Loss calculation of Fittings (Elbow and Tee)
In this application, the study of loss calculation for fittings is shown. Regarding Loss coeffcients
definition, there are several approaches includes Crane method and 3-K method. In order to
understand the differences, the comparion on loss coefficients and pressure drop is performed
in this document.
References:
- Flow of Fluids Through Valves, Fittings, and Pipe, Crane Valves North America,
Tecnical Paper No. 410M. 1979, p A-29
- Ron Darby, Chemical Engineering Fluid Mechanics 2nd edition, Marcel Dekker, 2001
- Ron Darby, Correlate pressure drop through pipe fittings, Chemical Engineering,
-New York- Mcgraw Hill Incorporated then Chemical Week Publishing Llc-,
vol 106, 1999.
0. Common equations
In this section, the commonly-used equations are defined.
Table of equations
Eq d table
2
r$v
Pressure drop
Eq "Pdrop" d dp = K$
Reynolds number
Eq "Reynolds" d Rey =
Flow equation
Eq "Flow" d v =
2
r$v$Dh
m
mflow
A$r
1. Constant (e.g. using ASHRAE's database)
The simplest approach of the loss coefficient K definition is using the constant parameter.
For example, by referring ASHRAE's database, the value of loss coefficient can be obtained
for the target type of fittings.
Table of loss coefficient
Loss d table
Constant
Loss "Const" d K = C
2. Crane method
The crane method is introduced in Crane Technical Paper 410.
Friction factor
4$0.0625
Loss "Crane_ft" d fT =
Dh
log10 3.7$ e
Loss coefficient
Loss "Crane" d K = Kc$fT
2
3. 3-K (Darby) method
The 3-K method is developed by Darby. In the reference book, the method is introcued as the
most accurate methods for all Reynolds numbers and fitting sizes.
Loss "3K" d K =
Loss coefficient
K1
Rey
C Ki$ 1 C
Kd
0.3
Dh_in
4. Comparion
In this section, 3 approaches are compared for loss coefficients and pressure drop. And, the
commont parameters are defined as follows.
Geometrical parameters
Hydraulic diameter
Dh d 0.15 m
Dh_in d
Dh
inch
= 5.906
Properties (Fluid is Air)
Density
r d ThermophysicalData:-Property DMASS, P = 101325 Pa, T = 300 K, "Air"
r = 1.177
kg
3
m
Viscosity
m d ThermophysicalData:-Property viscosity, P = 101325 Pa, T = 300 K, "Air"
K5
m = 1.854 # 10
Pa$s
Additionally, in order to make plotting easier, the following equations are prepared.
Constant
Coeffconst d Loss "Const"
Eqconst d eval Eq "Pdrop" , Coeffconst
Crane method
Coeffcrane d eval Loss "Crane" , Loss "Crane_ft"
Eqcrane d eval Eq "Pdrop" , Coeffcrane
3-K method
Coeff3K d eval Loss "3K" , Eq "Reynolds"
Eq3K d eval Eq "Pdrop" , Coeff3K
4-1. Case : 90° Elbow (r/D = 1.5)
As the first case, 90° Elbow (r/D = 1.5) is selected for this section. The value for the constant loss
coefficient is adjusted based on the result of Crane and 3-K method.
Constant
Coefficient
C d 0.23
Crane method
Pipe roughness
e d 0.0018$
Coefficient
Kc d 14
inch
m
K5
$m = 4.572 # 10
3-K method
Coefficients
K1 d 800
Ki d 0.071
Kd d 4.2
m
Plot data
Constant
pK_const d plot rhs Coeffconst , 0 ..5, legend = "const", color = red
pdp_const d plot rhs Eqconst , v = 0 ..150$
m
, legend = "const", color = red
s
Crane method
pK_crane d plot rhs Coeffcrane , 0 ..5, legend = "crane", color = blue
pdp_crane d plot rhs Eqcrane , v = 0 ..150$
m
, legend = "crane", color = blue
s
3-K method
pK_3K d plot rhs Coeff3K , v = 0 ..5$
m
, legend = "3-K", color = green
s
pdp_3K d plot rhs Eq3K , v = 0 ..150$
m
, legend = "3-K", color = green
s
Plot of loss coefficients
plots:-display pK_const, pK_crane, pK_3K, gridlines, axes = "framed",
labels = "Velocity [m/s]", "Loss coefficient [-]" , labeldirections = horizontal, vertical
=
0.9
Loss coefficient [-]
0.8
0.7
0.6
0.5
0.4
0.3
0
1
2
3
Velocity [m/s]
const
crane
3-K
4
5
Plot of pressure drop
plots:-display pdp_const, pdp_crane, pdp_3K, gridlines, axes = "framed",
labels = "Velocity [m/s]","Pressure drop Pa " , labeldirections = horizontal, vertical
=
Pressure drop [Pa]
3000
2000
1000
0
0
50
100
150
Velocity [m/s]
const
crane
3-K
4-2. Tee
For the case of Tee (diverging), there are 2 braches, Run-though and Through-branch. The value
for the constant loss coefficient is adjusted based on the result of Crane and 3-K method.
