James P Houghton

# Cooking with Gas vs Electric Stovetop

01 Oct 2013

Allison and I just moved into a new apartment, and one of the (many) things we're working out how to use in the new space is the electric stovetop. Our old apartment had a gas range, and it was pretty easy to control the rate of heat flow into the pan, and thus the approximate pan temperature. We'll have to change our control behavior for the new electric setup, but how?

Let's build a model of an empty cast iron skillet on a gas burner, modeling the heat in the pan (in Joules) in excess of what was present with the pan at room temperature, and look at the temperature of the pan in excess of room temperature (in Kelvins, or degrees Celsius). We'll model heat outflow to be proportional to the temperature, via some coefficient that represents the combined effect of radiation and (probably smaller) convection to the room. For now we just make up a reasonable number.

Here are our assumptions:

Mass = 1kg
Specific Heat = 450 Joules/kg*K
Outflow Coefficient = 5 Watts/K
Max Flame Size = 1000 Watts
Starting Excess Heat in Pan = 0 Joules
Starting Control Setting = .5

And the equations which describe the system:

Flames = Max Flame Size * Control
Excess Heat in Pan = integral[Flames-Outflow]
Outflow = Excess Temperature * Outflow Coefficient
Excess Temperature = Excess Heat in Pan / Thermal Mass
Thermal Mass = Specific Heat * Mass

When we simulate leaving the burner on at 50% of maximum, the temperature profile of the pan looks like:
As the temperature rises, the rate of heat outflow increases until it matches that of the heat inflow due to the flames, and the temperature stabilizes at this point.

Its a little hard to see the 'Flames' line, as its right at the top of the graph at 500 Watts.

Now this isn't the full system as the cook isn't included. The cook forms a positive feedback loop between the temperature of the pan and the gas control. We'll assume that the cook checks the temperature of his pan every thirty seconds, and adjusts the control slightly if the pan is too hot or too cold. The system then becomes:

Check = 0-1 pulses every 30 seconds, for 1 second
Gain = .005 /Kelvin
Desired Pan Excess Temperature = 75 Kelvins

IF THEN ELSE( Excess Temperature > Desired Excess Temperature :AND: Control > 0,
Gain*(Desired Excess Temperature -Excess Temperature), 0) +
IF THEN ELSE( Excess Temperature < Desired Excess Temperature :AND: Control < 1,
Gain*(Desired Excess Temperature -Excess Temperature), 0), 0)

With this control loop in place, the cook adjusts the gas flow until his pan reaches about the right temperature:

He is able to do so with a small amount of overshoot, and a little wobbly oscillation at the end:
If we change the frequency with which the cook adjusts the temperature, intuition would tell us that checking more frequently would improve the response, but this is actually not the case, as his impatience causes him to overcorrect, and his oscillations are worse.

Fair enough.

Now lets look at an electric cooktop. There is an additional thermal stock in the presence of the heating coil, and a secondary transfer process between the coil and the pan, and an additional heat transfer process:

For the sake of comparison, let's assume that the maximum heat transfer rate remains the same - 1000 Watts, and that we use the same (30 second interval) control process. We'll say that the transfer coefficient between the element and the pan is twice that between the pan and the room (which seems reasonable - radiative heat transfer should be about the same, convection should drop, and we'll add conductive heat transfer - but not that much, as the physical connection is poor.) The heating coils are made of some combination of nickel, iron, and chromium, which have relatively similar specific heats (440, 450, 460 J/kgK respectively), so we'll just assume the middle value.

Max Watts = 1000 Watts
Element Mass = .25 kg
Element Specific Heat = 450 Joules/kg*K
Transfer Coefficient = 10 Watts/K

The structural additions for accounting heat and temperature parallel those of the pan, and the addition of heat transfer :

Transfer = Transfer Coefficient*(Excess Temperature in Pan - Excess Temperature in Element)

What we see is that the additional heat stock introduces a lag between the element temperature and the pan temperature. Here the element reaches maximum temperature at about 150 seconds, when the heat is turned down, but the pan continues to get hotter for another 30 seconds or so.
As the lag is on the same order as the delay in control feedbacks, the oscillations are stronger than with a gas cooktop and take longer to die out. (Notice the change of x scale)

These lags and overshoots make it harder to maintain a desired temperature on electric stovetops than gas ones, all else being equal.

Now as to how I should modify my control strategy for cooking on the new stove, I could take a lesson from our earlier experiment varying the time between temperature checks. If I grow in patience and only adjust the temperature every 60 or 90 seconds, I should see smaller overshoot and less dramatic oscillation:
The problem is, it's hard to be patient when the asparagus is burning!

References:
Thermal Mass: http://www.engineeringtoolbox.com/specific-heat-metals-d_152.html