### Correlations of Pressure Drop in Electronics

```Thermal Analysis
Correlations of Pressure Drop
K=0.2065+0.1549xCv
in Electronics Packaging
Calculating pressure drop is one of the most basic and
important aspects of any thermal analysis. Knowing the
pressure drop in a system is key to determining how
much fluid is available for the cooling process. This is
true for heat sinks, heat exchangers, telecom chassis
or any other device using some form of liquid or gas.
-0.4224
In 1839, experiments by Hagen showed that pressure
drop in laminar flow was proportional to flow rate, but
for turbulent flow it is roughly proportional to-0.4224
the square
2
of the flow velocity.
K=0.2065+0.1549xCv
L V
∆P
hf = f
=
The pressureD
drop 2
for a laminar flow in a tube was 1 ρU12
2
g
found by Hagen to be:
K=0.2065+0.1549xCv-0.4224
∆P
8µLG
L V2
∆P
h
f == f D4 2
∆P
1
2
K ρU1
=1
g
R
-0.4224 2
K=0.2065+0.1549xCv
Pressure drop can be divided into two fundamental
2
L3/4 V1/4
Where G is the volumetric
flow rate, and μ is the
categories: frictional and dynamic. The pressure drop
∆P
8µLG
µ D-4.75G1.75
∆P=0.241Lρ
hf =viscosity.
f
dynamic
from frictional effects is the dominant part of the overall
1
D 2g
ρU
∆P =
4
pressure drop. Dynamic pressure drop owing to
K2 =
The Darcy-Weisbach
equation for turbulent flow
R
momentum changes is typically small, unless the fluid
pressure drop in a tube can
L V2 be shown as:
velocity reaches Mach 0.2. Dynamical changes can
∆P
4A
8µLG 1/4 -4.75 1.75
h
be attributed to entrance effects, sudden expansion
1
Dh∆P=0.241Lρ
=f = f D 3/4
2gµ D
G
ρU1
or contraction, elbows, valves, etc.
2
∆P
P =
4
K
R
A fundamental formula for pressure drop is the
This relation shows that pressure drop in turbulent flow
Darcy-Weisbach equation. This correlates head loss
varies by∆P
the power
of 1.75 of velocity. L is the length
8µLG
=the
4A
3/4µ 1/4 -4.75 1.75
due to friction in a pipe of any cross section, factoring K of
pipe.2
∆P=0.241Lρ
D
G
ρV
/2g
∆P
=
D
=
4
in velocity and pipe dimensions for either laminar or
h
K=
Ran equivalent diameter, called
P pipes
For noncircular
turbulent flow. Head loss is simply the pressure drop
A1
-0.4224
the hydraulic2diameter, Dh, can be defined as:
K=0.2065+0.1549xC
divided
by the density, which is equivalentv
to a column
V∆P f1
of that fluid exerting the same pressure as the
4A
∆h
=
+3/4ΣK
µ1/4D-4.75G1.75
U1
∆P=0.241Lρ
K
loss
pressure drop.
2
D=h =
2g
D
ρVP/2g
(
This equation is stated as:
(
)
Where A is the cross section and P is the wetted
2
perimeter.
( ) (( ) )
A1
L1
∆P
4A
V
f1
L V
2
2
1
∆P
A1
f
=VA2
∆P
A
DKhdynamical
∆h
+ aΣK
2
hf = f
== =
-1+
1-drop,
+ coefficient, K, U1
comp
=1Forloss
pressure
loss
ρV
/2g
P
1
1
2
2
2g
D
D 2g
C2vρU1
=
,
A1
A2
D
ρU1
A
2
is defined
as:
Vchannel
A1
A1
∆P
Where f is the Darcy friction factor, L is the length of the
L
U1
1
V2 f1
K
=
8µLGof the pipe and V is the velocity. ∆h V= 2
pipe, D is the diameter
+
ΣK
U1
/2g D
lossρV
2g
comp
The∆P
friction
=factor can4be taken from the Moody Chart, C =
,
which plots the friction
R coefficient as a function of the Kv= 1.4
A1
V
A1
2 density and g is the gravitational
Reynolds number for both laminar and turbulent flow,
Where ρ is channel
theV
fluid
f1
and for smooth and rough pipes.
constant. =
1
∆h
+ ΣKU1
U1
loss V
2g
∆P=0.241Lρ3/4µ1/4D-4.75G1.75
comp D
Cv =
,
V
A1
2
(
)
(
)
usa | T: 781.769.2800 www.qats.com
(
Page 8
g
3/4µ 1/4 -4.75 -0.4224
2
∆P=0.241Lρ
D vG1.75
K=0.2065+0.1549xC
2
LG
4A K = 1.4
L V2
P
h =
f
THermal
Analysis
4
Dh =
f
ρ3/4µ1/4D-4.75G1.75
2g
1
2
∆P
ρV2/2g
8µLG
K=
ρU12
=
( ) ( )
A2
2
-1+ 1-
A1
A1
(
)
V
f1
∆hloss =
+ ΣK
3/4µ 1/4 -4.75 1.75
2g
D
∆P=0.241Lρ
D
G
-0.4224
+
U1
( )
f1
∆P
A2
=11
2
ρU1
A1
2
D
U2
A2
1
2
Where V is the fluid velocity.
