Calculation of Gas Density and Viscosity

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PDH Credits:

2 PDH Course No.: GDV101

Publication Source:

Harlan Bengtson, PhD, PE “Calculation of Gas Density and Viscosity”

Release Date: 2017

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Flow Measurement in Pipes and DuctsDr. Harlan H. Bengtson, P.E.

COURSE CONTENT

1. Introduction

This course is about measurement of the flow rate of a fluid flowing under pressure in a closed conduit. The closed conduit is often circular, but also may be square or rectangular (such as a heating duct) or any other shape. The other major category of flow is open channel flow, which is the flow of a liquid with a free surface open to atmospheric pressure. Measurement of the flow rate of a fluid flowing under pressure, is carried out for a variety of purposes, such as billing for water supply to homes or businesses or, for monitoring or process control of a wide variety of industrial processes, which involve flowing fluids. Several categories of pipe flow measurement devices will be described and discussed, including some associated calculations.

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2. Learning Objectives

At the conclusion of this course, the student will

• Be able to calculate flow rate from measured pressure difference, fluid properties, and meter parameters, using the provided equations for venturi, orifice, and flow nozzle meters.

• Be able to determine which type of ISO standard pressure tap locations are being used for a given orifice meter.

• Be able to calculate the orifice coefficient, Co, for specified orifice and pipe diameters, pressure tap locations and fluid properties.

• Be able to estimate the density of a specified gas at specified temperature and pressure using the Ideal Gas Equation.

• Be able to calculate the velocity of a fluid for given pitot tube reading and fluid density.

• Know the general configuration and principle of operation of rotameters and positive displacement, electromagnetic, target, turbine, vortex, and ultrasonic meters.

• Know recommended applications for each of the type of flow meter discussed in this course.

• Be familiar with the general characteristics of the types of flow meters discussed in this course, as summarized in Table 2 in the course content.

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3. Types of Pipe Flow Measurement Devices

The types of pipe flow measuring devices to be discussed in this course are as follows:

I) Differential pressure flow meters

a) Venturi meterb) Orifice meterc) Flow nozzle meter

ii) Velocity flow meters – pitot / pitot-static tubes

iii) Variable area flow meters - rotameters

iv) Positive displacement flow meters

v) Miscellaneous

a) Electromagnetic flow metersb) Target flow metersd) Turbine flow meterse) Vortex flow metersf) Ultrasonic flow meters

4. Differential Pressure Flow meters Three types of commonly used differential pressure flow meters are the orifice meter, the venturi meter, and the flow nozzle meter. These three all function by introducing a reduced area through which the fluid must flow. The decrease in area causes an increase in velocity, which in turn results in a decrease of pressure. With these flow meters, the pressure difference between the point of maximum

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velocity (minimum pressure) and the undisturbed upstream flow is measured and can be correlated with flow rate.

Using the principles of conservation of mass (the continuity equation) and the conservation of energy (the energy equation without friction or Bernoulli equation), the following equation can be derived for ideal flow between the upstream, undisturbed flow (subscript 1) and the downstream conditions where the flow area is constricted (subscript 2):

Where: Qideal = ideal flow rate (neglecting viscosity and other friction effects), cfs

A2 = constricted cross-sectional area normal to flow, ft2

P1 = upstream (undisturbed) pressure in pipe, lb/ft2

P2 = pressure in pipe where flow area is constricted to A2, lb/ft2

β = D2/D1 = (diam. at A2)/(pipe diam.)

ρ = fluid density, slugs/ft3

A discharge coefficient, C, is typically put into equation (1) to account for friction and any other non-ideal factors, giving the following general equation for differential pressure meters:

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Where: Q = flow rate through the pipe and meter, cfs

C = discharge coefficient, dimensionless

All other parameters are as defined above

Each of the three types of differential pressure flow meters will now be considered separately.

Venturi Meter: Fluid enters a venturi meter through a converging cone of angle 15o to 20o. It then passes through the throat, which has the minimum cross-sectional area, maximum velocity, and minimum pressure in the meter. The fluid then slows down through a diverging cone of angle 5o to 7o, for the transition back to the full pipe diameter. Figure 1 shows the shape of a typical venturi meter and the parameters defined above as applied to this type of meter. D2 is the diameter of the throat and P2 is the pressure at the throat. D1 and P1 are in the pipe before entering the converging portion of the meter.

Figure 1. Venturi Meter Parameters

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Due to the smooth transition to the throat and gradual transition back to full pipe diameter, the head loss through a venturi meter is quite low and the discharge coefficient is quite high. For a venturi meter the discharge coefficient is typically called the venturi coefficient, Cv, giving the following equation for a venturi meter:

The value of the venturi coefficient, Cv, will typically range from 0.95 to nearly one. In ISO 5167 ( ISO 5167-1:2003 – see reference #2 for this course), Cv is given as 0.995 for cast iron or machined venturi meters and 0.985 for welded sheet metal venturi meters meeting ISO specifications, all for Reynold’s Number between 2 x 105 and 106. Information on the venturi coefficient will typically be provided by venturi meter manufacturers.

Example #1: Water at 50o F is flowing through a venturi meter with a 2 inch throat diameter, in a 4 inch diameter pipe. Per manufacturer’s information, Cv = 0.99 for this meter under these flow conditions. What is the flow rate through the meter if the pressure difference, P1 – P2, is measured as 8 inches of Hg.

Solution: The density of water in the temperature range from 32o to 70oF is 1.94 slugs/ft3, to three significant figures, so that value will be used here. A2 = πD2

2/4 = π(2/12)2/4 = 0.02182 ft2. β = 2/4 = 0.5. Converting the pressure difference to lb/ft2: P1 – P2 = (8 in Hg)(70.73 lb/ft2/in Hg) = 565.8 lb/ft2. Substituting all of these values into equation (3):

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Orifice Meter: The orifice meter is the simplest of the three differential pressure flow meters. It consists of a circular plate with a hole in the middle, typically held in place between pipe flanges, as shown in figure 2.

Figure 2. Orifice Meter Parameters

For an orifice meter, the diameter of the orifice, d, is used for D2 (giving A2 = Ao), and the discharge coefficient is typically called an orifice coefficient, Co, giving the following equation for an orifice meter:

The preferred locations of the pressure taps for an orifice meter have undergone change over time. Previously the downstream pressure tap was preferentially located at the vena-contracta, the minimum jet area, which occurs downstream of the orifice plate, as shown in Figure 2. For a vena-contracta tap, the tap location depended upon the orifice hole size. This link between the tap location and the orifice size made it difficult to change orifice plates with different hole sizes in a

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given meter in order to alter the range of measurement. In 1991, the ISO-5167 international standard came out, in which three types of differential measuring taps were identified for orifice meters, as illustrated in Figure 3 below. In ISO-5167, the distance of the pressure taps from the orifice plate is specified as a fixed distance or as a function of the pipe diameter, rather than the orifice diameter as shown in Figure 3.

