Smoothie Thermodynamics

I've been drinking lots of smoothies lately, and I've developed a small annoyance. Right when I start blending, nothing happens. No matter how hard I push down, my immersion blender just whirls away with blades spinning but no chopping. "What's wrong, Blender? Have you been talking to the printer again?" Just when I'm ready to give up, it starts dicing up strawberries like Michael Myers does teenage babysitters.¹ What gives? Must I needlessly waste energy for two minutes before any delicious goodness gets chopped up? Perhaps my blender needs to warm up first? While I like the visual of my blender doing calisthenics and some light stretching before going to work, I'm actually speaking quite literally. How much of the frozen fruit in the smoothie does my blender melt before it begins chopping?

In the 1840s, James Prescott Joule showed that mechanical energy and heat were different forms of the same thing. To do this, Joule tied a string from a falling weight to a paddle submerged in water. As the weight fell, it pulled the string which turned the paddle which stirred the water. During this process, the temperature of the water rose. By making careful measurements, Joule was able to show that the mechanical energy lost by the falling weight was gained by the water as heat.

The illustration of Joule's experiment features a paddle submerged in a water tank on the left and a falling weight on the right.   A meter stick measures the distance the weight falls, which can be used to find the mechanical energy lost.

Much like Joule's experiment, my immersion blender has a rotating "paddle," i.e., the blade. In this case, the paddle is turned by electrical energy rather than mechanical energy. As the blade turns, it stirs the liquid in the smoothie and heats it. Let's assume the 2.0 cm long blade weighs 1.0 grams and takes 0.5 seconds to reach its top angular velocity of 10,000 rpms.² Using dimensional analysis, we find this costs about 1.0 Watt of mechanical power. If I blend for two minutes, I'll gain 120 Joules of thermal energy.

My smoothie might have 250 grams of ice in the form of frozen strawberries and other fruits. The heat given off by the blender will melt some of this ice. The heat required to melt a solid is called the heat of fusion. The heat of fusion for ice is 334 Joules per gram. Since two minutes of blending only provides 120 Joules of heat, I will only melt about one-third of a gram of the ice in the container. That's about one one-thousandth of the total ice. It's much more likely the warmer air in the room is melting my smoothie. If I need the air to melt some of the smoothie before blending, I'd be better off just waiting two minutes rather than wasting energy by needlessly running the blender.

[1] I am, of course, referring, to Michael Myers the character who brutally murders his victims in the slasher classic Halloween, not Michael Myers the actor, who brutally murders comedy in The Love Guru.
[2] Within an order of magnitude, this rotational speed is typical of what you find in immersion blender advertisements.

Aaron Santos is a physicist and author of the books How Many Licks? Or How to Estimate Damn Near Anything and Ballparking: Practical Math for Impractical Sports Questions. Follow him on Twitter at @aarontsantos.