Be 301 Low-Pressure Manometer Spring 2001

Be 301 Low-Pressure Manometer Spring 2001

FINAL PROJECT

GROUP NUMBER T3

Jehann Biggs

David Kim

Anju Mathew

Naomi Sta. Maria

TITLE

Designing a Low-Pressure Manometer

DATE SUBMITTED

May 9, 2001

ABSTRACT

A U-tube manometer was designed and constructed to measure pressure drops to precisions of 0.0033 psig for flow in a long, horizontal tubing. The objective was to improve on the current pressure transducer, which is incapable of measuring pressures below 0.0033 psig. The design of the manometer was an inverted U-tube filled with a manometer fluid with density less than water (naptha, r = 0.75 g/mL). With our proposed design, the apparatus could ideally measure pressure differences as low as 0.0004 psig. When Re*f vs. Re was plotted, a y-intercept of 68.99 + 10.67(95%CI) was achieved, which is more accurate than the value when using the pressure transducer, 98.12 + 17.95 (95%CI). The manometer successfully outperformed the transducer by a factor of two in measuring pressure drops of near-Poiseullian flow.

BACKGROUND

The U-Tube Manometer

The simple U-tube manometer has been used to measure pressures for many years because of its simplicity and accuracy. Manometers measure pressure by balancing the pressure forces directly against the liquid column[i]. The ideal U-Tube manometer has a few characteristics that allows it to make simple, accurate measurements:

1)  It requires no calibration

2)  It is cost efficient

3)  It is useful for measuring slowly varying pressures

4)  It is able to measure low changes in pressure

Another method of measuring pressure in the horizontal tube is using a pressure transducer. Pressure transducers, similar to the one used in the lab, produces a quantitative electrical signal in response to rapid changes in pressure of fluid flowing in a tube. The transducer used in the lab has the ability to only pressures low as 0.0033 psig, which is not accurate enough to measure the pressures at the low flow rates. The monometer we have calculated will make up for the transducers shortfall and we have calculated that our best manometer design will be sensitive pressure measurements as low as 0.0004 psig.

The Manometer Fluid

A light or heavy density material is used in the U-tube depending on the magnitude of the pressure difference to be measured. For large pressure differences, a very dense liquid such as mercury can be used and for smaller pressure differences, fluids whose densities closely match the flowing fluid can be used. The design and orientation of the manometer with respect to the tube is a factor in determining the type of fluid to be used. In addition, the fluids should be immiscible, not readily mixed with other fluids, have good wetting characteristics and be able to form an observable meniscus.

Application of Fluid Statics

The understanding and application of fluid statics is necessary in the design of the manometer model. The general equation for fluid statics is shown below.

p1 + rgz1 = p2 + rgz2 (1)

For a U-shaped glass tube that is partially filled with a liquid and that one end of the tube is open to the atmosphere (Figure A), while the other is connected to the container whose pressure is to be measured. Applying equation (1) to the manometer fluid (whose density is ) between the fluid surfaces at 2 and 1:

where we have noted that the pressure equals the atmospheric pressure . The pressure is not necessarily the same as the pressure of the fluid at the center of the container if the fluid is a liquid. To account for this difference, we apply equation (1) to the container fluid (of density ) between points 3 and 2:

If the container fluid is a gas, then its density is much less than that of the manometer liquid , and and are substantially equal. On the other hand, if the container fluid is a liquid then the pressure difference may contribute significantly to the determination of the container pressure and should not be disregarded.

Figure A: A U-tube manometer used to measure the pressure of a fluid in a container.[ii]

Hagen-Poiseuille’s Law

The Hagen-Poiseulle’s law (4) is derived from differential analysis of fluid flow through a straight, horizontal cylindrical tube using the conservation of linear momentum. An expression for the velocity profile for parabolic flow is then obtained. Poiseulle’s law (5) relates the pressure drop Dp to the steady, laminar, fully-developed flow rate Q for a Newtonian fluid.

