Heat Conduction Equation

This blog is intended to summarize the Heat Conduction Equation and its derived forms in different analyses. Prerequisite to this article is basic knowledge in Heat Transfer and therefore explanation of the used symbols in the equations, as well as explanation of formulas is excluded from this writing.

Heat conduction in a medium is a dimensional process, depending on the number of dimensions in which it acts. It also varies with time, however, based on the assumptions of steady state (when the temperature does not change with time), or unsteady state (transient), one can have two approaches.
The following flowchart is intended to help distinguishing the states and to select the best suited case with formulas provided.

Steady: no change with time at any point,
Transient: variation with time and position.
(Lumped system: variation with time but not with position.)
Figure 1. Conduction main flowchart
In the Steady State Conduction the temperature is independent of the time change. One can further distinguish one dimensional heat transfer, where the temperature change varies only in one direction. We can talk about multiple dimensions as two- and three-dimensions, which are calculated in a different manned.
In case of 1D conduction, the function depends only on the x dimension, therefore the partial derivatives are replaced by ordinary derivatives.

General form of Heat Conduction Equation (HCE), in rectangular coordinates is called: Fourier- Kirchoff equation:
The general HCE equation can be derived into different special forms, depending on the assumptions and the used boundary conditions.

Boundary and initial conditions for HCE:
There are 4 main boundary conditions used or HCE, these are:

  1. Temperature of the surface (TS) at any time is given,
  2. Heat flux (qS) on the boundary at any time is given,
  3. Fluid temperature distribution (Tf) around the system and convective heat transfer (h) between the system and the fluid is given:
  4. Balance of heat fluxes on both sides of the boundary is given, (applied between 2 solids of different thermal conductivity)
Figure 2. Steady State Conduction flowchart

Figure 3. Transient Conduction flowchart

Figure 4. Transient Conduction computation flowchart

·         Yunus A. Cengel – Heat Transfer – A practical approach, second ed., 2003

·         Heat Transfer lecture notes – Maciej Jaworski, Warsaw, 2016


Hyperbolic and diagonalizable matrices


When dealing with system of Partial Differential Equations (PDE) one may have to determine if the matrix A, satisfying the equation is whether hyperbolic or not. In order to determine if a system of PDE is hyperbolic, one must determine if the matrix is diagonalizable.

To open full text: click here

Reference: Advanced Computational Fluid Dynamics, Lecture notes - Jacek Rokicki, 2014


Finite Difference method

In this post, you can see how the analysis of the accuracy of the given finite-difference formula is achieved for a first order derivative case.
In order to solve ODE problems or Partial Differential Equations (PDE) by system of algebraic equations, there are certain methods available. The Finite Difference method is probably the oldest numerical method that is used.
Figure 1. Numerical solution flowchart
It is recommended to choose a uniformly distributed grid size, having the size of X and Y components the same, due to memory limitations.

Suppose that function U(x) is given as such:
Figure 2. Selection of points on a function
One would like to estimate the first derivative of the function U(x) at some point x(j). The value of the neighboring nodes are given: uj = u(xj), uj+1, uj-1, where xj = j*h.
Figure 3. 2D final difference grid
Having a differential equation for a 2D, compressible flow, non-viscous, non-stationary:
One takes the definition of the first derivative:
If the discretization is small enough (Δx), it will approximate the value of the function as:

Similarly to Equation (1.2) one can propose different algebraic formulas for determining the determinant for a given point of the function.
It is important to notice that Equations (1.2) and (1.4) are 1st order accurate, meanwhile (1.5) is 2nd order accurate.
Figure 4. Different approximations
From the above plot, it is clearly visible that out of the 3 different formulas for finding the derivative at a given point Xj, the line that is closest to the tangent point at that point is III. This is the so called central difference and is more accurate that the other, forward difference one. The errors can be determined simply by the Taylor expansion.
Similarly to Equation (1.3) the following algebraic equations can be written:
One can conclude that the finite difference formula has order of accuracy n and is proportional to hn for small values of step size h. The central difference is 2nd order accurate and higher order terms are resulting in lower accuracy, therefore the 2nd order formula works best for calculating ODE-s and PDE-s.
Other method for deriving finite difference formulas (with different accuracy) for a given differential order remains a problem to solve.

