Why a Tumbler Keeps Cold Longer Than Hot

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The flask does not treat them differently. The ice does, and you can prove it from the two numbers printed on the same listing, because the insulation cancels exactly out of the ratio between them.

Three of our own records publish both figures. A 36 ounce tumbler states more than 36 hours cold and more than 13 hours hot, a ratio of 2.77. A 64 ounce one states 30 and 12, a ratio of 2.50. A 10 ounce mug states 12 and 6, a ratio of 2.00. None of those ratios is a measure of how good the vacuum is. Each one is a readout of how much ice the maker put in the test.

Why the cold trace is flat and the hot trace is a curve

A hot drink has nothing to spend except its own temperature. Heat leaks out, the temperature falls, and because the leak is proportional to the gap between the drink and the room, the fall is an exponential decay that slows as it goes. Write the leak rate as UA, a conductance in watts per kelvin that bundles the whole vessel into one number. Then the time to fall from a starting temperature to a threshold is the mass times the specific heat, divided by UA, multiplied by the natural logarithm of the two temperature gaps.

A cold drink with ice in it does something categorically different. While any ice remains the temperature cannot move, because every joule arriving through the wall is spent melting rather than warming: 334,000 joules per kilogram, at a fixed zero degrees. So the cold hold time is simply the melt energy divided by the leak rate, with no logarithm in it at all. That is the flat line in the drawing.

The insulation cancels, and that is the whole answer

Write the two times out and divide one by the other.

The cold time is the mass of ice times 334,000, divided by UA times the room temperature above zero. The hot time is the mass times 4,186 divided by UA, times the logarithm term. Divide, and UA appears once in each numerator and once in each denominator and vanishes. What is left is the ice mass times 334,000, over the total mass times 4,186 times the room temperature times the logarithm.

Put ordinary numbers in it. Room at 22 degrees, a hot fill of 95 degrees, a threshold of 60 degrees for still drinkably hot. The logarithm of 73 over 38 is 0.6529. So the denominator constant is 4,186 times 22 times 0.6529, which is 60,140, and 334,000 divided by 60,140 is 5.55. The ratio of cold hours to hot hours equals 5.55 multiplied by the fraction of the fill that was ice.

Run our three products backwards through that. The 2.77 ratio implies an ice fraction of 50 percent. The 2.50 implies 45 percent. The 2.00 implies 36 percent. Those are all plausible test fills, which is the reassuring part, and they are all statements about a test protocol rather than about a cup. Two tumblers with identical vacuum walls would publish different ratios if one was tested half full of ice and the other a third full. Two tumblers with wildly different walls would publish the same ratio if they were tested the same way.

So the pair of numbers cannot be used to compare brands, and the comparison most people reach for is empty.

What the hot figure can tell you, which the cold one cannot

There is one number in each listing that is clean, and it is the hot one, because it contains no latent heat. Invert the hot formula and you get UA, the whole vessel's conductance, in watts per kelvin. That is the figure that actually describes the cup.

A Coldest 36 ounce tumbler with a handle and a coffee lid publishes triple wall vacuum construction, 36 plus hours cold and 13 plus hours hot, three lids and a stated fully leak proof seal. At 36 ounces, which is 1.065 kilograms of water, 13 hours implies a UA of 0.062 watts per kelvin. Model the body as a cylinder about two and a bit times as tall as it is wide and you get roughly 0.12 square metres of boundary, so about 0.50 watts per square metre kelvin. Details in our Coldest 36 ounce tumbler review.

A Hydraful 64 ounce insulated tumbler publishes 18/8 stainless with double wall vacuum insulation, 30 hours cold and 12 hours hot, and a 2 in 1 leak proof lid. At 1.893 kilograms, 12 hours implies a UA of 0.120 watts per kelvin over roughly 0.18 square metres, which is 0.67 watts per square metre kelvin. Its record is honest that the time ratings are the maker's stated figures and are best with pre-chilling. See our Hydraful 64 ounce review.

And a YOME 10 ounce ceramic lined insulated mug publishes 304 stainless, a ceramic lining, 6 hours hot and 12 hours cold. At 0.296 kilograms, 6 hours implies a UA of 0.037 watts per kelvin over roughly 0.056 square metres, which is 0.68 watts per square metre kelvin. Full record in our YOME ceramic lined mug review.

Line those three up per unit of surface and the ordering is 0.50, 0.67, 0.68. The one that is published as triple wall is a third better per square metre than the two published as double wall, and the two double wall vessels land on effectively the same figure despite one being six times the size of the other and having a ceramic lining. That is as close to a measurement as published figures allow, and it is worth more than either raw hour count.