(4)

(5)

Friction Factor and Head Loss

Pressure losses occurring in internal flows are results of friction. The HL term actually represents the decrease (loss) in mechanical energy between points 1 and 2, and in general contains both major and minor losses. In Equation (2) we see that pressure changes result from velocity changes, elevation changes and friction losses. For incompressible case of flow though a straight, horizontal tube of constant diameter, equation (2) becomes HL = (p1-p2)/rg.

Knowing pressure change is known to depend on pipe diameter D, average velocity U, length L, viscosity m, density r, and wall roughness, these parameters have been combined to empirically define the friction factor from the HL, as shown below (6).

(6)

Analytically the friction factor f for laminar flow is f = 64/Re, this is derived from Poiseulle’s Law from the following derivation from Lab experiment 2.

(7)

where Q = Vave A = UpR2, n = m/r, D=2a

DESIGN

Figure 1: Manometer Design.

The above figure describes the setup of the manometer. The yellow section represents naptha, which is less dense than water. The calculation for the difference in pressure between Pa and Pb shows that ∆P=(ρH20 – ρnaptha)gh, so the height difference can be maximized by decreasing the difference in density between water and the manometer fluid.

Figure 2: Experiment Setup

The above figure shows the manometer connected to the long horizontal tube as in the experiment. The image on the right shows actual prototype, using lighter fluid (napthalene) as manometer fluid.

MATERIALS

·  Tygon ID tubing diameters (¼”and 3/16”)

·  Flow cutoff valve

·  U-shaped manometer

o  Moving Fluid: Water r = 1000 kg/m3

o  Manometer Fluid: Naptha r = 750 kg/m3

·  Water tank

·  Flow restrictor

·  Various connectors, stoppers, and Teflon® tape

METHODS

The actual manometer prototype was constructed using ¼” ID tubing attached in a U-shape around a meter stick. The top of the tubing had an open connector which allowed for the easy pouring of naphthalene and also allowed for the easy removal of air bubbles. After naphthalene had been added and the air bubbles removed, the connector was closed with a rubber stopper. The ends of the manometer were connected to the horizontal tubing via connectors and sealed with Teflon® tape. The manometer was supported by a ring stand.

Measurement of the pressure-flow relationship in a straight, horizontal tube was made using an apparatus that consists of a reservoir fluid tank with supports elevated connected to Tygon ID tubing connected to a flow cutoff valve then to a U-shaped manometer then continuing on to a Tygon ID tubing cut to a determined length with a flow restrictor, to a collection bucket. In developing a protocol, parameters such as flow rates, density of moving fluid, density of manometer fluid, and U-shaped manometer set-up were tested for greatest efficacy.

In this determination of the pressure-flow relationship, we used naphthalene as the manometer fluid and water as the moving fluid. Various sizes (diameter) of Tygon tubing were used as another variable, such as 1/4in and 3/16in. The pressure differential was directly controlled through flow rates, and we conducted at least 10 trials for each size tubing.

First, the Dh in the manometer at zero PSIG was noted (static pressure). The needle valve was opened with exit blocked, and the height of the water above the manometer was measured and converted to pressure, with the relationship that water exerts 1 PSIG per 27.68 inches. The U-shaped manometer was connected above the flow tube. For a schematic diagram, please refer to the Design section. The manometer, which looks like an inverted U, contained manometer fluid with density close to but slightly less than the density of the moving fluid. Ten trials were conducted and average Dh (pressure height) and 95% confidence intervals were determined.

Prior to the start of the experiment, the needle valve was opened again to eliminate air bubbles out of the tubing by obtaining a slow flow at first. Then, ten series of pressure-flow measurements were conducted by opening the cutoff clamp a given amount, recording the pressure output as Dh (+0.01cm) and the flow rate from recording the volume differences and dividing by the time length of accumulation. Q = (Vfinal – Vinitial)/Dt. The height of the water in the tank was kept constant by continuously refilling the tank with the emptying fluid into the collection-graduated cylinder after each trial. For very low flow rates, the graduated cylinder was placed so that the lip of the cylinder touched the flowing water droplets thereby ensuring a constant flow. The water flow from the tank is driven by gravity and controlled by a needle valve. Bubbles were minimized in the tube since they increase the Darcy friction factor and other losses. The same protocol for pressure-flow measurements were used for the all other parameters stated initially.

