AN
ENGINEERING VIEW OF AQUARIUM SYSTEMS DESIGN: PUMPS AND PLUMBING
by
SANJAY JOSHI, Ph.D., NATHAN PADEN & SHANE GRABER
Sponsored in part by:
Every
aquarist at one time or another has had to deal with plumbing issues
such as sizing of pumps, selecting pipe size, determining whether the
pump can be upgraded without changing the returns in the overflow box,
maintaining water flow while controlling velocity so that it does not
blow off the coral tissue, etc.
A
lot of this information is available in the form of “rules of thumb”
as well as established fluids engineering formulas and data.This is an attempt to explain the basics of the devices used to
create and manage water flow and to provide a better understanding of
the principles involved as well as trying to consolidate the relevant
information for an aquarist in one single document.
In
addition to providing the theory, formulas and tables of relevant data,
for practical use we also provide an Excel
spreadsheet that incorporates all the useful information in a
useable form, without the requirement that the user understand the math
and formulas needed for solving the plumbing design problems.
Pump
Basics
Pumps
are the most common devices used to move water through the filters,
skimmers, and create circulation in the tanks.The most common type of pump used is called the centrifugal pump.The centrifugal pump is basically a rotary machine and comprises
3 main elements.
The
impeller - rotating element
The
volute- the casing inside which the impeller rotates
The
motor - imparts the rotation to the impeller
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The
pump functions by converting the rotational energy of the motor to
kinetic energy of the liquid by accelerating the liquid from the
center of the impeller to the outer via centrifugal force.The amount of energy imparted depends on the velocity of
the liquid at the tip of the impeller.Similar to rotating an object tied to a string, the velocity
will be higher if the speed of rotation is higher or the diameter
of the impeller is larger. The volute of the pump harnesses this
kinetic energy by creating a resistance to the flow and slowing it
down, this results in the creation of pressure energy.So in reality a centrifugal pump only creates flow, and the
resistance to this flow is what creates the pressure. Flow is
typically measured in GPH (Gallons per Hour) or GPM (Gallons per
Minute).
The
impeller is driven by a motor and is usually connected to the
motor in 2 ways. Direct coupling to the motor via a shaft –
called direct drive pumps,magnetically
coupled.
Pumps
with the impellers directly connected to the motor via the shaft require
the use of a mechanical seal, which are prone to failure and end up with
leaks. The magnetic coupled pumps avoid the seal problem by using
magnets to drive the impeller, thus allowing the creation of a
centrifugal pump that does not require a mechanical seal. For reef
aquarium applications the best choice is typically the magnetically
driven centrifugal pump and most of the common brands are of this
design.
The
kinetic energy that is created by the pump is often measured as head.Head refers to the height of a liquid column which the pump could
create using the kinetic energy that is generated by the pump.If the discharge of the pump is pointed straight up into the air,
it will pump the fluid to a certain height – the maximum head or shut
off head.This is usually determined by the speed of the motor and outside
diameter of the pumps impeller.
The
amount of fluid the pump moves is measured by the flow rate in GPH or
GPM. The flow rate can be converted to velocity of fluid as follows:
(1)
Velocity
= feet/sec
GPM
= Gallons per Min
D
= Inside diameter of pipe in inches
From
this we can see the first important observation about velocity and pipe
diameter.Doubling the pipe diameter will decrease the velocity by a factor
of 4.
Pumps
are rated by flow rate, head, and power consumption.When designing an aquarium system we are concerned with X amount
of flow at Y amount of head.Each pump will have its own relationship between head and flow
rate, depending on the pump design, and this information will typically
be displayed on the pump performance curve. For any pump, the flow rate
will reduce as the amount of head increases.
Pipes
and Piping Systems
The
flow of the pump is channeled through pipes and the piping system
comprising the pipes, fittings, control valves, etc.Solving fluid flow problems requires the use of a few basic
equations. The first one is the simple law of conservation of mass,
where the flow rate between any 2 sections is conserved.
