Article pubs.acs.org/jchemeduc
Plastic Models Designed To Produce Large Height-to-Length Ratio Steady-State Planar and Axisymmetric (Radial) Viscous Liquid Laminar Flow Gravity Currents Harvey F. Blanck* Department of Chemistry, Austin Peay State University, Clarksville, Tennessee 37044, United States ABSTRACT: Naturally occurring gravity currents include events such as air flowing through an open front door, a volcanic eruption’s pyroclastic flow down a mountainside, and the spread of the Bhopal disaster’s methyl isocyanate gas. Gravity currents typically have a small heightto-distance ratio. Plastic models were designed and constructed with a restricted horizontal liquid flow using a series of connected cells that produced high viscosity laminar liquid flow planar and radial gravity currents. Because these gravity currents had a large height-to-distance ratio, they were easy to observe. Incremental transfer equations were used in spreadsheets to calculate the expected gravity current profiles. The calculated profiles were in good agreement with the liquid heights in the model’s cells at steady state. The steady-state profile for the planar viscous laminar flow gravity current was an x axis parabola.
KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Computer-Based Learning, Liquids
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MicroSoft PowerPoint
ravity currents, which are the result of horizontal gravitationally forced spreading, are very common and include events such as a pyroclastic flow down a mountainside during a volcanic eruption,1 the spread of methyl isocyanate gas in the Bhopal disaster, and sequestered carbon dioxide spreading in deep water.2 Gravity currents typically have a small heightto-distance ratio. Plastic models were designed and constructed with a restricted horizontal liquid flow, which resulted in large enough height-to-distance ratios that the gravity currents can be easily observed.
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Graphs for a series of rows from a spreadsheet can be displayed as a series of slides using MS PowerPoint so that they look like the models in action. (For an easy way to generate the slides see ref 3). Spreadsheet Equations
The spreadsheet equations can be used for a variety of initial and boundary conditions for laminar flow gravity currents.
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APPLICATIONS OVERVIEW
PLANAR GRAVITY CURRENT MODELS
Hole Model Design
Model Construction
The hole model (Figure 1) design is similar to this author’s model for diffusion emulation in which there are a series of cells, but, instead of having one connection, there are numerous holes in each partition.5
The models for planar gravity currents are easier to construct than the model for an axisymmetric gravity current. Construction details for acrylic models are described in the online notes of a previous article3 and in ref 4. If these notes concerning the use of acrylic shavings dissolved in methyl ethyl ketone are followed, the “cementing” will go smoothly. The tools required are not costly. Student construction of either of the planar models should be a rewarding project for both high school and college students. Model Demonstration
For a classroom demonstration, the slit model performs very well with saturated sugar solution at room temperature. In a simple arrangement, the solution can be hand poured into the model and the output collected in a container in a sink. The output can be directed by a trough along the back of the model so that it will flow into the same collection container as the overflow. © 2011 American Chemical Society and Division of Chemical Education, Inc.
Figure 1. Hole model for a planar (two-dimensional) gravity current. Cells are 2.5 cm × 2.5 cm × 20 cm. Published: December 12, 2011 234
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Operation of the Model
This model and the two other models described in this article were designed primarily for observing steady-state conditions in which fluid flow causes no change in liquid height in the cells with time. The mathematical and spreadsheet descriptive calculations that will be discussed later require laminar flow between the cells. To achieve laminar flow, which is associated with a low Reynolds number, the flow must be rather low. In these models, this is achieved by using a concentrated aqueous sucrose solution. It has a relatively high viscosity, affords easy clean up of the models and spills, and has no safety concerns. It was necessary to cool the sucrose solution to about 15 °C to ensure laminar flow.
Figure 4. Output flow for a series of seven rectangular cells at steady state. A region’s area, which is composed of a rectangle and small triangle, is depicted by similar shading or color and represents the flow out of the previous cell. For example, area 1 is the output from cell 0. Areas 1−7 are all equal. The height, Hn, of each rectangle is the liquid height in that cell.
