# Setting up Fick's Law with Internal heating

• jonthebaptist
In summary, the conversation discusses the use of Fick's Law with internal heating to solve for the flow of heat on a resistor. There is a disagreement about the correct mathematical formulation, with some suggesting the use of Fourier's Law instead. The conversation also mentions the use of COMSOL to model heat flow on parts with Joules internal heating. Ultimately, it is suggested to consider using a control volume and vector analysis to arrive at a sensible differential equation for heat flow.
jonthebaptist
I am solving for the flow of heat on a resistor, say it has a cross-section in the x-z plane and is extended along the y-axis. Fick's Law with internal heating is simply
$\frac{\partial T}{\partial t}=A\nabla^{2}T+BQ$
My PDE text gives Q as power delivered per unit volume. So substituting in the electrical power delivered divided by the differential volume element, I get
$\frac{\partial T}{\partial t}=A\nabla^{2}T+B\frac{I^{2}\rho}{dx^{2}dz^{2}}$
What I did makes sense to me physically, but I am not sure if it makes sense mathematically as a PDE with a term having differential elements instead of derivatives.

My advice would be to choose a control volume, V and look for the heat entering and leaving this control volume, use some vector analysis and it will give you a sensible differential equations. The PDE you write down is wrong mathematically.

Can you really apply Fick's laws to heat? From what I understand, the laws concern concentrations of actual objects. I would think you could measure energy density that way, considering it it also dependent somewhat on position and time.

I would personally use Fourier's Law and other traditional heat transfer equations to measure heat flow.

@hunt_mat Thanks for confirming my suspicion about my PDE

@timthereaper I believe Fick's laws were discovered empirically from diffusion experiements, but Asmar's PDE text gives a derivation for a PDE describing heat flow on a rod based on Fourier's law and arrives at Fick's law.

Either way, I notice that COMSOL has a module that models the physics of heat flow on parts with Joules internal heating, so someone has figured this out. I'll think about it a little more and see if I come up with anything...

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Hmm...I'll have to look at that text. You've piqued my interest.

john, the last term in heat conduction equation should be J^2*rho. This is the general form for the local heat production (and you don't need the derivatives). J is the current density (charge per time per area), and rho is the resitivity (resistance*length). However, if the cross section and rho are constant through the wire, then J is uniform everywhere and you can substitute I=J*A. The term becomes I^2*rho/A - no derivatives.

My first take was to use $J^{2}\rho$, but that only works if you are modeling heat flow in the one dimension longitudinal along the resistor, so all transverse flow effects are ignored. I need to account for those effects for what I'm doing, which is why I am taking my infinitesimal volume element to have infinitesimal lengths in all three dimensions.

## 1. How does internal heating affect Fick's Law?

Internal heating refers to the production of heat within a system due to chemical reactions or other sources. This can affect Fick's Law, which describes the diffusion of particles across a concentration gradient. Internal heating can increase the temperature of the system, leading to faster diffusion rates and potentially altering the concentration gradient.

## 2. How can Fick's Law be set up to account for internal heating?

Fick's Law can be modified to account for internal heating by adding a term for the heat flux in the equation. This term takes into account the heat generated by the system and its impact on the diffusion process. The modified equation is known as the Heat-Affected Fick's Law.

## 3. What factors can affect the accuracy of Fick's Law with internal heating?

There are several factors that can affect the accuracy of Fick's Law with internal heating. These include the temperature of the system, the type and concentration of the diffusing particles, the size and shape of the system, and any external factors that may impact the diffusion process.

## 4. Are there any limitations to using Fick's Law with internal heating?

Yes, there are limitations to using Fick's Law with internal heating. This equation assumes that the system is at equilibrium, meaning that the concentration gradient is constant over time. It also assumes that the diffusion process is driven solely by concentration differences and not affected by other factors such as temperature or pressure.

## 5. How is Fick's Law with internal heating used in scientific research?

Fick's Law with internal heating is commonly used in scientific research to study diffusion processes in various systems, such as biological tissues, polymers, and nanoparticles. It has applications in fields such as materials science, pharmacology, and environmental science, where understanding diffusion rates is crucial for predicting and controlling processes.

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