(2)
The
second is the energy equation between any 2 sections of a pressurized
pipe. The energy equation between any two sections of a pressurized pipe
can be written as
(3)
Where:Z = elevation of centerline of pipe relative to an arbitrary
datum
P
= pressure on centerline of pipe
γ
= Specific Weight of Fluid
V
= average flow velocity
Hf
= head loss due to friction
Hm
= minor losses
As
the water flows through the pipe and piping system it encounters
resistance, primarily due to the following 3 elements: resistance
due to the elevation it has to raise the water – called static
head, or static head loss resistance due to friction with the walls
of the pipe – friction loss resistance due to the fittings and
valves used in the piping system.
The
cumulative effect of this resistance is to reduce the resulting flow
at the outlet. This cumulative resistance is often measured in terms
of head or pressure loss, and also called the total dynamic head (TDH)
of the piping systems.
To
determine the flow parameters at the outlet we need to compute the
TDH.Let us look at each one of the components separately:
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Resistance
due to elevation – Static Head
The
pump is used to move the liquid from a lower point to a higher point.The difference between the height of the liquid levels at the
output and input of the pump is the static head.
Friction
Loss in Pipes
As
the fluid flows through the pipe, friction on the side walls of the pipe
create resistance to the flow.Two
approaches are used to estimate the friction loss:
Using
published tables
Using
the governing equations
1)
Using Published Tables
Tables
indicating friction loss are available at several web sites, one such
table is published below as Table 1.These tables are typically derived using the empirical formula
called the Hazen-Williams formula. This formula is as follows:
(4)
Where:Hf = Friction loss in feet of head
L
= Length of pipe in feet
Q
= flow rate in GPM
D
= Nominal pipe ID in inches
C
= friction factor from Hazen-Williams
The
value of C is critical for computation and relates to the roughness of
the interiorwall of the pipe. One of the reasons that you find that tables
are not identical is due to the fact that different values of C are used
for the same material. For PVC pipe the value of C ranges from 140 to
150.
The
Hazen Williams formula has a limited range of validity and is only valid
for turbulent flow with reasonable velocity [Ref. C.P. Liou]. However,
for the range of typical values in a reef aquarium the formula is quite
valid.
Table
1: A Published Table for Friction loss in Schedule 40 PVC
Friction
Loss Per 100 Feet of SCH 40 Plastic Pipe
Nominal Pipe Diameter
GPM
1/2"
3/4"
1"
1 1/4"
1 1/2"
2"
3"
4"
1
2.08
0.51
-
-
-
-
-
-
2
4.16
1.02
0.55
0.14
0.07
-
-
-
5
23.44
5.73
1.72
0.44
0.22
0.066
0.015
-
7
43.06
10.52
3.17
0.81
0.38
0.11
0.021
-
10
82.02
20.04
6.02
1.55
0.72
0.21
0.03
-
15
-
42.46
12.77
3.28
1.53
0.45
0.07
-
20
-
72.34
21.75
5.59
2.61
0.76
0.11
0.03
25
-
-
32.88
8.45
3.95
1.15
0.17
0.04
30
-
-
46.08
11.85
5.53
1.62
0.23
0.06
35
-
-
-
15.76
7.36
2.15
0.31
0.08
40
-
-
-
20.18
9.43
2.75
0.41
0.11
45
-
-
-
25.1
11.73
3.43
0.51
0.17
50
-
-
-
30.51
14.25
4.16
0.61
0.16
60
-
-
-
-
19.98
5.84
0.85
0.22
70
-
-
-
-
-
7.76
1.13
0.31
75
-
-
-
-
-
8.82
1.28
0.34
80
-
-
-
-
-
9.94
1.44
0.38
90
-
-
-
-
-
12.37
1.8
0.47
100
-
-
-
-
-
15.03
2.18
0.58
2)
Using the Governing Equations
This
approach is based on using the Darcy-Wiesenbach equation and is valid
for all types of flow. This approach is the one most commonly used in
software packages for fluid flow analysis. The Darcy-Weisbach method is
generally considered more accurate than the Hazen-Williams method.Additionally, the Darcy-Weisbach method is valid for any liquid
or gas; Hazen-Williams is only valid for water at ordinary temperatures
(40 to 75o F).The Hazen-Williams method is very popular, especially among civil
engineers, since its friction coefficient (C) is not a function of
velocity or pipe diameter.Hazen-Williams is simpler than Darcy-Weisbach for calculations
where you are solving for flowrate, velocity, or diameter.
