Entries Categorized as 'Uncategorized'

Solving 1D Heat Transfer Problems with Matricies

March 5, 2017

Say, for example, we have a furnace with an interior temperature of 1560 K (hot enough to melt copper) while the outside temperature is a balmy 300 K (27 Celcius). The wall of the furnace is 1 meter thick (it’s a big furnace). Can we figure out how the temperature changes inside the wall as goes from hot on the inside to cool on the outside?

Figure 1. Five nodes (points of interest) where we will calculate the heat flow through a furnace wall.

Figure 1. Five nodes (points of interest) where we will calculate the heat flow through a furnace wall.

To start with, we’ll look at the temperature at five points (call them nodes): three are in the wall and two at the edges (see Figure 1). We’ll say the temperature at node 1 which is next to the furnace is T1, and the temperature at the outer edge of the wall is the 5th node (T5) so:

  • T1 = 1560 K
  • T5 = 300 K

Now we need to find the temperature at nodes 2, 3, and 4.

Heat Flow at Equilibrium

For each of the interior nodes, we can consider how things are at equilibrium. Heat is always moving through the wall, but as some point in time the heat flowing from the furnace into the wall will be equal to the heat flowing out of the wall, and for each node the heat flowing in will equal the heat flowing out.

The heat flow (Q) from one location to another is driven by the difference in temperature (ΔT): heat flows from high temperature to cooler temperatures (which makes ΔT negative). However, you also need to consider the distance the heat is traveling (Δx) since, given the same temperature drop, heat will flow faster if the distance is short. So, it’s best to consider the temperature gradient, which is how fast the temperature is changing over distance:

\text{Temperature gradient} = \frac{\Delta T}{\Delta x}

How fast heat flows is also mediated by the type of material (different materials have different thermal conductivities (K)), so our heat flow equation is given by the equation:

Q = - K \frac{\Delta T}{\Delta x}

Consider Node 2

HEAT IN: Heat is flowing into Node 2 from Node 1, so:

Q_{in} = - K \frac{T_2 - T_1}{\Delta x}

HEAT OUT: Heat flows out to Node 3 so:

Q_{out} = - K \frac{T_3 - T_2}{\Delta x}

At equilibrium the heat into the node equals the heat out:

Q_{in} = Q_{out}

so:

- K \frac{T_2 - T_1}{\Delta x} =  - K \frac{T_3 - T_2}{\Delta x}

Which we can simplify a lot because we’re assuming the thermal conductivity of the wall material is constant and we’ve (conveniently) made the spacing between our nodes (Δx) the same. Therefore:

T_2 - T_1 =  T_3 - T_2

And now we solve for temperature at T2 to get:

T_2 =  \frac{T_1 +T_3}{2}

Finally, we can break the fraction into separate terms (we need to do this to make it easier to solve using matricies as you’ll see later) and start using decimals.

T_2 =  0.5 T_1 + 0.5 T_2

If we do the same for all the internal nodes we get:

T_3 =  0.5 T_2 + 0.5 T_3

T_4 =  0.5 T_3 + 0.5 T_5

You should be able to see here that the temperature at each node depends on the temperature of the nodes next to it, and we can’t directly solve this because we don’t know the temperatures of the interior nodes.

A System of Equations

Let’s collect all of our information to get a system of equations:

  • T1 = 1560 K
  • T2 = 0.5 T1 + 0.5 T3
  • T3 = 0.5 T2 + 0.5 T4
  • T4 = 0.5 T3 + 0.5 T5
  • T5 = 300 K

Now to rewrite this as a matrix. We start by putting all terms with variables on the left and all the constants on the right.

  • T1 = 1560 K
  • -0.5 T1 + T2 – 0.5 T3 = 0
  • -0.5 T2 + T3 – 0.5 T4 = 0
  • -0.5 T3 + T4 – 0.5 T5 = 0
  • T5 = 300 K

Now to matrixize:

\begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ -0.5 & 1 & -0.5 & 0 & 0 \\ 0 & -0.5 & 1 & -0.5 & 0 \\ 0 & 0 & -0.5 & 1 & -0.5 \\ 0 & 0 & 0 & 0 & 1  \end{bmatrix} \begin{bmatrix} T_1 \\ T_2 \\ T_3 \\ T_4 \\ T_5  \end{bmatrix} = \begin{bmatrix} 1560 \\ 0 \\ 0 \\ 0 \\ 300 \end{bmatrix}

This you can solve on paper, and, if you’d like, check your answer using Alex Shine’s Gaussian Elimination solver, which gives step by step output (although it’s a little hard to follow). I used Alex’s solver to get the result:

Solution
Variable 1 = 1560.0
Variable 2 = 1245.0
Variable 3 = 930.0
Variable 4 = 615.0
Variable 5 = 300.0

This problem is set up to be easy to check because the results should be linear (if you plot temperature versus distance through the wall you’ll get a straight line). It will also give you whole number results up to 10 nodes.

Next Steps

Any Number of Nodes

This procedure for setting up the matrix give the same basic equations no matter how many nodes you use, because as long as the distance between the nodes (Δx) and the thermal conductivity (K) are constant, the equation for each internal node (Ti) will be:

[math] T_i = \frac{T_{i-1} + T_{i+1}}{2}

and the matrix will continue to just have three terms along the diagonal.