In ISO-5167, an equation for the orifice coefficient, Co, is given as a function of β, Reynolds Number, and L1 & L2, the distances of the pressure taps from the orifice plate, as shown in Figures 2 and 3. This equation, given in the next paragraph can be used for an orifice meter with any of the three standard pressure tap configurations.

Figure 3. ISO standard orifice meter pressure tap locations

The ISO-5167 equation for Co , (also available in reference #3 for this course, U.S. Dept. of the Interior, Bureau of Reclamation, Water Measurement Manual), is as follows: Co = 0.5959 + 0.0312 β2.1 - 0.1840 β8 + 0.0029 β2.5(106/Re)0.75

+ 0.0900(L1/D)[β4/(1 - β4)] - 0.0337 (L2/D) β3 (5)

Where: Co = orifice coefficient, as defined in equation (4), dimensionless

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L1 = pressure tap distance from upstream face of the plate, inches

L2 = pressure tap distance from downstream face of the plate, inches

D = pipe diameter, inches

β = ratio of orifice diameter to pipe diameter = d/D, dimensionless

Re = Reynolds number = DV/ν = DVρ/µ, dimensionless (D in ft)

V = average velocity of fluid in pipe = Q/(πD2/4), ft/sec (D in ft)

ν = kinematic viscosity of the flowing fluid, ft2/sec

ρ = density of the flowing fluid, slugs/ft3

µ = dynamic viscosity of the flowing fluid, lb-sec/ft2

As shown in Figure 3: L1 = L2 = 0 for corner taps; L1 = L2 = 1 inch for flange taps; and L1 = D & L2 = D/2 for D – D/2 taps. Equation (5) is not intended for use with any other arbitrary values for L1 and L2.

There are minimum allowable values of Reynolds number for use of equation (5) as follows. For flange taps and (D – D/2) taps, Reynolds number must be greater than 1260β2D. For corner taps, Reynolds number must be greater than 10,000 if β is greater than 0.45 and Reynolds number must be greater than 5000 if β is less than 0.45.

Fluid properties (ν or ρ & µ) are needed in order to use equation (5). Tables or graphs with values of ν, ρ, and µ for water and other fluids over a range of temperatures are available in many handbooks and fluid mechanics or thermodynamics textbooks, as for example, in reference #1 for this course. Table 1 shows density and viscosity for water at temperatures from 32o F to 70o F.

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Table 1. Density and Viscosity of Water

Dynamic Kinematic

Temperature, o F Density, slugs/ft 3 Viscosity, lb-s/ft 2 Viscosity, ft 2 /sec

32 1.94 3.732 x 10-5 1.924 x 10-5

40 1.94 3.228 x 10-5 1.664 x 10-5

50 1.94 2.730 x 10-5 1.407 x 10-5

60 1.938 2.334 x 10-5 1.204 x 10-5

70 1.936 2.037 x 10-5 1.052 x 10-5

Example #2: What is the Reynolds number for water at 50oF, flowing at 0.35 cfs through a 4 inch diameter pipe?

Solution: Calculate V from V = Q/A = Q/(πD2/4) = 0.35/[π(4/12)2/4] = 4.01 ft/s. From Table 1: ν = 1.407 x 10-5 ft2/s. From the problem statement: D = 4/12 ft. Substituting into the expression for Re: Re = (4/12)(4.01)/(1.407 x 10-5)

Re = 9.50 x 10 4

Example #3: Use equation (5) to calculate Co for orifice diameters of 0.8, 1.6, 2.0, 2.4, & 2.8 inches, each in a 4 inch diameter pipe, with Re = 105, for each of the standard pressure tap configurations: i) D – D/2 taps, ii) flange taps, and iii) corner taps.

Solution: Making all of these calculations by hand using equation (5) would be rather tedious, but once the equation is set up in an Excel spreadsheet, the repetitive calculations are easily done. Following is a copy of the Excel spreadsheet solution to this problem.

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D - D/2 Taps:

D, in d, in L1, in L2, in β Re Co

4 0.8 4 2 0.2 100000 0.597264 1.6 4 2 0.4 100000 0.603274 2 4 2 0.5 100000 0.609244 2.4 4 2 0.6 100000 0.617794 2.8 4 2 0.7 100000 0.62939

Flange Taps:

D, in d, in L1, in L2, in β Re Co 4 0.8 1 1 0.2 100000 0.597224 1.6 1 1 0.4 100000 0.602044 2 1 1 0.5 100000 0.605794 2.4 1 1 0.6 100000 0.609564 2.8 1 1 0.7 100000 0.61095

Corner Taps:D, in d, in L1, in L2, in β Re Co

4 0.8 0 0 0.2 100000 0.597254 1.6 0 0 0.4 100000 0.601984 2 0 0 0.5 100000 0.605344 2.4 0 0 0.6 100000 0.608034 2.8 0 0 0.7 100000 0.60673

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Note that Co is between 0.597 and 0.63 for all three pressure tap configurations for Re = 105 and β between 0.2 and 0.7. For larger values of Reynolds number Co

will stay within this range. For smaller values of Reynolds number, Co will get somewhat larger, especially for higher values of β.

Example #4: Water at 50o F is flowing through an orifice meter with flange taps and a 2 inch throat diameter, in a 4 inch diameter pipe. What is the flow rate through the meter if the pressure difference, P1 – P2, is measured as 8 inches of Hg.

Solution: Assume Re is approximately 105, in order to get started. Then from the solution to Example #3, with β = 0.5: Co = 0.60579.

From Table 1, the density of water at 50oF is 1.94 slugs/ft3. A2 = πD22/4 =

π(2/12)2/4 = 0.02182 ft2. β = 2/4 = 0.5. Converting the pressure difference to lb/ft2: P1 – P2 = (8 in Hg)(70.73 lb/ft2/in Hg) = 565.8 lb/ft2. Substituting all of these values into equation (4):

Check on Reynolds number value:

V = Q/A = 0.3624/[π(4/12)2/4] = 4.152 ft/sec

Re = DV/ν = (4/12)(4.152)/(1.407 x 10-5) = 9.836 x 104

This value is close enough to 105, so that the value used for Co is ok.

