Where Did the Water Go? Boyle's Law and Pressurized Diaphragm

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Where Did the Water Go? Boyle’s Law and Pressurized Diaphragm Water Tanks James Brimhall Physics Department, West Virginia State University, Institute, WV 25112-1000 Sundar Naga* Chemistry Department, West Virginia State University, Institute, WV 25112-1000; *[email protected]

Boyle’s law is one of the major topics of discussion in general chemistry and high school chemistry. Several experiments that students can do in the laboratory to “discover” Boyle’s law (1–3) and demonstrations that teachers can perform in the classroom to illustrate this law (4–8) have been published in this Journal. It is also important that the students are exposed to how Boyle’s law can be effectively applied in various real-life situations. DeLorenzo has showed the applicability of this law in gaining an insight into the mysterious behavior of the so-called Loch Ness monster as well as into the ability of the gannet, a sea bird, to dive for food in the northern Atlantic and southeastern Pacific Oceans (9). DeLorenzo has also indicated how this law can be applied to opening a beverage can so that the liquid does not rush out (10). Here we present a unique, real-life application of Boyle’s law to pressurized diaphragm water storage tanks. Domestic versions of these tanks are used throughout the world, typically placed in a basement or in a small enclosure outside. Because they are often out of sight, the interesting operating characteristics of pressurized diaphragm water storage tanks are little known and under-appreciated. An internal flexible butyl or vinyl diaphragm permanently separates the air and the water within these tanks. In this presentation, Boyle’s law is applied to the enclosed air within such a tank, which is the tool for understanding its water-supplying characteristics. Manufacturers’ web sites can provide technical and descriptive information about these tanks (11, 12). The majority of deep well pumps that furnish water for domestic consumption utilize a pressure monitoring switch whose purpose is to keep the operating gauge pressure either between 20–40 psi or between 30–50 psi (1 psi = 6,895 Pa = 0.0680 atm). Another available switch maintains a range of 40–60 psi. There are advantages to each choice, but ei-

ther the 20–40 psi switch or the 30–50 psi switch is commonly used in many domestic situations. An important practical issue is maximizing the quantity of the so-called “drawdown” water, which is the total volume of water that can be drawn from a tank before the pump is again activated by the switch to keep the water-line pressure within the desired range. This issue is important for two reasons. First, failure to maximize the drawdown water can lead to excessive cycling of the well pump, which can lead to its premature failure. Second, users might be tempted to compensate for the lack of maximization of drawdown water by installing a larger tank. Both of these unnecessary situations can be avoided by maximizing the quantity of drawdown water. Besides depending on the range limits of the pressuremonitoring switch, this maximization also depends upon properly setting the “precharge” air pressure, which is the air pressure initially present within the tank prior to the introduction of any water. Tanks are usually precharged at the factory, but the internal pressure can also be adjusted after the tank is installed. In practice even the maximum amount of drawdown water is small compared with the volume of the tank itself; Boyle’s law (PV is constant) provides insight as to why this is the case. Theoretical Basis To aid in the understanding of such tanks with the help of Boyle’s law, we consider the internal air—separated from the water by a flexible, nonleaking membrane— to be ideal. Thereby, one can analyze the pressure and volume of the air within the tank, and also predict the amount of drawdown water. Figure 1 shows a tank and its membrane at four stages in the water-usage cycle. These pictorial representations provide accurate indications of conditions within the tank, be-

Figure 1. Cross-section diagram of a diaphragm tank (see ref 11 for further details).

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Vol. 84 No. 3 March 2007

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Journal of Chemical Education

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cause in actual usage, even at the hold cycle, the tank is not substantially full of water. Let us consider the 30–50 psi switch mentioned previously as an example: the lower pressure of 30 psi, at which the pump is activated, is designated as pmin and the higher pressure of 50 psi, at which the pump is disengaged (the hold cycle in Figure 1), is represented by pmax. In practical usage, the values of pmin and pmax are given as gauge pressures rather than absolute pressures. The pressure to which the tank is precharged, that is, initially charged in the absence of any internal water, is designated as pcharge and this pressure can be measured with any common tire gauge by using the tire valve mounted on the tank. The sea level atmospheric pressure patm of 14.7 psi must be added to any gauge pressure ( p) measured in psi to obtain an absolute pressure usable with Boyle’s law. An absolute pressure will be indicated by P (i.e., uppercase) and will represent ( p ⫹ patm ), that is, gauge pressure plus atmospheric pressure where patm ⫽ 14.7 psi for this discussion. The following analysis allows prediction of the percentage of the tank volume occupied by the drawdown water as a function of Pmin, Pmax, and Pcharge. Let V1water be the actual volume of water in the tank when the pressure inside is Pmin, and V2water be the volume of water at Pmax. The volume of drawdown water is then (V2water ⫺ V1water ) and the fractional drawdown water volume, expressed as a percentage, is:

