Miscellaneous

Electrical resistivity & conductivity

The formula to use : resistance = resistivity * Length / Area

If one uses the same units for L and A , the resistivity is expressed as "Ohms.L" (L = length unit)
For instance Ohm.mm or Ohm.cm or Ohm.m or even Ohm.in
and the product : Ohm.L * L / L^2 gives Ohm as a result.

From time to time, because of the definition ("resistance per unit of volume"),
one will find "Ohms/cm3" instead of Ohm.cm
This is obviously a mistake.

To settle, there's nothing better than an example. Let's take the copper resistivity at 20°C :
17.2 10^ -6 Ohm.mm or 1.72 10^ -6 Ohm.cm or 17.2 10^ -9 Ohm.m or 677 10^ -9 Ohm.inch

Another trap : a table may choose different units. For copper, I've seen : 0.0172 Ohms.mm2/m
This is the same value as seen before but it has to be divided by 1000 to become homogeneous.

Resistivity is sometimes defined as the resistance of a mil-foot (a wire 1 ft long with a uniform section of 0.001 inch in diam.)
If you apply the previous figures, you should find 10.346 Ohms for copper.
Do it as an exercise !

As for conductivity, it is of course the reciprocal of resistivity.
Let's take a resistivity of 17.2 10^ -6 Ohm.mm : the conductivity will be 58 139.5 / (Ohm).(mm)
The unit "1 / Ohm" was originally called the Mho. Therefore : 58 139.5 Mho/mm.
It is still used sometimes.

About 30 years ago, the name Siemens replaced "Mho". If we convert the whole line above, we get for copper :
58 139.5 S/mm or 581 395 S/cm or 58 139 535 S/m or 1 477 105 S/inch

Conductivity is mainly used for compounds with a higher resistance, like electrolytic solutions.
Example : very pure water at 20°C will have a conductivity of 4.2 microSiemens/m or 42 nanoS/cm or 4.2 nanoS/mm
( And 4.2 microS/m is equivalent to 238 095 Ohm.m or 23.8 MegaOhm.cm)
Note : add 1 ppm of NaCl and the conductivity goes to 2 microSiemens/cm (equiv. to 0.5 MOhm.cm) !

Viscosity

Dynamic (or absolute) viscosity

Imagine a liquid pressed in a thin layer between two plates : if one tries to move one plate versus the other, the liquid may exert a resistance. To achieve a constant speed, a constant force will be needed.

To simplify, consider 2 layers of liquid slipping on each other, each one "attached" to its plate. They are separated by a surface dS and the total thickness of the fluid is dN. The difference of speed between them is dV. If the dynamic viscosity is VIS, the force needed is dF, equal by definition to :

dF = VIS * dV * dS / dN

hence, the dynamic viscosity will be expressed as : VIS = dF * dN / (dV * dS)

in units, we have : Force * Length / (Length/Time * Surface)

Let's put that in good order. Remains : Force * Time / Surface
or, in SI units : newtons * sec / m^2 - also called poiseuille (Pl - not yet BIPM approved)
As 1 N / m^2 = 1 pascal, we can also express it in Pa.s

For the sake of history, there are other ways to express the dynamic viscosity.
Remember the first law of dynamics : F = ma
hence a force may be replaced with : Mass * Length / Time^2
and VIS may be expressed in : Mass * Length * Time / (Time^2 * Surface)
Some order again, and we find : Mass / (Time * Length)
or, for instance : kg / (sec * m) (which, in SI, has the same value as the N*sec/m^2)
This is obviously the kg mass and not the kg force !