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Flow Nozzle Meter: The flow nozzle meter is simpler and less expensive than a venturi meter, but not quite as simple as an orifice meter. It consists of a relatively short nozzle, typically held in place between pipe flanges, as shown in Figure 4.

Figure 4. Flow Nozzle Meter Parameters

For a flow nozzle meter, the exit diameter of the nozzle, d, is used for D2 (giving A2 = An), and the discharge coefficient is typically called a nozzle coefficient, Cn, giving the following equation for a flow nozzle meter:

Due to the smoother contraction of the flow, flow nozzle coefficients are significantly higher than orifice coefficients. They are not, however as high as venturi coefficients. Flow nozzle coefficients are typically in the range from 0.94 to 0.99. There are several different standard flow nozzle designs. Information on

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pressure tap placement and calibration should be provided by the meter manufacturer.

5. Velocity Flow Meters – Pitot / Pitot-Static Tubes

Pitot tubes (also called pitot-static tubes) are an inexpensive, convenient way to measure velocity at a point in a fluid. They are used widely in airflow measurements in ventilation and HVAC applications. Definitions for three types of pressure or pressure measurement are given below, because understanding them helps to understand the pitot tube equation. Static pressure, dynamic pressure and total pressure are defined below and illustrated in figure 5.

Static pressure is the fluid pressure relative to surrounding atmospheric pressure, measured through a flat opening, which is in parallel with the fluid flow, as shown with the first U-tube manometer in Figure 5.

Stagnation pressure is the fluid pressure relative to the surrounding atmospheric pressure, measured through a flat opening, which is perpendicular to and facing into the direction of fluid flow, as shown with the second U-tube manometer in Figure 5. This is also sometimes called the total pressure.

Dynamic pressure is the fluid pressure relative to the static pressure, measured through a flat opening, which is perpendicular to and facing into the direction of fluid flow, as shown with the third U-tube manometer in Figure 5. This is also sometimes called the velocity pressure.

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Figure 5. Various Pressure Measurements

Static pressure is typically represented by the symbol, p. Dynamic pressure is equal to ½ ρV2. Stagnation pressure, represented here by Pstag, is equal to static pressure plus dynamic pressure plus the pressure due to the height of the measuring point above some reference plane, as shown in the following equation.

Where the parameters with a consistent set of units are as follows:

Pstag = stagnation pressure, lb/ft2

P = static pressure, lb/ft2

ρ = density of fluid, slugs/ft3

γ = specific weight of fluid, lb/ft3

h = height above a specified reference plane, ft

V = average velocity of fluid, ft/sec

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(V = Q/A = volumetric flow rate/cross-sectional area normal to flow)

For pitot tube measurements, the reference plane can be taken at the height of the pitot tube measurement, so that h =0. Then stagnation pressure minus static pressure is equal to dynamic pressure, or:

The pressure difference, Pstag - P, can be measured directly with a pitot tube such as the third U-tube in Figure 5, or more simply with a pitot tube like the one shown in Figure 6, which has two concentric tubes. The inner tube has a stagnation pressure opening and the outer tube has a static pressure opening parallel to the fluid flow direction. The pressure difference is equal to the dynamic pressure ( ½ ρV2 ) and can be used to calculate the fluid velocity for known fluid density, ρ. A consistent set of units is: pressure in lb/ft2, density in slugs/ft3, and velocity in ft/sec.

Figure 6. Pitot Tube

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For use with a pitot tube, equation (9) will typically be used to calculate the velocity of the fluid. Setting (Pstag – P) = ∆P, and solving for V, gives the following equation:

In order to use Equation 10 to calculate fluid velocity from pitot tube measurements, it is necessary to be able to obtain a value of density for the flowing fluid at its temperature and pressure. For a liquid, a value for density can typically be obtained from a table similar to Table 1 in this course. Such tables are available in handbooks and fluid mechanics or thermodynamics textbooks. Pitot tubes are used more commonly to measure gas flow, as for example, air flow in HVAC ducts, and density of a gas varies considerably with both temperature and pressure. A convenient way to obtain a value of density for a gas at known temperature and pressure is through the use of the Ideal Gas Law.

The Ideal Gas Law, as used to calculate density of a gas is as follows:

Where: ρ = density of the gas at pressure, P, & temperature, T, slugs/ft3

MW = molecular weight of the gas, slugs/slug-mole (The average molecular weight typically used for air is 29.)

P = absolute pressure of the gas, psia

T = absolute temperature of the gas, oR (oF + 459.67 = oR)

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R = Ideal Gas Law constant, 345.23 psia-ft3/slug-mole-oR

But, you may ask, this is the Ideal Gas Law, so how can we use it to find the density of real gases? Well …. the Ideal Gas Law is a very good approximation for many real gases over a wide range of temperatures and pressures. It does not work well for very high pressures or very low temperatures (approaching the critical temperature and/or critical pressure for the gas), but for many practical, real situations, the Ideal Gas Law gives quite accurate values for density of a gas.

Example #5: Estimate the density of air at 16 psia and 85 oF.

Solution: Convert 85 oF to oR: 85 oF = 85 + 459.67 oR = 544.67 oR

Substituting values for P, T, R, & MW into Equation 11 gives:

ρ = (29)[16/(345.23)(544.67)] = 0.002468 slugs/ft 3

Example #6: A pitot tube is being used to measure air velocity in a heating duct. The air is at 85 oF and 16 psia. The pitot tube registers a pressure difference of 0.22 inches of water (Pstag – P). What is the velocity of the air at that point in the duct?

Solution: Convert 0.023 inches of water to lb/ft2 (psf) (conversion factor is: 5.204 psf/in of water):

0.023 in of water = (0.023)(5.204) psf = 0.1197 psf

Air density at the given P & T is 0.002468 slugs/ft3 from Example #5.

Substituting into equation (10), to calculate the velocity, gives:

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6. Variable Area Flow Meter - Rotameters

A rotameter is a ‘variable area’ flow meter. It consists of a tapered glass or plastic tube with a float that moves upward to an equilibrium position determined by the flow rate of fluid going through the meter. For greater flow rate, a larger cross-sectional area is needed for the flow, so the float is moved upward until the upward force on it by the fluid is equal to the force of gravity pulling it down. Note that the ‘float’ must have a density greater than the fluid, or it would simply float to the top of the fluid. Given below, in figure 7, are a schematic diagram of a rotameter and a picture of a typical rotameter.