The
main governing equation for friction loss is Darcy’s equation
(5)
Where:Hf = friction loss in feet of head
f = dimensionless friction factor
L = Pipe Length in FEET
D = Pipe Inside Diameter in FEET
V = Flow Velocity in FEET PER SECOND
g = Gravitational Constant = 32.2 feet per second squared
From
this equation it is quite clear that the flow velocity has a big impact
on frictional losses. From equation 1 we know that the flow velocity is
inversely proportional to the square of the pipe diameter. So if we
reduce the pipe diameter by ½, we increase the flow velocity by a
factor of 4 and hence also increase the friction loss by a factor of 16.
To
apply this equation we need to determine the friction factor f, which is
the complicated part.The
steps are as follows:
We
first need to determine the type of flow which is typically determined
from the Reynold’s number, which rates the type of flow in the pipe:
Laminar, Turbulent or Transitional flow.The Reynold’s number is a dimensionless number and is
calculated as follows
(6)
D=
diameter of pipe in Feet
V
= velocity in Feet/sec
= kinematic viscosity of
the fluid being pumped
The
kinematic viscosity is the ratio of the fluid’s density and the fluids
absolute viscosity. The kinematic viscosity changes according to
temperature, see table (www.pump.net)
Table
2 : Kinematic Viscosity of Fresh Water
Temp
(degrees F)
Kinematic
Viscosity of fresh water
70
1.0265
x 10-5 ft2/sec
75
9.6199
x 10-6 ft2/sec
80
9.0363
x 10-6 ft2/sec
Kinematic
Viscosity is the ratio of the fluid's density and the fluid's absolute
viscosity. These values are for FRESH water. For saltwater multiply the
kinematic viscosity of water by 1.024 (or whatever your planned specific
gravity will be).
This
Reynolds Number will tell us whether a particular flow is laminar, in
the transition zone, or is turbulent, and the following chart gives the
generally accepted ranges for these flows:
Table
3: Relationship between Reynolds Number and Type of Flow
Reynolds
Number
Flow
Type
RD
< 2300
Laminar
2300
< RD < 4000
Transitional
4000
< RD
Turbulent
For
most of our application the flow through the pipes will generally be
turbulent.
Once
the type of flow is determined, the next step is to calculate the
friction factor.
The
Moody Equation is used to calculate the dimensionless friction factor f.
and originates from a paper published by Lewis F. Moody in 1944.Typically the values can be read off a Moody diagram (see http://www.mestudent.com/fluids/moody.htm
for a Moody Chart) which is created using the Colebrook-White formula
(7)
Where:f is the friction factor
ε
= roughness factor (generally .000005 ft for PVC pipe.)
D
= diameter in inches
Re
= Reynold’s number
The
equation is difficult to solve in close form since f appears on both
sides of the equation.It has to be solved numerically.
The
Swamee-Jain approximation can be used to calculate the friction factor
under certain conditions (where ε/D < .02 and Re >
3000),and
gives results within 3% of the results obtained from the Moody diagram.
It has the advantage of being easily programmed in a computer or
calculator. For most of the aquarium applications the flow is turbulent.
The Swamee-Jain approximation for the friction factor is as follows:
(8)
Another
alternate formula given by Haaland as follows can be used and the result
varies less than 2% of the Moody Chart. [Reference].
(9)
Using
this, as an example let us compute the friction loss in 100ft of 1”
PVC pipe with a flow rate of 30 GPM for saltwater at 80oF.
Given:L =100ft
D
= 1.049” (A 1” pipe has dimension of 1.049”)
Q
= 20 GPM
μ
= 9.0363 x 10-6 ft2/sec * 1.024 = 9.2521 x 10-6
ft2/sec
Velocity
= .4085*30/(1.049)2 = 11.136 ft/sec
The
friction factor f computed using Swamme-Jain equation = 0.01926
The
total friction loss Hf =39.71
The
value according to the published charts (based on Hazen-Williams) for
PVC schedule 40 pipe is 46.08 and is more conservative than the result
obtained using the governing equations.
