BASIC CALCULATION FOR PROCESS ENGINEER
One of the most basic calculations performed by any process engineer, whether in design
or in the plant, is line sizing and pipeline pressure loss. Typically known are the flow
rate, temperature and corresponding viscosity and specific gravity of the fluid that will
flow through the pipe. These properties are entered into a computer program or
spreadsheet along with some pipe physical data (pipe schedule and roughness factor) and
out pops a series of line sizes with associated Reynolds Number, velocity, friction factor
and pressure drop per linear dimension. The pipe size is then selected based on a
compromise between the velocity and the pressure drop. With the line now sized and the
pressure drop per linear dimension determined, the pressure loss from the inlet to the
outlet of the pipe can be calculated.
Calculating Pressure Drop
The most commonly used equation for determining pressure drop in a straight pipe is
the Darcy Weisbach equation. One common form of the equation which gives pressure
drop in terms of feet of head is given below:
The term
is commonly referred to as the Velocity Head.
Another common form of the Darcy Weisbach equation that is most often used by
engineers because it gives pressure drop in units of pounds per square inch (psi) is:
To obtain pressure drop in units of psi/100 ft, the value of 100 replaces L in Equation
2.
The total pressure drop in the pipe is typically calculated using these five steps. (1)
Determine the total length of all horizontal and vertical straight pipe runs. (2) Determine
the number of valves and fittings in the pipe. For example, there may be two gate valves,
a 90o elbow and a flow thru tee. (3) Determine the means of incorporating the valves and
fittings into the Darcy equation. To accomplish this, most engineers use a table of
equivalent lengths. This table lists the valve and fitting and an associated length of
straight pipe of the same diameter, which will incur the same pressure loss as that valve
or fitting. For example, if a 2” 90o elbow were to produce a pressure drop of 1 psi, the
equivalent length would be a length of 2” straight pipe that would also give a pressure
drop of 1 psi. The engineer then multiplies the quantity of each type of valve and fitting
by its respective equivalent length and adds them together. (4) The total equivalent
length is usually added to the total straight pipe length obtained in step one to give a total
pipe equivalent length. (5) This total pipe equivalent length is then substituted for L in
Equation 2 to obtain the pressure drop in the pipe.
See any problems with this method?
Relationship Between K, Equivalent Length and Friction Factor
The following discussion is based on concepts found in reference 1, the CRANE
Technical Paper No. 410. It is the author’s opinion that this manual is the closest thing
the industry has to a standard on performing various piping calculations. If the reader
currently does not own this manual, it is highly recommended that it be obtained.
As in straight pipe, velocity increases through valves and fittings at the expense of
head loss. This can be expressed by another form of the Darcy equation similar to
Equation 1:
When comparing Equations 1 and 3, it becomes apparent that:
K is called the resistance coefficient and is defined as the number of velocity heads
lost due to the valve or fitting. It is a measure of the following pressure losses in a valve
or fitting:
•
•
•
•
Pipe friction in the inlet and outlet straight portions of the valve or fitting
Changes in direction of flow path
Obstructions in the flow path
Sudden or gradual changes in the cross-section and shape of the flow path
Pipe friction in the inlet and outlet straight portions of the valve or fitting is very small
when compared to the other three. Since friction factor and Reynolds Number are mainly
related to pipe friction, K can be considered to be independent of both friction factor and
Reynolds Number. Therefore, K is treated as a constant for any given valve or fitting
under all flow conditions, including laminar flow. Indeed, experiments showed1 that for
a given valve or fitting type, the tendency is for K to vary only with valve or fitting
size. Note that this is also true for the friction factor in straight clean commercial steel
pipe as long as flow conditions are in the fully developed turbulent zone. It was also
found that the ratio L/D tends towards a constant for all sizes of a given valve or fitting
type at the same flow conditions. The ratio L/D is defined as the equivalent length of the
valve or fitting in pipe diameters and L is the equivalent length itself.
In Equation 4, ƒ therefore varies only with valve and fitting size and is independent of
Reynolds Number. This only occurs if the fluid flow is in the zone of complete
turbulence (see the Moody Chart in reference 1 or in any textbook on fluid
flow). Consequently, ƒ in Equation 4 is not the same ƒ as in the Darcy equation for
straight pipe, which is a function of Reynolds Number. For valves and fittings, ƒ is the
friction factor in the zone of complete turbulence and is designated ƒt, and the equivalent
length of the valve or fitting is designated Leq. Equation 4 should now read (with D being
the diameter of the valve or fitting):
The equivalent length, Leq, is related to ƒt, not ƒ, the friction factor of the flowing fluid
in the pipe. Going back to step four in our five step procedure for calculating the total
pressure drop in the pipe, adding the equivalent length to the straight pipe length for use
in Equation 1 is fundamentally wrong.