The height of the float as measured by a graduated scale on the side of the rotameter can be calibrated for flow rate of the fluid being measured in appropriate flow units. A few points regarding rotameters follow:

Because of the key role of gravity, rotameters must be installed vertically Typical turndown ratio is 10:1, that is flow rates as low as 1/10 of the

maximum reading can be accurately measured. Accuracy as good as 1% of full scale reading can be expected. Rotameters do not require power, so they are safer to use with flammable

fluids, than an instrument using power, which would need to be explosion proof.

A rotameter causes little pressure drop. It is difficult to apply machine reading and continuous recording with a

rotameter.

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Figure 7. Rotameter Schematic diagram and typical example

7. Positive Displacement Flow Meters

Positive displacement flow meters are often used in residential and small commercial applications. They are very accurate at low to moderate flow rates, which are typical of these applications. There are several types of positive displacement meters, such as reciprocating piston, nutating disk, oval gear, and rotary vane. In all of them, the water passing through the meter, physically

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displaces a known volume of fluid for each rotation of the moving measuring element. The number of rotations is counted electronically or magnetically and converted to the volume which has passed through the meter and/or flow rate.

Positive displacement meters can be used for any relatively nonabrasive fluid, such as heating oils, Freon, printing ink, or polymer additives. The accuracy is very good, approximately 0.1% of full flow rate with a turndown of 70:1 or more.

On the other hand, positive displacement flow meters are expensive compared to many other types of meters and produce the highest pressure drop of any flow meter type.

8. Miscellaneous Types of Flow meters

In this section several more types of flow meters for use with pipe flow will each be described and discussed briefly.

a) Electromagnetic flow meters

An electromagnetic flow meter (also called ‘magnetic meter’ or ‘mag meter’) measures flow rate by measuring the voltage generated by a conductive fluid passing through a magnetic field. The magnetic field is created by coils outside the flow tube, carrying electrical current. The generated voltage is proportional to the flow rate of the conductive fluid passing through the flow tube. An external sensor measures the generated voltage and converts it to flow rate.

In order to be measured by an electromagnetic flow meter, the fluid must have a conductivity of at least 5 µs/cm. Thus, this type of meter will not work for distilled or deionized water or for most non-aqueous liquids. It works well for water, which has not been distilled or deionized and many aqueous solutions.

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Since there is no internal sensor to get fouled, an electromagnetic flow meter is quite suitable for wastewater, other dirty liquids, corrosive liquids or slurries. Since there is no constriction or obstruction to the flow through an electromagnetic meter, it creates negligible pressure drop. It does, however, have a relatively high power consumption, in comparison with other types of flow meters.

b) Target flow meters

With a target flow meter, a physical target (disk) is placed directly in the path of the fluid flow. The target will be deflected due to the force of the fluid striking it, and the greater the fluid flow rate, the greater the deflection will be. The deflection is measured by a sensor mounted on the pipe and calibrated to flow rate for a given fluid. Figure 8 shows a diagram of a target flow meter.

Figure 8. Target Flow Meter

A target flow meter can be used for a wide variety of liquids or gases and there are no moving parts to wear out. They typically have a turndown of 10:1 to 15:1.

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c) Turbine flow meters

A turbine flow meter operates on the principle that a fluid flowing past the blades of a turbine will cause it to rotate. Increasing flow rate will cause increasing rate of rotation for the turbine. The meter thus consists of a turbine placed in the path of flow and means of measuring the rate of rotation of the turbine. The turbine’s rotational rate can then be calibrated to flow rate. The turbine meter has one of the higher turndown ratios, typically 20:1 or more. Its accuracy is also among the highest at about + 0.25%.

d) Vortex flow meters

An obstruction in the path of a flowing fluid will create vortices in the downstream flow if the fluid flow speed is above a critical value. A vortex flow meter (also known as vortex shedding or oscillatory flow meter), measures the vibrations of the downstream vortices caused by a barrier in the flow path, as illustrated in figure 9. The vibrating frequency of the downstream vortices will increase with increasing flow rate, and can thus be calibrated to flow rate of the fluid.

Figure 9. Vortex Flow Meter

e) Ultrasonic flow meters

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The two major types of ultrasonic flow meters are ‘Doppler’ and ‘transit-time’ ultrasonic meters. Both use ultrasonic waves (frequency > 20 kHz). Both types also use two transducers which transmit and/or receive the ultrasonic waves.

For the Doppler ultrasonic meter, one transducer transmits the ultrasonic waves and the other receives the waves. The fluid must have material in it that will reflect sonic waves, such as particles or entrained air. The frequency of the transmitted beam of ultrasonic waves will be altered, by being reflected from the particles or air bubbles. The resulting frequency shift is measured by the receiving transducer, and is proportional to the flow rate through the meter. A signal can thus be generated from the receiving transducer, which is proportional to flow rate.

Transit-time ultrasonic meters, also known as ‘time-of-travel’ meters, measure the difference in travel time between pulses transmitted in the direction of flow and pulses transmitted against the flow. The two transducers are mounted so that one is upstream of the other. Both transducers serve alternately as transmitter and receiver. The upstream transducer will transmit a pulse, which is detected by the downstream transducer, acting as a receiver, giving a ‘transit-time’ in the direction of flow. The downstream transducer will then transmit a pulse, which is detected by the upstream transducer (acting as a receiver), to give a ‘transit-time’ against the flow. The difference between the upstream and downstream transit times can be correlated to flow rate through the meter.

The components of a transit-time ultrasonic flow meter are shown in figure 10. One of the options with this type of meter is a rail-mounted set of transducers, which can be clamped onto an existing pipe without taking the pipe apart to mount the meter. It could be used in this way to check on or calibrate an existing meter, or as a permanent installation for flow measurement. Ultrasonic flow meters are also available with transducers permanently mounted on an insert that

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is mounted in the pipeline, much like other flow meters, such as an electromagnetic flow meter.

Like the electromagnetic flow meter, ultrasonic meters have no sensors inside the pipe nor any constrictions or obstructions in the pipe, so they are suitable for dirty or corrosive liquids or slurries. Also, they cause negligible pressure drop.

Figure 10. Transit-time Ultrasonic Flow Meter

9. Comparison of Flow Meter Alternatives

Table 2 shows a summary of several useful characteristics of the different types of pipe flow meters described and discussed in this course. The information in Table 2 was extracted from a similar table at the Omega Engineering web site: http://www.omega.com/techref/table1.html . The flow meter characteristics summarized in Table 2 are: recommended applications, typical turndown ratio (also called rangeability), pressure drop, typical accuracy, upstream pipe diameters (required upstream straight pipe length), effect of viscosity, and relative cost.

