Upload
vukhanh
View
234
Download
2
Embed Size (px)
Citation preview
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Fluid and Particulate systems 424521 /2018
EXTERNAL FLOWS /FLOW IN POROUS STRUCTURES
Ron ZevenhovenÅA Thermal and Flow Engineering
3F
luid
& P
artic
ulat
eS
yste
ms
4245
21 /
201
0Fl
uid
& P
artic
ulat
eSy
stem
sÅ
A 4
2452
1 /
201
8
3.1 Flow around objects, drag forces, lift forces
Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
RoNz 2februari 2018
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Flow Around BodiesF
luid
& P
artic
ulat
eS
yste
ms
4245
21 /
201
0Fl
uid
& P
artic
ulat
eSy
stem
sÅ
A 4
2452
1 /
201
8
Basic concept
carlosscar wFP
212121 ,0, ppwwzz
Fw
wFP
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
februari 2018 Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
5/52
Fluid flow around objects In the cases of
– an object moving through a fluid– a fluid flow around an object
the velocity difference generates forces Forces acting parallel to the flow direction are drag
forces; forces acting perpendicular to the flow direction are lift forces
The flow field around an object can be divided in two parts: the boundary layer, where the viscous forces are active, and the free-stream velocity (or the stagnant surrounding fluid) P
ictu
re: h
ttp://
ww
w.w
eird
richa
rd.c
om/im
ages
/forc
es.jp
g
PTG / PRC
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
februari 2018 Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
6/52
Fluid flow over a surface
CRBH83
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Flow over a flat plate
F
Consider air flowing above a thin plate in x-direction with the undisturbed velocity w∞. Due to viscosity (fluid friction), the air layer closest to the plate has w = 0, and a velocity gradient in y-direction appear. The presence of the plate is hence “felt” up to a distance δv in y-direction; a boundary layer occur. Beyond, w∞ remains unchanged.
Since the air layers have a velocity gradient but are still attracted to each other, one layer will try to drag an adjacent layer along. Consequently, the plate is affected by a (drag) force F in the air flow direction. The friction drag force Fdivided by the area of the plate gives the shear stress.
februari 2018 RoNz 7Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
• For flow along a flat plate, the forces on the plate are frictionforces. The shear stress on each side of the surface is
with (laminar) boundary layer thickness δ and relative velocity vr
• The drag force on each side of a plate with length L and width b is then given by
• The pressure ½ρv2 is known as THRUST (sv: stöt)
Flow around a flat plate /1
5
fluid
fluidr 103η
xρv for ,
rfluid
½
fluid
fluidrrfluidwyyx v½ρ
η
ρxv.
δ
vηττ
5
fluid
fluidrL
2
103η
ρLv Re for
rfluid
½
fluid
fluidr
L
wDdrag
v½bLη
Lρv
dxbFF
331
0
.
Picture: BMH99
PTG / PRC
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Flow around a flat plate /2
This defines the (length-averaged) drag coefficient CD as
where A (m2) is the area (one side) of the plate For turbulent cases, experimental results give
For a flat surface with a laminar region followed by a turbulent region, a ”composite” drag composition can be calculated with
For a flate plate perpendicular to a fluid the drag coefficient equals CD~2 , largely independent of Re-number
9L
72.58
L10D
7L
51/5L
D 10Re10 for )log(Re
0.445 C ;10Re10 for
Re
0.074 C
5L
LD 103Re for
Re
1.33 C withvACF rDD
2½
Re
1740
Re
0.074 C
L1/5L
D Picture: KJ05
Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
februari 2018 9/52
PTG / PRC
Transition regionRe ~ 5×105
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Calculation of total drag
F
2
2
Dprj
w
CAF
w∞
CD: drag coefficient
Aprj: projection area (area normal to the direction of the flow, datum area)
w∞: undisturbed fluid velocity
februari 2018 RoNz 10Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
februari 2018 Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
11/52
For a general surface area A ┴
(m2) perpendicular to the flow, the drag force is
FD = CD·A┴·½ρvr2
(where ½ρvr2 is actually the pressure
difference between the front and the back of the object)
with drag coefficient CD
With increasing Re-numbers, boundary layer separationoccurs, and a wake region (sv: köl(vatten)) ariseswhere kinetic energy is only partlyconverted into pressure
Picture: KJ05
Flowaround a cylinder
Flow around cylinders, spheres /1PTG / PRC
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Flow around a cylinder
Fw∞
a
Consider air flowing across a cylinder. Air will attach to the cylinder and boundary layer and a friction drag force occur, as in the case of the plate.
