that when you've got ice in a drink, it's the specific heat part of the equation that dominates: T3, that of the ice, will never be reached by the drink and will never be left by the ice throughout the period of the experiment. Thus your entropy equation is inapplicable for the experiment you wish to run on it.
And as far as me saying "all else being equal" that's how you address a given claim.
No, that's how you pick an inappropriate model. For the third time, at the scale you're dealing with all the stuff you're ignoring matters more than the stuff you're focusing on. And I'm going to have to appeal to authority here: I spent three years of my life being tested on thermodynamics, and I spent ten years of my life applying fluid mechanics. You looked up an equation on Wikipedia. I'd say "take my word for it" but I've done a lot more than that. Let me restate it in nice, scientific terms:
IF: the system under investigation is characterized by
A) initial liquid volume within an order of magnitude of initial solid volume
B) investigated thermal behavior within the transition temperature of the system
C) a primary characteristic of study is the effect of solid geometry on melt speed
THEN: Effects of convection and transition temperature will dominate the system while fundamental equations of entropy will not appreciably impact the model.
You linked to experimental investigation of this. They did one run, but at least they did a run. They found a difference. Based on your equation, they shouldn't have. That in and of itself should demonstrate that your approach is flawed: theoretically, it shouldn't matter; experimentally, it did: QED, the theory is wrong.
And there's no point in comparing A to B if you're going to have a bunch of other factors that are different.
Well, step back for a minute: Your initial argument was:
The transient response is trivial, so let's move to steady-state.
My initial argument was that steady-state is irrelevant (you agreed with this, and restated your argument) and that the transient response is not only what you care about, it's messy as fuck. You circled back to
What I meant was the state that the liquid and the solid are the same temperature, while the ice is melting.
And again - that's not an "entropy" equation. That's a convection and specific energy equation.
So we're back to where we started: you want this to be simple, but it just isn't. That's why I'm saying "you're wishing it so." I've described a bunch of considerations based on an engineering background. You've described a bunch of reasons why they don't matter because you want things to be simple:
As for the formula being the correct one, it is. I learned it in my high school Chem 1 class my freshman year, and it is still used today. In fact, you'll notice the same formula in the Enthalpy of Fusion wiki link.
You want to use what you learned in high school chemistry but refuse to acknowledge that I might have learned something in 500-level fluid mechanics.
And that's where the problem isn't that your understanding is "different" it's that you won't accord me the respect of acknowledging that I might just know more about this than you do. You aren't trying to understand my arguments, you're trying to nullify them. You aren't trying to apply my insights to your problem, you're arguing they're invalid. See, check this out:
In fairness, yeah, one could argue that all those Newtonian physics formulas are dead wrong over-simplifications now that one has taken a relativity course, but for validating a claim that there is a difference in A to B it should be a perceivable difference.
You linked to this yourself:
I've now spent a thousand words explaining why those curves don't line up, and explaining the dismal science in putting math to that gap.
Your response is "but they have to line up, I took chemistry in high school." So:
| The small stone and large stone dropped off a tower may not accelerate at exactly 9.8m/s/s and may not hit at exactly the same time, but most people would agree it's close enough; the small stone does not fall faster.|
Drop a pound of water and a pound of snowflakes. Which one hits the ground faster?
I had to do the math to explain why. You didn't. So when I say "they're different, here's why" I can offer an explanation as to why the observed experiment does not align with vf=vi+at. Meanwhile, you've got "snow isn't slower, you're observing it wrong."
There are only so many ways I can point this out. I've hit my limit. Frankly, the fact that you're rubbing high school chemistry in my face three layers deep just shows how little respect you accord me, and makes me regret trying in the first place.
Happy new year.