To download full text in PDF: click here
Reference: Computational Fluid Dynamics, Lecture notes - Jacek Rokicki, 2014


How to create Half-Section View in NX Unigraphix Drafting

Demonstration the problem with Section View
The following blog is describing how the one can create break-out section views for engineering drawings, using Siemens UNIGRAHIX NX software.
I personally find NX Drafting a long, time taking process therefore it's always good to have some tricks handy. (Click on images to obtain full size.)

The problem:

If the one would like to create a half section of an object on different projections, there is a tool called "Section View" which can create a view from any parent drawing view but it will require always to have both views shown.

This can be done by simply clicking at the desired point (with Dynamic Section Line definition) or by creating a sketch for the section line (with Select Existing). This second option requires a sketch that can be created with the "Section Line" option.

However, any of these options will make the section and add the cutting lines, arrows and lettering, which is unwanted. Any half-projection is requiring a different and complex method.→ The question arises, how to make a proper half-section then?

Step-by-step solution:

Simple Assembly

(For the demonstration, a simple composite wall, with integrated sleeve and bolt is assembled. This is just a simple, quick model, no guarantee of correct alignment of elements and dimensions.)

1. Place a top view and a side view.

2. Activate the view that you would like to have half sectioned.

3. Draw a circle/Studio Spline around the area that encloses all the components that you would like to cut in half. Then Finish Sketch.

4. Select Break-Out Section option:

- "Select view for break-out creation": Select the drawing frame of the view where you want the section to be.

- "Define base point - Select object to infer point": In this step, you need to use the other view, and select a point through which the cut will be done on the other view. (in most cases it lies on a symmetry plane or a center of a circle)

- "Define the extrusion vector or continue to accept the default - Select objects to infer vector": You may accept or reverse the vector in which direction the cut will be made. An orange arrow shows this on the view where the base point was selected. If you accept direction, continue with next step without clicking anything here. If not, reverse vector.

- Click on "Select Curve" and "Select break line near start": then you need to select the previously drawn circle or spline curve.

- Click Apply and wait for the half section to be loaded.
Half-section after Break-Out View

5. Now you can delete the other view that was only needed for selecting the "Base point". This way you can only have one view and not like in the case of Sections with arrows.

Note that after completing a break-out section, no modification is possible. The same is true after deleting the second view, it has to be added again and start the whole process from the beginning.

Next part deals with correcting the view for engineering drawing compatibility. (Here is where your theoretical knowledge comes in, since any CAD software you're using, won't tell you how to do so. You as an engineer/designer need to know the correctness of your drawings!)

Adding and removing hatching on object:

It may seem in some cases that there is no hatching on a surface. If you look carefully, you will find a tiny line that is part of hatching but the distance is too large between the hatching lines. In such case, changing the distance of the hatching lines will solve the issue.

To get rid of hatching where it is not needed:

(e.g. Bolts, pins, washers, shafts should never be hatched.)

Use: Section in View:

- Select View: Drawing frame- Select Object: I recommend right-mouse-clk on the object and "Select from list..." that is hatched and select proper one corresponding to the object. In this case, you will always select the right object, not the one that NX offers first.- Action field: Select "Make Sectioned"- Ok/Apply- Right clk on drawing and update view. Only then you will see the change.

Possible problems that might occur:

According to NX detailed description, the one can check the following things to solve the problem:

  Recommended Actions

    o  Choose Ignore to restore the view to the state it was in
       prior to the update.  This is the DEFAULT action.
    o  Choose Suppress to convert the view to manual update.
    o  Choose Delete to delete the view and all associated objects.

