Heat and scale
In the following, I will use the link
between size and heat quantity exchanged by an object to
explain some phenomenons you may have observed such as why a
child is getting cold faster than an adult. The "thread" will
be to answer the question: why a child does not sweat while
doing an physical effort, in contrast to an adult (this, under
some constraints to be explained later (same heat quantity
products by the effort <> same physical effort
(identical speed) for mass ~ h³)).
Scale and spatial quantities.
To better understand the notions
explained in the following, nothing is better than
experimentation. So I strongly recommand you to get some chil
games such as wood cubes or building bricks made of plastic
before to read this document.
Let's start with the influence of the
scale on an object. In this aim, we could use a sphere (like
in another document), however to help to understand, I am
going to use some kind of building block
Let's take a block to be used as a
standard. We would like, for example, to obtain a block whose
size is "twice" the original, in the sense that all its edges
are twice longer.
If its edges length are a, b and c, so
the new block would have its edges 2a, 2b and 2c large. By
extension, every distance between any two points would be
twice as large; a point situated at the center of the vertical
edge is at c/2 (respectivelyt c) from the nearest summits.
d' = 2.d
With the help of the Pythagoras' theorem,
it is not difficult to see that. For example, the distance
between the farest two summits (They are situated on the
diagonal) is, (a²+b²+c²)1/2 for the
standard block and becomes (4a²+4b²+4c²)1/2
in the other case. In consequence, there is indeed a factor
two between these distances.
This factor is the number of time we
need to carry forward the block to obtain the wished size.
Let's proceed on the same way to obtain
the factor that links the large block's surfaces to the
standard block. Let's filled each side and let's count the
number of sides we have used .
We can notice each side has surface
four times larger than the initial side. It is normal since
the surface of a rectangle is the product of two lengths; if
these are enlarged with the same factor, then the surface
increase with the square of this factor.
d'1 = 2.d1 , d'2
=> s' = d'1.d'2 = 2.2.d1.d2
If you have understood the process
until now, then you certainly have already found the solution
in the case of the volume.
The large block is formed of eight
standard blocks, from where the volume is:
v' = 2³.v
iv Generalisation: scale factor
This reasoning is not peculiar only to
a two time increase of the sizes; it is also valid with a
three, four, etc. time enlargement. That works as well with a
reducing or for non integer factors. If h is this scale factor,
then in general we get:
d' = h.d s' = h².s
v' = h³.v
So, if two objects are different by their size but not
their shape, then their dimensions would be like theses
A child and an adult are such two objetcs. Their sizes are
different, but their shapes, to some tiny rough estimates, are
identical; an adult is not the stretched version of a child
and in the opposite, a child is not a flattened adult.
As we now have solved the problem of
the size, let's work on the heat one. We assume an house hold
a post office. Customers go in and out to sent or collect mail.
The moves in or out of customers form a flux. The mail move is
also a flux. We can measure the importance of this flow by
counting the number of customers moving through the door. But
number of moving customers and flow are not the same thing;
the flow doesn't depend of the house size, in contrary of the
customers number. The link between both will be explain later
A flux is counted through a surface.
That could be a piece of a surface as a door, or a close surface
surrounding a volume such as an house. The sign of the flow
shows if the volume is filling up or emptying.
Flux through a piece of a surface Flux through a close surface "Filling" flux
ii Exchange and equilibrium
A piece of a surface define two regions
between which the flux carries elements from one region to the
other. There is an exchange between thoses regions.
This exchange can yet be done between
two close surfaces. If the flux from the region 1 to the
region 2 is equal to the flux from 2 to 1, or if the flux in
is equal to the flux out, then the total flux is nil and there
In UK, streets are made of houses
identical to each other (the street is built in once, rows of
bricks after rows all along the street ;o) ). Sometime it is
possible to walk from one to the other through a common door.
If we want to know, the flux of persons between the houses and
the street, it is enough to count only the people walking through
the main doors and to compare this output with the crossed
surface. The common wall flux becomes invisible. On the
opposite, if we separate an house with a wall, the inside flux
can appear. This is true even when the equilibrium is not
iii Link with the scale
In an standing alone house, a flux of
persons through walls with an opening can happen. If two
such houses are adjacent and share an opening in the common
wall, a flux can cross this wall. However it is now invisible
from the outside. This wall doesn't count any more in the
calculation of the output with the outside. If we do so on the
scale of a street, it is so many walls that disappear of the
balance. To sum up this, we only need to add the output of
each house with the outside. It is clear that the output of
twin houses is doubled on the front compare with an house
standing alone. But the flux of each house stays the same;
here is the link between output and flux: the flux is the
output per unit of surface (intensive, extensive). If the
flux is the same through all the walls of each house, then to
know the number of persons in transit with the street, we
only need to add all the outside surfaces.
