Index

System of units

Ver: 1
  Polyflow does not refer to any particular system of units. It is however strongly recommended to use a coherent system of units based on a unique definition of basic length, mass and time units (and temperature in non-isothermal problems). This note will help the user to describe his or her problem in a consistent system of units.

The first section is devoted to the three systems of units commonly used in fluid mechanics : the MKS system, the cgs system, and the british system. Conversion factors between these systems have been summarized.

In the second section, units of the material parameters in use in Polyflow are described, together with conversion factors.

In the third section, we indicate typical values for various material data. Four materials have been selected: a typical polymer, a typical molten metal, molten glass and air.

This allows the user to evaluate the order of magnitude of all material data.

The last section presents a list of several non-dimensional numbers, together with their physical meaning. They are useful to evaluate the physic of the problem being solved.

  THREE SYSTEMS OF UNITS Once units for length, weight, time and temperature have been defined, the units of all related material data can easily be obtained. The table hereafter summarizes those basic units. For the UK system, the inch has been selected as the units for length.
 
                    \ unit
      quantity \
 MKS
cgs
UK
length
m
 cm
 in / ft
weight
kg
g
lb
time
s
s
s
temperature
°C / K
°C / K
°F / R
  Conversion factors for length :
1 m = 100 cm = 39.37 in
1 cm = .3937 in = .01 m
1 in = .0254 m = 2.54 cm
1 ft = .3048 m = 30.48 cm

Conversion factors for mass :
1 kg = 1000 g = 2.2046 lb
1 g = .0022046 lb = .001 kg
1 lb = .4536 kg = 453.6 g

 Conversion factors for temperature :
1 K =     1 °C -     T (K) =     t (°C) + 273.15
1 °C =     1 K -     t (°C) =     T (K) - 273.15
1 R =     1 °F -     T' (R) =     t' (°F) + 459.67
1 °F =     1 R -     t' (°F) =     T' (R) - 459.67
1 °F =     5/9 °C -     t' (°F) =     32 + 1.8 t (°C)
1 °C =     1.8 °F -     t (°C) =     5/9 * [ t' (°F ) - 32 ]

VARIABLES AND MATERIAL DATA

A. Variables in Polyflow

Polyflow uses variables (fields) defined in terms of space, velocity, temperature, pressure, extra-stress, heat fluxes and flow rate. In the following table, we indicate the exponent of the elementary unit used to derive the unit of the variable.
 
field
mass
length
time
temper
space
velocity
temperature
pressure
deform. rate
extra-stress
heat flux
vol. flow rate
flow rate (2D)
force
energy
power
0
0
0
1
0
1
1
0
0
1
1
1
1
1
0
-1
0
-1
0
3
2
1
2
2
0
-1
0
-2
-1
-2
-3
-1
-1
-2
-2
-3
0
0
1
0
0
0
0
0
0
0
0
0

In MKS, cgs and UK systems, this means :
 
field
MKS
cgs
UK
velocity
force
pressure
extra-stress
energy
power
m/s
N
Pa
Pa
J
W
cm/s
dyne
dyne/cm²
dyne/cm²
erg
erg/s
in/s
lbf
psi (lbf/in²)
psi (lbf/in²)
BTU (in lbf)
HP

The conversion factors between those units are summarized hereafter.
velocity : 1 m/s = 100 cm/s = 39.37 in/s
1 cm/s = .3937 in/s = .01 m/s
1 in/s = .0254 m/s = 2.54 cm/s
force : 1 N = 10+5 dynes = .22481 lbf
1 dyne = .22481 10-5 lbf = 10-5 N
1 lbf = 4.4482 N = 4.4482 10+5 dynes
pressure : 1 Pa = 10 dyne/cm² = 1.4504 10-4 psi
1 dyne/cm² = 1.4504 10-5 psi = .1 Pa
1 psi = 6894.7 Pa = 68947 dynes/cm²
energy 1 J = 10+7 ergs = 9.4783 10-4 BTU
 1 erg = 9.4783 10-11 BTU = 10-7 J
1 BTU = 1.055 10+3 J = 1.055 10+10 ergs
power 1 W = 10+7 ergs/s = 1.3405 10-3 HP
1 erg/s = 1.3405 10-10 HP = 10-7 W
1 HP  = 746 W = 746 10+7 ergs/s

B. Units for material data All material data must be entered in a system of units which is consistent, in particular with the system of units used for the variables.
 
data
symbol
mass
length
time
temper
density
viscosity
consistency
gravity
heat capacity
therm. conduc.
therm. expan.
relax. tim
therm. convec.
radia. coeff.
surf. tension
slip. coeff.
Arrhenius coe.
Arrhenius app.
r0
fac
fac
gx-gy-gz
Cp
k
beta
facr
alpha
rad
gamma
fslip
alpha
alpha
1
1
1
0
0
1
0
0
1
1
1
1
0
0
-3
-1
-1
1
2
1
0
0
0
0
0
-2-e
0
0
0
-1
n-2
-2
-2
-3
0
0
-3
-3
-2
-1+e
0
0
0
0
0
0
-1
-1
-1
1
-1
-4
0
0
1
-1
The conversion factors between those units are summarized hereafter.
 