  Possible Corrective Actions

    o  Validate and fix the model
    o  Set view to manual update
    o  Set view to reference
    o  Recreate view
    o  Remove view
    o  Suppress view update
    o  Fix invalid section lines
    o  Validate layer settings
    o  Unsuppress suppressed solid bodies
    o  Re-establish section segment associativity in
       Edit Section Line


Starting a 2-stroke (chainsaw) engine after running out of fuel

Starting a 2-stroke (chainsaw) engine** after running out of fuel

“Troubleshooting” your chainsaw

Everyone should and probably have heard that engines should never run out of fuel completely for many reasons. One of the reasons is that the last drops of fuel can bring in unwanted contaminants but other, more sensible reason is that air gets into the combustion circuit. (Even if you think about human organs, air bubble is an unwanted and can be fatal if it is present in the veins. Similarly, it can be "fatal" for engines too.) Imagine a running engine that just runs out of fuel and for a few seconds that engine will be running without a mixture. This is serious to the metal elements.

Now that I have mentioned some of the "don't-s", I can tell a solution on, how to resolve the problem if it happened. It can even take place after a long time of not being used, when you're trying to start your 2-stroke engine.
Main components under the cover (Note: not all parts are listed.)
Eventually the steps for a cold engine start are:
  1. Engage the chain brake when the chainsaw is started.
  2. Press the air purge repeatedly until fuel begins to fill the bulb. (The bulb need not be completely filled.)
  3. Pull the choke lever to full extent.
  4. Start throttling.
  5. Push the choke control to “half choke” as soon as the engine fires.
  6. Pull the starting chord until the engine starts.
What you should never do in case, it’s run out of fuel:
  • Never press the purge! By doing this, air will be pumped from the empty tank to the carburetor, that will be completely full of air and it will be difficult to get it to work.
  • Start pulling in the engine: This will move the pistons up and down without any lubrication and mixture. This can result in huge damage again!
Steps to resolve the problem:
  • Add fuel :)
  • Pull out the choke to full extent (fast idle),
  • Start pulling the starter handle 3-6x,
  • Press the air purge ~6x, 
  • Air purge: Press the air purge repeatedly until fuel begins to fill the bulb. The bulb need not be completely filled.
  • Start pulling the starter again.
  • Press air purge again to release air.
  • Do you see fuel in the purge?
    • Yes. Great, keep pulling in, the engine will start soon.
    • No. Try the following steps:
Try 1.:
  • Start pressing the purge.
  • Open the fuel tank, and while open, start pressing again. This will actually help the air to flow out, but not all. If there's too much of air, it'll not resolve the problem. But in some cases it may help. (Pay attention to have enough fuel while doing this and don’t end up sucking in air.)
Fuel lines
Try 2.:
  • Remove the cover which holds the purge button.
  • Look for the two fuel lines connected to the purge. (Picture above) One is coming from the carburetor, the other one is going to the fuel tank.
Fuel flow diagram
It may seem that the fuel flows to the purge (if you hold upside-down the engine and press a hundred times) but it'll never go beyond it, therefore this line will not be useful.
  • Try to remove the other line (the one coming from the carburetor). This will not flow, since it's the part, which is filed with air.
  • Now try to add manually fuel into this line. The best way is to use a syringe, because it can fit in the 2-3 mm inner diameter pipe.
  • Once you've added enough fuel (20-30 ml), you'll hear the air leaving the carburetor, and now you're ready to put back the pipe.
  • Start pressing the purge a few times.
  • (Now you should see fuel flowing through the purge.)
  • Assemble back the cover and do your normal starting procedure.
  • Success?

    • Yes.
    • No success? I'm afraid you need to contact a mechanic.

Engine characteristics:

Manufacturer: McCulloch
Type: CS 340

Cylinder displacement
Idle speed
Ignition system
Spark plug
Champion RCJ 7Y

Fuel tank capacity
Oil pump capacity at 8,500 rpm
Oil tank capacity

** About 2-stroke engines:

The Ideal Otto Cycle is used in all internal combustion engines. This is why it is important to know well the p-V (pressure-volume) diagram for Otto cycle. The one who is interested in the topic, is invited to read at:

Used reference: 
  • McCulloch - CS 340 Chainsaw User manual


International Conference „Young Visegrad 2016”