Minor
Losses
For
any pipe system, in addition to the friction losses, there are
additional losses called minor losses (although in our case these
losses may far exceed the friction loss).These losses arise due to the pipe entrance and exit, sudden
expansion or contraction, bends, elbows, and other fittings, valves,
etc.
Here
again the empirical method can be used to determine the resistance
due to the fittings. The friction loss in the table 4 below is
expressed in terms of equivalent feet of straight pipe. What this
means is that the amount of friction created by the fittings is the
same as that of the specified straight pipe.We can add this length of straight pipe to the total pipe
length in the system and compute the friction loss.
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Table
4: Friction Losses in PVC fittings
Friction Loss in PVC Fittings = Equivalent Feet of Straight Pipe
Pipe
Size
1/2"
3/4"
1"
1
1/4"
1
1/2"
2"
3"
4"
90
ELBOW
1.5
2
2.25
4
4
6
8
12
45
ELBOW
0.75
1
1.4
1.75
2
2.5
4
5
GATE
VALVE
0.3
0.4
0.6
0.8
1
1.5
2
3
TEE-Flow-RUN
1
1.4
1.7
2.3
2.7
4.3
6.3
8.3
TEE-Flow-Branch
4
5
6
7
8
12
16
22
Another
more acceptable approach is to use the loss coefficient of each entity
in the plumbing system (often measured experimentally and correlated to
pipe flow parameters).
The
head loss across each fitting is then obtained using this equation:
Where:Hm =head loss in feet
K =loss coefficient
V =velocity in Ft per Second
g = gravity 32.2 ft/s2
The
resistance coefficients (K) for the valves and fittings are available as
tables. A good source is for these coefficients is www.pump.net.
Table 5 reproduced below can be found at http://pump.net/frictiondata/friclossfittings.htm
.In addition to providing the K values, this table also has a
column for L/D.This L/D can also be used directly in equation (5) to calculate
the friction loss for each fitting. One piece of data missing from table
5 is the K or L/D values for coupling or union fittings which are a
common item in aquarium plumbing. Typically quoted values of K for
coupling/union range between 0.02-0.04, with L/D=2.
Table
5: Friction Loss Coefficients for Pipe Fittings
Friction
Losses in Pipe Fittings
Resistance Coefficient K (use in formula hf = Kv²/2g)
The
system performance curve is the mapping of head required to produce flow
in a given system. A system includes all the pipes, fitting and devices
the fluid must flow through and represents the frictional and static
loss the fluid experiences.
The
operating point of the plumbing system is the point at which the pump
performance curve intersects the system curve (see Figure 1).This typically requires an iterative solution to determine the
point of intersection, and becomes tedious by hand.Selecting a pump requires that these calculations be repeated for
different pumps to determine the operating point for the different
pumps. Also if different plumbing systems are to be evaluated this would
require that the different systems performance curves are generated.
A
use of spreadsheet or computer software is
required. There are several commercially available software packages
that will do this, however these are not free.A sample spread sheet is included along with this article. It is
my hope that this will eventually evolve over time to become a complete
plumbing system design tool for reef hobbyists.
EXAMPLE
DESIGN SCENARIO
To
help in understanding the concepts presented, consider a simple
circulation loop to be designed for an aquarium system shown in figure 2
below:
The
design problem can take one of several forms:
Given
the pump and size of piping system, fittings, valves, and plumbing
schematic, what will be the resulting flow and velocity at the output?
Given
a desired flow rate and velocity at output, what size pump and piping
schematic should be used?
Given
the piping system, what is the best pump to use for the job?
.
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Scenario
#1
Let
us assume for starters that the system designer has determined that the
pump to be used is an Iwaki 55 RLT with a 1” outlet and inlet, and the
piping system used a PVC Schedule 40 pipe of nominal diameter 1.”We need to know what the flow rate and velocity will be at the
output in the tank. Given the inter relationships between friction and
flow the problem needs to be solved in an interactive manner.We start with a guess as the initial solution.A good initial guess may be made by just using the static head of
the system and the pump performance curve.Let’s start with a guess of say 780GPH or 13 GPM.
