*By Larry Henke*

*Summary*: Water treatment system pressure losses are important to building engineers, water treatment designers and the customers that have to live with them. They can result in slower flows and/or unexpected changes in temperature or improper water treatment at worst. Predicting expected losses in advance of manufacture and installation can aid treatment specialists in design and can minimize consequences, especially in commercial/industrial applications. Here are guidelines and recommendations for estimating pressure losses for typical softener and filter applications.

Pressure is the energy required to move water through a system. It’s composed of the potential energy—often called “head”—and represents the force of gravity plus kinetic energy, which is a function of the velocity of the water through the pipe or vessel plus the pressure. Bernoulli’s Theorem relates kinetic and potential energy and flow in fluid physics. The theorem is a special case of the Law of the Conservation of Energy and can be expressed in several forms, whether the problem under consideration is focused on the kinetic energy or velocity, the pressure at a certain point or the change in height (potential energy). Pressure and head are the same, although head is usually stated in feet and pressure in pounds per square inch (psi), i.e., 231 feet of head equals 100 psi.

The energy or pressure is reduced as water flows through a system. Pressure loss can be calculated by subtracting the resistance imposed by the nature of water flow, such as turbulence or viscosity, piping restrictions and friction within the piping components and the media from the initial pressure. Losses are introduced by each component of the system including the pipe, each bend or turn, change in pipe or fitting diameter, valves, media and the distributor or riser piping. Let’s look at each of these factors independently and then illustrate how to combine them for a total “pressure loss” or “head loss.”

The total pressure loss through a system is the sum of the individual pressure losses. When water is split into two or more parallel flows the calculation is more complicated but can still be performed. For the purposes of this article, however, we’ll ignore parallel flows and discuss only the main sources of pressure loss: pipe and fittings, control valves, the media bed and under-drain or distributor internals.

**Pressure drops—pipe**

All water flow is either laminar (where streamlines can be imagined that are continuous and smooth) or turbulent (where the water flow forms a series of vortices and swirls). As velocity increases, turbulence increases. Surface effects, along with other interference in the flow introduced by bends, constrictions or converging flows also form turbulence. Turbulence can be imagined as water molecules that are moving contrary to the general flow of the fluid. The faster the water moves the greater the turbulence, as more molecules move “backward” (or sideways) in opposition to the general flow. The sum of friction, turbulence and velocity pressure losses generate an overall pressure loss. The use of an index called a Reynolds Number offers engineers a way of expressing the conditions of laminar and turbulent flow.

In most cases, however, for pipe flows, empirical measurements have been used to confirm the theory and to establish constants. Charts available in plumbing code guides will give expected losses through pipe and fittings of the size and type for most applications. Engineers have established the flows at which water should be allowed to move through pipe to minimize corrosive and turbulent effects. The maximum velocity varies from 5-to-10 feet per second, depending on the pipe material and the application. In some cases, such as at filter or softener inlets, still lower velocities are required. Seldom are higher velocities allowed because of the rapid increase in pressure loss as the velocity increases and because of the hazard of hydraulic shock.

The following formula is helpful in calculating velocities through a pipe:

Velocity in feet per second equals Q (flow in gallons per minute) multiplied by 0.408 (a conversion factor) and divided by d2 (pipe diameter in inches squared).

**Pressure drops—fittings**

Pressure losses through pipe fittings are commonly expressed in equivalent length of pipe at certain specified velocities. Different fittings and valves induce different losses, and the same fitting (such as a “T” or elbow) may have different losses depending on the placement in the pipe system. Ts where two streams flow together will differ from two streams that converge (see Figure 2). Valves also provide different losses depending on the style (ball/butterfly/gate) and the placement within a system.

Several formulae have been developed for pipe and fitting calculations. While they have several differences in assumptions, they all relate the velocity of water to the conditions, such as the comparative smoothness or roughness or the pipe wall, the radius of turns, and the nature of connections and branching. Most of the formulae are of the form DP = ½ KV², where V is the velocity of the water and K is a constant unique to the situation. DP, or “delta P,” represents the change in pressure. The pressure drop is related to the square of the velocity, which means a very small change in velocity results in a large change in pressure. A chart for typical residential fittings showing pressure drop in psi, which should be added to the total pipe length, is shown in Figure 3, and should be added to the total pipe length.

In the case of constrictions or in reductions in pipe size, such as with a venturi, pressure losses can be dramatic. Venturis are a special case for when the flow is suddenly restricted, the velocity increases and the pressure drops at that point. This drop in pressure is exploited to create a suction for air and other gases or fluids. The brine eductor in a water softener employs this principle.