221
0 wpp The flow will then divide into an upper and a lower flow, and the pressure starts to decrease as the air velocity increases. The boundary layer detaches from the cylinder at the separation point c, and a wake is formed in the rear. The pressure in the rear will hence be lower compared to the front, and an additional drag force, pressure drag, will further increase the total drag force acting on the cylinder.
Additionally, the undisturbed air has a pressure p∞ and a velocity w∞, at a point a before it acts on the cylinder. The air flow at the cylinder is stopped the stagnation point b, where the pressure increases according to
februari 2018 RoNz 12Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Detachment of Boundary layerStarts atRe ~ 10,Complete atRe ~100
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Friction and Pressure Drag
februari 2018 RoNz 13Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Pressure,forces
Shear, tangentialforces
Stress tensor (cartesian) τxxτyyτzz τxyτyxτxzτzxτyzτzy
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Friction and Pressure Drag
februari 2018 RoNz 14Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Picture: KJ05
τ forceshear P,force pressure with
cossin
sincos
AsurfaceAsurface
lift
AsurfaceAsurface
drag
dAdAPF
dAdAPF
Pressure and shear stresses acting on a cylinder (103 < ReD < 105).
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
F w∞
Pressure drag and separation point location
F w∞
Low w∞ (Re<2ꞏ105), early separation, high pressure drag.
High w∞ (Re>2ꞏ105), late separation (transition to turbulent flow), low pressure drag.
februari 2018 RoNz 15Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
februari 2018 Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
16/52
Flow around cylinders, spheres /2
For spherical particlesthe drag coefficientequals
For flow at Re <0.1 around a sphere, the relation CD=24/Re
follows also from Stokes’ law
Fdrag = 3πηvrd
for a sphere with diameter d and relative velocity vr = vsphere-vflow
Picture: KJ05
5
D
32
D
D
D
10Re800 for
0.44 C
800Re2 for
Re6
11
Re
24 C
2Re0.2 for
Re16
31
Re
24 C
0.2 or 1Re forRe
24 C
Picture: http://www.school-for-champions.com/science/friction_changing_fluid.htm
PTG / PRC
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Flow around cylinders, spheres /3
Roughness effects on the drag coefficient of a sphere.
februari 2018 Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
RoNz 17
Picture: KJ05
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
februari 2018 Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
18/52
Boundary layer separation examples
Pic
ture
: http
://w
ww
.aer
ospa
cew
eb.o
rg/q
uest
ion/
aero
dyna
mic
s/q0
215.
shtm
l
Pic
ture
: http
://w
ww
.str
uctu
ral.n
et/n
ews/
Med
ia_c
over
age/
med
ia_p
lant
_ser
vice
s.ht
ml
PTG / PRC
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
februari 2018 Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
19/52
A smooth (a) and roughened (b) ball entering water at 25 °C
CR83
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Drag coefficient CD
9.1, 9.2februari 2018 RoNz 20Åbo Akademi University - Värme- och Strömningsteknik
Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Lift - why?
The principle of conservation of angularmomentum demands that the angularmomenta of the two rotational patterns tobe equal and opposite.
9.3
w
p
februari 2018 RoNz 21Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Picture:KJ05
Pressure and shear acting on an airfoil.
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Example: drag on a flat plate
An advertising banner (1 m x 20 m) is towed behind an aeroplane at 90 km/h, in air at 32°C.
Calculate the power (in kW) needed to pull the banner.
