Basic Flow equations

1. Dw
Dt
w
t
( w grad ) w g
1
grad p 2
w
u
t
u
u
x
v
u
y
w
u
z
X
1
p
x
(
2
u
x 2
2
v
y2
2
w
z 2
)
D w
dt
w
t
(wgrad) w g
1
grad p
18
(1)
(2)
(3)
2 Basic flow equations
In this chapter the complete flow equations are presented (1,2,4). They are not directly usable
for calculations, but all usable flow equations, both theoretical and empirical, have been
derived from them (3,4). These equations have a very general validity and the restrictions that
are necessary to deduce usable formulas have been given.
After these equations a number of dimensionless numbers (5) are presented together with
comments on in which area (heat transport, flow dynamics etc) they are used.
2.1 Navier - Stokes equation
Vectorized form with
= Laplace operator ( = /x + /y + /z )2 2 2 2 2 2 2
w = velocity
g = force (normally only gravity force)
p = density
= pressure
= /
= viscosity defined by = u/y
= shear stress at wall in flowing medium.
In cartesian coordinates this is in the x-direction
where u, v, and w are the velocities in x, y and z direction, respectively,
X = force in x-direction (usually = 0). The similar equations for y- and z-direction
are not presented here.
2.2 Euler's equations for friction less flow
(derived from 2.1 by taking w = 0)2
the vector form can be developped to cartesian coordinates which gives, for w(u,v,w)

2. u
t
u
u
x
v
u
y
w
u
z
gx
1
p
x
v
t
u
v
x
v
v
y
w
v
z
gy
1
p
y
w
t
u
w
x
v
w
y
w
w
z
gz
1
p
z
w 2
2
P U constant
where P
P2
P1
dp
U inner energy
w 2
2
p
g z constant
w 2
2
p g z constant
w 2
2g
p
z constant
w 2
2g
p
19
(4.a)
(4.b)
(4.c)
(5)
(6)
(7)
(8)
(9)
2.3 Bernoulli equation
(from Euler's equations by integrating along one streamline for steady (not varying in
time) flow)
Let P = p/ (incompressible flow, = const) and U = -g • z (gravitation as force) gives the
following wellknown expression
or
Let • g = and the following results
= the height from which a particle at rest must fall, by influence of gravity, to reach the
velocity w, i.e. the velocity height.
= the height to which a liquid pillar reaches when influenced by the pressure p against
gravity, i.e. pressure height.
z = the height of one point on the streamline, relative to a defined level.

3. t
w 2
2
P U f(t)
t
div (w 2
) 0
t
(u)
x
(v)
y
(w)
z
0
A1 w1 1 A2 w2 2 A w
A1 w2 A2 w2 A w
Re
inertia force
viscous force
u l
20
(10)
(11)
(12)
(13)
(14)
(15)
In general form for potential flow Bernoulli's equation is written
where is the potential function of the velocity field w = - grad , i.e. u = /x, v = /y,
w = /z and f(t) is a arbitrary time function
U and P see above.
2.4 Continuity equation
and in cartesian coordinates
for time-independant flow in pipes (/t = 0)
where A and A are perpendicular planes in point 1 and 2, respectively, w och w are1 2 1 2
mean velocity through each plane and and are the densities at the points.1 2
For constant density this gives for pipe flow
2.5 Dimensionsless numbers
Reynolds number (Flow inside and around bodies)
= density
u = velocity
1 = characteristic length
= viscosity.