**Pressure drops—control valves**

Pressure losses through multi-port control valves are commonly expressed as a Cv, or C-V, index. The Cv is determined by the manufacturer, often empirically, and offered as a graph of the pressure loss at different flow rates in the different operating positions of a valve. Thus, there can be a different Cv for service flow, backwash flow and rinse flows. It’s the flow at which a 1-psi pressure loss is incurred. The Cv expresses the pressure loss as a function of the flow rate as in the following formula (see Figure 4):

If you know the Cv of a valve, and the flow rate, the pressure loss is calculated as (see Figure 5):

Many of the graphs supplied by the manufacturers are straight-line graphs, just as in the case of the graphs for pipe flow and fitting losses. It’s important to note that they’re “semi-log” or “log-log” graphs—that is, the straight line is a result of the graph having been drawn on a paper that has logarithmic scales on one or both axes. This is done to make the scales easier to read and interpret. A logarithmic graph, however, means a very small change on one scale may result in a very large or exponential change on the other.

Most valve Cv graphs for multiport valves will show separate curves for service, backwash and rinse positions with separate lines, since in such a valve the water is diverted through different chambers and ports for each of these functions.

**Pressure drops—media beds**

Media flow characteristics are a function of the media shape, size and specific gravity (or weight). Factors included in the calculation of flows through a deep bed of media are water temperature, bed surface area and water velocity. Bed depth also plays a role; however, bed loss is usually shown for a 1-foot bed depth and deeper beds can be extrapolated from that figure.

Other researchers have approached the problem of water flow through granulated media in columns with varying assumptions. Some assume uniform spheres, others allow for irregular shaped media. One common model treats the media as a large number of tiny, tube-like channels. Particles are thought to travel through a lengthy path where some can become attached to the media granules. The more convoluted—or tortuous—the path, the more opportunity for a particle to attach.

As the water is divided up into streams, each small micro-location becomes a “laminar” flow region, where there is little or no turbulence. The particle then attaches to the grains through a poorly understood combination of electrostatic, molecular forces and adhesion.

In the case of a water softener, a calcium or magnesium ion approaches the region, contacts the resin, attaches to a site releasing a sodium ion and is then moved to the internal structure. Although the capture mechanism is different, the water flow characteristics within a softener are similar to that of a filter

Unless your application is unusual, and loss through the bed is critical, the use of graphs supplied by the media manufacturer should be sufficient to approximate headloss. The following graph (see Figure 6) is one from Rohm and Haas, although other manufacturers have similar graphs. Note that it too is a “log-log” graph, thus producing a straight line for an exponential function.

**Pressure drops—distributors**

Beneath most resin or sand beds in softeners and filters are support beds of coarse media, such as gravel. Within the bottom support beds are under-drains or bottom distributors. Most bottom distributors are slotted to prevent resin or media from entering the riser pipe. Although the slots vary in width and length, a simple measurement can determine the total area open for water flow. While there are factors that should be included—such as the change in flow rate at more distant points of a lateral, for example—most of these can be safely neglected for our purposes.

Empirical formulas can be used for screen or distributor determination, but it’s important to note that simply matching the exposure area of the distributor to the cross section of the pipe won’t limit the pressure loss, and that once again, the pressure loss is a function of the square of the velocity. Still, knowing the area can be helpful in selecting a distributor. As an example, the following distributor *configurations were measured to produce the areas as indicated:*

**For a residential water softener distributor the area of the combined slots in a distributor is:**

- A standard basket on ¾-inch pipe equals 0.65 in²; ¾-inch pipe has 0.44 square inches of cross section,
- A basket attached to a 1.05-inch pipe equals 1.17 in²; a one-inch pipe has 0.78 square inches of cross section, and
- A slotted ¾-inch riser pipe equals 1.10 in²; a ¾-inch pipe has 0.44 square inches.

**For commercial softener distributors:**

- A 2-inch basket (for a 24-inch tank) equals 3.78 in²; a 1 ½-inch riser pipe has 1.76 square inches of cross section,
- A ¾- × 8-inch, 8-leg hub-and-lateral equals 10.08 in²; a 2-inch riser pipe has 3.14 square inches of cross section, and
- A 5/8- × 10-inch, 8-leg hub-and-lateral equals 5.94 in²; a 1 1/4-inch riser has 1.23 square inches of cross section.

