"for any given shape, the ratio of surface area to volume increases with decreasing size"
- Thread starter teuchter
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free spirit
more tea vicar?
I'm fairly sure that's true.
it's why smaller hot water tanks have proportionally bigger heat losses than bigger ones, as the surface area is bigger in proportion to it's volume.
it's why smaller hot water tanks have proportionally bigger heat losses than bigger ones, as the surface area is bigger in proportion to it's volume.
And why Tesco have one large freezer warehouse for all their stores rather than several regional ones. I think it is Tesco anyway.
Bigger the fridge the more efficient it is!
DownwardDog
Riding a Brompton with a power meter.
1m Diameter Sphere:
Volume: 0.52m3
Surface Area: 3.13m2
Surface Area to Volume Ratio: 5.98
10m Diameter Sphere:
Volume: 523.50m2
Surface Area: 313.10m2
Surface Area to Volume Ratio: 0.60
Empiricism suggests that NS is correct.
Volume: 0.52m3
Surface Area: 3.13m2
Surface Area to Volume Ratio: 5.98
10m Diameter Sphere:
Volume: 523.50m2
Surface Area: 313.10m2
Surface Area to Volume Ratio: 0.60
Empiricism suggests that NS is correct.
quimcunx
imprimeo, lamino, distribuo
Reading the New Scientist after being out drinking?
It is true. Surface area is increasing with the square of the radius. Volume is increasing with the cube of the radius. The ratio of surface area to volume decreases with increasing radius.
The propensity of different parts of a system to scale at different rates leads to highly unintuitive outcomes. Scale a mouse to the size of an elephant, and it collapses - the strength of its legs increases with the square of the radius of its legs, while its mass increases with the cube of the effective radius of its body. There is a critical scale factor at which its legs cannot support its mass, and it falls over.
It's also why renewable power systems, which currently appear to work in their various toy applications (they currently supply less than 1% of energy demand) will never supply any significant fraction of global power demand - linear power increases require geometric resource increases (renewable energy sources are areal systems, where hydrocarbon energy sources are point systems). Geometric scaling of inputs (land, materials, manufacturing energy) quickly reaches the exhaustion point of the first critical resource. We are seeing that now with the critical rare earth elements that are essential and unsubstitutable components of the manufacturing process.
The propensity of different parts of a system to scale at different rates leads to highly unintuitive outcomes. Scale a mouse to the size of an elephant, and it collapses - the strength of its legs increases with the square of the radius of its legs, while its mass increases with the cube of the effective radius of its body. There is a critical scale factor at which its legs cannot support its mass, and it falls over.
It's also why renewable power systems, which currently appear to work in their various toy applications (they currently supply less than 1% of energy demand) will never supply any significant fraction of global power demand - linear power increases require geometric resource increases (renewable energy sources are areal systems, where hydrocarbon energy sources are point systems). Geometric scaling of inputs (land, materials, manufacturing energy) quickly reaches the exhaustion point of the first critical resource. We are seeing that now with the critical rare earth elements that are essential and unsubstitutable components of the manufacturing process.
Pickman's model
Starry Wisdom
Unsubstitutable as things currently stand, would I think be more accurate
teuchter
je suis teuchter
My erroneous empiricism had gone like this
1m Diameter Sphere:
Volume: 0.52m3
Surface Area: 3.13m2
Surface Area to Volume Ratio: 5.98
1cm Diameter Sphere:
Volume: 0.52cm3
Surface Area: 3.13cm2
Surface Area to Volume Ratio: 5.98
But I can now see that I was not comparing apples with apples.
I had also confused a ratio with a function, or something, I think.
True. "Currently" being the point when substitutes needed to have been found. We have never undergone a substitution of our primary energy source at the point of its saturation (our current status). Since substitution requires an energy margin in excess of current demand to power the substitution process, the point is rather significant.
bi0boy
Power User
Yes, nonsense.
A balloon has the same surface area and shape regardless of whether you blow it up really big or just a little bit.
It's not exactly bollocks. The ratio itself remains constant - e.g. for a sphere it's 4 * pi * r² : 4/3 * pi * r³ - but the magnitude of that ratio increases.
dylanredefined
Not a house elf a tiger
Meltingpot
Living in our pools we soon forget about the sea
I've just looked this up on the Internet. The general case is prettty hard to prove because it's difficult to find an easy way of expressing surface area as a function which would be applicable no matter what the 3-dimensional shape. I wasn't in fact able to find a proof of the general case online.