It also tells you what the ceramic lining is not doing. A lining a fraction of a millimetre thick, in contact with the liquid on one side and the steel on the other, adds a resistance that is negligible against the vacuum behind it, and it sits nowhere near the gap where radiation crosses. So it changes the surface your coffee tastes, which is a real reason to want it, and it does not appear in the heat budget. The arithmetic agrees: that mug's conductance per square metre is indistinguishable from the plain steel 64 ounce one.

Why small cups are worse, and the exponent our own guide missed

Our guide to how vacuum insulated tumblers work says that "bigger tumblers hold temperature longer, simply because there is more drink to change". That is right, and it has a derivable exponent. Mass scales with volume, so with length cubed. Surface area scales with length squared. Hold time is mass over area, so it scales with length, which is the cube root of capacity. Double the capacity and geometric similarity predicts only 26 percent more time.

Test it on our own figures. Going from 36 to 64 ounces is a factor of 1.78 in capacity, so the cube root predicts 1.21 times the hot hold: 13 hours should become about 15.7. The published Hydraful figure is 12. It is three quarters of what pure scaling predicts, which is another way of saying its conductance per square metre is worse, which is what the table above already found. The scaling rule and the UA calculation agree, independently, from different published numbers. That is the useful kind of cross check.

Then the part that penalises small vessels beyond the scaling. A lid is sized to a human mouth, not to the vessel, so it does not shrink when the cup does. On the 36 ounce body a lid of about 80 millimetres across is roughly 4 percent of the total boundary. On the 10 ounce mug the same lid is about 9 percent. And the lid is not vacuum walled on any of these: it is moulded plastic with a hole you drink through. So the smaller the cup, the larger the share of its boundary that is the one part with no vacuum behind it.

One thing our own guide says that contradicts itself

That same guide states, of the three heat paths, that "a vacuum insulated tumbler blocks all three". Two paragraphs later it says the difference between a good flask and a poor one is "the quality of the vacuum, the thickness and grade of steel, and often a copper lining that reflects radiant heat". Both sentences cannot be true. If the vacuum blocked radiation there would be nothing for a copper lining to reflect.

The second sentence is the correct one. A vacuum has no molecules, so it stops conduction and convection almost completely, and it is perfectly transparent to radiation, which needs no medium at all. That is why the inner wall is mirrored, and why the figure that separates a good flask from a merely shiny one is emissivity. Put a number on it. Two surfaces of bare brushed stainless at an emissivity of about 0.15 facing each other across the gap give an effective emissivity of 0.081, and between a zero degree inner wall and a 25 degree outer one that is about 11 watts per square metre. Copper plating at an emissivity near 0.04 gives an effective 0.020, which is about 2.7 watts per square metre. The plating is worth roughly a factor of four on the only path the vacuum does not touch. Our guide's own explanation was right and its summary sentence was wrong, and the summary is the part people read.

The honest limit

Every UA above rests on assumptions nobody publishes: a fill volume, an ambient temperature, a starting temperature and a threshold temperature for the words hot and cold. I used 95 degrees down to 60 at a 22 degree room. If the real threshold in these tests is 50 degrees rather than 60, every UA above rises by a factor of 1.47 and every ice fraction with it. What does not change is the ordering, because the same assumption is applied to all three, and what does not change at all is the cancellation, which is exact arithmetic rather than an estimate. Not one listing we carry publishes a hold time methodology, a fill volume, an ambient temperature or an emissivity.

And one product on the site declines to publish even the hours. A PROCHAIN insulated stainless steel shaker bottle publishes double wall vacuum insulation, 304 steel, a built in threaded mixer and a twist lock lid, and for retention says only that it "keeps drinks cold for hours". There is nothing to invert there, which is worth saying plainly rather than treating the absence as neutral. What that listing does publish is a detachable powder base, and our record is fair that it is heavier than plastic and a single serving size. See our PROCHAIN insulated shaker review.

Which leaves the free technique, and it beats the spec gap outright because it operates on the one term you control. The hold time is energy divided by leak, so add energy. Pre-chill the empty vessel with cold water or a few cubes for a minute and tip it out, so the steel is not a warm mass you have to cool with the drink you came for. Then fill it with ice rather than with a cold drink, because the drawing above shows exactly what ice is worth: every gram of it is 334 joules of standing still, against the 4.19 joules per degree that chilled water offers. For a hot drink, do the reverse and pre-heat with boiling water.

Two more that cost nothing. Fill it to the top, because air you have to cool is energy spent on nothing. And keep the most closed lid you have on it between sips, not because the lid leaks warmth in the abstract but because an open hole lets water vapour move, and water carries roughly 2,450 joules per gram in and out as it evaporates or condenses. One gram an hour through an open sip hole is a loss of 0.68 watts, which on the 36 ounce tumbler above is eleven times the vessel's whole conductance per degree of room difference. The hole is not a small detail. Our guide's advice to "use the most closed lid you have between sips" was right, and this is the reason it is right.