One Dimensional Slab Heat Transfer Taylor Expansion in Glasstone

In summary: Taylor's paper "An Introduction to the Taylor Series" provides a derivation for the higher order terms.
  • #1
terryphi
59
0
Hi There,

I came across the following passage in Sam Glasstone's 'Nuclear Reactor Engineering'

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See where I underlined in red that taylor series expansion? I don't understand how (dt/dx)_(x+dx) is equal to that.

I know it's a Taylor series expansion, but where did the x+dx go?
 
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  • #2
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
  • #3
Well,

I know it's the incremental form of the Taylor series:
http://geophysics.ou.edu/solid_earth/readings/taylor/taylor.html [Broken]

But I have no clue how to derive this incremental form.
 
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  • #5
The differential is supposed to be small, so we can neglect higher order terms in the Taylor series expansion since squaring a small number will give you an even smaller number. So you have heat entering and exiting your differential control volume,

That is Net Heat Flow:

qx- qx+dx.

See the heat enters at position x and leaves at a position x + a very small amount (dx). Now if we do a Taylor series expansion

qx+dx= qx +(dqx/dx)*dx

Take this expression and substitute into the above expression for net heat flow.

qx-qx+dx=qx-qx+(dqx/dx)*dx

Then:
qx-qx+dx=(-dqx/dx)*dx

No you can't cancel those two dx terms because one is a partial differential and the other is total. The one in parentheses is the partial differential.

Now we can describe qx using Fourier's law

qx=-kA(dT/dx).

Substitute this into our above expression

-d/dx*(-k *dydz*dT/dx)*dx

Net heat flow in x direction is now

d/dx*(kdT/dx)*dxdydz

The area in the x direction is given by y times z, similarly in the y direction area will be given by x and z

If you do those equations for the y and z directions and then divide the whole equation by dxdydz you will obtain the heat equation.

I assume you were ok with the other terms in the heat equation. If not I'm happy to explain the derivation of those as well
 
  • #6
terryphi,

I looks like a Taylor series expansion for the first derivative instead of the function itself.

The Taylor series expansion for function "f" about x = 0 is:

f(x) = f(0) + df/dx dx + 1/2 d2f/dx2 (dx)^2...

In this case, the function f is dT/dx; so all the derivative are one degree higher.

Greg
 

1. What is "One Dimensional Slab Heat Transfer Taylor Expansion in Glasstone"?

One Dimensional Slab Heat Transfer Taylor Expansion in Glasstone is a mathematical model used to analyze heat transfer in a one-dimensional slab. It is based on the Taylor Series Expansion method and was developed by Samuel Glasstone in his book "Heat Transfer". This model is commonly used in engineering and physics to predict temperature distribution and heat flux in solid materials.

2. How does the Taylor Series Expansion method work?

The Taylor Series Expansion method is a mathematical technique used to approximate a function by expressing it as an infinite sum of terms. In the case of heat transfer, the method is used to approximate the temperature distribution in a solid material by expanding it into a series of polynomials. This allows for a simplified analysis of heat transfer in a one-dimensional slab, making it easier to solve complex problems.

3. What are the assumptions made in One Dimensional Slab Heat Transfer Taylor Expansion in Glasstone?

The main assumptions made in this model are: 1) steady-state heat transfer, meaning that the temperature distribution in the slab does not change over time, 2) one-dimensional conduction, where heat transfer only occurs in one direction and there is no heat transfer in the other directions, 3) uniform material properties, such as thermal conductivity and heat capacity, throughout the slab, and 4) constant heat generation, meaning that there is no heat source or sink in the slab.

4. What are the limitations of One Dimensional Slab Heat Transfer Taylor Expansion in Glasstone?

One Dimensional Slab Heat Transfer Taylor Expansion in Glasstone is a simplified model and has some limitations. It assumes a one-dimensional heat transfer, which may not be accurate for complex geometries. It also assumes uniform material properties, which may not be true for all materials. Additionally, the model does not take into account transient heat transfer, which is important for time-dependent processes.

5. How is One Dimensional Slab Heat Transfer Taylor Expansion in Glasstone used in real-world applications?

This model is commonly used in engineering and physics to analyze heat transfer in solid materials, such as in the design of building insulation, cooling systems, and electronic devices. It can also be used to predict temperature distribution and heat flux in materials during manufacturing processes, such as in welding and casting. However, it is important to note that this model should be used with caution and its limitations should be considered when applying it to real-world applications.

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