4. Fr
inertia force
gravity force
u 2
l g
Ar
bouyant force
viscous force
g l
u 2
1
Fr
Ri
bouyant force
turbulent force
g h
u 2
Gr
(inertia force) (bouyant force)
(viscous force)2
g l 3
T 2
2
u/ l g
21
(16)
(17)
(18)
(19)
Froude number (Flow with surface waves and air jets with temperature differences)
(Sometimes is defined as Fr)
u = velocity
l = length
g = gravity force.
Archimedes number (Air jets with temperature differences)
g = gravity force
l = length
= density difference between object (e.g. air jet) and fluid (e.g. surrounding air)
= density (surrounding air)
= viscosity.
Richardson number
= density
h = length unit (usually vertical)
= density difference for each length unit
u = flow velocity
g = gravity force.
When Ri is nearing 0, the effect of the bouyant force is diminishing.
Grashof number (Free convective heat transfer)
= density
g = gravity force
l = length
= volume expansion coeff (= 1/T for ideal gases)
T = temperature difference
= viscosity.

5. Kn
free mean path for molecules
characteristic length
l1
d/2
Le
thermal diffusivity
mass diffusivity
a
D
D Cp
Sc
Pr
D / Cp
Nu
total heat transfer
conduktive heat transfer
/l
l
Num
mass transfer
mass diffusivity
D/l
l
D
22
(20)
(21)
(22)
(23)
(24)
Knudsen number (Movement of particles in air)
1 = free mean pathlength for air molecules (see 1.5)1
d = particle diameter
Lewis number (Water vaporization)
D = diffusion coefficient
= density
C = specific enthalpy (const.pressure)p
= heat conductivity
a = heat diffusivity.
Lewis relation means that Le = 1, i.e. a = D, which is nearly exact for water vapor in
air and small mass transfer velocities, then
Nusselt number (Heat transfer during flow)
= heat transfer coefficient
l = length
= heat conductivity.
Nusselt number for mass transfer (= Sherwoods number)
= mass transfer coefficient
l = length
D = diffusion coefficient.

6. Pe
heat convection
heat conduction
u l
a
u l Cp
( Re Pr)
Pe
mass transfer
mass diffusivity
l u
D
Pr
momentum diffusivity
thermal diffusivity
Cp
a
Pe
Re
Sc
Le
23
(25)
(26)
(27)
Péclet number for heat transfer (Convective flow)
= density
C = specific heat capacity (constant pressure)p
u = velocity
l = length
= heat conductivity
a = heat diffusivity = /( • C ).p
Péclet number for mass transfer (Aerosols)
l = length
u = velocity
D = diffusion coefficient.
Prandtl number (Heat transfer with flow)
= viscosity
c = specific heat capacity (constant pressure)p
= heat conductivity
= /
a = heat diffusivity = /( • C ).p

7. Sc
momentum diffusivity
mass diffusivity
D
( Le Pr)
St
heat transferred to fluid
heat transported by fluid
Cp u
Nu
Re Pr
Stk
stop distance
characteristic length
d 2
u Cc
18 L
S
L
d 2
Cc
18
24
(28)
(29)
(30)
Schmidt number (Mass transfer)
[= Prandtl number for mass transfer = Colburn number]
= viscosity medium 1
= density medium 1
D = diffusion coefficient for medium 2 in medium 1.
Stanton number (Convective heat transfer)
= heat transfer coefficient
= specific heat capacity
u = velocity
St = /8 when Pr = l (compare part 7:2 where has the nomination ).
Stoke number (Particles in air)
= viscosity
u = velocity
= density
g = gravity force
l = length
S = u • = particle stop-distance
= particle relaxation time =
L = characteristic length (usually radius of cylinder or sphere).
C = Cunningham factor, see 16.1c

8. jv
Num
Re Pr
Pr 2/3
u Cp
Pr 2/3
js

9. m
Re Sc
Sc2/3
u
Sc2/3
u Cp
u
8
25
(31)
(32)
(33)
j-factor for heat transfer
Compare part 7:2.
j-factor for mass transfer
Compare part 7:2.
For fully developped flow (in pipes, over plane surfaces or around cylinders etc) j andv
j are only depending on Re. The same goes for the pressure loss coefficient , whichs
means that Colburn's hypothesis is j = j = /8. (Compare part 7:2 where the pressurev s
loss coefficient has the nomination ).
If Pr = l and Sc = l, then Le = l, which results in