When will replacing a thermostat improve cooling?

[Hack]

70NitrousEater - you have a lot of patience! I don't think some people will listen to logic though. I didn't read every post, but I read enough to see that you explained everything very well and nobody believed you.

Oh well, what are you going to do?

Boo Boo Foo, nice of you to jump in and lend a hand - I wish I would have seen this thread earlier. It sucks when one guy who knows what he's talking about is outnumbered in a thread.

Yes, everything that 70NitrousEater and Boo Boo Foo said is 100% correct.

Edit: I'm not going to even try to explain in my own words. I think the logic has already been laid out very clearly. If you don't get it, I'm not going to be able to help.

 

[Max Power]

Actually, we don't know if either side knows what they are talking about. Nitrous is going just as much on theory as the rest of us.

Regardless of the attempted physics explanations, I have seen cars overheat due toi a lack of a stat a number of times in the trade. I have seen just as many times where people pulled the stat for an overheating problem, and the problem was elsewhere.

I have theories on why it happens, but I can't say they are anything more than that. I just know what I have seen.

 

[Route666]

You know what, after thinking about it too much, I agree with you Nitrous lol.

The only difference between thermo'd and non thermo'd but with a faster flow rate is that thermo'd will have a higher difference in temperature between incoming and outgoing coolant, whereas the non-thermo'd will have a more consistent temperature.

One cools more per unit of coolant, but one flows more coolant.

I guess a good analogy is diffs, say a 3.00 and a 4.10. Both can take a car 100 miles in the same time, but one spins more times to do it.

 

[D.Hearne]

It isn't a parabolic graph with an ideal operating flow per minute area in the middle. Heat exchange is a function of exposure to cooler mediums - period. If there is a greater flowrate, greater amounts of coolant are being exposed to the radiator's cooler surfaces per given time frame. In short, radiator dwell time is an old wive's tale. An unimpeded coolant system exposes the ENTIRE coolant to an overall greater heat exchange, rather than just a PART of it.

The dwell time in the radiator isn't an old wives tale. If you were proficient in physics, you'd know that the longer the coolant stays in the radiator, the more airflow it's exposed to, that in turn, draws off more heat. And that's not physics, just plain old common sense.

Essentially, in almost every American and European vehicle made since 1960, an unimpeded coolant system has almost always been capable of overcooling an engine unless the vehicle is driving around in Death Valley ambient temperatures.

And here again, this is also not true. I've seen TOO many 60's cars that had vastly inefficient cooling systems for that statement to hold water. It used to be a common occurance to see someone sitting on the side of the road with an overheated engine in the 60's. You rarely see it now, that the automakers have improved the cooling systems. Bottom line here is, run a car with an above average cooling system without a T-Stat, and it'll take forever and a day to reach operating temps. Run one with a below average system without a T stat and it'll overheat evertime. You can try your physics formulas in designing a system, but it still takes real world operation to find out if your calculations were accurate.

 

[67coupe351w]

Does anybody actually know roughly the amount of time it takes for the biggest chunk of energy to be lost?

If water sat in a radiator for 1 hour and you took the temperature of it then waited 1 more hour and took the temperature, they would be almost the same temperature (within a few degrees). This is because the most drastic loss of energy occurs immediatly when the Delta T is the biggest, as the temperature goes down the delta T is smaller and the rate of heat transfer is significantly reduced.

Now I'm wondering, at what point is this point of diminishing returns in a car's radiator, 1 second? 30 seconds? 1 minute? 2 minutes? more?

The more airflow you have through the radiator the smaller this time will be.

Also, I think its something around 1/2 the heat given up by a heat exchanger is done in the first 1/3 of its length. (heat exhanger = radiator)

 

[Max Power]

Does anybody actually know roughly the amount of time it takes for the biggest chunk of energy to be lost?

If water sat in a radiator for 1 hour and you took the temperature of it then waited 1 more hour and took the temperature, they would be almost the same temperature (within a few degrees). This is because the most drastic loss of energy occurs immediatly when the Delta T is the biggest, as the temperature goes down the delta T is smaller and the rate of heat transfer is significantly reduced.

Now I'm wondering, at what point is this point of diminishing returns in a car's radiator, 1 second? 30 seconds? 1 minute? 2 minutes? more?

The more airflow you have through the radiator the smaller this time will be.

