Numerical Modeling of Water Blocks

[Aesik]

Time to give this subject it's own thread. If you need to catch up on some of the other discussion, check out a good chunk of it here: http://forum.oc-forums.com/vb/showthread.php?s=&threadid=73775

First off I think it would be best to give a short background on how my particular numerical model works, which happens to be a finite difference model. A finite difference (fd) model takes the physical model and divides it up into several nodes. Generally speaking, the more nodes, the higher the accuracy, at a price. The computational power it takes increases dramatically with added nodes. Also since I am doing my current model in Excel, I have to have a cell for each node and I'm quickly running out of cells to use. I'm going to have to switch my model from being horizontal in Excel to being vertical (there are more rows than columns available.)

An fd model calculates the value of each node by comparing it to the neighboring nodes and calculating its value. All nodes have to be 'seeded' with a close enough guess at the final value in order to get convergence. The program goes through iteration after iteration until all of the nodal values meet a pre-defined convergence criteria, ie if at iteration 3004, a certain node has a value of 1.0010 and at iteration 3005 the value is 1.0014 and the convergence criteria is set at .0005, the node has 'converged' becuase it changed less than the criteria.

To apply this to a water block model, the boundary conditions must be applied to the model. These conditions consist of certain nodes haveing heat applied to them, some nodes feel the effect of convection, etc. Setting up the model is the easy part, getting the boundary conditions right is a whole other story and has caused more than a few mental breakdowns through the years. If the boundary conditions are not set up right and if the nodes are not seeded properly, convergence may never be reached.

The model I'm currently working with consists of nodes representing the block thickness and length along the channel. Input power from the cpu is represented by some of the nodes having a heat generation effect. In other words, these particular nodes are the ones adding the heat into the block. The nodes in contact with the moving fluid have several factors that have to be calculated seperate from the fd model. These include the temperature of the fluid at ever point along the path and the convective heat transfer coefficient. These numbers are calculated in a seperate spreadsheet by knowing the channel dimensions, fluid properties, flow rate, viscosity, etc. etc. etc. Also input into the fd model is the conductivity of the block material and the distance between each node.

Still with me? I know this may seem boring as hell, but it's important to have at least a minor grasp on how the model works in order to understand what it is capable of.

Now in another thread, BillA asked: "how are you modeling the channel being continously 'folded' about itself ?
and the heat source being radially applied 'across' the channel with shared walls (at an ever greater distance) ?" and "can you do a sensitivity calc to look at the incremental effect of adding a revolution, and another, etc
- each trial having the total channel resized for a constant head loss (as that is how the pump 'sees' the wb)"

Well, this is the tricky part. Each node is calculated by knowing its own boundary conditions, then taking into account the state of its immediate neighbors. For a straight channel, this is a piece of cake because a node's neighbors are just the ones right before it and the ones right after it. When a channel 'folds' or 'spirals' about itself, it brings a whole new factor into what each nodes neighbors are. Instead of just having the ones before and after, there are the ones that now neighbors because the channel snakes back on itself. This is where it gets tricky on deciding exactly what nodes affect what nodes and how to determine the weighting of each neighboring node. It can be done, it's just very tedious and time consuming and often requires alot of trial and error to get the nodal weighting right.

So long answer made short, yes, I can model the effects of adding extra spirals or length to a channel, it just a royal pain in the arse. But, to me it's a pain worth enduring and one I'm willing to bear.

The main object of this exercise is to come up with a model that will allow for changing certain variables such as channel dimensions, input power, flow rate, base plate thickness, etc. to see how each of these changes could change the performance of a block. It has never been my goal (with this particular model) to be able to exactly predict the results of specific blocks. While this is certainly possible to a certain degree of accuracy, it is still a 2-D model simulating a 3-D world. As I work towards a more intricate and sophisticated model, real world results will be more accurately predicted. Just remember that that is not the goal of this model, just a side benefit.

I'll be working with some commercial CFD (computational fluid dynamics) code shortly in particular FloWorks and FloTherm to see if and how they may be beneficial to our cause. These programs are best suited to model the flow itself in 3-D and they do a very good job at that. What I'm interested in seeing is if they can accurately model/predict the effects of the flow past the surface of a block.

 

[Aesik]

Almost forgot to answer another question from the other thread; Nihili asked "Very thin in the center, getting thicker towards the edges. Is there any way for your model to handle such a case Aesik?"

Yes, that's very possible. It just requires setting up the nodes to physically model such a geometry. It gets a little trickier because nodal 'neighbors' aren't as symmetric, but still very possible.

 

[Hoot]

I have closed the original thread that spawned this one and created the new gemini water block thread. That is to encourage people interested in either direction to participate in the new threads. If you had a numerical modeling unanswered question from the original thread, please copy it to your clipboard and re-submit it here.