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Table 2. Summary of Flow Meter Characteristics

Typical Typical Upstream Flow meter Recomm. Turndown Pressure Accuracy Pipe Effect of Relative

Type Application Ratio Drop % Diameters Viscosity Cost

Orifice clean liquids 4:1 medium + 2 - + 4 10 - 30 high lowgases of full scale

Venturi clean liquids 4:1 low + 1 5 - 20 high mediumgases of full scale

Flow Nozzle clean liquids 4:1 low to + 1 - + 2 10 - 30 high mediumgases medium of full scale

Pitot Tube clean liquids 3:1 very + 3 - + 5 20 - 30 low lowgases low of full scale

Rotameter clean, dirty 10:1 medium + 1 - + 10 none medium lowliq. & gases of full scale

Positive clean liquids 10:1 high + 0.5 none high mediumDisplacemt. gases of flow rate

Electro- clean, dirty 40:1 none + 0.5 5 none highmagnetic conductive of flow rate

liq. & slurries

Target clean, dirty 10:1 medium + 1 - + 5 10 - 30 medium mediumliquids, of full scale

& slurries

Turbine clean liquids 20:1 high + 0.25 5 - 10 high highgases of flow rate

Vortex clean, dirty 10:1 medium + 1 10 - 20 medium highliq. & gases of flow rate

Ultrasonic dirty liquids 10:1 none + 5 5 - 30 none high(Doppler) & slurries of full scale

Ultrasonic clean liquids 20:1 none + 1 - + 5 5 - 30 none high(transit-time) gases of full scale

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10. Summary

There are a wide variety of meter types for measuring flow rate in closed conduits. Twelve of those types were described and discussed in this course. Table 2 in section 8, summarizes a comparison among those twelve types of flow meters.

11. References

1. Munson, B. R., Young, D. F., & Okiishi, T. H., Fundamentals of Fluid Mechanics, 4th Ed., New York: John Wiley and Sons, Inc, 2002.

2. International Organization of Standards - ISO 5167-1:2003 Measurement of fluid flow by means of pressure differential devices, Part 1: Orifice plates, nozzles, and Venturi tubes inserted in circular cross-section conduits running full. Reference number: ISO 5167-1:2003.

3. U.S. Dept. of the Interior, Bureau of Reclamation, 2001 revised, 1997 third edition, Water Measurement Manual, available for on-line use or download at: http://www.usbr.gov/pmts/hydraulics_lab/pubs/wmm/index.htm

4. LMNO Engineering, Research and Software, Ltd website. Contains equations and graphs for flow measurement with venturi, orifice and flow nozzle flowmeters. http://www.lmnoeng.com/venturi.htm

5. Engineering Toolbox website. Contains information on flow measurement with a variety of meter types. http://www.engineeringtoolbox.com/fluid-flow-meters-t_49.html

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Calculation of Gas Density and Viscosity

Harlan H. Bengtson, PhD, P.E.

COURSE CONTENT 1. Introduction The density and/or viscosity of a gas is often needed for some other calculation, such as pipe flow or heat exchanger calculations. This course contains discussion of, and example calculation of, the density and viscosity of a specified gas at a given temperature and pressure. If the gas temperature is high relative to its critical temperature and the gas pressure is low relative to its critical pressure, then it can be treated as an ideal gas and its density can be calculated at a specified temperature and pressure using the ideal gas law. If the density of a gas is needed at a temperature and pressure at which it cannot be treated as an ideal gas law, however, then the compressibility factor of the gas must be calculated and used in calculating its density. In this course, the Redlich Kwong equation will be used for calculation of the compressibility factor of a gas. The Sutherland formula can be used to calculate the viscosity of a gas at a specified temperature and pressure if the Sutherland constants are available for the gas. It will be discussed and used in example calculations. Another method for calculating the viscosity of air at a specified temperature and pressure will also be presented and discussed. Some of the equations that will be discussed and illustrated through examples are shown below.

2. Learning Objectives Upon completion of this course, the student will

• Be able to calculate the density of a gas of known molecular weight at a specified temperature and pressure at which the gas can be treated as an ideal gas.

• Be able to calculate the compressibility factor for a gas at a specified

temperature and pressure, using the Redlich-Kwong equation, if the molecular weight, critical temperature and critical pressure of the gas are known.

• Be able to calculate the density of a gas at a specified temperature and

pressure for which the gas cannot be treated as an ideal gas, if the molecular weight, critical temperature and critical pressure of the gas are known.

• Be able to calculate the viscosity of a gas at a specified temperature if

the Sutherland constant for the gas is known and the viscosity of the gas at a suitable reference temperature is known.

• Be able to calculate the viscosity of air at specified air temperature

and pressure.

• Be able to make all of the calculations described in these learning objectives using either U.S. or S.I. units.

3. Topics Covered in this Course I. Calculation of Ideal Gas Density II. Calculation of Real Gas Density III. Calculation of Gas Viscosity by Sutherland’s Formula

IV. Calculation of Air Viscosity at Specified Temperature and Pressure V. Summary VI. References 4. Calculation of Ideal Gas Density The typical form for the Ideal Gas Law is: PV = nRT The parameters in the equation with a consistent set of units are as shown below: P is the absolute pressure of the gas in psia V is the volume of the gas in ft3 n is the number of slugmoles in the gas contained in volume, V R is the ideal gas law constant, 345.23 psia-ft3/slugmole-oR T is the absolute temperature of the gas in oR The mass of the gas can be introduced into the equation by replacing n with m/MW, where m is the mass of the gas contained in volume V in slugs, and MW is the molecular weight of the gas (slugs/slugmole). The Ideal Gas Law then becomes: PV = (m/MW)RT Solving the equation for m/V, which is the density of the gas gives: ρ = m/V = P(MW)/RT With P, R, and T in the units given above, the gas density will be in slugs/ft3. Note that 1 slug = 32.17 lbm, so if you want the gas density in lbm/ft3, the value in slugs/ft3 should be multiplied by 32.17. S.I. Units: If working in S.I. units, the equations remain the same with the following units:

P is the absolute pressure of the gas in kPa V is the volume of the gas in m3 n is the number of kgmoles in the gas contained in volume, V R is the ideal gas law constant, 8.3145 kg-m/kgmole-K T is the absolute temperature of the gas in K With these units for P, V, R, and T, the gas density will be in kg/m3. Critical Temperature and Pressure: As noted in the Introduction, in order to use the Ideal Gas Law to calculate a gas density, the gas temperature should be high relative to its critical temperature and the gas pressure should be low relative to its critical pressure. Table 1 gives critical temperature, critical pressure and molecular weight for 16 gases in U.S. units. Table 2 provides the same in S.I. units. Table 1. Critical Temperature and Pressure and Molecular Weight -U.S.

