Is there an ODE with only time dependence and constant position?

[williamrand1]

Hi all I am new to this forum. I need a little help on a project I am doin. The heat equation is a pde with dependence on time and position, what i want to know is is there a ode which the dependence is only on time, constant position?

Any help would be great thank

 

[nasu]

Hi all I am new to this forum. I need a little help on a project I am doin. The heat equation is a pde with dependence on time and position, what i want to know is is there a ode which the dependence is only on time, constant position?

Any help would be great thank

The equation depends on the problem. What is the problem you have in mind?

 

[MrMatt2532]

This would come up in the case of using a lumped capacitance model, which is a good assumption for small Biot numbers.

You could say something like:
rho*V*cp*dT/dt=Q, where Q might be a function of temperature, i.e. Q=h*A*(T-Tinf)

 

[williamrand1]

The equation depends on the problem. What is the problem you have in mind?

A polysilicon fuse on an integrated circuit blows, i have to set up an ode which describes the temperature over time.

 

[williamrand1]

This would come up in the case of using a lumped capacitance model, which is a good assumption for small Biot numbers.

You could say something like:
rho*V*cp*dT/dt=Q, where Q might be a function of temperature, i.e. Q=h*A*(T-Tinf)

Thanks ill look that up..

 

[nasu]

A polysilicon fuse on an integrated circuit blows, i have to set up an ode which describes the temperature over time.

So it's heated by a current flowing through it, is this correct?
The problem is that when you say "temperature", it's not clear what temperature. The temperature is not the same throughout the fuse. That's the reason the ODE has both time and position derivatives. The rate of change in temperature depends on the spatial distribution of temperature in the piece.

Now it may be good enough to assume/approximate some distribution or even simpler, assume that the temperature in is approximately the same throughout the fuse. In this case you can estimate the temperature change versus time by writing the thermal energy balance and using Newton's law of cooling to write the heat lost through the surface.

 

[williamrand1]

So it's heated by a current flowing through it, is this correct?
The problem is that when you say "temperature", it's not clear what temperature. The temperature is not the same throughout the fuse. That's the reason the ODE has both time and position derivatives. The rate of change in temperature depends on the spatial distribution of temperature in the piece.

Now it may be good enough to assume/approximate some distribution or even simpler, assume that the temperature in is approximately the same throughout the fuse. In this case you can estimate the temperature change versus time by writing the thermal energy balance and using Newton's law of cooling to write the heat lost through the surface.

Yea a current is flowing through it, a pd of 4v applied across the fuse and it blows in a few hundreds of a nanosecond. I was thinking the same thing but we were asked to set up n solve ode's describing the temperature. Say if i take the temperature to be approximately the same throughout the fuse and use Newton's law of cooling for the heat loss, then i would have something like dQ/dt= an equation for the increase heat - heat loss. I think this is right, is the equation I am looking for related to the current??

 

[nasu]

Yea a current is flowing through it, a pd of 4v applied across the fuse and it blows in a few hundreds of a nanosecond. I was thinking the same thing but we were asked to set up n solve ode's describing the temperature. Say if i take the temperature to be approximately the same throughout the fuse and use Newton's law of cooling for the heat loss, then i would have something like dQ/dt= an equation for the increase heat - heat loss. I think this is right, is the equation I am looking for related to the current??

The current determines the heat input ("increase heat" in your post above).
If the voltage is constant, you can write the heat input (per unit time) as Pin=V*I.
Unfortunately the current depends on the resistance of the fuse and that may change quite a lot with temperature. So that will be an extra difficulty. The power input won't be a constant but a temperature dependent function.

 

[MrMatt2532]

It sounds to me like you can assume the fuse reached a steady state operating temperature before it blew, and then after it blows the Joule heating will no longer exist due to a short circuit, so you can just model the problem as I showed in my first post.

If you need to figure out the steady state temperature/initial temperature you can just say that the Joule heating equals the heat lost from convection (Newton's law of cooling).

 

[nasu]

It sounds to me like you can assume the fuse reached a steady state operating temperature before it blew, and then after it blows the Joule heating will no longer exist due to a short circuit, so you can just model the problem as I showed in my first post.

If you need to figure out the steady state temperature/initial temperature you can just say that the Joule heating equals the heat lost from convection (Newton's law of cooling).

Why does it blow up if is in a steady state?

 

[MrMatt2532]

Maybe it got a current spike for a few nanoseconds and then blew. A few nanoseconds is enough time to blow, but not enough time to heat up significantly. No point over-complicating a simple problem.

You really need more information anyways if you are going to go down that route. If the problem said that voltage ramped up from 0 V to 50 V and then blew, then you could go down that route. Without that information, there is no reason to assume something like that.