Also, I think its something around 1/2 the heat given up by a heat exchanger is done in the first 1/3 of its length. (heat exhanger = radiator)

Don't forget to factor in the variable of the efficiency of the radiator and what it is made out of. I believe aluminum will give you a vastly different number than brass.

There are just so many variables in so many cooling systems that I don't think there is one answer. Example: Ford used the same radiator for 302s and 351s in 1969, and upgraded the AC cars with 351s to a big block sized radiator.

Is it safe to say that a 351-4v has the same cooling requirents as a 302 2-v? I wouldn't think so, but it appears they use the same radiator. It's just one of many variables in considering system capability. remember, Ford wanted to use one radiator for as many applications as possible to save cost.

 

[ForceFed70]

Nitrous, it CAN be worse because with a restriction the coolant can get hotter in the engine and cooler in the radiator, making the delta T when it reaches the engine or radiator greater than it would be if it were running around faster, and all lukewarm, thus energy transfer happens quicker and for longer.

Thus the system's ability to transmit energy is greater.

I agree with this statement...somewhat. BUT, look at it in regards to a car that's about to overheat. Even tho there is no thermostat in place, the temperature of the coolant is still 200* or more, thus your arguments about delta T are not relevant. Delta T is going to get as high as absolutely possible before the engine actually overheats. That's one of the reasons I'm saying that removing the thermostat won't cause it to overheat.

 

[68torino]

Sad very sad.

It is too bad some people jump into a conversation without reading all the posts. And sad that they jump in without much in the way of experience and blast people speaking from experience and try to equate their knowledge with ancient medicinal practices.

Boo Boo Foo Have you ever worked up a good sweat and jumped into a pool? I have. I have also sat in cool water not moving and slowly heated up. So I do know that I speak from experience.

My whole point that some of you in your rush to sound like you know something is this:
When you make a blanket statement such as this one.

Quote by 70_Nitrous_Eater " I'm saying that removing the thermostat won't cause it to overheat."

Someone sometime with very little knowledge or experience will remove their thermostat or someone else's and said engine may go into overheating. That person that removed the stat or their friend or family member could remove the radiator cap and recieve third degree burns.

In my thirty five plus years of motoring I have run across vehicles from the sixties and seventies that have overheated due to not having a t-stat. Some where mine, some where people I know. I have a friend who worked at a station in Murdo SD that made a lot of money towing in vehicles that had overheated. In SOME of the cases the comment from the owner was "I thought removing the thermostat would allow the water to go through faster and cool faster." This would be true in almost every case IF the radiator/airflow combo was sized large enough to accomodate the extra flow.

And to have one's experiences likened to archaic medical practices from the past would almost cause one to laugh at the person making the statement, except for the sadness I feel for the person's lack of intelligence.

 

[10secgoal]

OK... bolding mine... I came to this thread late, but if there's one thing I am an expert in, it's physics. Not necessarily all things Mustang, but Physics is definitely my bag. Hence, the bolding above is worth commenting on because it isn't exactly true. And here's why...

It's a fundamental law of physics that the rate of heat exchange between one medium and another is determined by the initial difference in temperature between those two mediums. In your example above, your assumption is that the pool water is initially quite high to begin with (which is not unfair seeing as how you did describe the day as being a hot one). However, consider the poor victims of the Titanic disaster who were thrown into the Atlantic after the ship sunk that night. The ambient water temperature that night was just 2C apparently, and as such, most of the victims died within 4 minutes - regardless of whether they tried to stay perfectly still. And the reason? The rate of heat exchange was incredibly great due to the permanent state of the ocean's temperature that night.

And my point? You can't have your cake and eat it too... if we agree that a thermostat restricts flow as an engine goes through it's warmup cycle, we can also agree that freeing up that restriction allows heat transfer to take place AFTER that restriction is removed. By extension, it's impossible to argue that freeing up that restriction EVEN MORE will somehow result in greater water temperatures once again. It isn't a parabolic graph with an ideal operating flow per minute area in the middle. Heat exchange is a function of exposure to cooler mediums - period. If there is a greater flowrate, greater amounts of coolant are being exposed to the radiator's cooler surfaces per given time frame. In short, radiator dwell time is an old wive's tale. An unimpeded coolant system exposes the ENTIRE coolant to an overall greater heat exchange, rather than just a PART of it.