Steady-State Behavior of the Hole Model for Planar Spreading
The steady-state laminar flow result for a saturated sucrose solution at 15 °C is shown in Figure 2. A steady stream of
triangle in which the width of the next area must be wider than the previous area. This is equivalent to drawing a large right triangle and dividing it up into the correct number of equal areas such that there is no remainder (Figure 4). Because all slices of the large triangle have the same slope at the top and the same area, it can be shown that
Hn + 1 = sqrt(Hn2 + H12 − H0 2)
(1)
where Hn is the length of each vertical line of the large triangle and the height of the liquid in that numbered cell (i.e., the height of the rectangle). H0 is the height of the liquid in cell 0. Because there are 16 steps in the plastic hole model down to zero, then the quantity H12 − H02 in eq 1 is equal to −(H02/16). If H0 is assigned a value of 100, then H02/16 equals exactly 625. The calculated values for the steady-state liquid heights are shown graphically in Figure 5. If the cell numbering is reversed, the calculated relative liquid heights exactly fit an x axis parabola. Superimposing the graph of these values (Figure 5) onto the photo of the model at steady state (Figure 2) reveals excellent agreement (Figure 6), which shows that a planar laminar flow gravity current at steady state has a parabolic profile.
Figure 2. Steady state for the 16 cell hole model showing a planar, laminar flow, viscous gravity current. The solution has a little food coloring to improve visibility.
liquid may be seen entering the model on the left end just to the right of an overflow channel. Triangle Method for Calculation of Expected Steady-state Liquid Heights in Cells for a Planar Laminar Flow Gravity Current
If the flow is laminar, the flow from one cell to the next is proportional to the pressure head. Flow out of a cell will increase at a uniform rate with an increase in liquid depth until the liquid height in the next cell is reached. From this point to the bottom of the cell, the output flow will be constant because the pressure head remains constant. In Figure 3, the relative size of the flow is depicted by the length of the arrows. At steady state, the output flow from each cell and hence the area defined by the set of arrows must be equal (Figure 3B).
Figure 5. Steady-state liquid heights calculated using eq 1. Figure 3. Flow between cells. Arrow length is proportional to flow at that level: (A) cell 1 before steady state and (B) cell 1 at steady state.
Calculation of Expected Liquid Heights for Planar Spreading Using Spreadsheet Repeated Transfers
If these equal flow areas of connected cells are placed adjacent to each other as in Figure 4, the result is a large
Another way to calculate the expected profile for the hole model is to use a spreadsheet. Using spreadsheet columns to 235
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The steady-state level for the slit model using a saturated aqueous sucrose solution at normal room temperature had a parabolic profile identical to that of the hole model at 15 °C. (The flow was not laminar using just water.) Flow was judged to be laminar when increasing the viscosity did not affect the liquid levels at steady state. Being able to use a saturated aqueous sucrose solution at room temperature rather than having to cool the solution is a significant improvement.
Figure 6. Superposition of the body of Figure 5 (transparent version) onto Figure 2 (less the spigot).
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represent model cells and rows to represent uniform time increments, liquid height values can be slowly changed using an incremental transfer equation (eq 2) as was done successfully with this author’s diffusion model.3 In eq 2, the spreadsheet cell B8 value, for example, will be the new height of the liquid in model cell 1 after the eighth spreadsheet row iteration.6 (Spreadsheet column A is model cell 0.)
AXISYMMETRIC (RADIAL) GRAVITY CURRENTS MODELS
Calculation of Expected Liquid Heights for Radial Spreading Using Spreadsheet Repeated Transfers
For a model with cylindrical partitions (Figure 8A), the spreadsheet equations for radial spreading from a central cylindrical source are of the following form:6
B8 =B7+((A7‐B7)/2*(A7‐B7)+(A7‐B7)*B7‐(B7‐C7)/ 2*(B7‐C7)‐(B7‐C7)*C7)*0.001
(2)
B7 is the liquid height in model cell 1 after the seventh spreadsheet row iteration. A7-B7 is the height of the triangle formed by the arrows in Figure 3. Because the length of an arrow is proportional to the pressure head, the base of the triangular area is the pressure head at that depth, A7-B7, times a proportionality constant, 0.001. Thus, the area of the triangle formed by the set of arrows is (A7-B7)/2*(A7-B7)*0.001. The rectangular flow area input to model cell 1 from model cell 0 is the arrow length (A7-B7)*0.001 times the height of the liquid in model cell 1, B7. The next two terms are the triangular and rectangular output areas from model cell 1 to model cell 2 and are calculated in the same way. The proportionality constant, 0.001, is a transfer (flow intensity) factor and is adjusted so that the iterations are stable. The first and last spreadsheet cells are set at 100 and 0, respectively. After 2500 rows, the steady-state liquid heights for planar spreading agree to four significant figures with those calculated using the triangle method discussed in the previous section. After 7000 rows, they agree to eight significant figures. Therefore, it is concluded that this spreadsheet method produces the same results as the triangle method’s eq 1.