FV =

(V2water

− V1water ) 100% V

(1)

where V is the total volume of the tank, usually expressed in gallons in the U.S. The values of V1water and V2water appearing in eq 1 depend on the particular switch in use, as well as on the precharge pressure Pcharge, which, under ordinary circumstances, can have any value, 1 atm or greater. It needs to be realized that irrespective of the pressure switch in use, water cannot enter the tank unless the pressure in the water line exceeds Pcharge. For analyzing the system, it is convenient to separate the range of precharge pressures into three adjoining regions in terms of Pmin and Pmax. Region 1 Patm ≤ Pcharge ≤ Pmin Region 2 Pmin ≤ Pcharge ≤ Pmax Region 3 Pcharge > Pmax where Patm refers to the absolute atmosphere pressure. Using eq 1, the percentage volume of drawdown water in each of the three regions can be determined.

Region 1 ( Patm ≤ Pcharge ≤ Pmin) In this region of Pcharge, the initial conditions are that the tank contains no water (i.e., Vair ⫽ V ) and the (absolute) inner air pressure of the tank equals Pcharge. For this situation, applying Boyle’s law gives (2)

PchargeV = C1

For the pressure of Pmin to be reached within the tank, a volume of water designated as V1water must be pumped in. Boyle’s law for that configuration gives Pmin (V − V1water ) = C1 426


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(3)

where the quantity (V ⫺ V1water ) represents the volume of air in the tank at Pmin. Combining eqs 2 and 3, the volume of water V1water in the tank at pressure Pmin is given by:

V1water = V 1 −

Pcharge Pmin

(4)

If the pump continues to add water, the pressure of the air inside the tank increases, and the pump cycles off when Pmax is reached. The corresponding volume of water at Pmax is designated as V2water. Boyle’s law is then satisfied by

Pmax (V − V2water ) = C1

(5)

where the quantity (V ⫺ V2water ) is the volume of air in the tank at Pmax. Combining eqs 5 and 2, the volume of water in the tank V2water at pressure Pmax is: V2water = V 1 −

Pcharge

(6)

Pmax

Using eqs 4 and 6, the volume of drawdown water can be written as:

Vdrawdown = V2water − V1water = V

Pcharge Pmin

1−

Pmin Pmax

(7)

from which:

FV =

Pcharge Pmin

1−

Pmin 100% Pmax

(8)

Since Pmin and Pmax are constants for any given pressure switch, it is evident from eq 8 that %Vdrawdown is a linear function of Pcharge with positive slope. According to eq 8, the maximum value of the drawdown volume in Region 1 would occur when Pcharge is maximum, which occurs in this region as Pcharge approaches Pmin. Hence, FVmax for Region 1 =

1−

Pmin 100% Pmax

(9)

Region 2 ( Pmin ≤ Pcharge ≤ Pmax) In this region V1water always equals zero since Pcharge is greater than the activation pressure Pmin of the switch. As stated previously, no water enters the tank until the pressure in the water line exceeds Pcharge. Since V1water ⫽ 0 throughout this region, according to Boyle’s law PchargeV = C 2

(10)

where, as mentioned previously, V is the volume of the tank. If water is then pumped in until the pump cycles off at Pmax, Boyle’s law requires Pmax (V − V2water ) = C 2

(11)

Using eqs 10 and 11 and recalling that V1water ⫽ 0 throughout this region, the volume of water V2water in the tank at


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pressure Pmax, as well as the volume of drawdown water, are given by: Vdrawdown = V2water − V1water = V2water = V 1 −

Pcharge

(12)