Answer: The power P needed can be related to the drag force byP = FDꞏvr, where vr = 90 km/h = 25 m/s and FD = CDꞏAꞏ½ρvr
2, with CD depending on the Re-number
At 32°C (assuming p = 1 atm) ρair = 1.161 kg/m3 and η = 1.875×10-5 Pa.s. With a length scale L = 20 m → Re = 3.1×107
Calculating with gives CD = 0.0024
With surface (2 sides!) A = 40 m2 this gives FD = 35 N and P = 875 W = 0.875 kW
Re
1740
Re
0.074 C
L1/5L
D
Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
februari 2018 RoNz 22
PTG / PRC
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
3.2 Flow through or along tubebundles
Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
RoNz 23februari 2018
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Flow in tube heat exchangers(outside the pipes)
Latitudinal direction
Longitudinal direction
Staggered orderIn-line order
februari 2018 RoNz 24Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
parallel
perpendicular
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Flow in the longitudinal directionCross section, in-line order
daꞏd
bꞏd
The pressure loss for this case can becalculated by the theory for flow in non-circular ducts.
dbad
1
4h
9.4
Cross section, staggered order
bꞏd
aꞏd
cꞏdd
februari 2018 RoNz 25Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Hydraulic diameter
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Flow in the latitudinal direction
The pressure loss can be calculated according to
2
2max
Rloss
wNp
In-line order Staggered order
wmax: maximal velocity
NR: how many times the maximal velocity appear
ζ: dimensionless flow resistance coefficient
februari 2018 RoNz 26Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Flow in the latitudinal direction
daꞏd
bꞏd
bꞏd
aꞏd
cꞏdd
In-line order
Staggered order
februari 2018 RoNz 27Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
In-line order
daꞏd
bꞏd
w∞
L
1max, a
aww a
For the control volume
The maximal velocity is reached between two pipes
februari 2018 RoNz 28Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Staggered order
bꞏd
aꞏd
cꞏdd
w∞
The maximal velocity can occur at two places
februari 2018 RoNz 29Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Staggered order
bꞏd
aꞏd
cꞏdd
w∞
Maximal velocity in the vertical passage
L
1max, a
aww a
februari 2018 RoNz 30Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Staggered order
bꞏd
aꞏd
cꞏdd
w∞
L
Maximal velocity in the diagonal passage
12max, c
aww c
februari 2018 RoNz 31Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Staggered orderac ww max,max,
222
21 dcdbda
2221 bac
The maximal velocity in the diagonal passage is achieved if
Utilizing the theorem of Pythagorasbꞏd½ꞏaꞏd
cꞏdd
w∞gives
which also can be written as 112
a
aw
c
aw
The expression of c inserted in the inequality gives 1221 ab
The maximal velocity in the diagonal passage is hence achieved if
1221 ab
Otherwise, the maximal velocity occurs in the vertical passage
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
How many times the maximal velocity appears
daꞏd
bꞏd
w∞
L
For in-line order
as many times NR as the number of pipe rows NB
NR = NB
februari 2018 RoNz 33Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
How many times the maximal velocity appears
For staggered order when the maximal velocity occurs in the vertical
passage when
bꞏd
aꞏd
cꞏdd
w∞
L
awmax,
1221 ab
as many times NR as the number of pipe rows NB
NR = NB
februari 2018 RoNz 34Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
How many times the maximal velocity appears
cwmax,
bꞏd
aꞏd
cꞏdd
w∞
L
For staggered order when the maximal velocity occurs in the diagonal
passage when 1221 ab
one time less NR than the number of pipe rows NB
NR = NB - 1
februari 2018 RoNz 35Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018 2
2max
Rloss
wNp
Re
turblam
Re f
ff
dw
maxRe
),( bafflam
),(turb baff
Re),,(Re baff
from diagram
from diagram
from diagram
9.5
Dimensionless flow resistance coefficient
flam=f(a,b)
More correction factors are needed whenNB < 5, or if density, temperature or viscositychange significantly
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Dimensionless flow resistance coefficient
fRe=f(Re)
fturb= f(a,b)
februari 2018 RoNz 37Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
3.3 Excercises 9
Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
RoNz 38februari 2018
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Exercises 9
9.1 A wire grating is placed in water (5ºC), which is flowing with a velocity of 4 m/s. The wire grating is made of steel wire that has a diameter of 2 mm, and the mesh has a length of 50 mm. Calculate the drag per m2 of the wire grating.