Table 2. Critical Temperature and Pressure and Molecular Weight -S.I.

Example #1: a) Calculate the density of air at -17 oF and 20 psig, assuming that the air can be treated as an ideal gas at those conditions. b) Is it reasonable to assume ideal gas behavior for air at -17 oF and 20 psig? Solution: a) The absolute temperature and pressure need to be calculated as follows: Tabs = -17 + 459.67 oR = 442.67 oR and Pabs = Pg + Patm = 20 + 14.7 = 34.7 psia. Substituting values into the ideal gas law (using 28.97 as the MW of air) gives: ρ = MW*P/(R*T) = 28.97*34.7/(345.23*442.67) = 0.00658 slugs/ft3

If desired, the density can be converted to lbm/ft3 by multiplying by the conversion factor, 32.17 lbm/slug. ρ = (0.000658 slugs/ft3)(32.17 lbm/slug) = 0.2116 lbm/ft3 b) The gas temperature (-17 oF) is much greater than the critical temperature of air (-220.9 oF) and the gas pressure (34.7 psia) is much less than the critical pressure of air (547 psia), so it would be reasonable to assume ideal gas behavior for air at this temperature and pressure. Example #2: a) Calculate the density of air at -20oC and 100 kPa gauge pressure, assuming that it can be treated as an ideal gas at those conditions. b) Is it reasonable to assume ideal gas behavior for air at 20 oC and 100 kPa guage? Solution: a) The absolute temperature and pressure need to be calculated as follows: Tabs = 20 + 273.15 K = 283.15 K and Pabs = Pg + Patm = 100 + 101.3 = 201.3 kPa abs. Substituting values into the ideal gas law (using 28.97 as the MW of air) gives: ρ = MW*P/(R*T) = 28.97*201.3/(8.3145*283.15) = 2.48 kg/m3 b) The gas temperature (10 oC) is much greater than the critical temperature of air (-140.5 oC) and the gas pressure (201.3 kPa abs) is much less than the critical pressure of air (3773.4 kPa abs), so it would be reasonable to assume ideal gas behavior for air at this temperature and pressure. (Note that the critical temperature was converted from the 37.25 atm value in the table with the conversion factor, 101.3 kPa/atm.) Spreadsheet Use for the Calculations: These calculations are rather straight-forward and not too difficult to do by hand, but they can be done very conveniently with an Excel spreadsheet set up to calculate gas density with the Ideal Gas Law. Figure 1 shows a screenshot of an Excel worksheet with the solution to Example 1 (a) and Figure 2 shows a screenshot with the solution to Example 2 (a).

Figure 1. Screenshot of Solution to Example #1 (a)

Figure 2. Screenshot of Solution to Example #2 (a)

5. Calculation of Real Gas Density In some cases, the Ideal Gas Law cannot be used to calculate the density of a gas because its temperature is too close to its critical temperature and/or its pressure is too close to its critical pressure. In that case, if the compressibility factor, Z, can be determined at the gas temperature and pressure, it can be used to calculate the gas density with the following equation: ρ = MW*P/(Z*R*T) The compressibility factor for a gas is, in general, a function of its reduced temperature (TR) and reduced pressure (PR), where reduced temperature is the absolute gas temperature divided by its absolute critical temperature and reduced pressure is the absolute gas pressure divided by its absolute critical pressure. Graphs, tables and equations are available for determining the compressibility factor at specified values for TR and PR. The Redlich-Kwong equation will be used in this book as a means of calculating Z as a function of TR and PR. Calculation of the compressibility factor of a gas from the Redlich-Kwong equation is rather awkward and time consuming to do by hand, but a spreadsheet can be set up to conveniently make the necessary calculations. The equations and calculation procedure are as follows: The Redlich-Kwong compressibility factor, Z, is calculated as the maximum

real root of the equation: Z3 – Z2 – qZ – r = 0, where r = A2B and

q = B2 + B – A2, with A2 = 0.42747PR/TR2.5 and B = 0.08664 PR/TR

To find the maximum real root, first the parameter C is calculated, where:

C = (f/3)3 + (g/2)2, with f = (-3q - 1)/3 and

g = (-g/2 - C1/2)1/3 + 1/3

If C > 0, then there is one real root,

Z = (-g/2 + C1/2)1/3 + (-g/2 - C1/2)1/3 + 1/3

If C < 0, then there are three real roots, given by:

Zk = 2(-f/3)1/2cos[(φ/3) + 2π(k - 1)/3] + 1/3

With k = 1, 2, 3 and φ = cos-1{[(g2/4)/((-f3)/27)]1/2}

These equations and this method of calculating a value for the compressibility factor are described at: www.polymath-software.com/ASEE2007/PDF1.pdf The only input parameters needed to calculate the compressibility factor of a gas by the Redlich-Kwong method are the temperature and pressure at which the compressibility factor is to be calculated, along with the gas molecular weight, critical temperature and critical pressure. However, there are 8 additional parameters introduced (B, A2, r, q, f, g, C, and φ) and there are also quite a few steps in the solution, so it is not a trivial calculation to carry out by hand. An Excel spreadsheet is a very convenient tool to use in carrying out this calculation, as illustrated in Example #3 and Example #4. Example #3: a) Calculate the density of air at -17 oF and 20 psig, using the compressibility factor calculated by the Redlich-Kwong method described above. Assume local atmospheric pressure is 14.7 psi. b) Compare the results from part (a) with the density calculated assuming ideal gas behavior in Example #1. Solution: a) The reduced temperature and reduced pressure can be calculated, using the gas temperature and pressure and the critical temperature and critical pressure of the gas, as follows: TR = (-17 + 459.67)/(-220.9 + 459.67) = 1.854 PR = 34.7/547 = 0.06344