The issues are maximum water pump flow rate, radiator efficiency, variable flow rate resistance, and most importantly, ambient air temperatures. The last two points are vitally important in the context of overheating. Essentially, in almost every American and European vehicle made since 1960, an unimpeded coolant system has almost always been capable of overcooling an engine unless the vehicle is driving around in Death Valley ambient temperatures. The assumption here is that the coolant system is perfectly healthy in all of it's parts. If that assumption is true, then the REAL role of a thermostat is actually to prevent overcooling, not overheating.

To that end, this is why modern Formula One cars overheat so quickly if they're sitting still on the grid for too long at the start of a race. The issue is not the thermostat, nor is it flow rate - rather, it's the fact that the still air in the radiator pods heats up to a point where vastly inefficient heat transfers are taking place.

Well, I will agree that a t-stat can keep a motor from overcooling, or overheating, depening on the efficiency of the radiator compared to the motor output. But when someone asks if it will cure their hot running car, then no, it will over heat. But for most, who do not have a 500 dollar radiator, the stocker will have to do. And putting in a higher t-stat helps for keeping it cheap.


I also don't see how you can say that the dwell time is a wifes tale. When the thermostat is closed, coolant is not moving thru the radiator. So it is not being moved to cooler medium. It is just sitting in the radiator, cooling as the air passes through the radiator fins, where the longer it sits there, the closer it becomes to the ambient air temp. I mean, to say the heat is being exchanged only by the surface ( it only does, actually exchange thru the surface), and has nothing to do with time, is saying you could pass the all of the coolant thru the radiator in a second, and it would be cool enough to maintain the motor's temperature. I see this as possible, but the radiator would have to be as big as our car.

 

[Hack]

Actually, we don't know if either side knows what they are talking about. Nitrous is going just as much on theory as the rest of us.

Regardless of the attempted physics explanations, I have seen cars overheat due toi a lack of a stat a number of times in the trade. I have seen just as many times where people pulled the stat for an overheating problem, and the problem was elsewhere.

I have theories on why it happens, but I can't say they are anything more than that. I just know what I have seen.

I believe you.

I've been trying to explain it, and I can't. All I can say is that the cooling system is complicated, and people usually remove the thermostat for a reason.

 

[10secgoal]

And to have one's experiences likened to archaic medical practices from the past would almost cause one to laugh at the person making the statement, except for the sadness I feel for the person's lack of intelligence.

Now now, not everyone is a car buff. There is a difference between ignorance and intelligence

 

[Hack]

The dwell time in the radiator isn't an old wives tale. If you were proficient in physics, you'd know that the longer the coolant stays in the radiator, the more airflow it's exposed to, that in turn, draws off more heat. And that's not physics, just plain old common sense.

That is incorrect. And it's been explained. Edit: I should say that your statement is correct, but it doesn't matter. It's not the amount of heat that is pulled from a specific "lump" of water that matters. It's the total amount of heat transferred from the contents of the radiator to the outside air.

The hotter the water in the radiator is, the faster heat comes out. If the radiator was closed off so no water could go in or out, the water would keep cooling until it reached ambient temperature. That would be the maximum heat you could get out of that water. At that point no more heat could come out. The rate at which heat comes out of the water decreases as the water temperature approaches ambient. This means that the maximum heat transfer will take place when the water in the radiator is at its maximum possible temperature. The hottest water in the system is the water coming out of the engine.

I hope that helps you. As I said, the physics have already been explained, probably better than I can do it.

I believe that you've seen cars without thermostats overheat. I don't believe that the missing thermostat is the cause. There was something else going on. A car with everything working properly that has the thermostat removed will merely run at a cold temperature, wear out faster, and sludge up.

 

[jbuening]

So what you just said was that a T-stat has nothing to do with overheating an engine. A car with a below average system will overheat with or without a T-stat. Why do you think people are putting in High volume water pumps and larger aluminum radiators? If the flow volume of the coolant in the motor cause the temps to increase, then why would anyone produce a high volume pump? I have a pretty hefty coolant system with a 28"x19" aluminum radiator with an aluminum high volume water pump. My temps are still higher than i want them to be when idling in traffic cause i still have the stock flex fan. The temperature of the engine is determined by the weakest link in the coolant system. In my case, it wouldn't be a missing T-stat but it is the amount of air flow through the radiator due to my stock flex fan.

