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Chap 4
Piping design
1 Pressure drop and friction loss in laminar flow
-
A common parameter used in laminar flow and especially in turbulent flow is the
fanning friction factor
-Pressure drop in the pipe (L) depends on , , v,
- The pressure drop can be obtained from the following expression,
Pf = 4f
(L/D) ( v2/2)
N/m2
f: Fanning friction factor
-
Define Skin friction factor
Ff= Pf/
-
= 4f ( L/D)( v2/2)
( J/kg)
For laminar flow only, use Hagen Poiseuille equation
Pf = 32
L v /D2 = 4f (L/D)( v2/2)
f= 16( / Dv
) = 16/Re
Re<2100
2 Pressure drop and friction factor in turbulent flow
-
As in laminar flow, the friction factor in turbulent flow depends also on the
Reynolds number.
It is not possible to predict theoretically the Fanning friction factor, f, in turbulent
flow.
The friction factor is determined experimentally and depends also on surface
roughness of the pipe ( ).
f= f (Re,
/D)
=Surface roughness of the pipe
/D= relative roughness
3 Pressure drop and friction factor in flow of gases.
P = 4f L G2/
D2
G (kg/m2.sec): Flow of gas per unit area.
4. Friction losses in expansion, enlargement, contraction and pipe fittings.
f= 4f (L/D)v2/2 + Kexp(v2/2) + Kcont(v2/2) + Kfitting(v2/2)
f = [4f (L/D)+ Kexp + Kcont + Kfitting] (v2/2
Kexp= (1- A1/A2)2 enlargement coefficient
Kcont= 0.55 (1- A2/A1) Contration coefficient
Kfitiings : obtained from tables
5 Friction loss in non circular conducts
-
Use Equivalent diameter which is defined as follows
Deq = 4 ( cross sectional area/ wetted perimeter)
For a pipe
Deq = 4( R2/
-
For an annular space
Deq = 4(
-
r0 = 2R= D
R2-
R /2
R
-
2
For a rectangular duct
Deq = 4 ab/ 2(a+b) = 2ab/(a+b)
R ) = 2R2 -2R1 = D2-D1
CHAPTER 5
Flow around submerged bodies
1 Introduction
- Flow of fluids outside immerged bodies appears in many chemical engineering and
industrial applications and other processes.
-Example, flow past spheres, packed beds, drying, filtration.
- Force exerted by the fluid on the solid on the direction of flow is called skin or
wall drag.
- If the fluid is not flowing parallel to the surface, but must change directions to
pass around a solid body (spheres), significant additional frictional losses will
occur and this is called form drag
-
In many cases, however, immersed body is a blunt-shaped solid which presents
various angles and various streamlines, additional force is exerted by the fluid on
the body.
2. Drag coefficient
- Geometry of the immersed body (solid) is the main factor to determine drag force
exerted on the body.
- Correlations are the same as Reynolds Number
- Drag coefficient, Cd, is defined as,
Cd = Fd/Ap
v/2
Fd: Total drag force (N)
Ap: area (m2)
Cd: Dimensionless
: Density of fluid
For a sphere Ap= Dp2/4
For a cylinder Ap=LDp
Dp: Sphere diameter perpendicular (m)
Fd= Cd
-
Ap
Reynolds Number for a given solid immersed in a flowing fluid,
Re= Dp.vo. /
= GoDp/
Go= vo.
3. Flow past sphere, long cylinder, disks
- For each particular shape of object and orientation of the object with the direction of
the flow, a relation of Cd vs Re exists.
-
For laminar flow (low Reynolds number less than 10), the experimental drag
force for the sphere is the same as the theoretical, given by Stoke’s law equation;
Fd= 3
-
Dp vo
solving for Cd,
Cd=24/Dpvo / = 24/Re
- Variation of Cd with Re is complicated
Chapter 6
Flow through porous media
1 Laminar flow in packed beds
-
Packed bed or packed column are important in chemical engineering and other
process engineering.
Packing materials in the bed may be spheres, irregular particles and shapes.
-
The void fraction in a packed bed is defined as follows;
= volume of voids in bed
Total volume of beds (voids + solids)
-
The specific surface of a particle
Av =Sp/Vp (m-1)
Sp: Surface area of a particle in m2.
Vp: Volume of a particle in m3
For a spherical particle
Av = 6/Dp
-
Dp= diameter of the sphere (m)
For a packed bed of nonspherical particles, the effective diameter Dp is,
Dp=6/av
- Since (1 -
)= volume fraction of particles in bed,
Av(1- ) = a = (1 - ) 6/Dp
-
a= ratio of total surface area in the bed to total volume of bed. (void plus particle
volume)
-
The average interstitial velocity in the bed is v(m/sec) and is related to the
superficial velocity v’, based on the cross section of the empty container
-
V’=
v
-
Define the hydraulic radius
Rh= (cross sectional area available for flow)/wetted perimeter
= void volume available for flow/ total wetted surface of solids
= volume of voids/volume of bed
Wetted surface/volume of bed
-
=
/a
Combining the above equations,
Rh= Dp/ 6(1- )
-Equivalent diameter
-
D= 4 rh
For a packed bed, the Reynolds number is given by,
Re = 4rh v /
= 4 Dp v’
/ 6(1- )
Re = 4 Dp v’ / 6 (1- 0
-
v’= superficial velocity
-
Ergun defined Re as above but without 4/6
Re= Dpv’
/ (1- )
= DpG’/(1- 0
G’= v’
-
For a laminar flow, the Hagen Poiseuille equation is given a sfollows
P = 32
rh=
-
v
L/ D2 = 32
(v’/ ) L/ (rh) = 72
Dp/ 6(1- )
Experimental data shows that the constant =150
Blake –Kozeny equation for laminar flow
P= 150 v’
Dp
<0.5
Re <10
L (1-
)
v’ L (1- )2/
Dp2
2. Turbulent flow in packed beds
- Same procedure
P = 3 F (V’) L
Dp
-
For high turbulent flow, friction factor = cte
Experimental data 3f= 1.75, Re>1000
P= 1.75 (v’) L
Dp
-
Ergun proposed addition of laminar and turbulent equations (low-intermediate
and high Re)
Arranging
P
-
Dp
= 150 =1.75
L (1- )
Red
This equation can be used for gases and can be arranged as follows
P
=150 +1.75
3. Shape factors and mixtures of particles
-
Many particles have irregular shapes
Define shape factor or sphericity
= Surface area of a sphere having same volume as the particle
Actual surface area of a particle
Sphere Sp= Dp2
- Any particle
Vp=
= Dp2/Sp
Dp3/3
Sp/vp =
Dp2/
= 6/ Dp
Dp /6
- av= Sp/vp = 6/ Dp
A= 6 ( 1- )/
- For a cylinder
-
L=D
=1.0
For a mixture of particles of various sizes, use mean specific surface area
Avm =
xi.avi
xi=volume fraction
-
-
Dpm = 6/avm =
Darcy’s empirical law for laminar flow
P = fn( 1/ L,
-
Purely viscous flow
V’= q’/A =- K P/
L
v’: Superficial velocity (cm/sec)
q’: flow rate (cm3/sec)
A: Empty cross section (cm2)
; viscosity Cp
k: Darcy coefficient ( cm2.Cp/sec-atm) = Darcyconstant