A spreadsheet screenshot is shown in Figure 3 with the calculation of the compressibility factor and the density for air at -17 oF and 20 psig. Note that the required user inputs (in the blue cells) are the gas molecular weight, temperature and pressure and the critical temperature and critical temperature of the gas. The inputs shown are for air at -17 oF and 20 psig (34.7 psia). The worksheet makes quite a few calculations in the yellow cells, starting with calculation of the reduced temperature (TR) and reduced pressure (PR). The calculated values of TR and PR are used to calculate the values of the 8 constants (B, A2, r, q, f, g, C, and φ). If C > 0, then the spreadsheet calculates the single real root of the equation, Z3 – Z2 – qZ – r = 0, (which is the value of the compressibility factor, Z) using the equation shown above (Z = (-g/2 + C1/2)1/3 + (-g/2 - C1/2)1/3 + 1/3). If C < 0, then the spreadsheet calculates the three real roots of the equation, using the equation given above and the maximum of those three roots is the value for the compressibility factor, Z. For Example #3, as shown in the screenshot, C > O and this results in a value of 0.997 for the compressibility factor, Z, and air density of 0.00659 slugs/ft3 = 0.21214 lbm/ft3. b) The value calculated for Z is very close to 1 and thus the calculated value of air density is quite close to that calculated with the Ideal Gas Equation in Example #1 as shown below: Ideal Gas Law: ρ = 0.00658 slugs/ft3 = 0.2116 lbm/ft3 Redlich-Kwong: ρ = 0.00659 slugs/ft3 = 0.2121 lbm/ft3 % difference = (0.2121 – 0.2116)/0.2121 = 0.24 %

Figure 3. Screenshot of Solution to Example #3 (a)

Note that for accurate calculations with the Redlich-Kwong equation, it is recommended that PR be less than half of TR. In the Figure 3 spreadsheet screenshot the ratio PR/0.5TR is calculated to facilitate checking whether this requirement has been met.

Example #4: a) Calculate the compressibility factor and density of air at -129 oC and 20 bar, using the Redlich-Kwong method. b) Compare the results from Part (a) with the density calculated assuming ideal gas behavior. . Solution: a) The reduced temperature and reduced pressure can be calculated as follows: TR = (-129 + 273.15)/(-140.5 + 273.15) = 1.087 PR = (20 bar)(0.98682 atm/bar)/37.25 atm = 0.5299 A spreadsheet screenshot is shown in Figure 4 with the calculation of the compressibility factor and the density for air at -129 oC and 20 bar pressure. The user inputs are shown in the cells at the upper left . Calculation of the reduced temperature and reduced pressure is shown below the user inputs and calculation of the various constants is shown in the cells at the right. The results, shown at the bottom of the screenshot are: Z = 0.845 and air density = 57.17 kg/m3 b) The value of 0.845 calculated for Z shows that the calculated value of air density will be somewhat different than that calculated with the Ideal Gas Equation. The results, shown at the bottom of the screenshot, are: Ideal Gas Law: ρ = 48.325 kg/m3 Redlich-Kwong: ρ = 57.170 kg/m3 % difference = (57.170 – 48.325)/57.170 = 15.5 %

Figure 4. Screenshot of Solution to Example #4 (a)

6. Calculation of Gas Viscosity by Sutherland’s Formula The Sutherland Formula provides a means for calculating the viscosity of a gas if the value of the Sutherland’s Constant is known for that gas along with a value of the viscosity of that gas at some reference temperature. Sutherland’s Formula is:

The parameters in this equation are: T is the temperature of the gas, K

To is a reference temperature, K

µo is the viscosity of the gas at To in any units

µ is the viscosity of the gas at T in the same units as µo

C is the Sutherland constant for the gas, K

Table 3 gives values of the Sutherland constant, the temperature range for that constant and the viscosity at three reference temperatures for 25 gases. These values can be used together with Sutherland’s formula to calculate the viscosity of any of the gases in the table at any temperature within the specified temperature range. References #2, #3, and #4 are sources for the information in Table 3. Note that the reference viscosity and reference temperature closest to the gas temperature should be used for the Sutherland’s Formula calculation.

Table 3. Constants for Sutherland’s Formula

Example #5: Calculate the viscosity of methane at 110oF using Sutherland’s formula and values from Table 3.

Solution: Converting 110 oF to oC gives: (110 – 32)/1.8 = 43.33 oC. Thus the reference temperature closest to the gas temperature is 50 oC, so from Table 3, we will use to = 50 oC and µo = 0.0120 cP. Also from Table 3, the Sutherland’s Constant for methane is 169. The gas temperature and reference temperature must be in K (degrees Kelvin) for use in Sutherland’s Formula to calculate the gas viscosity. The temperature conversions can be made as follows:

Gas temperature = (110 + 459.67)/1.8 = 316.5 K Reference temperature = (50 + 273.15) = 323.15 K

Now, substituting into Sutherland’s Formula to calculate the viscosity of methane at 110o F gives: µ = 0.012[(323.15 + 169)/(316.5 + 169)][(316.5/323.15)1.5] = 0.0118 cP Converting to typical U.S. units of lb-s/ft2 gives: µ = (0.0118)(2.08854 x 10-5) = 2.462 x 10-7 lb-s/ft2 Calculation of a gas viscosity using Sutherland’s Formula can conveniently be done using an Excel spreadsheet. Figure 5 shows a screenshot of a spreadsheet solution to Example #5. Note that, in the screenshot, the given information is entered into the blue cells on the left side of the screenshot and the spreadsheet makes the calculations in the yellow cells to make the necessary temperature conversions, calculate the gas viscosity and convert to typical U.S. units. Calculations in S.I. units would be very similar, but the unit conversions used for Example #5 wouldn’t be needed, because the values in Table 3 have units of oC, K, and cP. This type of calculation is illustrated in Example #6.

Figure 5. Screenshot of Solution to Example #5

Example #6: Calculate the viscosity of methane at 60o C using Sutherland’s formula and values from Table 3. Solution: The reference temperature closest to the gas temperature is 50 oC, so from Table 3, we will use to = 50 oC and µo = 0.0120 cP. Also from Table 3, the Sutherland’s Constant for methane is 169. The gas temperature and reference temperature must be in K (degrees Kelvin) for use in Sutherland’s Formula to calculate the gas viscosity. The temperature conversions can be made as follows: Gas temperature = (60 + 273.15) = 333.15 K Reference temperature = (50 + 273.15) = 323.15 K

Now, substituting into Sutherland’s Formula to calculate the viscosity of methane at 60o C gives: µ = 0.012[(323.15 + 169)/(333.15 + 169)][(333.15/323.15)1.5] = 0.0123 cP 7. Calculation of Air Viscosity at Given Temperature and Pressure With the following equations, it is possible to calculate the viscosity of air for specified values of the air temperature and pressure. The source for these equations is Reference #6 at the end of the book. air viscosity = (1.2867/107)[ηo(Tr) + ∆η(ρr)] lb-sec/ft2 ηo(Tr) = 0.128517Tr + 2.60661Tr

0.5 - 1.0 - 0.709661Tr-1

+ 0.662534Tr

-2 - 0.197846Tr-3 + 0.00770147 Tr

-4 ∆η(ρr) = 0.465601ρr = 1.26469ρr

2 - 0.511425ρr3 + 0.274600ρr

4 Tr = T/238.5 (T in oR) ρr = ρ/0.6096 (ρ in slugs/ft3) Note that ρ is the density of the air at the specified temperature and pressure, which can typically be calculated as described in Section 4 for ideal gas behavior. If the air cannot be treated as an ideal gas for the specified temperature and pressure, then the density would need to be calculated as described in Section 5 above.