There’s a significant variation from distributor to distributor, and the type and size should be considered when designing softening or filtering equipment. In addition, flow characteristic through a slotted pipe is different than the same flow rate through an open pipe of equal area (see Other Equations). Care in selecting whether to use a suspended basket (usually a tapered, screened cylinder), a hub-and-lateral (a center hub with slotted “spokes” reaching into the media) or a herringbone (a crossed pattern of screened tubes) distributor, and in the selection of screens with sufficient area, will help limit pressure losses at this point in the system.

**Conclusion**

Flow rate and pressure drops are an important consideration in the design of filters and softeners. While the calculations for any specific application may be complex and difficult, use of manufacturer supplied graphs, Cv values and a careful accounting of all piping and fitting components, along with attention to the under-drains and internal piping can offer a close approximation in most cases. A few minutes spent evaluating the expected losses can prevent problems after installation.

**References**

- Metcalf and Eddy, Wastewater Engineering, McGraw-Hill, New York, 1991.
- Feynman, R.P., R.B. Leighton and M. Sands, Lectures on Physics (40 and 41), Addison-Wesley Publishing Company, Reading, Mass, 1964.
- Ives, K.J., “Rapid Filtration,” Water Research, Vol.4, pp. 201-223, 1970.

*About the author*

*Larry Henke has more than 20 years experience in the water treatment industry and is technical director at the Robert B. Hill Co. in St. Louis Park, Minn., near Minneapolis. He’s a graduate of the University of Minnesota and is a member of the American Water Works Association, National Ground Water Association and WC&P’s Technical Review Committee. He can be reached at (612) 925-1444, (612) 925-1471 (fax) or email: lrhenke@email.msn.com*

Figure 1.

V = 0.408 Q

d2

Figure 2. “T” or elbow fittings may have different pressure losses depending on placement in the pipe system; shown are two Ts where two streams flow together will differ from two streams that converge.

Figure 3. A chart from Fleck controls for typical residential fittings showing pressure drop in psi, which should be added to the total pipe length.

Figure 4.

Q2 = Cv2 × DP

Figure 5.

DP = Q2

Cv2

Figure 6. The use of graphs supplied by the media manufacturer—this one from Rohm and Haas—should be sufficient to approximate headloss.

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Other Equations for Calculating Pressure Loss

Pressure and flow characteristics through various media have been studied by civil engineers for years. Over 100 years ago, Henry Darcy began studies of the flow through sand column, and established a formula:

Q/A = K[DH / DL]

In this equation, K is a constant and isn’t the same as any other K. Q is the flow rate, A the surface area, H the head loss and L the bed depth.

The Kozeny equation for service flow rates:

h/L = kµ/rg • (1 – e)²/e³ • (a/v)²V

Where:

• h equals head loss in depth of bed L

• g equals acceleration of gravity, a constant

• e equals porosity

• a/v equals grain surface area equals specific surface S, which equals 6/d for spheres and 6/yde for irregular grains, where y equals sphericity, and “de” equals grain diameter of a sphere of equal size

• V equals superficial velocity above bed, which equals flow rate/area

• µ equals absolute viscosity of fluid

• r equals mass density of fluid

• k equals dimensionless Kozeny constant, usually close to 5.

The total losses through media are then related to the velocity of the water through the bed, along with its viscosity (which is itself related to the temperature), the shape and density of the media, and the bed depth.

In 1952, Blake Ergun generated the following equation:

Dh/DL = k1m¤rg •[(1 – e)²/e³] • (S)²V + k2 1/g •[(1 – e)²/e³] • (S)²V2

You’ll notice the equation is very similar to the Kozeny equation, but with a second term. The Ergun equation assumes a more turbulent flow, and is generally preferred with higher flow rates, such as in condensate polishers. For irregularly shaped media, such as sand, garnet or anthracite, the determination of S is difficult. In the case of uniform beads of resin, however, S is fairly straightforward. In the case of spherical media, such as resin, it can be assumed that packing is uniform. Uniform packing offers the most compact media packing, thus flows will be different. This equation then is good for most low- to medium-flow softener and filter applications.

One formula for pressure loss through a screen or distributor is of the form:

DP = K1K2K3 V²/2g

Where again the K’s are constants related to specific conditions. K1 is a clogging constant (which will possibly change as a run continues), K2 is the shape of the bars or screen wires, and K3 is the openings between wires or bars. Other empirical formulae can be used for screen or distributor determination, but it’s important to note that simply matching the area of the distributor to the pipe won’t limit the pressure loss and, once again, the pressure loss is a function of the square of the velocity. Still, knowing the area can be helpful in selecting a distributor.

—Larry Henke