International Conference „Young Visegrad 2016”:
Can the Visegrad Group become a serious partner on the international arena?
On Thursday, 13th October 2016, you are invited to attend a series of public panel debates regarding foreign and economic policy in the V4 at SGH Warsaw School of Economics. The debates and evening networking session are addressed to everyone interested in the topic, and are part of the international conference “Young Visegrad 2016” coordinated by SGH Foreign Affairs Club.
The Visegrad Group has already reached its political goals. Member states’ accession to the EU and NATO was supposed to mark the end of the Group’s existence. At the time of Poland’s consecutive presidency experts wonder if the Visegrad Group is actually needed. If it is still needed, where can its new formula be found? Is it possible to say that on the 25th anniversary of the Group’s founding its idea has become just a cliché?
These and other questions will be raised on 13th October 2016 during public panel debates organised within „Young Visegrad 2016”, an international conference combined with a V4 talks simulation. The panel debates open to anyone interested in the topic will take place at the Warsaw School of Economics and will be divided into segments: morning and afternoon panels, and an evening networking session.
Next, on 14th October 2016 Polish and foreign students will play the roles of representatives of respective V4 member states. They will participate in exclusive workshops and simulation sessions during which they will talk about the V4’s joint position on the migration crisis and Brexit.
The migration crisis and Brexit are a new reality faced not only by established European powers like France or Germany, but also by newer EU member states like Poland, Czech, Slovakia and Hungary. Advocating for their joint interest, the Visegrad Group members have a chance of gaining a stronger voice on the international arena. In the experimental environment of a talks simulation, like in a laboratory, the potential future cooperation between core states of Central Eastern Europe will be put to a test.
The agenda and speakers will be available on our websites: www.youngvisegrad.pl and www.fb.com/youngvisegrad. The debates will be held in English.
The project „Young Visegrad 2016” is cofinanced by its main partner, the International Visegrad Fund (www.visegradfund.org).

Feel welcome to join us, visit our website www.youngvisegrad.pl and event: https://www.facebook.com/events/1773643492902737/


Thesis topic evolution

Many of us, engineering students undergo the following dilemma when it comes to selecting the final thesis. In this blog you will read about how my topic has been evolved during the years, what ideas were circling my mind during the semesters, until the final version of the work was chosen. Because it's just not that simple to choose.

Looking back at my 5th semester (3rd year on BSc studies), I knew that I would like to do 'something related to Aircraft Design'. Later in that year, I become more interested in landing gears and I imagined doing some design related to them.
Optimized trapezoidal strut
In the 6th semester, I was to choose an Intermediate Engineering Project (or a so-called pre-engineering thesis). This was the time when I first had to choose a topic wisely because it could have an effect on my final thesis. I was looking for a supervisor, who could give me a task about landing gears and this is how I made a project on a static analysis and a preliminary design of a UAV's landing gear.

In the following semester, I was about to continue this project and make a dynamic analysis as well and with more calculations involved, however I resigned from this topic. This was the time when I had an opportunity to work for an aircraft manufacturing company, doing some practical work for them. I was interested in structural design and I signed up for FEA (Finite Element Analysis) of an agricultural aircraft's control surface. Just before beginning the work in October 2015, I was offered an actual task about the vertical tail which has undergone some modifications and its calculations were required by the company.

Buckling calculation of stringers on the
vertical tail (Extract from thesis)
In December, the topic of my thesis was called: Structural analysis of PZL-106BT Aircraft’s Vertical Tail with a CAD/CAE Based Multidisciplinary Process. This is when the fun began with the introduction to the Multidisciplinary process. As I started working on the thesis, the work became more and more specific. I changed the main focus about the calculated case, and renamed the topic to: Multidisciplinary Structural Analysis of the Vertical Tailplane of PZL-106BT Aircraft.

This wasn't the end however, as I kept on working till May 2016, when the final topic became clear. The thesis is called: Finite Element Analysis of the Vertical Tailplane of PZL-106BT Aircraft with a CAD/CAE Based Multidisciplinary Process.
Flowchart of software compatibility for FEA (Extract from Thesis)

This would be all of my engineering thesis' evolution over the last two years. On the following timeline, this evolution is shown.
Others with similar studies might have similar workflow but for those who can't decide easily, I advise to try many things to find out about you real interests and to enjoy the work you have chosen. All the best for your future work.

More about the thesis will be presented in another article, or at special request.