Step
1: Compute the friction head
Calculate
the Reynold’s number using formula, and the kinetic viscosity of
saltwater at 80o F
Re
= 45600
Compute
the friction factor f, either by using the Moody Chart or Swamee-Jain
approximation, using =.000005
ft for PVC pipe
f
=.0214
Using
Darcy-Wiesenbach equation compute the friction head
Hf
= .5316 ft
Step
2: Compute the Minor Losses
Follow
the path of the water flow all the way from the input to the output, and
tabulate all the entities that would contribute to the minor losses,
along with their K value.
Pipe Size
1”
# of ball valves
2
0.14
Gate Valve
0
0
# of Elbows - 90
3
2.07
# of Elbows-45
0
0
# of Couplings/Union
6
0.18
#of Swing Check Valve
1
1.2
Sudden Expansion
0
0
pipe Exit
1
1
Pipe Entrance
0.5
0.5
Total K
5.09
Minor Head Loss
1.776
Using
the basic energy equation:
The
total dynamic head (TDH) = Static Head + Friction Head + Minor Head loss
TDH
= 5 ft + .7615 ft + 1.776 = 7.03 ft
From
the pump chart we can see that the flow rate at TDH of 7.03 ft is 16GPM
or 960GPH, which is higher than the initial guess.So we would have to increase the guess and resolve. Solving this
iteratively gives a flow rate of 15.9 GPM(954 GPH) at a TDH of 8.412 ft with an output velocity of 5.9
ft/sec.The change in TDH in this case did not affect the flow much since
this is a pressure rated pump.
Scenario
#2
Now
let us consider the same problem but using 1.5” Schedule 40 PVC pipe.An additional fitting, a male adaptor 1” to 1.5” will need to
be added.In this case the flow will be 1025 GPH at a TDH of 5.66 ft.By upsizing the pipe from 1” to 1.5” the TDH was reduced by
2.752’ or nearly 1/3rd.This is really amazing considering that 5’ of the TDH is pure
static head height so the friction and minor losses using the 1.5”
pipe is only .66’!This would make a much more significant difference on a pump with
a flatter performance curve.Clearly upsizing the pipe to one size higher will result in more
flow at lower velocity – at 2.69 ft/sec.By increasing the size of pipe that is used you get a
higher flow rate with a lower velocity using the same pump.
This
is using an Iwaki 55 pump which does fairly well in most pressure rated
applications.Let’s say you want to use a pure circulation pump like the
Ampmaster 3000 but would still like to use 1” plumbing.Solving for the final flow rate using the governing equations you
would end up with a flow rate of 1,516 gph at a TDH of 11.89 ft with an
output velocity of 9.37 ft/sec.Increasing the size of the plumbing to 1.5” for the ampmaster
would increase the flow rate to 2,475 gph at a TDH of 7.93 ft with an
output velocity of 6.49 ft/sec.Nearly 1,000 gph additional flow rate and because of the larger
sized plumbing the velocity has actually been reduced.
Practical
Issues and use of this Spreadsheet
The
proper plumbing setup can be as important if not even more important
than the pump or pumps that you end up choosing.The calculations provided are intended to be a tool to help you
determine the best plumbing and pump setup for you.
The
biggest key to designing the best plumbing setup is to match up the
right size plumbing to the amount of flow you want to run through
it.When designing your plumbing setup we personally would not
recommend exceeding a velocity over 4 ft/sec.In my experience, you will experience more problems from the
higher velocities than by providing the same flow and using
additional outlets into the tank and larger sized plumbing acting as
feeders to the outlet to keep the velocities under 4 ft/sec.
While
not exact, this basically amounts to keeping your flow rates to
3,000 gph when using 2” pipe, 1,500 gph using 1.5” pipe, 700 gph
using 1” pipe and 400 gph using ¾” pipe.These are the final flow rates not the pumps rating at 1’
of head.If you try to limit your flow rates to those mentioned, the
friction losses you are experiencing should be relatively low.Trying to cram water through the plumbing at higher
velocities is typically what causes the higher friction losses.
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One
problem is thinking that if the pump you are using has a ¾” or 1”
inlet or outlet that is the size of plumbing that should be used.Another common misconception that many people have is that 90
degree elbows create an extreme amount of head loss.I’ve heard several people mistakenly quote that you need to
count one foot of head loss for every 90 degree elbow in your plumbing
setup.The wonderful thing about using these calculations is that you
can easily compute the differences in flow under different designs.
