What is Your Zero

What is Your Zero?

By Jeff Sines, Product Engineer at Engineered Software, Inc.

In all aspects of life, numbers are used to describe the amount of something, whether it’s the cost of a product, the distance between two cities, how fast you are driving, or how old your children are. But a number is meaningless unless it has two things: a unit and a zero reference datum. In science and engineering, miscommunication, confusion, and costly mistakes can occur when numbers are used without proper attention to units or their relationship to a reference point.

In September 1999, the $325 million NASA Mars Climate Orbiter disintegrated in the upper atmosphere of Mars because the navigation software produced output in Imperial units of pound-seconds (lbf·s) instead of SI units of newton-seconds (N·s). This error caused Trajectory Correction Maneuvers to place the orbiter on a path to enter the Mars atmosphere at an altitude of 60 km instead of the intended altitude of 225 km. A minimum altitude of 80 km was required for safe orbit around Mars.

Not all miscommunication is as costly as the Mars Climate Orbiter. However, expensive mistakes are commonly made here on Earth.  When designing, installing, operating, and maintaining piping systems these mistakes often result in lost production, off-quality product, excessive downtime, environmental excursions, and re-work. It is crucial that communications be as concise as possible for all who are involved in any aspect of operating a process system, particularly those involved in extremely hazardous fluids and energy levels.

Figure 1. Artist conception courtesy of sanfrancisco.cbslocal.com.

Fluid Property Units

There are numerous fluid properties that are used in the design and analysis of piping systems: pressure, temperature, density, viscosity, and specific heat capacity to name a few. Each property has a numerical value and unit that gives meaning to that property and allows it to be used to determine the operating conditions for the system. Understanding what the unit means, and when a unit conversion must be applied, is crucial to the appropriate use of the property.

For example, when is a “foot” not a measure of distance? The pump industry characterizes the performance of a pump by how many feet of Total Head it produces. Head and feet in the study of fluid flow does not refer to distance or parts of your body, but instead is used to quantify the specific energy, or energy content per unit weight (or mass), of the fluid.

{Head=\frac{Energy}{Unit\ Weight}=\frac{ft\bullet lb}{lb}=ft}

But even the definition of head has some nuances that are not obviously apparent and must be taken into account in order to properly use the numerical values associated with it.

A key concept with head is that its numerical value is referenced to the density of the fluid under consideration. One foot of head of a given fluid is not the same energy content as one foot of head of a different fluid. Consider two site glasses containing two different fluids separated by a liquid/liquid interface at Points 1 and 2, as shown in Figure 2. In order for the two fluids to be in equilibrium at the interface, the total fluid energy at Point 1 (H1 due to the height of the column and density of Fluid 1) must be equal to the total fluid energy at Point 2 (H2 due to the height of the column and density of Fluid 2).

But it’s obvious that H1 does not equal H2, so how can the two values represent the same amount of fluid energy? To reconcile this apparent discrepancy, it’s important to understand that H1 quantifies the amount of fluid energy at the interface in reference to the density of Fluid 1, while H2 quantifies the same amount of fluid energy but uses the density of Fluid 2 as the reference.

If the two fluids are in equilibrium with no flow, the static pressure at Point 1 must equal the static pressure at Point 2.

 {P_1=\rho_1 g H_1=P_2=\rho_2 g H_2}

{\rho_1 g H_1=\rho_2 g H_2}
{H_1=\big(\frac{\rho_2}{\rho_1}\big) H_2}

This density compensation adjusts one fluid’s numerical value of head so that it can be directly compared to another fluid’s head, in essence putting them on the same zero reference datum.

 

Figure 2. Fluid Pressure Head is a function of column height and density.

An example where this distinction may have an impact on the sizing and selection of a pump can be seen in Figure 3. A pump must add sufficient energy to the fluid to overcome two aspects of fluid flow: energy lost due to friction (Dynamic Head Loss) and Static Head due to elevation and pressure differences between where the fluid is pumped to and where it is pumped from.

 {\text{Total Head}=\text{Dynamic Head Loss}+\text{Static Head}} 

 {\text{Static Head}=\Delta\text{Pressure Head}+\Delta\text{Elevation Head}} 

The pump itself does not know how much of its Total Head is used to overcome the Head Loss or how much goes to compensate for the Static Head. The pump “feels” a pressure at its inlet due to the configuration and Head Loss leading into its inlet, and feels a pressure at its discharge due to the configuration and Head Loss of the system at its outlet. In other words, the piping system tells the pump how much Total Head it must produce at a given flow rate.

Figure 3. Systems with different values of Static Head modeled in PIPE-FLO® Professional.

The static head that Pump 1 must overcome in System 1 can be calculated:

 {\text{Static Head of System 1}=(110\ ft-10\ ft)=100 \ ft} 

In System 2, a heat exchanger located at the 25 foot elevation heats up the water from 60 °F to 200 °F, resulting in a change in the density of the fluid from 62.38 lb/ft3 to 60.14 lb/ft3. Does Pump 2 feel the same amount of Static Head as Pump 1?