Figure 8. Cell arrangements for a radial model: (A) cylindrical and (B) wedge.
B5 =B4+(((A4‐B4)/2*(A4‐B4)+(A4‐B4)*B4)*2 *B$3‐((B4‐C4)/2*(B4‐C4)+(B4‐C4)*C4)*2* C$3)*0.006/(C$3∧2‐B$3∧2)
(3)
In this example, spreadsheet row 3 contains the radius of the model cell’s partition closest to the center. The input radius to cell 1 has a radius of 1; cell 2 input radius is 2; and so forth. The terms in eq 3 are similar to those in eq 2 except that the partition’s liquid contact area and the model cell’s horizontal liquid surface area must be considered. If cell 0 is kept at a constant level and the exit level at zero, the spreadsheet transfer equations result in the steady state shown graphically in Figure 9.
Slit Model
A model having a laminar flow at room temperature would be more convenient. A model was made using slits rather than holes (Figure 7). A spacer is used at the top and the bottom of each slit.4
Figure 7. (A) Slit model for a planar gravity current. (B) Top view diagram of a slit model.
Figure 9. Spreadsheet transfer results for steady-state axisymmetric (radial) spreading. 236
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Because curved partitions make construction difficult, a spreadsheet transfer equation was developed for a wedge (Figure 8B) with a 20° (arbitrary) angle at the apex to see how similar the two might be. The equation is6
B5 =B4+(((A4‐B4)/2*(A4‐B4)+(A4‐B4)*B4) *0.353*B$3‐((B4‐C4)/2*(B4‐C4)+(B4‐C4) *C4)*0.353*C$3)*0.006/(B$3*0.1765+C$3 *0.1765)
(4)
Partition contact area and the cell’s horizontal liquid surface area must be considered as in eq 3. The 0.353 factor is 2 times the tangent of 10°. It was surprising to find that the resulting spreadsheet calculations for a wedge model were identical to those for a cylindrical sector model. A closer look at eqs 3 and 4 shows that they will produce the same results. This was good news because it is much easier to construct a model with straight partitions. Wedge Model for Radial Spreading
The wedge model was constructed with numerous holes in uniformly spaced straight partitions. On the basis of the spreadsheet calculations, it seemed unnecessary to make all the partitions the same height. Even so, the model in Figure 10 has
Figure 11. Wedge model with a laminar flow axisymmetric gravity current at steady state: (A) view of the setup and (B) view of model.
Figure 12. Superposition of the body of Figure 9 (transparent version) onto Figure 11B.
To demonstrate the versatility of these spreadsheet equations, the same transfer equation used for steady-state planar liquid spreading in one direction was used to calculate liquid heights for planar liquid spreading of a fixed amount of liquid in two directions. The results at various stages of spreading are shown in Figure 13.
Figure 10. Wedge model for a radial gravity current.
about 1400, 5/32 in. holes in the 0.080 in. thick acrylic plastic. The holes were drilled using a step bit, which produces a smooth bore hole. A 15 °C saturated aqueous sucrose solution produced a radial, laminar flow steady-state gravity current (Figure 11). Superimposing the spreadsheet calculated graph from Figure 9 onto the liquid flow photo in Figure 11B shows them to be in good agreement (Figure 12).
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EXTENSION OF SPREADSHEET CALCULATIONS TO NON-STEADY-STATE LIQUID SPREADING All of the models described in this article produce a laminar flow at steady state, but in the early stages, the flow is not always laminar. However, the spreadsheet equations are based on laminar flow, and will therefore produce a profile for nonsteady-state laminar flow conditions. Spreadsheet emulations of this sort were previously successfully employed to accurately describe all this author’s plastic model diffusion emulations for both steady-state and non-steady-state situations.3
Figure 13. Various stages for planar spreading of a fixed volume of liquid on a solid surface from a plane using spreadsheet transfer equations if only gravitational and viscous forces are considered. 237
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spreadsheet results can be displayed as bar graphs so that they look very much like actual models3 and the spreadsheet equations have an advantage in that they can be used for a variety of situations without the difficulty of constructing models. It is hoped that this article will encourage introduction to and discussion of gravity currents in undergraduate chemistry classes.
