# Calculating Maximum Current in Metal

1. Oct 7, 2011

### AutoNub

Hello! A co-worker recently asked me how to calculate the maximum current through a titanium cylinder. I immediately realized there were probably many variables, likely including voltage, length, cross sectional area, and electrical conductivity.

I used wikipedia to see what I could find and it lists the electrical resistivity as being (420nΩ*meters - @20° celsius). We know electrical conductivity is defined as the inverse of the electrical resistivity, thus the conductivity must be 420nΩ(-1)*m(-1) at 20° celsius. Good so far?

So algebraically, can we solve for the maximum current by using the following equation?

[(Conductivity)*(Voltage)*(Cross-Sectional Area)]/[(Length)]=[(Current)]

If the above is a valid equation, does it merely give us a permissable current or is it truly the maximum current at a 20°C temperature? Also, let's say we have a variable temperature... for example, if our titanium cylinder is being cooled by ocean water onboard a ship, and the ship is constantly traveling. What is the rate of change based on temperature? If it is linear, we should be able to further develop our equation to account for it.

Any ideas or contributions would be highly appreciated. Thanks!

2. Oct 7, 2011

### f95toli

"Maximum current" is not a well defined quantity. What can be considered "maximum" really depends on how hot you are prepared to let you cylinder get.

I guess one "maximum" would be the amount of current you could pass through it in ambient conditions with no forced cooling without the metal melting, but that would probably lead to a maximum where the cylinder is glowing which (presumably) is not what you want.

3. Oct 7, 2011

### AutoNub

Let's pick a temperature value as a maximum temperature tolerance. Say, 105°C. Would there be a way we could incorporate this variable into our equation to give us the maximum current based on a maximum temperature allowance of 105°C?

4. Oct 8, 2011

### f95toli

You need to figure out the cooling rate. This is quite tricky since much of the cooling will be due to convection, especially for something like a cylinder.

Hence, if you are asking if one could write down the equation the answer is yes(it would be a PDE). All you need is the heat equation with Joule heating as a heat source.
However, I seriously doubt you could ever get anywhere if you actually wanted to solve this equation analytically, this is something best done using a FEM solver such as Comsol or Ansys.