Because the density of the fluid from the outlet of the heat exchanger to the liquid surface in the Product Tank in System 2 is less than System 1, the pressure felt at the 25 foot elevation due to this 85 foot column of water will be less than the pressure at the same location in System 1. The Static Head felt by Pump 2 must include a portion of the system that is density compensated because the water flowing through the pump is at 60 °F:

 {\text{Static Head of System 2}=(25\ ft-10\ ft)+\big(\frac{60.14}{62.38} \big)(110\ ft-25\ ft)=96.9 \ ft} 

This difference of 3% in Static Head between System 1 and System 2 may not be significant enough to result in over-sizing the pump in System 2, especially if a design margin of 15-25% is added prior to pump selection. But every unnecessary foot of head means that the pump is adding more energy than the system needs, which results in additional flow or additional throttling that must occur at a control valve to dissipate the energy. That equates to higher energy cost, more wear and tear on the control valve, a greater susceptibility to cavitation and choked flow and potentially more downtime for maintenance and equipment replacement.

Gauge or Absolute?

An important qualifier for fluid pressure is the distinction between absolute and gauge pressure. This distinction establishes the zero reference for values of pressure.

 {P_{absolute}=P_{atm}+P_{gage}} 

Absolute pressure (in units of psia, kPa a, bar a) uses the pressure that would exist when all atoms and molecules are removed from a given volume (absolute zero pressure) as the reference point. This condition is physically impossible to attain on Earth, but makes a good reference value for pressure. Pressure below absolute zero cannot be achieved. For example, many fluid properties, like vapor pressure and critical pressure, are tabulated using absolute pressure in units.

Gauge pressure (in units of psig, kPa g, bar g) uses the local atmospheric pressure as the zero reference. The most common type of pressure instrument compares the fluid pressure to the atmospheric pressure. Pressure below atmospheric (but above absolute zero) is referred to as a vacuum. A vacuum can be quantified as a negative value of gauge pressure but is typically expressed in units of inches of water column for slight vacuum, and inches of mercury for strong vacuum.

More often than not, when plant personnel are discussing pressure, “the pump discharge pressure is 125 psi,” the distinction between absolute and gauge is not made, but instead is assumed to mean gauge pressure because that’s where the pressure value is read from. Often, an abbreviated form of the pressure unit is used: “the pump discharge pressure is 125 pounds.” As long as everyone is on the same page, this lack of concise use of units won’t be a problem.

However, there are engineering calculations where the distinction between absolute and gauge pressure must be made. For example, when sizing and selecting control valves for compressible gas applications, the Pressure Drop Ratio (x) must be calculated from the inlet pressure (P1) and outlet pressure (P2) in absolute pressure units. The ISA 75.01 standard for sizing control valves defines the Pressure Drop Ratio as:

 {x=\frac{P_1-P_2}{P_1}=\frac{dP}{P_1}} 

The use of gauge pressure in the equation will result in the wrong value for the Pressure Drop Ratio. For example: Patm = 14.7 psia, P1 = 50 psig or 64.7 psia, P2 = 25 psig or 39.7 psia

Correctly using absolute pressure:

 {x=\frac{P_1-P_2}{P_1}=\frac{64.7-39.7}{64.7}=\frac{25}{64.7}=0.386} 

Incorrectly using gauge pressure:

 {x=\frac{P_1-P_2}{P_1}=\frac{50-25}{50}=\frac{25}{50}=0.5} 

An incorrect calculation for the Pressure Drop Ratio may make the difference between selecting the right control valve for the application or one that might result in choked flow at the valve.

Figure 4. Gauge and Absolute Pressure.

Differential Property or Property at a Given Location?

Understanding the distinction between units that refer to a fluid property at a given location in the piping system and the differential between two locations is also important to the concise communication and proper use of values and units. Often in engineering it’s not the value of a fluid property at a given location that’s important, but instead the difference between two points.

For example, when designing and analyzing a piping system, one of the first things that must be established is the zero reference datum from which all vertical elevation measurements are made. This reference plane may be at sea level, the grade of the facility, the keel of a ship, the bottom of a tank, the centerline of the pump impeller, or any random elevation of your choosing. To properly apply the Bernoulli Equation, Head Loss can be determined from the difference in the fluid’s total hydraulic energy between two points in the system.

 {Z_1+\frac{144P_1}{\rho}+\frac{v^2_1}{2g}=Z_2+\frac{144P_2}{\rho}+\frac{v^2_2}{2g}+h_L} 

The total hydraulic energy includes the fluid’s Elevation Head (Z), Pressure Head{\big(\frac{144P}{\rho}\big)}, and Velocity Head{\big(\frac{v^2}{2g}\big)}. The individual value of the Elevation Head at each point is not important as long as both elevations are measured from the same reference plane.

Know Your Zero!

Piping systems are designed, built, and operated to transport fluids through various process equipment to change their properties in order to make a product or perform work. The energy content of the fluid must be elevated to move the fluid through the system.

In the design and operation of a piping system, costly mistakes that result in lost production, off-quality product, excessive downtime, environmental excursions, and costly re-work can be prevented by solid engineering understanding and concise communication and use of units, equations, and concepts.

Don’t let your piping system burn up in the atmosphere of Mars! Know your zero!