Non-uniform Wall and Non-uniform Node Spacing

If the wall is not uniform then the thermal conductivity coefficient does not just drop out of the equation, so you’ll have to pay attention to the conductivity going into and out of each node. Same goes with the node spacing.

Alternative Solution Methods

Students could try:

  • writing their own matrix solvers.
  • setting up the system of equations in a spreadsheet program (they’ll need to change your program’s settings so that it uses its iterative solver).

Citing this post: Urbano, L., 2017. Solving 1D Heat Transfer Problems with Matricies, Retrieved September 26th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Emotional Words

August 10, 2016

The NRC Word-Emotion Association Lexicon (aka EmoLex) associates a long list of english words with their emotional connotations. It’s stored as a large text file, which should make it easy for my programming students to use to analyse texts.

David Robinson used this database to help distinguish tweets sent by Donald Trump from those sent by his staff (all on the same twitter account).

Emotional sentiment of Donald Trump's Twitter feed.  From Variance Explained

Emotional sentiment of Donald Trump’s Twitter feed. From Variance Explained

Citing this post: Urbano, L., 2016. Emotional Words, Retrieved September 26th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Bloom’s Taxonomy

May 17, 2016

From Wikimedia: https://commons.wikimedia.org/wiki/File:BloomsCognitiveDomain.svg

From Wikimedia: https://commons.wikimedia.org/wiki/File:BloomsCognitiveDomain.svg

Bloom’s cognitive taxonomy offers a useful model for defining learning objectives. You start with the basic knowledge of the subject that requires some memorization: fundamental constants like the speed of light; fundamental concepts like conservation of mass and energy; and basic equations like Newton’s laws. On the second level, you use these basic facts and concepts to extrapolate and generalize with questions like: is the Earth an open or closed system with respect to mass and energy? And then we can start to apply our knowledge and understanding to problem solving: determine the average temperature of the Earth based on conservation of energy. Finally, at the highest level, we can analyse our models and evaluate their advantages and disadvantages.

Citing this post: Urbano, L., 2016. Bloom's Taxonomy, Retrieved September 26th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Building a 3d Printer

July 1, 2014

Building the printer.

Building the printer.

I took three students to a workshop were we’re building a 3d printer. It’s run out of the Whitfield School. We spend today putting together the electronics to run the five motors we need to get the thing to work, and starting to put the frame together.

The frame is based on the RepRap Prusa i3 (via DIY Tech Shop), and the plastic parts that hold the rods, electronics and other metal bits together are 3d printed themselves.

Students learn to solder.

Students learn to solder.

The motors is run by an Arduino, which is great because I’ve been thinking about using one to operate the doors to the chicken coops.

Notes on hardware

Microcontroller

  • Arduino Mega 2560 R3
  • RAMPS Shield 1.4
  • RepRap StepStick Pololu A4988 (I think) stepper driver (plus heat sinks)

Filament: 1.75 mm PLA

Citing this post: Urbano, L., 2014. Building a 3d Printer, Retrieved September 26th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Academic Geometry

April 17, 2014

I very much like the geometries in this picture. Even if he was studying pre-Calculus.

Working in the hall.

Working in the hall.

Citing this post: Urbano, L., 2014. Academic Geometry, Retrieved September 26th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Teaching Coding

April 14, 2014

Vicki Davis has a nice compilation of resources for teaching coding to kids of all ages. Of the fifteen things she lists, the ones I’ve used, like Scratch and the Raspberry Pi have been great.

Ms. Douglass.

Citing this post: Urbano, L., 2014. Teaching Coding, Retrieved September 26th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

How Much Homework?

March 15, 2014

As noted previously, the Finns have no homework, while the South Koreans have a lot. Yet these two countries’ educational systems are ranked 1 and 2 in the world. Misty Adoniou summarizes some of the research into the effectiveness of homework.

A key point: there are two types of homework, neither of which may be awfully useful:

  • Extra-practice: Which sometimes does not help a lot because often parents don’t have the expertise to give help when needed.
  • Creative extensions: Which students don’t necessarily need or enjoy because they’d prefer to come up with their interesting projects — if they did not have all the homework (or other distractions).

The type of homework I assign differs by subject. For science, I’ll often ask students to do reading assignments and make vocabulary cards before we cover a topic in class. It’s to give them a little preparation and, theoretically, allows us to do more higher-level, application type projects in class–this is the same as the idea behind flipped classrooms. For math, the objective is for students to get extra practice. Much of algebra and calculus relies on pattern recognition–when you can use integration by substitution for example–so some students benefit from extra practice after class.

The Dish

Citing this post: Urbano, L., 2014. How Much Homework?, Retrieved September 26th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

A Protocol for Viewing Chromosomes

January 16, 2014

My advanced biology students will be seeing if this protocol by David Frankhauser will work to stain and view chromosomes.

Citing this post: Urbano, L., 2014. A Protocol for Viewing Chromosomes, Retrieved September 26th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Creative Commons License
Montessori Muddle by Montessori Muddle is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.