Calculating Pressure Drop, The Correct Way
So how should we use equivalent lengths to get the pressure drop contribution of the
valve or fitting? A form of Equation 1 can be used if we substitute ƒt for ƒ and Leq for L
(with d being the diameter of the valve or fitting):
The pressure drop for the valves and fittings is then added to the pressure drop for the
straight pipe to give the total pipe pressure drop.
Another approach would be to use the K values of the individual valves and
fittings. The quantity of each type of valve and fitting is multiplied by its respective K
value and added together to obtain a total K. This total K is then substituted into the
following equation:
Notice that use of equivalent length and friction factor in the pressure drop equation is
eliminated, although both are still required to calculate the values of K1. As a matter of
fact, there is nothing stopping the engineer from converting the straight pipe length into a
K value and adding this to the K values for the valves and fittings before using Equation
7. This is accomplished by using Equation 4, where D is the pipe diameter and ƒ is the
pipeline friction factor.
How significant is the error caused by mismatching friction factors? The answer is, it
depends. Below is a real world example showing the difference between the Equivalent
Length method (as applied by most engineers) and the K value method to calculate
pressure drop.
An Example
The fluid being pumped is 94% Sulfuric Acid through a 3”, Schedule 40, Carbon Steel
pipe:
Mass Flow Rate, lb/hr:
Volumetric Flow Rate, gpm:
63,143
70
Density, lb/ft3:
112.47
S.G.
Viscosity, cp:
1.802
10
Temperature, oF:
127
Pipe ID, in:
3.068
Velocity, fps:
3.04
Reynold's No:
12,998
Darcy Friction Factor, (f) Pipe:
0.02985
Pipe Line ∆P/100 ft.
1.308
Friction Factor at Full Turbulence (ƒt):
0.018
Straight Pipe, ft:
31.5
Fittings
90o Long Radius
Elbow
Branch Tee
Swing Check Valve
Plug Valve
3” x 1” Reducer4
TOTAL
Leq/D1
Leq2, 3
K1, 2 = ƒt
(L/D)
Quantity
Total Leq
Total K
20
5.1
0.36
2
10.23
0.72
60
50
18
5
None
15.3
12.8
4.6
5
822.68
1.08
0.9
0.324
57.92
1
1
1
1
15.34
12.78
4.6
822.68
865.633
1.08
0.9
0.324
57.92
60.944
Notes:
K values and Leq/D are obtained from reference 1.
K values and Leq are given in terms of the larger sized pipe.
Leq is calculated using Equation 5 above.
The reducer is really an expansion; the pump discharge nozzle is 1” (Schedule 80)
but the connecting pipe is 3”. In piping terms, there are no expanders, just
reducers. It is standard to specify the reducer with the larger size shown
first. The K value for the expansion is calculated as a gradual enlargement with a
30o angle.
5. There is no L/D associated with an expansion or contraction. The equivalent
length must be back calculated from the K value using Equation 5 above.
1.
2.
3.
4.
Straight Pipe ∆P, psi
Total Pipe Equivalent Length ∆P,
psi
Valves and Fittings ∆P, psi
Total Pipe ∆P, psi
Typical Equivalent Length
Method
Not applicable
K Value Method
0.412
11.322
Not Applicable
Not applicable
6.828
11.322
7.24
The line pressure drop is greater by about 4.1 psi (about 56%) using the typical
equivalent length method (adding straight pipe length to the equivalent length and using
the pipe line fiction factor and Equation 1).
One can argue that if the fluid is water or a hydrocarbon, the pipeline friction factor
would be closer to the friction factor at full turbulence and the error would not be so great,
if at all significant; and they would be correct. However hydraulic calculations, like all
calculations, should be done in a correct and consistent manner. If the engineer gets into
the habit of performing hydraulic calculations using fundamentally incorrect equations,
he takes the risk of falling into the trap when confronted by a pumping situation as shown
above.
Another point to consider is how the engineer treats a reducer when using the typical
equivalent length method. As we saw above, the equivalent length of the reducer had to
be back-calculated using equation 5. To do this, we had to use ƒt and K. Why not use
these for the rest of the fittings and apply the calculation correctly in the first place?
Final Thoughts - K Values
The 1976 edition of the Crane Technical Paper No. 410 first discussed and used the
two-friction factor method for calculating the total pressure drop in a piping system (ƒ for
straight pipe and ƒt for valves and fittings). Since then, Hooper2 suggested a 2-K method
for calculating the pressure loss contribution for valves and fittings. His argument was
that the equivalent length in pipe diameters (L/D) and K was indeed a function of
Reynolds Number (at flow rates less than that obtained at fully developed turbulent flow)
and the exact geometries of smaller valves and fittings. K for a given valve or fitting is a
combination of two Ks, one being the K found in CRANE Technical Paper No. 410,
designated KΨ, and the other being defined as the K of the valve or fitting at a Reynolds
Number equal to 1, designated K1. The two are related by the following equation:
K = K1 / NRE + KΨ (1 + 1/D)
The term (1+1/D) takes into account scaling between different sizes within a given
valve or fitting group. Values for K1 can be found in the reference article2 and pressure
drop is then calculated using Equation 7. For flow in the fully turbulent zone and larger
size valves and fittings, K becomes consistent with that given in CRANE.
