# Heat equation and stainless-steel wire

• Physgeek64
In summary, the conversation discusses the calculation of steady-state current passing through a stainless-steel wire when the center of the wire begins to melt. Different equations and values are used to estimate the current, with a final approximation of I ≈ 100 A or more. The temperature dependencies of thermal conductivity and resistivity are also mentioned as factors to consider in the calculation.
Physgeek64

## Homework Statement

A stainless-steel wire is 0.1 mm in diameter and 1 m long. If the outside of the
wire is held fixed at 20◦C, estimate the steady-state current passing through the wire
when the stainless steel at the centre of the wire begins to melt.

## The Attempt at a Solution

The solution to the heat equation will be ##T=-\frac{Hr^2}{4\kappa} +c## The ln term disappears because the temperature must be finite at the origin.

H is the source term ##H=\frac{I^2R}{Al}= \frac{I^2\rho}{A^2}## where A is the c.s.a of the wire

Since the outer surface of the wire is held at 20 degrees
##c=293+\frac{Ha^2}{4\kappa} ## where a is the radius of the wire

##T=293+\frac{Ha^2}{4\kappa}-\frac{Hr^2}{4\kappa}##

Therefore the temperature at the centre of the wire is

##T=293+\frac{Ha^2}{4\kappa}##

setting ##T=1400## which we are told is the melting point of the wire we get

##1380 = \frac{I^2\rho a^2}{\pi^2 a^4 4\kappa}##

##I=\sqrt{\frac{4(1380) \pi^2 a^2 \kappa}{\rho}}##
However this isn't right but i can't see where I've gone wrong

Many thanks

I obtained a similar expression for the current ## I ##, (with a 2 instead of a 4), when I used a much simpler heat transfer model, where I assumed the ## \Delta T ## occurred in a planar type geometry in a distance ## a ## and the surface area was ## 2 \pi a L ##. Thermal conductivity ## \kappa \approx 50 ## W(/m K) , but I think at elevated temperatures ## \rho ## is likely to increase from ## 10^{-7 } ## to perhaps ##10^{-6} ## or larger. The radius ##a=.5 \cdot 10^{-4} \, m ##, so that your answer may be in the right ballpark. It is also a somewhat extreme case where there is a heat sink at the surface that can keep the temperature at 20 degrees Centigrade.

Last edited:
@Physgeek64 I confirm your equation. What value did you use for a?

Chestermiller said:
@Physgeek64 I confirm your equation. What value did you use for a?

##0.5 X 10^{-3}##

Physgeek64 said:
##0.5 X 10^{-3}##
10{-4}?

Physgeek64 said:
##0.5 X 10^{-3}##
Sounds OK overall.

-4?
sorry, yes!

Chestermiller said:
Sounds OK overall.
I wonder how I've got the wrong answer then

Physgeek64 said:
I wonder how I've got the wrong answer then
You get an answer where ## I \approx 100 \, A ## or more if you use ## \rho=10^{-7} ##. I think ## \rho ## is somewhat larger than that. See also post 2.

If you are not given the values of κ and ρ to use then I am not sure how you are supposed to solve this. From what I read, over the temperature range involved, κ increases from 15 to 30 W/mK, while ρ increases from 7x10-7 to 12x10-7 Ωm.
To take that into account you would need either to use numerical methods or to approximate these temperature dependencies with analytical functions and solve the resulting diffusion equation.

## 1. What is the heat equation and how does it apply to stainless-steel wire?

The heat equation is a mathematical model that describes the flow of heat through a material. It is based on the principle of conservation of energy and takes into account factors such as temperature, thermal conductivity, and heat transfer. Stainless-steel wire is a commonly used material in industrial and engineering applications due to its high strength and resistance to corrosion. The heat equation can be used to calculate the amount of heat that flows through a stainless-steel wire under different conditions, which is important for understanding its thermal properties and performance.

## 2. How does the thickness of stainless-steel wire affect its heat transfer capabilities?

The thickness of stainless-steel wire can significantly impact its heat transfer capabilities. Thicker wires have a larger surface area, which allows for more heat to be transferred. However, thicker wires also have a higher thermal resistance, meaning it takes longer for heat to travel through them. This balance between surface area and thermal resistance must be considered when designing a stainless-steel wire for a specific heat transfer application.

## 3. Can the heat equation be used to predict the temperature of stainless-steel wire?

Yes, the heat equation can be used to predict the temperature of stainless-steel wire under certain conditions. However, it is important to note that the heat equation is a simplified model and does not take into account all variables that may affect the temperature of a wire, such as external heat sources or changes in environmental conditions. Therefore, it should be used as a guide rather than an exact prediction.

## 4. How does the thermal conductivity of stainless-steel wire impact its heat transfer capabilities?

The thermal conductivity of a material is a measure of its ability to conduct heat. Stainless-steel has a relatively high thermal conductivity compared to other materials, meaning it is able to transfer heat efficiently. This is important for applications where heat needs to be dissipated quickly, such as in electrical components. However, it is also important to consider other factors, such as surface area and thermal resistance, when determining the overall heat transfer capabilities of stainless-steel wire.

## 5. Is there a limit to the amount of heat that can flow through stainless-steel wire?

Yes, there is a limit to the amount of heat that can flow through stainless-steel wire. This limit is determined by the wire's thermal conductivity, surface area, and thermal resistance. If the amount of heat being generated exceeds the wire's ability to transfer it, the wire may overheat and potentially fail. Therefore, it is important to carefully consider the heat equation and other factors when designing and using stainless-steel wire for heat transfer applications.

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