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"Optimum" heat sink fins  A Practical Example
Now, let's take a look at a practical example that illustrates the rules of thumb.
This sample situation has relatively low power dissipation, but tight thermal parameters. Here is a list of the constraints:
4 W power dissipation
12x19 mm power input area
45 mm wide, 60 mm long in the flow direction maximum (but can be smaller)
15 mm total clearance height including heat sink base thickness, installation clearance, and tolerances
40 °C ambient at 3000 m altitude
65 °C maximum temperature at the chip side of the input area (thermal slug)
air speed uncertain, but finite and somewhat measurable  estimate 1.3 m/s
$3 "gasp" cost limit (makes the program manager gasp when looking at the bill of materials  never mind that the chipset runs at least 200 times that)
Right off, we can identify several major issues: spreading resistance in the heat sink base, low total heat sink height, and low air speed. What do those things mean for the fin design? Let's look at the implications of each.
§ The small heat input area relative to the allowable base area means that the heat sink base should be thick compared with the total height. As an initial guess, 3 mm should be the minimum base thickness. That leaves 12 mm for the fin height, following the first rule of thumb, "Fill the available space."
§ Low total heat sink height, and from the previous bullet a maximum of 12 mm for the fin height, means the fins can be really thin, because they are so short. The second rule of thumb recommends using the thinnest fins possible; how thin they can be depends on the manufacturing technique.
§ Low air speed implies low heat transfer coefficient, so the fins can be really thin because worries about fin efficiency (i.e. cold fin tips) become less significant than with high air speed. It also means that we really have to pay attention to air heating  the temperature of the air increases as it removes the heat from the fin area.
So we can consider "very thin" fins at most 12 mm tall. Now, what is "very thin?" Let's look at the candidates. The "gasp" cost limit of $3 pretty much precludes anything fancy. So consider the entrylevel basic extruded heat sink. The thinnest fins (rule of thumb #2) easily made by extrusion usually run around 1.5 mm thick. How about a quick fin efficiency calculation? In a previous column, I recommended against the extra effort of fin efficiency calculation for extruded heat sinks because of the relatively thick fins produced by extrusion. I think that still applies here. So let's skip that calculation and focus instead on how many fins would be needed to do the thermal job, the second part of rule of thumb #1.
The number of fins for this extruded heat sink is going to be limited by the space between the fins. For the maximum number of fins, we are going to need the smallest possible space between them. Let's go for a relatively aggressive "aspect ratio" of 10, meaning that the fin height is about 10 times the spacing. For 12 mm fins, that means the minimum spacing is 1.2 mm. How many fins will fit on a 45 mm wide base? Follow me on this math  total width = (n)*fin thickness + (n1)*space. (We decided in the last paragraph that the fin thickness was 1.5 mm.) I get n=17 for an answer. So if 17 fins don't do the thermal job, we need to understand why.
Now, how do we know whether 17 fins will do the job? I am a bit worried, with such a small spacing of only 1.2 mm, that the airflow is going to get choked off. If that's true, then reducing the number of fins might actually make the heat sink work better. As a side bonus, fewer fins will be a bit less expensive too. (Extruded heat sink cost closely follows weight.) But back to airflow: let's estimate the performance with only 1 m/s speed instead of our previous estimate of 1.3 m/s, because of my worry about choking off the airflow. First, let's get the air temperature rise using an energy balance based on the total airflow (air speed times crosssectional area of 16 fin spaces) of 2.3 e4 m3/s. Using a 30% altituderelated air density penalty estimate for 3000 m, I get about 22 °C temperature rise from that air volume absorbing the total power dissipation. This sets off alarm bells in my head, because the specs say that there is only 25 °C allowable temperature rise overall. Good thing we didn't waste any time on fin efficiency calculation. Maybe fewer fins would be better. More airflow space between the fins would reduce the air temperature rise. But will there be enough heat transfer area? (Good thing the fins fill up the vertical space.) Let's estimate the convection performance of eight fins (from q=hADT). First we need the heat transfer coefficient, h, which we can obtain from the dimensionless Nusselt number. In a previous column about heat transfer in rectangular ducts, there was a table of Nusselt numbers. For fully developed laminar flow, the simplest case, Nu = 7.54. We'll need the hydraulic diameter  about twice the spacing for very tall spaces, but close enough for an estimate at this point. For 8 fins, the fin spacing over a 45 mm total width is 4.8 mm. Then the hydraulic diameter is about 9.6 mm. So from the definition of Nusselt number (hD/kair), the heat transfer coefficient comes to approximately 20 W/m2K. This seems to be within the typical range. The temperature rise due only to convection comes to about 17 °C.