Also stated above was the "over the years" i have seen cars overheat so they took the thermostat out. The car was already overheating because of a weak coolant system and not because of the thermostat. The motor with a capable coolant system will run way below preferable operating temps with the T-stat removed. The purpose of a thermostat is to block coolant flow in order to bring the operating temps up and then opens to maintain the operating temp. If the coolant system is unable to maintain the operating temp, then it is underdesigned.

 

[68torino]

So what you just said was that a T-stat has nothing to do with overheating an engine. A car with a below average system will overheat with or without a T-stat. Why do you think people are putting in High volume water pumps and larger aluminum radiators? If the flow volume of the coolant in the motor cause the temps to increase, then why would anyone produce a high volume pump? I have a pretty hefty coolant system with a 28"x19" aluminum radiator with an aluminum high volume water pump. My temps are still higher than i want them to be when idling in traffic cause i still have the stock flex fan. The temperature of the engine is determined by the weakest link in the coolant system. In my case, it wouldn't be a missing T-stat but it is the amount of air flow through the radiator due to my stock flex fan.

Also stated above was the "over the years" i have seen cars overheat so they took the thermostat out. The car was already overheating because of a weak coolant system and not because of the thermostat. The motor with a capable coolant system will run way below preferable operating temps with the T-stat removed. The purpose of a thermostat is to block coolant flow in order to bring the operating temps up and then opens to maintain the operating temp. If the coolant system is unable to maintain the operating temp, then it is underdesigned.

Bingo! Basically what I have been saying but maybe not as well. SOME vehicles from the sisties and seventies had poorly designed cooling systems(why give advice to a Classic 'Stanger to put in a shroud to help cooling?) and/or some other factor ie plugged due to too much stop leak, or dirty bug incrusted fins, bent fins or placing something in front of the radiator opening blocking air flow. Also the same conditions COULD cause a vehicle from the 80s and up to overheat.

When you remove the t-stat you have removed one link from the cooling system chain which Could result in overheating or Could result in overcooling. In effect you are relying on the water pump to be the temp. throttling device.

I still stand on my statement that making blanket statements that would in effect cover all engines and cooling systems and real world conditions to be irresponsible at best.

 

[r x F]

Wow, this thread needs some equations!

Let’s start with a little background on heat transfer basics.

Radiation
Conduction
Convection

Heat is transferred to the block from the combustion chamber via radiation and convection. Heat then moves via conduction from the combustion side of the block to the coolant side of the block. Next, heat is transferred via convection from the block to the coolant. Once in the radiator, heat is transferred from the coolant to the radiator metal via convection. Then the heat is conducted to the cool side of the radiator tubes. Lastly, air passing over the radiator convects the heat from the tubes.

The transfer we're concerned with is the convection from the block and to the radiator.

The equation for total heat flux for a given area via convection is:

q = h-bar*As*(Ts - T∞ )

Where,

q = total heat transfer rate, W
h-bar = average convection coefficient, W/m^2*K
As = area of the surface, m^2
Ts = surface temperature, K
T∞ = bulk fluid temperature, K

As is constant for our example (an engine block). So, heat transfer is related to h-bar and the temperature difference between the block and the coolant. The delta-T is straight forward, but h-bar is not. Let's delve into that now.


For a flat plate with fluid running parallel to it...(This is not a completely accurate representation of an engine block. The actual h-bar will be different, but the relationships will remain the same.)

h-bar = (k/L)*NuL

Where,

k = thermal conductivity, W/m-K
L = length of plate, m
NuL = Nusselt number, dimensionless

Thermal conductivity of the fluid and the length of the plate are constant. So, h-bar is directly related to the Nusselt nmber. Basically, the Nusselt number is related to how well the fluid is convecting heat. Bigger is better. This number is mostly independent of temperature. As I will show later, it is mainly related to mass flow rate (or velocity) of the fluid. As mass flow rate goes up, the Nusselt number goes up. As the Nusselt number goes up, the average convection coefficient goes up.

For a flat plate with turbulent fluid flowing parallel to it...