9.2 The total drag in the case when a fluid flows around a sphere can at low velocities (Rep<0.2) be described by the law of Stokes,
Derive an expression for the drag coefficient CD as a function of Re for fluid flow around spheres at low velocities.
9.3 The lift of an airplane wing was measured to 2000 N per m2 of the wing when the velocity of air below the wing was 120 m/s. Calculate the velocity of air above the wing.
februari 2018 Åbo Akademi University - Värme- och StrömningsteknikBiskopsgatan 8, FI-20500 Åbo / Turku Finland
RoNz 39
wdF 3
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Exercises 9
9.4 Derive the expression for dh for longitudinal flow in a tube heat exchanger that have the tubes in in-line order, and for a tube heat exchanger that have the tubes in staggered order.
9.5 Water of 80ºC is flowing in a tube heat exchanger in the latitudinal direction. The tubes have a staggered order (a=1.75, b=1.53, d=17 mm). At what velocity w∞ is the needed theoretical pump power equal to 1 kW per m3 of the heat exchanger?
februari 2018 Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
RoNz 40
dbad
1
4h
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
3.4 Flow in Porous Materials
Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
RoNz 41februari 2018
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Porosity
Total porosity εtot: volume of all pores / total volume
Effective porosity ε : volume of open pores / total volume
8.1februari 2018 RoNz 42Åbo Akademi University - Värme- och Strömningsteknik
Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Calculation concept
w
l
p~loss
Definition: Superficial flow (= fictive) velocity through a porous material
Acs is the whole cross sectional area, not only the area of the open pores.
Hypotheses:
2loss ~ wl
p
which is equivalent to laminar flow
which is equivalent to turbulent flow
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Law of Darcy and permeability
w
l
p~loss
Experiments have shown that agrees with flow in materials
with small pores when the velocity is low.
The relationship is called law of Darcy and can be written
wkl
p 1loss
The inverse value of the proportional constant k is called permeability, which means “how good a material let a fluid flow through it”.
8.2februari 2018 RoNz 44Åbo Akademi University - Värme- och Strömningsteknik
Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Often, vertical position changes must be considered:
)( 2121loss zzgppp
With l = z1-z2: )(
)(
21
2121
zz
zzgppkw
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Porosity and Permeability
Material Porosity ε Permeability k [m2]
loose sand 0.37 – 0.50 20 – 180 (ꞏ10-12)
soil 0.43 – 0.54 30 – 140 (ꞏ10-12)
sand (of stones) 0.08 – 0.38 0.0005 – 3 (ꞏ10-12)
lime stone 0.04 – 0.10 0.0002 – 0.0004 (ꞏ10-12)
brick 0.12 – 0.34 0.005 – 0.22 (ꞏ10-12)
leather 0.56 – 0.59 0.096 – 0.12 (ꞏ10-12)
fossil meal 0.37 – 0.49 0.013 – 0.05 (ꞏ10-12)
8.3
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Flow in packed beds
2loss ~ wl
p
When the pores are bigger and the velocity is higher, the flow will, at least partly, be turbulent. In such a case, following hypothesis is more agreeable than the law of Darcy.
A dimensionless flow resistance number K is used in the same way as ζ = 4 f in calculation of the pressure loss in pipe flow.
p
2loss
2 d
wK
l
p
The length quantity dp is defined as the diameter of an equivalent sphere, which have the same volume as the non-spherical particle that the bed is made up of.