At this point, I am saying "ouch." Why? For the maximum number of fins, air temperature rise dominates the performance because the fins take up so much air space. If we solve that problem by using fewer fins, there will be too little fin area. A little optimization exercise might be in order here to identify the bestperforming number of fins, but we haven't yet considered the spreading resistance, which I expect to be significant. So, maybe extrusion isn't the right technology for this application. Let's have a quick look at the next step up: folded fins glued onto a base. It might be slightly more expensive, but let's see where the performance lands. The heat sink ought to be lighter, since the fins will be much thinner, so the material cost will be less even if the assembly labor is a bit more.
Folded fins can be made in various thicknesses, but let's follow rule of thumb #2 and go for the minimum practical thickness (about 0.3 mm) so as to maximize the airflow. If we use, say, 16 fins (it has to be an even number), then I can cut the convection rise from the last calculation for eight fins approximately in half to about 9 °C. (Getting a new heat transfer coefficient is left as an exercise for the reader.... I'm estimating.) What about the air temperature rise? For 45 mm wide overall heat sink space minus 16*0.3 mm occupied by fins, we have 6.2e4 m3/s airflow. (I went back to the original estimate of 1.3 m/s because this design is quite open with the thin fins, thus very little flow restriction.) The air temperature rise then is 8 °C, or roughly comparable to the convection temperature rise. This feels much better. Using foldedfin construction really opens up the air space and is a nice feature in this application. Now, do we need to worry about fin efficiency causing the fin tips to be too cool? That would be the next thing to check if we weren't worried about spreading resistance. But at this point, I think it's safe to say that we would want to seriously explore the sourcing and pricing structure of such a heat sink before going any further with calculations. If you're the analytical type, you can fiddle with fin efficiency, spacing, and fin thickness. But none of those things is going to affect the cost much, and you can do that while you wait for the budgetary quote from the vendor. Or you can move on to rest of the work that's piling up in your inbox!
The possibly surprising conclusion of this little analysis is that foldedfin construction is suitable for this application even though the air speed is low. Actually, it's better than extrusion because the air speed is low. The reason that might be surprising is that in the old days, foldedfin heat sinks meant very tight fin spacing (and resulting high pressure drop) and expensive brazed construction. So they were only used for high performance applications. But by using modern thermal adhesives for the assembly, and (following rule of thumb #1) only enough fins to do the job, the cost, pressure drop and thermal performance combine for a solution that's right for this particular application .
***** For more informations about how to select a heat sink see that :
http://www.electronicscooling.com/R...5/jun95_01.htm
Thumbs Up 
Received: 0 Given: 0 
"Optimum" heat sink fins  A Practical Example
Now, let's take a look at a practical example that illustrates the rules of thumb.
This sample situation has relatively low power dissipation, but tight thermal parameters. Here is a list of the constraints:
4 W power dissipation
12x19 mm power input area
45 mm wide, 60 mm long in the flow direction maximum (but can be smaller)
15 mm total clearance height including heat sink base thickness, installation clearance, and tolerances
40 °C ambient at 3000 m altitude
65 °C maximum temperature at the chip side of the input area (thermal slug)
air speed uncertain, but finite and somewhat measurable  estimate 1.3 m/s
$3 "gasp" cost limit (makes the program manager gasp when looking at the bill of materials  never mind that the chipset runs at least 200 times that)
Right off, we can identify several major issues: spreading resistance in the heat sink base, low total heat sink height, and low air speed. What do those things mean for the fin design? Let's look at the implications of each.
§ The small heat input area relative to the allowable base area means that the heat sink base should be thick compared with the total height. As an initial guess, 3 mm should be the minimum base thickness. That leaves 12 mm for the fin height, following the first rule of thumb, "Fill the available space."
§ Low total heat sink height, and from the previous bullet a maximum of 12 mm for the fin height, means the fins can be really thin, because they are so short. The second rule of thumb recommends using the thinnest fins possible; how thin they can be depends on the manufacturing technique.
§ Low air speed implies low heat transfer coefficient, so the fins can be really thin because worries about fin efficiency (i.e. cold fin tips) become less significant than with high air speed. It also means that we really have to pay attention to air heating  the temperature of the air increases as it removes the heat from the fin area.
So we can consider "very thin" fins at most 12 mm tall. Now, what is "very thin?" Let's look at the candidates. The "gasp" cost limit of $3 pretty much precludes anything fancy. So consider the entrylevel basic extruded heat sink. The thinnest fins (rule of thumb #2) easily made by extrusion usually run around 1.5 mm thick. How about a quick fin efficiency calculation? In a previous column, I recommended against the extra effort of fin efficiency calculation for extruded heat sinks because of the relatively thick fins produced by extrusion. I think that still applies here. So let's skip that calculation and focus instead on how many fins would be needed to do the thermal job, the second part of rule of thumb #1.