L1
A1 we show the pressure drop for some simple geometries:
In the following,
U2
V4A
f1
comp
Sudden
expansion

D
=
+ ΣK C =
1
h U1
,
D
v V P
)
( ρV) /2g( )
channel
∆P
=
K=
1
ρU12
2
,
G
2
A2
The total∆P
loss of a complicated system can be calculated as:
A1 K = 1.4
R4
2
gv
5
D
∆P
2
A2 ∆P
A1
-1+ 1L1
A1 2
A2
V
U1 ∆h loss =
1
K = 1.4 2g
2
A1
1.75
Cv =
(
2
2
Sudden Contraction 
A1
f1
( ) ( )
∆P
A2 2 1
A1
f1L1 f2L2
1
A2
=1+ 2 1- 2
+
+
1
2
ρU1
A1
A2
D1
D2
2
+ U1
D L2
A1
)
f1
+ ΣK2
A2
D
L
U1
Vchannel
U2
1
,
A1
U1
A2
2
L1
L
Vcomp
L2
A2
L2
Introducing THE
BWT-104
1
A2
2
Benchtop Wind Tunnel
L
A1
U2
U1
1
Overall dimensions (L x W x H)
91 cm x 44 cm x 44 cm
A2 x 17.25”)
(36 x 17.25
2Dimensions
Test Section
50.8 cm x 44 cm x 10 cm
(20 x 17.25 x 4”)
L1
L2
Materials
Aluminum, Plexiglas
Flow Range
0 to 6 m/s (1200 ft/min)
A1
1
A2
2
Weight
14.5 kg (32 lbs.)
The BWT-104™ is a research quality, open loop,
benchtop wind tunnel for thermal characterization of
components, circuit boards and cooling devices such
as heat sinks, heat exchangers and cold plates. It
provides homogeneous flow, up to 6 m/s (1200 ft/min)
within its Plexiglas® test section, has 12 ports for probes
and sensors and can be operated on any axis, making
it ideal for laboratory environments.
Advanced Thermal Solutions, Inc. at 1-781-769-2800
or www.qats.com
L
Page 9
2
+
v
∆P=0.241Lρ3/4µ1/4D-4.75G1.75
Dh =
4A
P
K=
∆P
ρV2/2g
Thermal Analysis
-0.4224
(
A1
)
V2 boardf1channels 
K value in circuit
∆hloss =
+ ΣK
U1
2g
D
-0.4224
K=0.2065+0.1549xCv
U2
( () )( () )
( ) ( )
8µLG
1. White, F., Fluid Mechanics, McGraw-Hill, 1979.
K = 1.4
2. Blevins, R., Applied Fluid Dynamics Handbook, Van Nostrand
R
K = 1.4
Reinhold, 1984.
3. Azar, K., Electronic Cooling Theory and Application, Lucent
Technologies, 1994.
4. Kays, M. and London, A., Compact Heat Exchangers,
3/4 1/4 -4.75 1.75
∆P=0.241Lρ
G
Pressure drop for a heatµsinkD

S
α=
Third Edition, McGraw-Hill, 1984.
H
4A
∆P friction P
=f
Dh =
(
L
−3
10
Dh (S)
)
( )(
2
(
1.7012
−
(( )
(1 )
)
2
0.5 ρU
∆P
Kf == 96 1 −21.3553 +1.9467
ReρV /2g
α
α
α
3
+
0.9564
α
4
−
0.2537
α
5
)
ATS-619
A1
)
V2L 10 f1 0.5 (ρU2)
D (S) + ΣK
∆hloss =
U1
2g D
2
Uapp
2
A22
(U2
∆P expansion = ρ
1- σ - 1.0257σ +2.029σ - 1.0058)
∆P friction = f
A1
U1
−3
h
Vcomp
(
2
2
)
U2
1
L1
2
L1
total
L2
L2
U1
1
A2
= ∆P friction +∆P contraction +∆P expansion
A1
L
σ = Open channel area / Total frontage area
1
A2
2
Where S is channel spacing, H is the fin height, U is
velocity between fins, Uapp is the approach velocity,
and Dh(s) isLthe channel hydraulic diameter.
A2
2
Uapp
C∆P
=
, ( 1- σ2 - 0.4405σ2 +0.039σ - 0.4011)
= ρ
v contraction
Vchannel 2
A1
∆P
( ) (
L
References:
K value
for a sharp4corner turn
∆P =
G1.75
A2
Readers should note that the literature contains many
1
correlations
for pressure2drop that should be used with
caution. While these correlations can provide good first
order approximations for quick analyses, they may break
Where the volume fraction coefficient is defined by:
L1
L2
down in cases that do not fall into their range of
Vcomp
applicability. For example, some correlations will
Cv =
,2
reasonably predict the pressure drop across a heat
Vchannel
L2V
A1
2
∆P
A2 2 a bypass
f1 the fin density
2
1
∆Pf2L2
A2 2 1
A1
21- A1
flow +when
is=1moderate.
A2
A1
f
h∆P
=
f
= sink inA2
-1+
∆P
A1
f1L1
+ 2 11
f = D -1+
1+
U1
1
A2
2
1
1
2
2
=1+
1+
+
2
ρU1
A1
A2
D
ρU1
A1
A2
may fail
2 if the heat sink has2 3-4 mm tall,
1 Here,2Vcomp is the total
g volume
ρU1
A1
A2 occupied
D by the 2 12 ρU12 But they
A1
A2
D1
D2
2 components and V
closely packed, thick fins.
channel is the total volume of
the circuit board forming the channel.
2
from black anodized,
extruded aluminum, and
employs ATS’ patent
pending maxiGRIPTM
attachment solution.
The ATS-619 is just
16.25mm high with a
footprint of 22mm by
24mm (LxW) and has
a thermal resistance of
6.11◦C/W.
For further technical information,