dpV
V
februari 2018 RoNz 46Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Flow in packed bedsDarcy’s Law:
with permeability K
Kozeny - Carman equation:
Sv = specific surface = surface/volume
Sv = 6 /dp for a sphere with diameter dp
L
LS
pu
fluidv
2
3
15 )(
L
pKu
fluid
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Ergun equation
5.31
Re
3001
p3
K
The dimensionless flow resistance number K can be described by the Ergun equation,
Ψ: shape factor, which is the surface area of the equivalent sphere divided byThe surface of the actual particle that the bed is made up of (≤1)
Rep: Reynolds number for the particles
p
pRedw
ε: bed porosity
8.4
februari 2018 RoNz 48Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
Note: also fluidised beds for gas flowsbelow fluidisation velocity are packed bedsAT the point of minimum fluidisation_Δppacked bed = Δpfluidised bed
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
3.5 Excercises 8
Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
RoNz 49februari 2018
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Exercises 8
8.1 A 150 cm3 sample of a porous plastic material have a dry weight of 104 g. The weight of it, when soaked in water (10ºC), is 147 g (the air has first been removed with vacuum). The compact density of the material is 1150 kg/m3.a) Calculate the total porosity.b) Calculate the effective porosity.
8.2 Calculate the volumetric flow of water (20ºC) through the sand filter below. The cross sectional area of the filter is 3 m2 and the permeability of the sand is 20·10-12 m2.
februari 2018 Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
RoNz 50
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Exercises 8 8.3 The permeability is sometimes given in the unit of darcy = cP cm2 / (s atm).
What is the factor to convert darcy into m2.
8.4 a) Calculate the pressure loss when 3.0 m3/s of a gas (ρ=0.52 kg/m3, η=34·10-6 kg/ms) is flowing through a packed bed with a height of 0.60 m. The packed bed is made up of cylindrical catalyst particles with a length of 8 mm and a diameter of 5 mm. Experiments have showed that the porosity ε is 0.34. The cross sectional area of the bed is 3.8 m2.b) Calculate the superficial gas velocity w at which the pressure loss is equal to the value of (the bed gravity / cross sectional area). The compact density of the cylindrical catalyst particles is 2100 kg/m3.c) The particle bed starts to expand (→fluidized bed) when the gas velocity reach values over the velocity calculated in b). The pressure loss is beyond this superficial velocity constant. Calculate the bed porosity ε when the gas velocity is 50% higher than the value calculate in b).
februari 2018 Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland
RoNz 51
Flu
id &
Par
ticul
ate
Sys
tem
s42
4521
/ 2
010
Flui
d &
Par
ticul
ate
Syst
ems
ÅA
424
521
/ 2
018
Further reading BMH99: Beek, W.J., Muttzall, K.M.K., van Heuven, J.W. ”Transport phenomena”
Wiley, 2nd edition (1999) BSL60: R.B. Bird, W.E. Stewart, E.N. Lightfoot ”Transport phenomena” Wiley (1960) CR83: Coulson, J.M., Richardson, J.F., Backhurst, J.R., Harker, J.H. “Chemical
Engineering, Vol. 2 : Unit Operations” Pergamon Press, Oxford (1983) Chapter 3 CEWR10: C.T. Crowe, D.F. Elger, B.C. Williams, J.A. Roberson ”Engineering Fluid
Mechanics”, 9th ed., Wiley (2010) FZ98: L-S Fan, C Zhu “Principles of gas-solid flows” Cambridge Univ. Press (1998) KJ05: D. Kaminski, M. Jensen ”Introduction to Thermal and Fluids Engineering”, Wiley
(2005) Chapter 4, chapter 10 T06: S.R. Turns ”Thermal – Fluid Sciences”, Cambridge Univ. Press (2006) vD82 van Dyke, M. “An album of fluid motion”, The Parabolic Press, Stanford (CA)
(1982) Z13: R. Zevenhoven ”Introduction to process engineering / Processteknikens
grunder”, Chapter 6: Fluid mechanics: fluid statics, fluid dynamics. Course material ÅA (version 2013) available on line: http://users.abo.fi/rzevenho/PTG%20Aug2013.pdf
ÖS96: G. Öhman, H. Saxén ”Värmeteknikens grunder”, Åbo Akademi University (1996)
Ö96: G. Öhman ”Teknisk strömningslära” Course compendium ÅA (1996) sections8 and 9 (p. 27-34)
februari 2018 Åbo Akademi University - Värme- och StrömningsteknikBiskopsgatan 8, FI-20500 Åbo / Turku Finland
RoNz 52