Hoot

 

[Tecumseh]

There are a lot of good papers on HS design here:
http://www.coolingzone.com/Content/Library/Papers/index.html

They also have some calculators on this site.

A copy of the S. Lee "Calculating Spreading...." is there, too.

A quote from that paper needs analysis:

...depending on the magnitude of the average heat sink resistance, the spreading resistance may either increase or decrease with the base-plate thickness. If the heat sink resistance is sufficiently small, as in liquid cooled heat sink applications, the spreading resistance always increases with thickness, and an optimum thickness does not exist. On the other hand, if the heat sink resistance is large, as experienced with most air-cooled applications, the spreading resistance decreases with thickness and a finite optimum thickness exists.

The author didn't quantify the cross-over point from
sufficiently small to large.

Stating the obvious as Aesik pointed out in the old
gemini waterblock thread... If you have an air-cooled
HS, it's easy to see an optimal base plate thickness may
exist. Too thin a base and the heat flow to the fins
is restricted. Too thick and the heat never gets to the
fins. This implies an non-linear relation, a bell shaped
curve. Since the same physics must apply for both
the air-cooled and water cooled HSs, this means our
friendly neighborhood WB must always be on one
side of this curve for the S. Lee's quote above to be true.

OK, I'm rambling; it's late. Aesik, BillA, anybody...can you
help me out here?

 

[Aesik]

Tec, I dug up my copy of that paper and read it again. While he has many good points, he's left out a ton of detail that is needed in order to give creedence to his statements. The only time he even mentions water cooling is in that one paragraph that you posted. If I had a better idea of his derivation of equations, I might better understand where his statement comes from.

From my own calculations I can show how increasing thickness will reduce the temperature of the die, but only by very small amounts past about 3mm. I need to try a few more things before I come to any real conclusions.

I'm going to try to dig up S. Lee's other papers (probably have to head to the University Library) and see if he has published the actually process he used to derive his equations. Should be interesting.

When working on trying to analyze base plate thickness, I realized that my model was not accurate and even blew up at lower thicknesses. I had to decrease the element size to .00025m to get good convergence. Why this may not sound like much, what it does mean is that for every iteration, the model does nearly 100,000 calculations and it usually takes about 8000 or so iterations to properly converge. In other words, it takes a long time to run each model and I usually run 15 models (all different thicknesses) simultaneously. My poor 1050Mhz Pentium is getting a work out!

 

[Aesik]

After nearly killing my computer, I came up with this chart. This is for a block with channels .25" inches square and 10cm in total length. I varied the base plate thicknessfor both an aluminum and a copper block. It's easily seen that the aluminum has a definite 'sweet spot' at around 2.5-3.0 mm. When I was originally doing the copper, it seemed that it was just very gradually getting cooler, but after really stretching it out, it begins to show an optimum thickness at around 10mm thick. 10mm for copper may be the best spot, but after about 3mm thickness the best case improvement would only be about half a degree.

I thought this was pretty interesting to see. While these exact numbers will only hold true for this particular model and geometry, it does show how varying the thickness impacts the temperature. What is also interesting to note is that there is not that large of a difference between copper and aluminum at the 1.25-1.5mm mark, but past that the copper widens the gap rather quickly.

 

[Les56]

Aesik
I know it is early days and the model is specific but am surprised that for Cu the temp continues to fall in the 3mm to 10 mm range wheras for Al it rises .My expectations were that the "block" temperatures would follow the C/W data(for Typical CPU Die and Peltier(Waterblock) dimensions) presented by Billa in his thickness article* -- the decrease of C/W for Al is much greater than it is for Cu in this range . Would you think this is due to the choice of "spreading dimension" or my misplaced expectations.?
Apoligies for having nothing constructive but I am still getting my head round "heat-spreading resistance"(got confused in the Waterloo stuff a year ago when first read Billa,s article) and CFD is a bit new modern for me.

* http://www.overclockers.com/articles305/[/url

 

[Tecumseh]

Aesik, I don't understand your curves either. Maybe I'm
missing something, but I believe that the temps must
follow the C/W curves in all cases, and those curves
should follow a monotonic convergence to a single
value based on geometry and thermal resistance of
the metal.

My understanding is that the difference between
copper and aluminum for spreading resistance should
be a linear shift up in values for aluminum compared
to the copper.

What am I missing?

 

[Aesik]

Just a quick note since I don't have time right now to go in-depth.

C/W is a nothing more than total hack method of comparing HSF and water blocks. It has nothing to do with heat transfer physics, but rather is just a method to try to compare performance of cooling systems.

 

[Aesik]

Ok, I'm going to attempt to explain the graphs and my rather harsh earlier statement.