As an example of this kind of planar spreading, consider a plane of lower density liquid before spreading floating in a more dense second liquid such as in Figure 14A. If the density
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AUTHOR INFORMATION
Corresponding Author
Figure 14. Cross-sectional view of a gravitationally forced planar spreading of a less dense liquid in a higher density liquid: (A) before spreading and (B) during spreading.
*E-mail:
[email protected].
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ACKNOWLEDGMENTS I wish to thank the reviewers for their helpful suggestions. One reviewer suggested the possibility of using a glycerol/water solution to avoid the need to cool the sucrose/water solution for use in the hole models. Literature data shows that the viscosity of 95% (weight percent) glycerol(aq) solutions have a viscosity similar to 70% sucrose(aq) solutions. Glycerol(aq) solutions having 97% or more glycerol at room temperature appear to be a possible alternative to a saturated sucrose(aq) solution at 15 °C.
difference for the two liquids is small enough so that it spreads slowly and if only viscous and gravitational forces are considered, then spreadsheet eq 2 can be used with results similar to Figure 13. The portion below the surface will look like an amplified reflection of the portion above the liquid as in Figure 14B. If the end spreadsheet cell equations are set for zero output, then the spreading liquid will be contained, which eventually results in a layer with uniform thickness. Radial spreadsheet equations can be used in a similar way. These equations may also be used to show flow from the perimeter toward the center by altering the end spreadsheet cell equations and values.
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REFERENCES
(1) Simpson, J. E. Gravity Currents in the Environment and the Laboratory, 2nd ed.; Cambridge University Press: Cambridge, U.K., 1997. (2) Golomb, D.; Pennell, S.; Ryan, D.; Barry, E.; Swett, P. Environ. Sci. Technol. 2007, 41, 4698−4704. (3) Blanck, H. F. J. Chem. Educ. 2009, 86, 651−652. (4) To form uniform slits, bond the pieces in the following order. The piece for the top of a T is placed on a flat surface and the stem of the T is bonded to it using methyl ethyl ketone. The T is then bonded to the spacers and the spacers to a side of the model. The two halves are finally bonded together. Acrylic model construction techniques are available in a previous article’s JCE WebWare in this Journal.3 (5) Blanck, H. F. J. Chem. Educ. 2005, 82, 1523−1529. (6) Spreadsheet equations 2−4 are copied directly from the Microsoft Excel spreadsheet. (7) Huppert, H. E. J. Fluid Mech. 1982, 121, 43−58. (8) Spannuth, M. J.; Neufeld, J. A.; Wettlaufer, J. S.; Worster, M. G. J. Fluid Mech. 2009, 622, 135−144. (9) Blanck, H. F. J. Chem. Educ. 1999, 76, 1635−1638.
DISCUSSION The steady-state profiles for the planar and radial viscous gravity current models are in good agreement with profiles in published articles by Huppert7 and by Spannuth et al.8 These articles as well as the book by Simpson1 are excellent sources of differential equations useful in describing gravity currents. In low Reynolds number flow gravity currents, the dominant forces are associated with gravity and viscosity with the viscosity being high enough to produce a laminar flow. Results from the models show that such laminar flow profiles are determined by the pressure head generated by the gravitational field. However, in many gravity currents, the viscose force is often low compared to the inertial force, which results in a high Reynolds number and turbulent flow. Turbulence is often readily apparent at the leading edge of a spreading fluid. The models described here cannot be used to satisfactorily show turbulent flow gravity currents. In this author’s article concerning the excess warm water in the equatorial Pacific during an El Nino, this author stated that the warm water “wedge” had an appearance rather like a Gaussian curve.9 This was based on the TOPEX/Poseidon satellite data for the excess sea surface height and the NOAA Atlas buoy subsurface temperatures. Perhaps the profile could be better described as having some characteristics of a planar laminar flow gravity current as in Figure 14B. Unfortunately, gravity currents are usually not introduced in traditional undergraduate chemistry classes. It is not uncommon to discuss the layering of a less dense liquid on a higher density liquid in the classroom yet omit any discussion of the spreading process. Students should be cautioned about heavy vapors pouring out of a hood if air flow is inadequate, but miss an excellent opportunity to discuss gravity currents. These models allow laminar flow gravity currents to be easily and conveniently viewed in the classroom. It is perhaps important to reemphasize that the models in this article with liquid flowing through them are in fact gravity currents and are not models that emulate gravity currents. Although it is nice to see the models in action, 238
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