Example #7: Calculate the viscosity of air at 50oF and 40 psig at a location where atmospheric pressure is 14.7 psi. Solution: The absolute air temperature can be calculated as: T = 50 + 459.7 = 509.7 oR The absolute air pressure is: P = 40 + 14.7 = 54.7 psia The air density can then be calculated as: ρ = P(MW)/RT ρ = (54.7)(28.97)/[(345.23)(509.7) = 0.0090 slugs/ft3 Tr and ρr can now be calculated as follows: Tr = T/238.5 = 509.7/38.5 = 2.137 ρr = ρ/0.6096 = 0.0090/0.6096 = 0.00715 Now ηo(Tr) and ∆η(ρr) can be calculated with the equations shown above to give: ηo(Tr) = 2.878 and ∆η(ρr) = 0.00715 Finally, the air viscosity can now be calculated using the first equation shown in this chapter, to give: air viscosity = (1.2867/107)[ηo(Tr) + ∆η(ρr)] = (1.2867/107)(2.878 + 0.00715) = 3.713 x 10-7 lb-sec/ft2 As you may expect, this set of calculations also can be conveniently done with a properly set up spreadsheet. Figure 6 shows a screenshot of a spreadsheet solution to Example #6.

Only three user inputs are needed. They are the temperature of the air, the gage pressure of the air, and atmospheric pressure. These three inputs are entered in the three blue cells in the upper left part of the screenshot in Figure 6. The spreadsheet then calculates the absolute air temperature and pressure, the air density, Tr, ρr, ηo(Tr), ∆η(ρr), and finally the air viscosity at the specified temperature and pressure. Note that the calculated value for the air viscosity at 50oF and 40 psig shown as 3.17 x 10-7 lb-sec/ft2, the same as shown in the calculations above.

Figure 6. Screenshot of Solution to Example #7

For calculations in S.I. units some of the constants in the equations given above are changed. Specifically, the equations for Tr, ρr, and the air viscosity change, while the equations for ηo(Tr) and ∆η(ρr) remain the same. The S.I. equations with changed constants are: Tr = T/132.5 (T in K) ρr = ρ/314.3 (ρ in kg/m3) and air viscosity = (6.16090/106)[ηo(Tr) + ∆η(ρr)] Pa-s Example #8: Calculate the viscosity of air at 327oC and 20 bar gage pressure at a location where atmospheric pressure is 101.325 kPa. Solution: The absolute air temperature can be calculated as: T = 327 + 273.15 = 600.2 K The absolute air pressure is: P = (20)(100) + 101.325 = 2101.3 kPa The air density can then be calculated as: ρ = P(MW)/RT ρ = (2101.3)(28.97)/[(8.3245)(600.2) = 12.20 kg/m3 Tr and ρr can now be calculated as follows: Tr = T/132.5 = 600.2/132.5 = 4.529 ρr = ρ/314.3 = 12.20/314.3 = 0.0388 Now ηo(Tr) and ∆η(ρr) can be calculated with the equations shown above to give: ηo(Tr) = 5.003 and ∆η(ρr) = 0.01995

Finally, the air viscosity can now be calculated using the air viscosity shown above, to give: air viscosity = (6.16090/106)[ηo(Tr) + ∆η(ρr)] = (6.16090/106)(5.003 + 0.01995) = 3.095 x 10-5 Pa-s Figure 7 shows a screenshot of a spreadsheet with the solution to Example #8.

Figure 7. Screenshot of Solution to Example #7

8. Summary The density and viscosity of a gas at specified temperature and pressure can be calculated using the methods covered in this course. Gas density can be calculated using the Ideal Gas Law if the gas temperature is sufficiently greater than the critical temperature and the gas pressure is sufficiently less than the critical pressure. If the gas temperature and/or pressure are such that the Ideal Gas Law cannot be used then the gas density can be calculated using the calculated compressibility factor. In this course, use of the Redlich-Kwong equation of state to calculate the compressibility factor was presented and illustrated with example calculations. The viscosity of a gas at specified gas temperature can be calculated using Sutherland’s Formula as presented and illustrated with examples. Equations were also presented for calculation of the viscosity of air at specified temperature and pressure. Use of both U.S. units and S.I. units was presented and illustrated with examples for all of the calculations. 9. References 1. Redlich-Kwong Equation of State Calculations, www.polymath-software.com/ASEE2007/PDF1.pdf 2. Chapman, S. & Cowling, T.G., The Mathematical Theory of Non-Uniform Gases, 3rd Ed., 1970, Cambridge Mathematical Library 3. National Physical Laboratory, Kaye & Laby, Tables of Physical and Chemical Constants, Chapter 2.2, Subsection 2.2.3, http://www.kayelaby.npl.co.uk/general_pysics/2_2/2_2_3.html

4. Engineering Toolbox website: www.engineeringtoolbox.com/gases-absolute-dynamic-viscosity-d_1888.html. 5. Green, Don W. and Perry Robert H., Perry’s Chemical Engineers’ Handbook, 8th Ed, Table2-312, McGraw-Hill 6. Kadoya, K, Matsunaga, N, and Nagashima, A, Viscosity and Thermal Conductivity of Dry Air in the Gaseous Phase, J. Phys. Chem. Ref. Data, Vol 14, No. 4, 1985. https://srd.nist.gov/JPCRD/jpcrd283.pdf 7. Bengtson, Harlan H., Sutherland Formula Viscosity Calculator, an online blog article at www.EngineeringExcelTemplates.com. 8. Bengtson, Harlan H. Gas Compressibility Factor Calculator Excel Spreadsheet, an online blog article at www.EngineeringExcelSpreadsheets.com 9. Bengtson, Harlan H., Gas Property Calculator Spreadsheets, An Amazon Kindle E-Book.