density 1 kg/m³ = 10-3 g/cm³ = .062428 lb/ft³
1 g/cm³ = .062428 10-3 lb/ft³ = 10+3 kg/m³
1 lb/ft³ = 16.019 kg/m³ = .016019 g/cm³
viscosity 1 Pa s = 10 poises = .20886 10-1 lbf s/ft²
1 poise = .20886 10-2 lbf s/ft² = .1 Pa s
1 lbf s/ft² = 47.88 Pa s = 478.8 poises
gravity 1 m/s² = 100 cm/s² = 3.281 ft/s²
1 cm/s² = .03281 ft/s² = .01 m/s²
1 ft/s² = .30478 m/s² = 30.478 cm/s²
heat capa. 1 J/kg/K = 10+4 erg/g/K = .238846 10-3 BTU/lb/R
1 erg/g/K = .238846 10-7 BTU/lb/R = 10-4 J/kg/K
1 BTU/lb/R = 4.1868 10+3 J/kg/K = 4.1868 10+7 erg/g/K
therm. cond. 1 W/m/K = 10+5 erg/s/cm/K = .12489 lbf/s/R
1 erg/s/cm/K = .12489 10-5 lbf/s/R = 10-5 W/m/K
1 lbf/s/R = 8.0068 W/m/K = 8.0068 10+5 erg/s/cm/K
1 BTU/h/ft/R = 1.7307 W/m/K = 1.7307 10+5 erg/s/cm/K
therm. expan. 1 ./K = .5555 ./R
1 ./R = 1.8 ./K
therm. conv. 1 W/m²/K = 1000 erg/s/cm²/K = .038068 lbf/ft/s/R
1 erg/s/cm²/K = 3.8068 10-5 lbf/ft/s/R = .001 W/m²/K
1 lbf/ft/s/R = 26.269 W/m²/K = 26269 erg/s/cm²/K
1 BTU/ft²/h/R = 5.6782 W/m²/K = 5678.2 erg/s/cm²/K
TYPICAL PHYSICAL DATA FOR SOME MATERIALS We indicate hereafter typical values in MKS of selected material data for gases, fluids and solids. A. Gases Typical material data are indicated below, as well as the temperature dependence. The reference temperature is T = 300 K.

Molecular hydrogen H2 :
ro = 0.0818874 * ( T/300 ) ** -1 kg/m³
eta = 0.896 10-5 * ( T/300 ) ** 0.678 Pa s
k = 0.183 * ( T/300 ) ** 0.71 W/m/K
Cp = 14310 * ( T/300 )  ** 0.031  J/kg/K
beta = 1/Tref  /K

 Molecular nitrogen N2 :
ro = 1.137982 * ( T/300 ) ** -1 kg/m³
eta = 0.1782 10-4 * ( T/300 ) ** 0.678 Pa s
k = 0.0259 * ( T/300 ) ** 0.773 W/m/K
Cp = 1041 * ( T/300 ) ** 0.077 J/kg/K
beta  = 1/Tref /K

Air :
ro = 1.1764 * ( T/300 ) ** -1 kg/m³
eta = 0.1846 10-4 * ( T/300 ) ** 0.65 Pa s
k  = 0.0263 * ( T/300 ) ** 0.79 W/m/K
Cp = 1007 * ( T/300 ) ** 0.104 /kg/K
beta = 1/Tref /K

B. Fluids (inelastic simulations) Typical material data for water, molten metal, molten glass and molten polymer are indicated below.   Water
ro = 1000 kg/m³
eta = 0.001 Pa s
k = 0.589 W/m/°C
Cp = 4180  J/kg/°C
beta = 24 10-5 /°C

Molten glass
ro = 2000 kg/m³
eta = 20 Pa s
k = 20 W/m/°C
Cp = 2000 J/kg/°C
beta = 8 10-5 /°C

Molten Silicium
ro = 2530 kg/m³
eta = 7 10-4 Pa s
k = 43 W/m/°C
Cp = 950 J/kg/°C
beta = 110-5 /°C
Molten polymer

ro         = 1000         kg/m3
for the Bird-Carreau law :
    - eta0 = 100000     Pa s
    - tnat = 1               s
    - n = 0.4
for the Arrhenius law :
    - alpha = 2000       K
k = 0.5 W/m/°C
Cp = 2500 J/kg/°C
beta = 20 10-5 /°C

B. Fluids (viscoelastic simulations) Typical material data for a polymer are given hereafter.
 
ro = 1000 kg/m³
eta = 1000 - 10000 Pa s
lambda = 0.01 - 1. s
for the Giesekus model :
alpha                     = 0.1 - 0.3
for the Phan Thien-Tanner model :
xi = 0.1 - 0.3
epsilon = 0.01 - 0.1
C. Solids Typical material data for iron, copper and a solid polymer are indicated below.   Iron :
ro = 7870 kg/m³
k = 40 W/m/°C
Cp = 400 J/kg/°C

Copper :
ro = 8960 kg/m³
k = 360 W/m/°C
Cp = 400 J/kg/°C

Solid polymer :
ro  = 1200  kg/m³
k = 0.5 W/m/°C
Cp = 2000 J/kg/°C

NON-DIMENSIONAL NUMBERS Non-dimensional numbers inform about the relative importance of the terms found in the conservation equations. They are based upon typical quantities such as length ,time, velocity ,...

Let us call L or R a typical dimension, V a typical velocity, g a typical shear rate and dT a typical temperature difference.