Periodic heating of a glass of liquid

[klawlor419]

So my question is if I periodically heat some glass of liquid from an arbitrary source, hence providing a driving frequency for the system that will give rise to a phase lag between the temperature of the liquid and the incoming heat from the source, how can I show that there will be a possible range of phase lags between 0 and Pi/2? How can I get a plot of phase lag vs. frequency? I have an idea about how it will look, but I'm not sure how to get there.

 

[klawlor419]

phase lags in heating of liquid

So my question is if I periodically heat some glass of liquid from an arbitrary source, hence providing a driving frequency for the system that will give rise to a phase lag between the temperature of the liquid and the incoming heat from the source, how can I show that there will be a possible range of phase lags between 0 and Pi/2? How can I get a plot of phase lag vs. frequency? I have an idea about how it will look, but I'm not sure how to get there.

I posted earlier in general physics but thought maybe this questions more suited towards classical.

 

[Mapes]

Heat transfer follows the diffusion equation, not the wave equation. Why would the liquid temperature keep increasing after you remove the heat source? (Ignoring spatial variation.)

 

[klawlor419]

"Heat transfer follows the diffusion equation, not the wave equation. Why would the liquid temperature keep increasing after you remove the heat source? (Ignoring spatial variation.)"

Mapes thanks for the response. I'm not sure if this is correct, but just as you remove a driving force from a person pushing a swing and it takes time for that system to return to equilibrium so too does the glass of water. Dissipative forces due to maybe the specific heat of material are present. Let me know what you think

 

[Andy Resnick]

So my question is if I periodically heat some glass of liquid from an arbitrary source, hence providing a driving frequency for the system that will give rise to a phase lag between the temperature of the liquid and the incoming heat from the source, how can I show that there will be a possible range of phase lags between 0 and Pi/2? How can I get a plot of phase lag vs. frequency? I have an idea about how it will look, but I'm not sure how to get there.

What are the boundary conditions? (that is, where is the heat applied, what is the fluid geometry, etc.). Mikhailov and Ozisik's "Unified analysis of heat and mass diffusion" (Dover) provides an exhaustive analysis of problems like this.

 

[klawlor419]

I'm really just aiming to understand the simplest situation. Let's say that the heat is applied uniformly to the glass from all directions. What do you mean by fluid geometry?

 

[olivermsun]

You need to define what your input and output are. For example, you might have a bottom plate temperature U and a thermister temperature V taken near the top of the liquid.

Now assuming you can achieve some sort of "quasi steady state" oscillation in V, you can plot the (sinusoidal) record of V against the signal U and look at the constant phase lag. You can repeat this for a set of frequencies. If you are more interested in the step response instead of the response to periodic heating, you might try measuring for bursts and approximate the Laplace transform.

Since this is a glass of liquid, I don't think you can easily derive an solution for all (or even many) frequencies from first principles.

 

[berkeman]

So my question is if I periodically heat some glass of liquid from an arbitrary source, hence providing a driving frequency for the system that will give rise to a phase lag between the temperature of the liquid and the incoming heat from the source, how can I show that there will be a possible range of phase lags between 0 and Pi/2? How can I get a plot of phase lag vs. frequency? I have an idea about how it will look, but I'm not sure how to get there.

I posted earlier in general physics but thought maybe this questions more suited towards classical.

Please do not multiple post here. It is against the PF rules.

I've merged your two threads into one.

 

[Andy Resnick]

I'm really just aiming to understand the simplest situation. Let's say that the heat is applied uniformly to the glass from all directions. What do you mean by fluid geometry?

The geometry of the problem is needed to specify the boundary conditions. If you have a closed spherical container of fluid, apply a constant flux of heat over the entire boundary and neglect bouyancy (so the fluid doesn't flow and advect heat), and measure the temperature at a single fixed point at the center of the sphere, IIRC the temperature will follow a sigmoid-type curve with the final temperature reached when the heat flux in is equal to the heat flux out.