NuL = .037*Pr^.33*Re^.8 (This relationship is derived experimentally)

Where,

Pr = Prandtl number, dimensionless
Re = Reynolds number, dimensionless

Where,

Pr =(mu*cp)/k

Where,

mu = dynamic viscosity, cP (centipoise)
cp = specific heat, kJ/kg-K
k = thermal conductivity, W/m-K

Specific heat and thermal conductivity are constants. Viscosity is related to temperature, the colder the fluid, the higher the viscosity, and the higher the Prandtl number.

Re = (rho*u*L)/mu

Where,

rho = density, kg/m^3
u = free stream velocity, m/s
L = length of plate, m
mu = dynamic viscosity, cP

Density and length are constants, so the Reynolds number is a function of flow velocity and dynamic viscosity.

If you use some algebra, the Nusselt number simplifies out to the following,

NuL = (.037*cp^.33*L^.8*rho^.8*k^-.33)*(u^.8*mu^-.46)

All of the numbers in red are constants for any given case that you solve for. The numbers in green are variables.
What you are left with is an equation mostly related to flow velocity (mass flow rate). If velocity goes up, NuL goes up. The increase in flow velocity outweighs the increase in viscosity (which by itself will lower the Nusselt number) that might come about by a decrease in system temperature caused by increased flow rate or more efficient convection. Run the numbers if you don't believe me. The end-game is, increase in flow rate = increase in convection efficiency, regardless of temperature.

The other factor in the coolant heat transfer is the delta-T. The delta-T for a cooling system without a thermostat will be at some equilibrium point. Assuming the vehicle is at idle or some other constant load with a constant flow of air over the radiator...an increase in flow rate will decrease the coolant temperature. Whyso, you say?!

Let's go back to the original equation for convection.

q = h-bar As (Ts - T∞ )

The constant load assumption means q is constant. As is also constant. So, if you are at an equilibrium delta-T and you increase flow rate, you increase h (as I proved above). This requires that delta-T decrease to maintain the same q (heat transfer).

One thing you'll notice that doesn't appear in my derivations is time. Time is irrelevant in this analysis. Boo Boo Foo covered that part well enough.


My post isn't meant to state that running without a thermostat won't cause any vehicles to overheat. Things such as pump cavitation from too much flow can occur. This will cause the vehicle to overheat, but the owner will interpret the lack of a thermostat for the overheating. My post was simply meant to say that "dwell time" isn't true. If your car overheats without a thermostat/restriction, something else is at fault (that could be caused by high flow rate).

PS Boo Boo Foo, do you post on CC.com?

 

[DarkBuddha]

First, I'll admit I didn't read all the posts. Mainly because this is actually a pretty simple concept and there are too many cobbled statements in this thread...

Nitrous, I have to agree with others that removing a tstat can (notice I say "can", not necessarily "will") cause heat exchange problems. The answer is simple, but is complicated by the types of heat exchangers involved, including the types of medium carrying the heat, as well as the metals used for transferring the heat. There is a necessary amount of time required for any medium to absorb its maximum in heat. Equally, there is a necessary amount of time necessary to disperse its heat. Certain metals are better (or worse) conductors of heat. So the amount of time needed for a medium to reach its maximum heat saturation (without boiling) will be affected by the material being used to transfer heat to that medium.

So, with all that said, the amount of time necessary for water/coolant to absorb engine heat is probably longer than a radiator made of aluminum or cooper. So, I woud think that it would be beneficial to slow coolant going through the block enough to allow it to absorb its maximum in heat as long as the coolant doesn't boil. Once it makes it to the radiator, it should dissapate heat quickly enough to allow for a good bit of heat absorbtion on its next pass through the block. This is efficiency.

With that in mind, an engine needs to make an adequate amount of heat to have efficient combustion while not overheating either the coolant or the metals. So choosing a tstat should be a matter of choosing a temp that is reasonably below the overheating threshold, but high enough to allow the engine to reach efficient operating temp.

 

[Boo Boo Foo]

The dwell time in the radiator isn't an old wives tale. If you were proficient in physics, you'd know that the longer the coolant stays in the radiator, the more airflow it's exposed to, that in turn, draws off more heat. And that's not physics, just plain old common sense.

But it's also profoundly flawed common sense, and here's why. Yes, it's true that heat exchange is a function of temperature difference AND time exposure. However, the logic flaw which relates to the theory of "dwell time" relates to the heat exchange which takes place from the engine block INTO the engine coolant due to greater time exposure - that is the problem, not the other way around. The concept of dwell time is a myth because you can't get something for nothing with energy exchange... it's impossible to allow just a portion of an engine's coolant to reduce it's kinetic energy via the radiator without ALSO allowing a significant other portion of that engine coolant to overheat due to being exposed to the engine block for an unacceptable amount of time.