Knowing this, we may now consider the various equivalencies.
The old unit was the poise (Po), equal to 1 g / (sec * cm) or 1 dyne * sec / cm^2
Its subdivision, the centipoise (cPo) was used more often
(because the dynamic viscosity of water at 20°C or 68°F is about 1 centipoise)
Adapting the units, we find : 10 poises = 1000 centipoises = 1 N*sec / m^2
Therefore 1 centipoise = 1 mN.s/m^2 = 1mPa.s

Other units :
• Using the old kg force, we have : 1 kgf*sec / m^2 = 9.80665 N*sec / m^2
• 1 lbmass /(ft.sec) = 1 poundal.sec/sq.ft = 1.48816 N*sec / m^2
• dividing by 3600, 1 lbmass /ft.hr = 1/2419 N*sec / m^2 (or 413.4 micropoiseuilles)
• 1 lbforce.sec/sq.ft = 47.88 N*sec / m^2

Maybe you'll hear the name fluidity : it is the inverse of viscosity. An old unit was the rhe, equal to 1/poise

Relative viscosity

You may also encounter a specific viscosity, which is the ratio of a fluid viscosity to that of a standard fluid (maybe water, as in the relative viscosity) but in a more general way : it will be, for instance, the viscosity of a solute versus the solvant in a solution.

Kinematic viscosity

• 1 sq.ft/sec = 0.0929 m^2/sec
• the old cgs unit was called the stoke = 1 cm^2/sec = 1/10000 m^2/sec

Thermal units

Thermal conductivity

The notions of resistivity - resistance - conductivity - conductance are exactly the same.
One can even calculate the flow of heat from a changing source (e.g. with a sinusoidal temperature.)

There are however some differences : heat is much easier to lose than electricity and one cannot discount surface exchange (convection) and radiation, but in electricity we also have to cope with magnetic fields and thermo-ionic valves !

For now, let's look at conduction alone

• with Q = quantity of heat (can be expressed in : W or J/sec - J/h - kcal/h - kcal/sec - BTU/hr - etc.)
• S = surface through which the heat circulates (in m2, sq.ft., sq.in., square centimeters, etc.)
• d = thickness of the material through which the heat has to travel ( meters, inches, etc.)
• T = difference of temperature between the two ends, across "d" in other words (in °F or °C)

In a nutshell, the heat transmission increases with "k", "S" and "T", and decreases with "d".

If we apply the formula to a piece with given sizes, we get its conductance = k * S / d
and the formula becomes : Q = conductance * T
We can now define the resistance "R", equal to 1 / conductance
and we can write : T = R * Q ( compare to Ohm's Law : U = R * I)

As you guessed, there are lots of different values for "k" according to the units used.
The official SI units would be the watt, the meter, the square mater and the kelvin (or degree Celsius, don't worry : we only use differences of temperature at this stage !)

Applying these units to the formula, we see that "k" should be expressed in W / m * K

The table here under gives some equivalencies for most of the units human mischief could invent.
Just select the line with the unit you have and read the factor in the column you want.
Example : 1 BTU/hr.ft.°F is equal to 1.73 W/m.K
Note : - in "BTU/hr.ft.°F", the surface is in sq.ft. and the thickness in ft.
- in "BTU.in/hr.ft2.°F", the surface is still in sq.ft but the thickness is expressed in inches !

Resistance is generally expressed as R and its converse - conductance - is expressed as U (especially in the building industry where it's used to indicate the heat losses of a house.)

When different materials are stuck together, one has to calculate each thermal resistance and add them together to find the total resistance. R = R1 + R2 + ...

If one uses the thermal conductances, it has to become : 1/U = 1/U1 + 1/U2 + ...
Never add conductances !
Precision : you may add conductances if you have several possible routes for the heat, like for resistors in parallel in electricity. But the general problem is more often about sticking together different materials in series - like resistors in series in electricity.

Note : Conductance will obviously be expressed in "W/K" and resistance in "K/W".