RESULTS

Figure 3. Plot of Friction vs. Reynold’s Number

Figure 3 shows the Darcy friction factor versus Reynold’s number plot. The graph shows laminar flow and that friction is approximately inversely proportional to the Reynold’s Number.

Figure 4a. Plot of Reynold’s Number * Friction vs. Reynold’s Number

Using the calculated values of friction and Reynold’s Number, the product of the two variables was plotted against the Reynold’s Number. For Re<500, the product appears to be constant, while for Re>500, the product increases. Extrapolating a regression of the data to the y-intercept for Re<2000 yields an intercept of 98±18 (95%CI), as shown below in Figure 4b.

Figure 4b. Plot of Reynold’s Number * Friction vs. Reynold’s Number for Reynold’s number less than 500

Figure 5. Results using one pressure transducer from T3 experiment 2

Figure 5 shows results using one pressure transducer with a 98.12 ± 17.95 (95%CI) y-intercept to which the results from figures 4a and 4b were compared.

Table 1. Sample Trial Data for ¼” ID Tubing

Pressure drop values (in yellow highlight) show precision to 0.004 psig.

Table 2. Sample Entrance Length Effects Data from ¼” ID Trials

The laminar development length XL (m) for the flow to become fully developed is given by Boussinesq XL = 0.03ReD. Table 1 shows a sample data set for resulting entrance lengths due to different flows.

DISCUSSION

Poiseulle’s law states that for laminar flow, the Darcy friction factor f varies inversely with Re. Figure 3 exhibits an near-inverse relationship and therefore is misleading. A much steeper decrease should be seen if it were truly inversely related. Thus, Poisuellian flow was not observed.

Theoretically, for a long, horizontal tube the product Re*f must be a constant value of 64 for Re<2000 and increase with Re>2000. However, our data, as shown in Figure 4a, indicate an increase immediately after Re>500 which reconfirms non-Poiseullian flow.

In Figure 4b, an extrapolated regression of the graph fRe versus Re to the y-intercept for Re<500 yields an intercept of 68.99 + 10.67(95%CI). As the flow approaches zero, the data exhibits Poiseullian flow, which is theoretical, limiting flow. Equation 5.24 in H&B yields an intercept of 64, which in well within our confidence intervals.

Results from Experiment 2 using one pressure transducer, as shown in Figure 5, yielded a y-intercept of 98.12 ± 17.95 (95%CI). The confident limits do not include the theoretical number, 64, for Poiseullian flow. The electronic transducer does not measure accurate pressure difference for low pressure flows. In addition, Table 1 shows the apparatus measuring pressure differences as low as 0.0004 psig. Thus, the manometer is a better apparatus for measuring low pressure flows.

Recommendations:

The design and construction of a manometer may not be the time efficient for future laboratory experiments. However, certain design aspects of the manometer may be applied to Experiment 2. Instead of using just one pressure transducer located at the base of the water tank, the protocol may be improved by using two pressure transducers. The placement of the pressure transducers will be critical due to the effects of entrance and exit effect. The first transducer should be placed after a maximum calculated entrance length such as shown in Table 2. The second transducer is needed because the assumption of zero gage pressure at the outlet may be inaccurate due to exit effects, especially if a restrictor has been used. The second transducer should be placed near, but not at, the exit to avoid these effects.

In addition, when LabView software is used to monitor pressure when using electronic transducers. If the flow is very low, the fluid exits in small constant droplets. This produces a “noisy” signal because the pressure is similar to static pressure between droplets. We accommodated this by placing the lip of the graduating cylinder touching the droplet, thereby breaking surface tension and producing a constant flow.