Hence, the solution is to increase dwell time in a radiator while ALSO reducing dwell time within the engine block. The way to do this is to have a high flow rate which is unimpeded PLUS a multi-core radiator which causes the coolant to traverse the radiator for a sufficiently long time to shed it's kinetic energy.

In short, many vehicles have poor radiators - and if maximum flow rate STILL results in overheating, it has nothing to do with the thermostat.

 

[67coupe351w]

Wow, this thread needs some equations!

You Sir are a heat transfer god. That kind of walk through from someone with the mental capacity to put it all together was excatly what I was wanting to see...No matter which way turned out to be true.

Thank you!

You guys really shouldnt get your panties in a wad about my blood letting statement. I wasn't calling you ignorant or stupid I was just making the same point as Boo Boo Foo that what seems to be common sense might have a flaw in its chain of logic causing a diffrent conclusion from what could be descovered through experamentation (i.e. discover that the water pump is cavitating, or maybe some other varible). And that goes for what anybody thinks, unless its proven its just an opinion and you know that they say about those.

I'm sorry you think I'm ignorant 10secondgoal, and unintellegent 68torino. I'll stick to saying what I mean the first time and skip the analogies, theres a reason I didn't major in english.

 

[DarkBuddha]

First of all, none of us need a damn degree in physics to understand any of this stuff... just some common sense and a basic understanding of how the world works.

Now, as for:

The concept of dwell time is a myth because you can't get something for nothing with energy exchange... it's impossible to allow just a portion of an engine's coolant to reduce it's kinetic energy via the radiator without ALSO allowing a significant other portion of that engine coolant to overheat due to being exposed to the engine block for an unacceptable amount of time.

Hence, the solution is to increase dwell time in a radiator while ALSO reducing dwell time within the engine block. The way to do this is to have a high flow rate which is unimpeded PLUS a multi-core radiator which causes the coolant to traverse the radiator for a sufficiently long time to shed it's kinetic energy.

You're making a huge assumption that it is necessary for the fluid to remain in the radiator the same amount of time or longer in order to dissipate more heat than it can absorb in the block. The fact of the matter is that "dwell time" is not flawed nor irrelevant. As I mentioned before, you have to consider the types of materials involved in the heat exchanges. Aluminum and copper are better conductors of heat than iron, therefore the coolant can dissipate more heat quicker than than it collects in the block.

Consider the system as a set of dwell time: the dwell time of the coolant while it is in contact with the block, and the dwell time the coolant has with the radiator. It is not necessary for the 2 dwell times to be equal. In fact, since iron is a poorer conductor of heat than aluminum (or copper), then the coolant could spend approximately 3x the amount of time in the block than would be necessary for it to cool in the radiator. How could this be possible??? Well, as long as the combined length of the coolant passages in the block is longer (or has higher volume) than the radiator, it is possible for the coolant to spend more time in the block than in the radiator and still spend adequate time in the radiator to dissipate the same amount of heat.

Makes sense, right???

 

[65up2d8]

I read every post on this thread and could not decide who was right until Dark Buddha posted. What 70 Nitrouseater said made intuitive sense; for every increase in time the water is in the radiator cooling more there is an equal and opposite time water is in the block heating more. However, the replys by D. Hearne, 68 Torino, 10 secgoal, and others also made sense; just because the amount of time the water spent heating and cooling was equal either way did not necessarily seem to mean that the result wouldn't change.
I simply couldn't put it all together nearly as well as Dark Buddha has. Although, you do have equal and opposite TIMES spent cooling and heating either way, that does not mean you will get equal RESULTS. I won't even try to explain Dark Buddha's explanations as I would simply be plagiarizing his work.
I would, however, like to add something to Dark Buddha's analysis. In addition to the differences in metals making a significant difference, the radiator has airflow assisting it in cooling whereas the engine has no equal and opposite assistance in heating the water. Sure, the harder the engine works the more heat it produces, but that would be tempered to a large extent by the same airflow over the engine.
The most amazing part? Not one single equation was required; just a solid, common sense observation by Dark Buddha.