Darby3 expanded on the 2-K method. He suggests adding a third K term to the
mix. Darby states that the 2-K method does not accurately represent the effect of scaling
the sizes of valves and fittings. The reader is encouraged to get a copy of this article.
The use of the 2-K method has been around since 1981 and does not appear to have
“caught” on as of yet. Some newer commercial computer programs allow for the use of
the 2-K method, but most engineers inclined to use the K method instead of the
Equivalent Length method still use the procedures given in CRANE. The latest 3-K
method comes from data reported in the recent CCPS Guidlines4 and appears to be
destined to become the new standard; we shall see.
Conclusion
Consistency, accuracy and correctness should be what the Process Design Engineer
strives for. We all add our “fat” or safety factors to theoretical calculations to account for
real-world situations. It would be comforting to know that the “fat” was added to a basis
using sound and fundamentally correct methods for calculations.
NOMENCLATURE
D
d
ƒ
ƒt
g
hL
K
K1
KΨ
L
Leq
NRE
∆P
ν
W
ρ
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
Diameter, ft
Diameter, inches
Darcy friction factor
Darcy friction factor in the zone of complete turbulence
Acceleration of gravity, ft/sec2
Head loss in feet
Resistance coefficient or velocity head loss
K for the fitting at NRE = 1
K value from CRANE
Straight pipe length, ft
Equivalent length of valve or fitting, ft
Reynolds Number
Pressure drop, psi
Velocity, ft/sec
Flow Rate, lb/hr
Density, lb/ft3
REFERENCES
1. Crane Co., “Flow of Fluids through Valves, Fittings and Pipe”, Crane Technical
Paper No. 410, New York, 1991.
2. Hooper, W. B., The Two-K Method Predicts Head Losses in Pipe Fittings, Chem.
Eng., p. 97-100, August 24, 1981.
3. Darby, R., Correlate Pressure Drops through Fittings, Chem. Eng., p. 101-104,
July, 1999.
4. AIChE Center for Chemical Process Safety, “Guidelines for Pressure Relief and
Effluent Handling systems”, pp. 265-268, New York, 1998.
Reader / Author Question and Answers
1. "Could you please give me in layman terms a better definition for K values. I know
that K is defined as "the number of velocity heads lost"...But what exactly does that
mean???"
Well, I'll try to give you the Chemical Engineer's version of the layman answer. Velocity
of any fluid increases through pipes, valves and fittings at the expense of pressure. This
pressure loss is referred to as head loss. The greater the head loss, the higher the velocity
of the fluid. So, saying a velocity head loss is just another way of saying we loose
pressure due to and increase in velocity and this pressure loss is measured in terms of feet
of head. Now, each component in the system contributes to the amount of pressure loss in
different amounts depending upon what it is. Pipes contribute fL/D where L is the pipe
length, D is the pipe diameter and f is the friction factor. A fitting or valve contributes K.
Each fitting and valve has an associated K.
2. "It appears that the K values in CRANE TP-410 were established using a liquid (water)
flow loop. Is this K value also valid for compressible media systems? (Can a K value be
used for both compressible and incompressible service?)"
Crane also tested their system on steam and air. Now, this is where things get sticky. As
per CRANE TP-410, K values are a function of the size and type of valve or fitting only
and is independent of fluid and Reynolds number. So yes, you can use it in ALL services,
including two-phase flow. However, as I point out towards the end of my article, there is
now evidence that shows using a single K value for the valve and fitting is not correct
and that K is indeed a function of both Reynolds number and fitting/valve
geometry. I reference an article by Dr. Ron Darby of Texas A&M University which can
be found in Chemical Engineering Magazine, July 1999. Dr. Darby just published a
second article on the subject which can be found in Chemical Engineering Magazine,
April 2001.
I don't believe there is any question as to the proper way to use K values in pressure drop
calculations. The only question is whether industry will accept the new data.
3. "When answering my first question, you stated: 'Velocity of any fluid increases
through pipes, valves and fittings at the expense of pressure.' When you say this, you are
talking about compressible (gas) flow right? For example, in a pipe of constant area, the
velocity of a gas would increase as the fluid traveled down the pipe (due to the decreasing
pressure). However, the velocity of a liquid would remain constant as it traveled down
the same pipe (even with the decreasing pressure). Is this a correct statement?