Application to the problem
Also we can add stairs to this house.
Then there is a flux through the floors, but as it is an
inside flux, it is not used in the balance. Only the flux
through the sides is useful. It is the same if the street
houses have stairs.
Finaly, a building is a block of flats
that pass on to each others. Of course, we are not going to
consider the flux of persons through the outside walls, but
we can do this with sound, for example, or heat.
Thoses different examples show how the
output depends of the outside surface: it is the sum of the
flux over the whole surface. Now, surface depends of the
square of the scale. If the flux is the same for all flats,
then the building output will be in the same proportion as the
scale to the square.
i Temperature, source and flow
The problem deal with the heat transfert
versus the size. We have to clarify some notions related to
heat. Temperature is a measurement of the heat state of an
object. It is independant of the object size: two pans filled
with hot water at the same temperature don't make the water
warmer. As it is not dependant of the volume, it is said to be
intensive. On the contrary, the heat quantity stored by an
object depends of its size: two water pans need twice the
energy (gas, electricity, ...) to heat at the same temperature
than an only one. The amount of heat, or heat energy, is
an extensive quantity. In order to well understand, we can
draw an analogy with a water tank; the water level is the
temperature while the water volume is the heat energy.
Gathering tanks together, increases the amount of water but
not its level.
Let's back to our pans. To heat them up,
current circulates in a resistor, which produces heat. A resistor
is a heat source. There is resistors able to work in water. Our pan is then
fitted with its own heat source (this is usualy called an electric kettle).
If we gather several kettles, they are able to heat more water in the same time:
the amount of heat produced is an extensive quantity, but the source is not one (it is
in the same relation with the volume that the flow is with the surface).
Q = F.S
Drawing an analogy, we could fit our water tank with a
little rainy cloud; rain increase the water level and volume
in the tank.
Our object, now fitted with a temperature
and a heat energy, is going to exchange heat with its vicinity,
through its surface. This heat flow is whose we are interested
in here. As every flow, it is independant of the surface area
but is linked to the rate of flow through the surface.
If the tank is fitted with a tap, the
rate of flow depends of the number of tanks.
Simplistic example of an object in
i Effect of the scale factor: ratio v/s
(§ suivant très mal dit: faire une liste )
Let's sum up all this. Temperature
depends of the heat amount contained in the object, which is
proportional to the volume and depends of the amount of heat
produced by the heat source. That source also depends of the
volume, in opposite to the flow which is proportional to the
surface. So, if Qc is the heat energy produced and
Qf the one transmitted:
Qc ~ v and Qf
What effect does the scale have ? The
scale has an influence on those quantities through the volume
and the surface area. Remember,
they change like
v' = h³.v s' = h².s
So the ratio between those quantities,
proportional to v/s, change with the scale for:
Q'c / Q'f ~
v'/s' ~ h.v/s ~
h.Qc / Qf
This means that when only the size
increases, then the amount of heat produced increases faster
than the amount lost.
Before we conclude, we still have to
quickly understand how works the body in relation to heat.
(discussion des hypothèses et autres causes)
Body temperature is usually regulated at 37.2°C. To
achieve this, several means are used such as skin blood
circulation control or sweating which uses the very efficient
heat absorption of water evaporation.
Heat is produced in the whole body by
usual cells functioning, which burn 'sugar' to live. During
physical activities, muscles work more and then produce more
heat. We make the assumption that muscle mass is always in the
same proportion to the size (ie is related to the volume) and
that physical activity is the same. This allows us to assume
the heat source is a constant and independant of the size, and
so that the produced heat is proportional to the volume.
The conclusion is an evidence: an adult
produces more heat than he can lost, compared to a child. And
yet if he keeps this excess of heat, his temperature is going
to increase. This is why the body is protecting itself by
increasing the heat flow with sweating: water stores a lot of
heat and absorbs even more to evaporate. While evaporating,
water takes that heat away from the body, and cools the skin
We have just seen a child losts heat
faster than he produces it, compared to an adult. This is why
a child cool down faster.
In the contrary, a large size reduces
heat transferts. Then this allows to keep a constant temperature
more easily. This is true at the poles as well as on the
Equator: a polar bear has a size allowing him not to cool down
too quickly while an elephant reduces this way the risk to
overheat. It is to notice that both have a thickset shape that
is close to a sphere. This is normal for a sphere has the best
v/s ratio according to insulation. It is easy to notice this
with the following figure where the volume remains constant
while the surface increases from left to right.
1st September 2002
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