The number of fins for this extruded heat sink is going to be limited by the space between the fins. For the maximum number of fins, we are going to need the smallest possible space between them. Let's go for a relatively aggressive "aspect ratio" of 10, meaning that the fin height is about 10 times the spacing. For 12 mm fins, that means the minimum spacing is 1.2 mm. How many fins will fit on a 45 mm wide base? Follow me on this math  total width = (n)*fin thickness + (n1)*space. (We decided in the last paragraph that the fin thickness was 1.5 mm.) I get n=17 for an answer. So if 17 fins don't do the thermal job, we need to understand why.
Now, how do we know whether 17 fins will do the job? I am a bit worried, with such a small spacing of only 1.2 mm, that the airflow is going to get choked off. If that's true, then reducing the number of fins might actually make the heat sink work better. As a side bonus, fewer fins will be a bit less expensive too. (Extruded heat sink cost closely follows weight.) But back to airflow: let's estimate the performance with only 1 m/s speed instead of our previous estimate of 1.3 m/s, because of my worry about choking off the airflow. First, let's get the air temperature rise using an energy balance based on the total airflow (air speed times crosssectional area of 16 fin spaces) of 2.3 e4 m3/s. Using a 30% altituderelated air density penalty estimate for 3000 m, I get about 22 °C temperature rise from that air volume absorbing the total power dissipation. This sets off alarm bells in my head, because the specs say that there is only 25 °C allowable temperature rise overall. Good thing we didn't waste any time on fin efficiency calculation. Maybe fewer fins would be better. More airflow space between the fins would reduce the air temperature rise. But will there be enough heat transfer area? (Good thing the fins fill up the vertical space.) Let's estimate the convection performance of eight fins (from q=hADT). First we need the heat transfer coefficient, h, which we can obtain from the dimensionless Nusselt number. In a previous column about heat transfer in rectangular ducts, there was a table of Nusselt numbers. For fully developed laminar flow, the simplest case, Nu = 7.54. We'll need the hydraulic diameter  about twice the spacing for very tall spaces, but close enough for an estimate at this point. For 8 fins, the fin spacing over a 45 mm total width is 4.8 mm. Then the hydraulic diameter is about 9.6 mm. So from the definition of Nusselt number (hD/kair), the heat transfer coefficient comes to approximately 20 W/m2K. This seems to be within the typical range. The temperature rise due only to convection comes to about 17 °C.
At this point, I am saying "ouch." Why? For the maximum number of fins, air temperature rise dominates the performance because the fins take up so much air space. If we solve that problem by using fewer fins, there will be too little fin area. A little optimization exercise might be in order here to identify the bestperforming number of fins, but we haven't yet considered the spreading resistance, which I expect to be significant. So, maybe extrusion isn't the right technology for this application. Let's have a quick look at the next step up: folded fins glued onto a base. It might be slightly more expensive, but let's see where the performance lands. The heat sink ought to be lighter, since the fins will be much thinner, so the material cost will be less even if the assembly labor is a bit more.
Folded fins can be made in various thicknesses, but let's follow rule of thumb #2 and go for the minimum practical thickness (about 0.3 mm) so as to maximize the airflow. If we use, say, 16 fins (it has to be an even number), then I can cut the convection rise from the last calculation for eight fins approximately in half to about 9 °C. (Getting a new heat transfer coefficient is left as an exercise for the reader.... I'm estimating.) What about the air temperature rise? For 45 mm wide overall heat sink space minus 16*0.3 mm occupied by fins, we have 6.2e4 m3/s airflow. (I went back to the original estimate of 1.3 m/s because this design is quite open with the thin fins, thus very little flow restriction.) The air temperature rise then is 8 °C, or roughly comparable to the convection temperature rise. This feels much better. Using foldedfin construction really opens up the air space and is a nice feature in this application. Now, do we need to worry about fin efficiency causing the fin tips to be too cool? That would be the next thing to check if we weren't worried about spreading resistance. But at this point, I think it's safe to say that we would want to seriously explore the sourcing and pricing structure of such a heat sink before going any further with calculations. If you're the analytical type, you can fiddle with fin efficiency, spacing, and fin thickness. But none of those things is going to affect the cost much, and you can do that while you wait for the budgetary quote from the vendor. Or you can move on to rest of the work that's piling up in your inbox!
The possibly surprising conclusion of this little analysis is that foldedfin construction is suitable for this application even though the air speed is low. Actually, it's better than extrusion because the air speed is low. The reason that might be surprising is that in the old days, foldedfin heat sinks meant very tight fin spacing (and resulting high pressure drop) and expensive brazed construction. So they were only used for high performance applications. But by using modern thermal adhesives for the assembly, and (following rule of thumb #1) only enough fins to do the job, the cost, pressure drop and thermal performance combine for a solution that's right for this particular application .
***** For more informations about how to select a heat sink see that :
http://www.electronicscooling.com/R...5/jun95_01.htm
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