Achieving bigger popcorn through vacuum pump

[ianyappy]

I'm tutoring a high-school level student where one of the questions is asking why fitting a vacuum pump to a popcorn maker would cause the popped kernels to get bigger. For those unfamiliar with popcorn :), it's modeled as a sealed hulled with starch and moisture inside. Heat causes the moisture to mix with the starch to form a gel-like substance, and as the moisture turns to vapor/steam, the pressure inside builds up till it causes the hull to break. The gel-like starch gets pushed out and expands till it solidifies.

Reading a paper by the guy who developed this idea, he modeled the expansion of the water vapor inside the kernel as an adiabatic expansion

P_{Y}V_{0}^{\gamma}=C_0=\text{constant} \\<br /> \text{where} \\<br /> P_{Y}- \text{yield pressure}\\<br /> V_{0}-\text{initial volume of the kernel}\\<br /> \gamma-C_p/C_v<br />

Resulting manipulations show that the final volume of the kernel would depend on the surrounding pressure around the kernel, thus causing the popcorn to be bigger in a vacuum.

My question is regarding the above equation. Would the yield pressure not depend on the difference between the surrounding pressure and the internal pressure of the hull? I believe the rest of the terms in the equation (i.e. excluding pressure) do not depend on the surrounding pressure so is this correct or have I misunderstood something?

 

[Baluncore]

The corn will pop when the differential pressure exceeds the yield strength of the hull. In a vacuum, the pop will occur at a very slightly lower temperature than when in one atmosphere.

On popping, the reduction of pressure would cause all moisture in the corn to flash to steam.
This fixed mass of gas (steam) would have a final volume determined by the external pressure.

The mass of water in the corn will pre-determine the mass of steam produced.
In a vacuum that final volume could approach infinity.

Maybe you are not considering the steam explosion of the entire water mass. That will occur independently of the hull yield pressure and temperature.

 

[ianyappy]

Thank you for your reply, sorry for my lack of understanding but perhaps I should highlight the point that I'm confused about. In the given equation, my understanding is that all terms excluding pressure are the same regardless of surrounding pressure. However, since as you say that the pressure at which the corn will pop depends on the differential pressure (i.e. P_{\text{internal}} - P_{\text{surronding}}), isn't that inconsistent with the equation?

 

[Baluncore]

I believe that it depends on the precise definition of Py. Is Py the absolute internal pressure on bursting, or the hull pressure differential on bursting, Py = (Pinternal - Pexternal).

Looking at the linked article. “Let Py be the yield pressure at which the adiabatic expansion begins, and Vo be the initial unpopped volume of the kernel.” It is implied by the equation that Py is being used as the absolute internal pressure when the corn pops. The term “yield pressure” is therefore misleading because it suggests the differential pressure for failure of the hull. The hull will actually fail at a differential pressure of (Py – Po).

 

[ianyappy]

Which is my point actually, if the hull breaks at a fixed/constant differential pressure, how could an absolute pressure be used in the equation, given all other things are constant?

 

[Baluncore]

I believe the fundamental Pressure * Volume relationship in thermodynamics must be based on absolute pressures and absolute volumes, not differential pressures and differential volumes.
How do you define Py ?

 

[ianyappy]

I take Py as that from the paper, the absolute pressure.

 

[256bits]

Which is my point actually, if the hull breaks at a fixed/constant differential pressure, how could an absolute pressure be used in the equation, given all other things are constant?

If you notice he has estimated the yield pressure to be an approxiamate value of 200 times that of atmosphere pressure ( not an exact value since popcorn will not all pop at exactly 200, some at a lower pressure of 198 or less and some higher depending upon hull thickneess ). He did not give an error value to be used in his calculations. We can assume perhaps 2% for the majority of popcorn, so we could have a yield pressure of 196 to 204 atmospheres.
The error (0.5% ) of not subtracting the outside atmospheric pressure from the internal yield pressure thus becomes not that significant.