Pmax

Table 1. Theoretical Values for Common Pressure Switches

Pressure Switch Range, psi

Maximum Tank Volume Available as Drawdown Water, %

20–40 30–50 40–60

36.6 30.9 26.8

From eq 12 one obtains for Region 2:

FV =

1−

Pcharge Pmax

100%

(13)

Since Pmax is a constant for a given pressure switch, the function is again seen to be linear, this time with a negative slope. And as eq 13 shows, the maximum value of the drawdown volume arises when Pcharge is minimum in this region, which occurs when Pcharge ⫽ Pmin. Substituting Pcharge ⫽ Pmin into eq 13 results in: FVmax for Region 2 = 1 −

Pmin 100% Pmax

(14)

Interestingly, eq 14 for Region 2 is seen to be the same as eq 9 for Region 1, each representing the point of continuity between the two regions.

Region 3 ( Pcharge > Pmax) This region is perhaps the easiest to understand, for the switch keeps the pump permanently cycled off since Pcharge ⬎ Pmax. Hence, no water is pumped into the tank and no water is available to draw from the tank. Therefore,

Vdrawdown = V2water − V1water = 0 − 0 = 0 (15) throughout this region. Equations 8, 13, and 15 can be plotted to obtain the Boyle’s law-based theoretical graph of %Vdrawdown versus pcharge (recall pcharge ⫽ Pcharge ⫺ 14.7 psi). Figure 2A shows the graph for a pressure switch of pmin ⫽ 30 psi and pmax ⫽ 50 psi. Figures 2B and 2C illustrate the graphical characteristics for the cases of a 20–40 psi switch and a 40–60 psi switch, respectively. For any given pressure switch, the volume of drawdown water increases to a maximum at pcharge ⫽ pmin, decreases to zero at pcharge ⫽ pmax, and then remains at zero for larger values of pcharge. Thus, the value of pcharge that provides the maximum volume of drawdown water is simply pmin. For the three common pressure switches, Table 1 indicates the theoretical maximum percentages of tank volumes available as drawdown water. As shown in Table 1 and as seen from Figure 2A for the 30–50 psi pressure switch, the three-region graph reaches a maximum percentage drawdown volume equal to 30.9% of the tank volume (when pcharge ⫽ pmin ⫽ 30 psi). Note that for these three common switches, the maximum amount of drawdown water is only about one-third of the tank’s volume.

Figure 2. Theoretical volume of drawdown water vs the precharge diaphragm pressure for three switches: (A) 30–50 psi; (B) 20–40 psi; (C) 40–60 psi. Eqs 8, 13, and 15 were used for Regions 1, 2, and 3, respectively.

Experimental Results The experimental setup is shown in Figure 3. A commercial 30–50 psi switch was connected with a typical sixgallon (1 gal = 3.785 L) diaphragm tank. Standard water www.JCE.DivCHED.org

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Figure 3. The experimental setup used to obtain the data presented in Figures 4 and 5. The commercial 30–50 psi switch used in our experiment had a measured pressure range of 31–52 psi.


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pressure gauges were attached to the water line, as seen in Figure 3, to monitor the tank pressure. An ordinary tire gauge was used to measure pcharge (recall Pcharge ⫽ pcharge ⫹ 14.7 psi). Measurements of the volumes of water were made using ordinary plastic buckets from a hardware store. The buckets were graduated in pints (1 pt ⫽ 0.4732 L). A water faucet provided the source of water. It is important that the pressure of the available water source be greater than pmax of the pressure switch so the switch can cycle properly. Using the attached pressure gauges, the values of pmin and pmax for our particular switch were measured to be 31 and 52 psi. Measuring the volumes of draw-

down water as a function of pcharge produced the results shown in Figure 4. The volumes of water V1water and V2water, which were present at pmin and pmax, respectively, were also measured and compared to the volumes based on eqs 4 and 6 for Region 1 and eq 12 for Region 2 (V1water being zero in this region). Recall that V1water and V2water are each zero throughout Region 3. The results are shown in Figure 5. The rates at which water was fed into and bled from the tank were deliberately kept low in order to keep the pressure equalized within the system so the water gauges and water pressure switch would be subjected to equal pressures. Classroom Activities A number of activities might be initiated within a classroom setting using the analytical techniques and the ensuing results detailed in this manuscript.

Figure 4. The experimental and theoretical fractional volumes of drawdown water associated with the experimental setup shown in Figure 3.