Sorry for the confusion. Yes to both of your questions. If you look at the Bernoulli
equation, you will see that velocity cancels out for a liquid as long as there is no change
in pipe size along the way and pressure drop is only a function of frictional losses and a
change in elevation.
However, the K value of a fitting is still a quantifier of the head loss (frictional loss) in
that fitting and this head loss is still calculated as the velocity head of the liquid (V^2/2g).
So in essence, you still achieve a
liquid velocity at the expense of pressure loss; the velocity head just happens to be
constant. Read section 2-8 in CRANE TP-410. They define the velocity head as a
decrease in static head due to velocity.
The big thing is not to get too hung up on the definitions and just remember you can't
have flow unless you have a driving force and that force is differential pressure. Also, in
a piping system there is frictional losses which comes from the pipe and all fittings and
valves. The use of K is just a way of quantifying the frictional component of the fittings
and valves. You can even put the piping friction in terms of K by using fL/D for the pipe
and multiplying that by V^2/2g.
I hope this helps. If you are still confused, let me know and I'll just explain it again but I'll
try to do it in a different way. Sometimes, a concept just needs to be re-worded and I'm
willing to spend as much time on this as you need.
4. I'm reading the Crane Technical Paper #410 and I have the following
questions/comments:
Page 2-8 of TP 410 states that:
"Velocity in a pipe is obtained at the expense of static head". This makes sense and
Equation 2-1 shows this relationship where the static head is converted to velocity
head. However, there is no diameter associated with this. So is it correct to say based on
equation 2-1 that if you had a barrel of water with a short length of pipe attached to the
bottom that discharged to atmosphere, and in this barrel you had 5 feet of water (5' of
static head), the resulting water velocity would be 17.94 ft/sec (regardless
of the pipe diameter).
Maybe the real question is how do you use equation 2-1. Do you have to know the
velocity and then you can calculate the headloss? And why does equation 2-1 and
equation 2-3 seem to show headloss equaling two different things?
Also, why does it say that a diameter is always associtated with the K value, when as I
mentioned above there is no diameter associated with equation 2-1?
Maybe I'm trying to read into all of this too deeply, but I still do not feel that I fully grasp
what page 2-8 is trying to reveal.
You need a diameter to get velocity. Velocity is lenght/time (for example, feet/sec). Flow
is usually given in either mass units (weight/time or lb/hr for example) or in volumetric
units (cubic feet per minute for example). To get velocity, you need to divide the
volumetric flow by a cross sectional area (square feet). To get an area, you need a
diameter. So the velocity is always based on some diameter.
As I show in my paper, equation 2-1 is just the basis of the velocity head. To get the
frictional loss, you need to know the contribution of each component in the system; pipe,
fitting and valve. To get that contribution, you use 'K' (equation 2-2). Each component
has an associated 'K' value. You multiply the velocity head by the appropriate 'K' value.
Equation 2-3 is just another way of expressing the same thing. As you can see, this means
you can calculate a 'K' for a component such as a pipe using the formula fL/D as shown
in Equation 2-3. Again, I explain this in my paper so I would suggest you re-read it.
I would also suggest you look at the examples in CRANE. There are many of them in
Chapter 4.
'K' is associated with the velocity and therefore the diameter. Look at the values for 'K' in
CRANE (starting on page A-26). You will see that for the most part, K is a function of a
constant times the friction factor at fully turbulent flow. This friction factor changes with
pipe diameter as shown on page A-26. Again, re-read my paper and look at the examples
in Chapter 4.
Table of Minor Loss Coefficients (Km is unit-less): References Back to
Calculation
Fitting
Km
Valves:
Fitting
Km
Elbows:
Globe, fully open
10
Regular 90°, flanged
0.3
Angle, fully open
2
Regular 90°, threaded
1.5
Gate, fully open
0.15
Long radius 90°, flanged
0.2
Gate 1/4 closed
0.26
Long radius 90°, threaded
0.7
Gate, 1/2 closed
2.1
Long radius 45°, threaded
0.2
Gate, 3/4 closed
17
Regular 45°, threaded
0.4
Swing check, forward flow
2
Swing check, backward flow
infinity Tees:
180° return bends:
Line flow, flanged
0.2
Line flow, threaded
0.9
Flanged
0.2
Branch flow, flanged
1.0
Threaded
1.5
Branch flow, threaded
2.0
Pipe Entrance (Reservoir to Pipe):
Pipe Exit (Pipe to Reservoir)
Square Connection
0.5
Square Connection
1.0
Rounded Connection
0.2
Rounded Connection
1.0
Re-entrant (pipe juts into tank)
1.0
Re-entrant (pipe juts into tank)
1.0