Conduction - Heat Equation - Units Don't Add Up

Click For Summary
SUMMARY

The discussion centers on the application of the heat equation in the context of inertia friction welding, specifically addressing the challenge of unit consistency in heat input terms. Chris is working with a non-homogeneous partial differential equation (PDE) represented as ut - αuxx = (α/κ)*f(x,t), where the heat input is expressed in watts or watts/m² rather than the required watts/m³. This discrepancy leads to inconsistencies in the resulting temperature change rates (K/s). Chris later clarifies the use of Neumann boundary conditions for the problem formulation.

PREREQUISITES
  • Understanding of the heat equation and its applications in thermal analysis.
  • Familiarity with non-homogeneous partial differential equations (PDEs).
  • Knowledge of boundary conditions, specifically Neumann boundary conditions.
  • Basic principles of heat transfer and thermal conductivity.
NEXT STEPS
  • Research the formulation of non-homogeneous boundary conditions in heat transfer problems.
  • Study the implications of different units in thermal analysis, focusing on watts/m² versus watts/m³.
  • Explore numerical methods for solving non-homogeneous PDEs in heat conduction.
  • Investigate the relationship between heating rates, temperature profiles, and microstructure in welding processes.
USEFUL FOR

Engineers and researchers involved in thermal analysis, particularly those working with welding processes and heat transfer modeling. This discussion is beneficial for anyone looking to understand the complexities of applying the heat equation in practical scenarios.

chrissimpson
Messages
11
Reaction score
0
Conduction - Heat Equation - Units Don't Add Up!

Hi there

I have what I think/hope is a simple question:

I've been working on heat inputs and outputs in inertia friction welds and have managed to produce a net power term (W) as a function of time.

I now want to use that in the heat equation to come up with heating rates, temperature profiles and final cooling rates (these can then be related back to the final weld microstructure).

So, the equation I wish to use is:

ut-αuxx=(α/κ)*f(x,t) - a non-homogeneous pde

The problem I'm having with this is that my heat input term is in watts or watts/m^2, not watts/m^3. This is because my heat input is coming from one surface into a long/infinite length bar. When heat generation is in watts/m^3 you get consistent units of K/s throughout the equation. I don't!

Any ideas?

Cheers

Chris
 
Engineering news on Phys.org


I think that I may have made a mistake with my formulation of the problem; I actually have a non-homogeneous boundary condition, so:

ut=αuxx

with Neumann boundary conditions of:

ux(0,t)=Q/kA - with both sides having units of K/m

ux(L,t)=0


Does this seem like the correct way of looking at the problem?

Cheers

Chris
 

Similar threads

Help me heat up a powder coating oven (made out of an old steel cabinet)
  • · Replies 9 ·
Replies
9
Views
2K
Clarification about Heat Conduction equations
  • · Replies 1 ·
Replies
1
Views
2K
Max thermal conductivity required for heating elements
  • · Replies 3 ·
Replies
3
Views
3K
Exit flow rate in a Plate Heat Exchanger
  • · Replies 4 ·
Replies
4
Views
3K
Heat transfer through a MultiLayer Cylinder (find the Temperature inside)
  • · Replies 7 ·
Replies
7
Views
3K
Some guidance about heating calculations (homemade heat-pump)
  • · Replies 3 ·
Replies
3
Views
3K
Some questions on heat transfer
  • · Replies 5 ·
Replies
5
Views
2K
How Long Should My Double Pipe Heat Exchanger Be for a 22kW Capacity?
  • · Replies 2 ·
Replies
2
Views
2K
Heat Transfer through a cooking pot
  • · Replies 1 ·
Replies
1
Views
4K
Engine Finite Heat Release Model (With Heat Transfer) Help
  • · Replies 2 ·
Replies
2
Views
2K