4-2-1. Run-through
Constant
Coefficient
C d 0.27
Crane method
Pipe roughness
e d 0.0018$
Coefficient
Kc d 20
inch
m
K5
$m = 4.572 # 10
3-K method
Coefficients
K1 d 200
Ki d 0.091
Kd d 4.0
m
Plot data
Constant
pK_const d plot rhs Coeffconst , 0 ..5, legend = "const", color = red
pdp_const d plot rhs Eqconst , v = 0 ..150$
m
, legend = "const", color = red
s
Crane method
pK_crane d plot rhs Coeffcrane , 0 ..5, legend = "crane", color = blue
pdp_crane d plot rhs Eqcrane , v = 0 ..150$
m
, legend = "crane", color = blue
s
3-K method
m
, legend = "3-K", color = green
s
m
pdp_3K d plot rhs Eq3K , v = 0 ..150$ , legend = "3-K", color = green
s
pK_3K d plot rhs Coeff3K , v = 0 ..5$
Plot of loss coefficients
plots:-display pK_const, pK_crane, pK_3K, gridlines, axes = "framed",
labels = "Velocity [m/s]", "Loss coefficient [-]" , labeldirections = horizontal, vertical
=
0.55
Loss coefficient [-]
0.50
0.45
0.40
0.35
0.30
0
1
2
3
Velocity [m/s]
const
crane
3-K
4
5
Plot of pressure drop
plots:-display pdp_const, pdp_crane, pdp_3K, gridlines, axes = "framed",
labels = "Velocity [m/s]","Pressure drop Pa " , labeldirections = horizontal, vertical
=
Pressure drop [Pa]
3000
2000
1000
0
0
50
100
150
Velocity [m/s]
const
crane
3-K
4-2-2. Through-branch
Constant
Coefficient
C d 0.91
Crane method
Pipe roughness
e d 0.0018$
Coefficient
Kc d 60
inch
m
K5
$m = 4.572 # 10
3-K method
Coefficients
K1 d 500
Ki d 0.274
Kd d 4.0
m
Plot data
Constant
pK_const d plot rhs Coeffconst , 0 ..5, legend = "const", color = red
pdp_const d plot rhs Eqconst , v = 0 ..150$
m
, legend = "const", color = red
s
Crane method
pK_crane d plot rhs Coeffcrane , 0 ..5, legend = "crane", color = blue
pdp_crane d plot rhs Eqcrane , v = 0 ..150$
m
, legend = "crane", color = blue
s
3-K method
m
, legend = "3-K", color = green
s
m
pdp_3K d plot rhs Eq3K , v = 0 ..150$ , legend = "3-K", color = green
s
pK_3K d plot rhs Coeff3K , v = 0 ..5$
Plot of loss coefficients
plots:-display pK_const, pK_crane, pK_3K, gridlines, axes = "framed",
labels = "Velocity [m/s]", "Loss coefficient [-]" , labeldirections = horizontal, vertical
=
Loss coefficient [-]
1.2
1.1
1.0
0.9
0
1
2
3
Velocity [m/s]
const
crane
3-K
4
5
Plot of pressure drop
plots:-display pdp_const, pdp_crane, pdp_3K, gridlines, axes = "framed",
labels = "Velocity [m/s]","Pressure drop Pa " , labeldirections = horizontal, vertical
=
12000
Pressure drop [Pa]
10000
8000
6000
4000
2000
0
0
50
100
Velocity [m/s]
const
crane
3-K
150
0