 

[nasu]

My understanding was that you apply a constant voltage and the temperature increases continuously until the thing blows-up.
I don't know if this is the case, you are right. More info is needed.
Is the blowing-up a random event or it happens every time, shortly after you apply the 4 V?

 

[MrMatt2532]

I will add that in my two previous posts i was greatly exaggerating in regards to how long it takes the fuse to blow, and of course it would normally be increasing in temperature prior to blowing. It just sounded to me like the problem was more concerned with modeling what would happen after the fuse blew and not before.

My question for William is whether the problem is concerned with modeling the temperature before or after the fuse has blown or both? Also, similar to nasu's question, was the fuse at 4V for a long time and then randomly blew, or was this 4V much higher than the fuse was rated so it quickly heated up and then blew? I suppose my answers were assuming the former case.

 

[williamrand1]

My understanding was that you apply a constant voltage and the temperature increases continuously until the thing blows-up.
I don't know if this is the case, you are right. More info is needed.
Is the blowing-up a random event or it happens every time, shortly after you apply the 4 V?

The fuse is blown as an irreversible programming step. The fuse has an initial temperature before the current goes through it. The fuse blows ina few hundreds of a nanosecond. The ode i want to set up and solve is how the temp changes over time. Yea it is constant voltage. We are givin a melting temp so i assume the ode should only describe the temp from initial to melting.

 

[williamrand1]

I will add that in my two previous posts i was greatly exaggerating in regards to how long it takes the fuse to blow, and of course it would normally be increasing in temperature prior to blowing. It just sounded to me like the problem was more concerned with modeling what would happen after the fuse blew and not before.

My question for William is whether the problem is concerned with modeling the temperature before or after the fuse has blown or both? Also, similar to nasu's question, was the fuse at 4V for a long time and then randomly blew, or was this 4V much higher than the fuse was rated so it quickly heated up and then blew? I suppose my answers were assuming the former case.

Hi Matt

My last post to nasu explains the problem in more detail

 

[MrMatt2532]

Ok, this should be pretty easy to setup:

@t = 0: T = T_initial
and V = 4 V

Assuming a lumped capacitance model and neglecting heat loss, we have:

m*cp*dT/dt = V^2/R

You can assume cp and R to be constant or to factor in how they vary with temperature to be more accurate.

 

[nasu]

Ok, this should be pretty easy to setup:

@t = 0: T = T_initial
and V = 4 V

Assuming a lumped capacitance model and neglecting heat loss, we have:

m*cp*dT/dt = V^2/R

You can assume cp and R to be constant or to factor in how they vary with temperature to be more accurate.

You forgot dissipation through surface. That is harder to estimate. You can use Newton's law of cooling:
dQ/dT=-k*A*(T-Ta) where Ta is the ambient temperature, A is the surface area and k is a coefficient which depends on material and surrounding fluid.
It may be negligible in this case (very fast process).
If the temperature is high enough, the radiative dissipation may be important too.


@matt
What is this "lumped capacitance" that you keep mentioning? The equation you wrote is the simple balance of powers. What is the relationship with capacitance, lumped or not?

 

[MrMatt2532]

You forgot dissipation through surface. That is harder to estimate. You can use Newton's law of cooling:
dQ/dT=-k*A*(T-Ta) where Ta is the ambient temperature, A is the surface area and k is a coefficient which depends on material and surrounding fluid.
It may be negligible in this case (very fast process).
If the temperature is high enough, the radiative dissipation may be important too.


@matt
What is this "lumped capacitance" that you keep mentioning? The equation you wrote is the simple balance of powers. What is the relationship with capacitance, lumped or not?

Well I didn't forget the heat transfer, I said neglecting heat loss I came to that equation. The heat transfer will likely be negligible in comparison to the joule heating, so it should be safe to neglect. This might need to be checked, but using reasonable values, you will find h*A*delT << V^2/R in the case of a very short blown fuse time.

When we neglect spatial dependence, we are in essence lumping all of the mass together and the math turns out to be similar to a charging/discharging capacitor for convective cooling/heating, hence the "lumped capacitance model". It is a good assumption for low Biot numbers (when the external resistance to heat transfer is much greater than the internal resistance to heat transfer).
see here: http://en.wikipedia.org/wiki/Lumped_capacitance_model

 

[Redbelly98]

I just came across this thread. Interesting problem, here are my 2 cents:

Since we are interested in a time frame of hundreds of ns, it should be safe to ignore heat losses via conduction, convection, or radiation.

The tricky part is the change in resistance with temperature. An approximation for metals is that R is roughly proportional to absolute temperature, so I would use that to set up a reasonably accurate differential equation.

Hope that helps.

 

[Redbelly98]

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