Let's take a look at what this C/W number is and how it's calculated:

C/W=[T(CPU)-T(AMBIENT)]/Power

All this does is take a specific HSF or water block and compare how well it is cooling the cpu relative to the ambient temperature. Note the significant lack of ANY material properties or ANY relationship to the governing equations of heat transfer. So, as I said before, C/W is a total hack number that was brought about as an extemely simplistic way of taking test results of cooling devices and comparing them. Nothing more, nothing less, and certainly nothing describing the physics behind the device.

Now spreading resistance is a more scientific attempt to model emperical data to describe the ability of the material to move heat. What it is not is a comprehensive way to look at how a cooling device performs under certain conditions. IE it does not take into account how heat is being removed from the system nor even attempt to describe any kind of boundary conditions to the system.

Ok, so what does my numerical model do? It takes a specific (and at this point very simple) geometry, uses the exact physical material constants, applies numerically simulated boundary conditions to the model and calculates the resulting temperatures of discrete nodes within the system at equilibrium. BIG difference. What I am doing is modeling a system, what C/W and spreading resistance calculations do is attempt to describe a certain attribute of a system.

So, why do the graphs appear as they do? I found the shape and general appearance of the graphs to be quite predictable, I just didn't know before how the specific numbers would be played out. When the base is thinner, the energy has a much smaller cross sectional path that it can follow away from the heatsource, but has a shorter path to the convective fluid. As the base thickens, there is a much larger area over which to dissipate the energy, but a longer path to the fluid carrying it away. The bottom of the graph shows the point of balance between the two scenerios in which the heat is most effectively removed from the entire system.

Copper has a larger optimal thickness over aluminum because of its higher conductance. There is less of a delta T across the thickness of the base, and thus the temperature of the surface of the copper in contact with the convective fluid is hotter. This promotes a higher heat transfer rate into the fluid.

Each run of the model is now taking over 3 hours to complete because I've had to reduce the convergence criteria and the element size to get accurate results. But by later tonight I should have a graph showing the results for copper at 50, 100, 200 adn 300 gph. It's pretty interesting to look at and I think the graph itself will speak volumes for itself. I've started the calcs for aluminum, but it will be awhile before they finish and I can post them.

Phew...hope that all makes sense.

 

[Tecumseh]

Clearly the model should converge on a specific point for
each block. In the limit the blocks would be pure metal
and the water circulation wouldn't matter. I am kind of
surprised that your curves don't show this yet.

 

[Aesik]

Patience grashopper...you haven't seen the graph that I'll post later today

 

[BBigJ]

All of a sudden I'm feeling real good about putting a 1/4" thick base in my waterblock....

Keep the results coming Aesik, I'm fascinated to see what comes out.

 

[Tecumseh]

Here is a plot of base spreading resistance calculated
from the equation in the S. Lee paper "Calculating spreading
resistance in heat sinks." It is for copper with a source
size of 10x10mm and a heat sink base of 50x50mm.

If you have read this paper you will find the next to last
paragraph has conclusions that are not supported in
the article. You will notice from the graph that the
spreading resistance can both increase or decrease
with a specific R0, but not both. With all values of
R0 there is a monotonic convergence to a specific
C/W for a given geometry and material.

There are discussions going on in the background as
to a validity of this equation.

 

[nihili]

Here is a plot of base spreading resistance calculated
from the equation in the S. Lee paper "Calculating spreading
resistance in heat sinks." It is for copper with a source
size of 10x10mm and a heat sink base of 50x50mm.

If you have read this paper you will find the next to last
paragraph has conclusions that are not supported in
the article. You will notice from the graph that the
spreading resistance can both increase or decrease
with a specific R0, but not both. With all values of
R0 there is a monotonic convergence to a specific
C/W for a given geometry and material.

There are discussions going on in the background as
to a validity of this equation.

I hate to be a cretin, but what's RO?

nihili

 

[Tecumseh]

I hate to be a cretin, but what's RO?

nihili

R0 in the article is the average heat sink thermal resistance.
In other words it is the thermal resistance on the other
side of the base plate from the heat source. The base
plate is sandwiched between the source and the ultimate
HS, be it conventional or water cooled.

Is that clear?

 

[nihili]

R0 in the article is the average heat sink thermal resistance.
In other words it is the thermal resistance on the other
side of the base plate from the heat source. The base
plate is sandwiched between the source and the ultimate
HS, be it conventional or water cooled.

Is that clear?

Yup, I got it now, thanks

nihili

 

[Aesik]

Here is the chart for copper as promised...

 

[Hoot]

Aesik, I for one want to take the time and thank you for the obvious effort and time that is going into these graphs. Though I may sound at times like some Luddite, I use numerical modelling software for analyzing radio antennas and I believe in it because it works. Like a lot of people who do not necessarily speak up, I look forward to your graphs. I eagerly await the flowrate aspect you are crunching.

Hoot

 

[Aesik]

Ok, let's try this again....