What
about the head loss calculators that are available on several bulletin
boards on the web?One of the biggest weaknesses of the head loss calculators on the
internet is that they do not have a wide variety of plumbing fixtures to
choose from.In addition they ignore a large number of frictional elements in
the plumbing system – e.g., entry and exit, sudden expansion and
contraction. Many of them only have an option for the number of feet and
flow rate, diameter of pipe and the number of 45 or 90 degree elbows.In addition, with these calculators you still need to guess on
the flow rate, and all comparisons to the pump curves to determine the
operating point have to be performed manually.
Instructions
for Using the Spreadsheet
Attached
here as part of the article is a spreadsheet that should make it
much simpler for you to calculate head loss.If you are familiar with Excel you should be able to add your own pump curve, but providing detailed instructions on how to add additional pump curves is beyond the scope of this article.On the tab labeled “head loss” you first need to select
the pump you are going to calculate the friction loss for.In addition, Reefs.org has just introduced a detailed pump database and the database is accessible from the following URL: http://www.reefs.org/library/pumps.This database contains pump curves, power usage, dimensions, etc. for the majority of the pumps commonly used in the aquarium hobby.Using this article in combination with the database should make pump and plumbing selection much easier than before.
On the tab labeled “head loss” you first need to select the pump you are going to calculate the friction loss for.Once
you have selected the pump, you need to enter the diameter (in inches) of the pipe and the total length of pipe to be used.Next, the spreadsheet provides a place to input the number of
several different types of valves and fittings that could be used in
your setup.It also has a
spot to use if you are upsizing or downsizing the pipe at any point in
the setup using expansion or reduction couplings.Finally, you will enter the static head of the setup, which is
the vertical height that the water would need to be pumped.A closed recirculation loop would have zero static head.
Once
all of the information is filled in you need to click the “Solve”
button and you should get your answer.The spreadsheet does all of the circular equation solving for
you!
One
common problem that you might encounter is the fact that the
“Solver” is an add-in item in Excel.If you get an error message when trying to use the solver, you
will need to install it.On
the “Tools” menu in Excel, click on “Add-Ins”, then scroll down until
you locate the “Solver Add-In” option.Make sure that you have the box checked.If you don’t have the box checked beside the “Solver
Add-In” option then you should be able to simply check the box and
then click on “OK” and the Solver Add-In will be installed.Now you should be all set to figure out all sorts of
different scenarios.
One
limitation that the spreadsheet does have is that currently it will only
solve for a problem with one size of pipe.If you are going to use multiple sizes of pipe, your options are
to try to manually solve for the equation using both sizes of pipe (the
spreadsheet solver won’t work for this), or to try to manipulate the
information that you input to give you basically the same answer.One method of solving for multiple sizes of pipe would be to use
the friction tables and the head loss calculators found on different
websites.You would need to estimate your final flow rate and enter the
information on each sized pipe being used into the calculator using your
estimated flow rate.When you get your answer concerningfriction loss, you would need to add up the friction loss for each
size of pipe and compare that to the flow rate you originally used, and
see if that matches up to the pump curve.You would need to continue to solve for each pipe size until you
got an answer where the same flow rate was used for all sizes of pipe
and when the friction and vertical head loss was added together for all
sizes of pipe, the total head matched up with the flow rate that you used
on the pump's performance curve.There really isn’t an easy way to solve for friction loss using
multiple sizes of pipe, which would explain why the spreadsheet won’t
do it!
Conclusions
This
article presents an engineering view of plumping and plumbing system
design for reef aquariums, and provides a basis for evaluating plumbing
designs and selecting pumps.While the mathematics need not be understood by all readers, the
key findings based on the mathematics are important for any aquarium
system designer: increasing the diameter of the pipe reduces friction
and velocity, but not the flow rate. The operating point of the system
will occur at the intersection of the system and pump performance
curves. Reduce friction by proper sizing of pipes!
In
this article we only focused on a single loop design.While the use of multiple branches and splits can be derived from
these equations, these will be the subject of a future article.