Sorry, but I don't really get this. Suppose that you have a solar energy site with, say, 100 = 10x10 solar panels (the size and energy output of each don't matter but for the sake of this let's assume both are constant). If you simply quadruple the size of the panelled area, which would give you 400 (20x20) panels each with the same output as each of the ones in the 100-panel installation, wouldn't you be able to draw four times the amount of power in total as you could with the 100-panelled system? And so on up, no matter how many more panels you added?
Sorry, but I don't really get this. Suppose that you have a solar energy site with, say, 100 = 10x10 solar panels (the size and energy output of each don't matter but for the sake of this let's assume both are constant). If you simply quadruple the size of the panelled area, which would give you 400 (20x20) panels each with the same output as each of the ones in the 100-panel installation, wouldn't you be able to draw four times the amount of power in total as you could with the 100-panelled system? And so on up, no matter how many more panels you added?
Mrs Magpie
On a bit of break...
Your solar panel analogy doesn't work as not all surfaces will catch the sun. Think of two potatoes and the fat to cook them in. Cut one into skinny McD-type chip proportions. Cut the other potato into two for roasting. The chips have a greater surface area per 100gms than a roast potato. The chips will be more fattening.
teuchter
je suis teuchter
This makes sense to me and I also don't really get what Falcon is trying to say.
Meltingpot's not talking about quadrupling the volume of a solid shape with solar panels on its surface though - rather quadrupling a flat area of panels.
quimcunx
imprimeo, lamino, distribuo
Say an elephant is 100 times longer than a mouse. If you were to scale up a mouse to the same length as an elephant but keeping its mouse shape it's surface area would be 10k times as great and its volume and mass would be 1m times as great. Legs with 100 times scaled up muscle wouldn't be up to the job.
Other 'services' that this elephant sized mouse's body needs as a mouse would have to be scaled up proportional to either surface area or volume and mass. So if a mouse's skin has 100 touch sensory receptors an elephantmouse would need not 100x100 but 10,000 x 100 of the same because its surface area is 10k times. This has knock on effects on brain size for instance. 1m times as many cells need capillaries to service them too, which means the heart needs to be bigger than a simple 100 times a mouse heart.
Not sure what he means about the renewable energy though.
Other 'services' that this elephant sized mouse's body needs as a mouse would have to be scaled up proportional to either surface area or volume and mass. So if a mouse's skin has 100 touch sensory receptors an elephantmouse would need not 100x100 but 10,000 x 100 of the same because its surface area is 10k times. This has knock on effects on brain size for instance. 1m times as many cells need capillaries to service them too, which means the heart needs to be bigger than a simple 100 times a mouse heart.
Not sure what he means about the renewable energy though.
free spirit
more tea vicar?
The link between land area and power output is linear not geometric.
2 x land area covered by solar PV equals 2 x electricity generation
10 x land area covered by solar PV equals 10 x electricity generation
all other factors being equal.
actually it can be better than linear as the inverters for larger installations should be operating more efficiently, so a 50kW inverter will probably operate around 2-4% more efficiently overall than a 3-4kW inverter.
where you do get geometric scaling it works the opposite way round to the way you describe it, with Hydro or wind for example, the power generation is proportional to the cube of the swept area of the turbine blades (Pmax =½ηρAv3). Though this obviously isn't directly related to the land area side of things
stuff_it
Too skool for cool
Never mind that a lot of large scale solar electric projects are based on heating water in a tower rather than PVs.
free spirit
more tea vicar?
I'd not go as far as to say a lot. There are a few, but that side of things has been hit by the massive PV price reductions over the last couple of years, with cheap end panels now around 1/3 the price they were 2-3 years ago.
Personally I think that CST plants are much needed for their ability to store significant amounts of energy and generate into the night, but they also have significant issues with distribution infrastructure, and the requirement for direct sunlight to actually generate anything at all, making them not much good for the UK and northern Europe directly, only via a HVDC grid from plants in North Africa, which doesn't currently exist. I'm sure it will happen mind, just a bit more gradually than the current PV deployment rates in this country and most of Europe.
teuchter
je suis teuchter
Yes I get all that. Indeed that is what the New Scientist article was about. It was the renewables bit I don't get.
quimcunx
imprimeo, lamino, distribuo
No one appears to be getting that bit, including the renewable energy expert on the thread. I'm beginning to think s/he's wrong.
e2a: the article is about mousephants?
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