• The indirect approach, which is utilized for determining volumes of water within a tank, could be discussed as a technique to problem solving. This is an instance in which the inside of a tank is filled with (separated) air and water. The real purpose of the analysis is to understand and predict the volumes of water within the tank in various conditions. However, assuming the air within the tank obeys Boyle’s law, prediction of the volumes of air within the tank for a variety of conditions is straightforward. Since the remaining volume in the tank is filled with water, indirectly the desired information about water volumes can then be determined. This approach was used for all the equations and graphs presented in the paper. (Note eqs 3 and 11.) • The equations within the manuscript could be used to generate the three-region graph of percentage drawdown volumes versus the precharge pressures for a pressure switch with arbitrary values of minimum and maximum pressures. The maximum %V of water drawdown possible in that situation could also be determined. (Initially substitute for Pcharge , Pmin, and Pmax into eqs 8, 13, and 15; then use either eq 9 or 14, or interpolate the maximum value from the graph.) • A calculation could focus on the determination of the actual volume of water present in a tank at some point, for example, at the instant the well pump cuts off, given arbitrary values for the initial precharge pressure and the maximum and minimum pressure values of the pressure switch. (Substitute for Pcharge, Pmin, Pmax, and V, as needed, into eq 6, 12, or 15.) • Using the description in the Experimental Results section, a mock-up apparatus could be utilized to obtain data for comparison with the results predicted by this model. The data acquisition activity could be converted into a full-fledged laboratory exercise.

Summary Figure 5. The experimental and theoretical %V1water and %V2water associated with the experimental setup shown in Figure 3.

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As a practical matter, pressure variations within the system at the beginning of a start-up cycle are potentially quite different in Region 1 and Region 2. Throughout Region 2,


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as the tank pressure drops to pmin, the volume of water within the tank (V1water ) goes completely to zero. Therefore, the pressure within the external water line also drops to zero (with the possibility of air actually entering the water line), although the pressure within the tank remains at pcharge. Pressure shocks are then created in the water pipes when the pump cycles on and the line pressure jumps from zero to pcharge very quickly. In Region 1, however, there is always (some) water remaining within the tank as its pressure drops to pmin. The pressure in the tank and within the water line drops only to pmin (i.e., not to zero), thus providing a cushion against pressure shocks when the pump cycles back on. Precharge pressures of 1 or 2 psi less than the rated pmin of the switch are often recommended. This will ensure that the system actually operates in Region 1 where the pressure transitions are somewhat smoother than in Region 2. One can also understand why these tanks, when operating near maximum efficiency during summertime, only sweat about one-third of the way up the tank. This occurs because only about one-third of the tank’s volume becomes refilled with newly added in cool well water. We have used Boyle’s law to provide accurate and useful insight into the nuances of precharged diaphragm water tanks. We have also noted the importance of utilizing the correct

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precharge pressure consistent with the selected pressure switch in order to obtain the maximum drawdown volume of water. Otherwise, as shown by any of the three-region graphs, the actual drawdown water volumes could be significantly reduced from the maximum-possible percentages, which are already not very impressive. Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Richmond, T. G.; Parr, A. J. Chem. Educ. 1997, 74, 414. Lewis, D. L. J. Chem. Educ. 1997, 74, 209. Hermens, R. A. J. Chem. Educ. 1983, 60, 764. Hughes, E., Jr.; Holmes, L. H., Jr. J. Chem. Educ. 1993, 70, 492. Broniec, R. J. Chem. Educ. 1982, 59, 974. Davenport, D. A. J. Chem. Educ. 1979, 56, 322. Moeller, M. B. J. Chem. Educ. 1978, 55, 584. Miller, D. W. J. Chem. Educ. 1977, 54, 245. DeLorenzo, R. J. Chem. Educ. 1989, 66, 570. DeLorenzo, R. J. Chem. Educ. 1980, 57, 601. Myers Pressurized Tanks Web Page. http://www.femyers.com/pdf/ pdf.ws/ws%20brochure/tanks.pdf (accessed Dec 2006). Flotec Pressure Tanks Page. http://www.flotecwater.com/pdf/ Page_44_2004.pdf (accessed Dec 2006).


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Where Did the Water Go? Boyle's Law and Pressurized Diaphragm

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