Special notes on building insulation

You may also encounter "RSI" (hopefully more and more) - standing for "Resistance Système International" and obviously its converse - conductance - is expressed in W/m2.K
A simple arithmetics shows that 1 R = 0.1761 RSI
In other words, a wall with R20 is also a wall with 3.522 RSI.
Similarly, a wall with 1 RSI will have a conductance of 1 and a loss of 1 watt per square meter per degree (celsius or kelvin)
For the same temperatures (here -10°C and +20°C), we lose 30 W per square meter
If RSI = 3.522 then the loss is : 30/3.522 = 8.518 W per square meter.
Check up : 8.518 * 0.3048^2 * 3600 / 1054 = 2.7 BTU/hr.(sq.ft) - good !

Note : this is only part of the problem. The full thermal loss of a wall has also to take into account the input of heat into the wall and the loss to the outside air, through convection and radiation. See later.

An insulating material will be defined as (for instance) R 3.9 per inch, meaning its specific conductivity is 0.2564 BTU.in/hr.ft2.°F (and you need about 5 inches to get R 20)
Using the above table, we see that it corresponds to 0.03695 W/m.K
and the specific resistivity is : 1/0.03695 = 27 RSI per meter.
To complicate matters, people use quite often "RSI per millimeter" ! So we get : 0.027 RSI/mm
"R" maybe, but not quite "SI" !

Surface exchange

Convection is a function of the gas speed vs. the solid.
You'll find lots of more or less accurate formulas. Let's give some of them :

• k = 2 + 10 * sq.rt.(speed) (for metallic surfaces)
• k = 1 + 5 * sq.rt.(speed) (for bricks, etc.)
• k = 10 + 5 * sq.rt.(speed) (for bricks and high temp. gas - metallurgy !)

Radiation is also quite tricky. A scientist will say that a black body will emit heat according to the fourth power of its absolute temperature :

k = 5.67 * 10^-8 * T^4

k in W/m2.K4 and T in kelvin.
The problem lies with the "black body" : there are very few of them ! Generally, a solid will emit between 10 and 90 % of what a black body will do !

Anyway, to use that formula, you also have to consider the temperature of the surroundings, in this form :
k = 5.67 * 10^-8 * (T^4 - t^4)
with "T" for the absolute temp. of the emitter and "t" for the surroundings.
Example : a wall at 10°C in winter (-5°C outside) will emit a maximum of 71 W/m2 (work it by yourself)

In this case, we have two different forms of heat losses and the conductances for convection and radiation have to be added - not the resistances.
This is quite an interesting problem of fluxes - you'll have to use a node equation - that's fun !

While we are at it, I'll give a personal experimental formula for heat losses vs a surface temperature, in calm air at room temp. :
k = 0.00021 * t^3 - 0.05973 * t^2 + 23.1238 * t - 861.1
in W/m2 ("t" is expressed in °C here !)
Valid from ambient to 400°C and quite useful to estimate the loss of a furnace or a duct of hot gas inside a building.

The ISO "A" Paper Series

It all starts with the A0 size (841 x 1189 mm).
Fold a sheet in two and you get A1, etc.
The usual A4, used in the rest of the world as standard "letter" size, has 1/4 of the size of A0 and 1/16 of its area. (NB = 210 * 297 mm - always rounded off to the nearest mm.)

How is A0 defined ? These are the data :
- area = 1 square meter
- ratio of height to width : square root of 2

we can therefore write :
h = w * sq.rt.(2)
and surface = w * w * sq.rt.(2) = 1 000 000 mm2
Solving this equation, we find : w = 840.8964 mm and h = 1189.2071 mm
Rounding off to the nearest mm, we get 841 * 1189 mm

Paper weight

We find for instance :

• size "A" (letter size) with 8.5 x 11 inches
• size "C" (basic for printers) : 17 x 22 inches

Example : 20 lbs paper (equivalent to 75 g/m2)
weighs 20 x 453.59237 = 9071.85 g for 2000 letter sheets
Total area : 8.5 x 11 x 0.0254 x 0.0254 x 2000 = 120.645 m2
and we find : 9071.85 / 120.645 = 75.2 g/m2.

Dynamics variations