In the Laboratory
Convection in a Continuously Stratified Fluid Richard M. Heavers Department of Chemistry and Physics, Roger Williams University, Bristol, RI 02809-2921 Two centuries have passed since Count Rumford recorded in 1797 his observations on the motions of circulating dust particles in the bore of a large thermometer (1): “…and these particles which were intimately mixed with spirits of wine, on their being illuminated by the sun’s beams, became perfectly visible…and by their motions discovered the violent motions by which the spirits of wine in the tube of the thermometer was agitated… . On examining the motion of the spirits of wine with a lens, I found that the ascending current occupied the axis of the tube and that it descended by the sides of the tube. On inclining the tube a little, the rising current moved out of the axis and occupied the side of the tube which was uppermost, while the descending currents occupied the whole of the lower side of it.”
Rumford clearly recognized that the circulation was caused by density differences due to thermal expansion. The hot liquid, being less dense, tended to rise; while the cold liquid, being more dense, tended to sink. In subsequent experiments, Rumford investigated the effects of stewed apples, eiderdown, and other fibrous materials on retarding thermal circulation in liquids. The term “convection”, here applied to density-driven fluid motion in a gravitational field, was first introduced by William Prout in 1834 (2). Convection is a very complex process, and there is no general equation to describe it. Consequently, only simple notions of rising and sinking fluid are presented in introductory physics and physical chemistry texts. These conceptual images are not reinforced by a quantitative discussion. The present paper offers a remedy for this situation. A simple, inexpensive, physical chemistry experiment is used to demonstrate how a mathematical model can be applied to convection in a salt-stratified fluid that is heated from below. In part I of the experiment, a mixing procedure is
Figure 1. Stratified fluid heated from below.
used to produce a linear salt concentration profile in a beaker of water at room temperature. The salt concentration (and fluid density) is greatest at the bottom of the beaker. In part II, the beaker containing the salt solution is heated on a hot-plate. The salt stratification limits vertical circulation in a manner somewhat analogous to the eiderdown in Rumford’s experiments. A well-mixed bottom layer forms soon after heating begins (Fig. 1), and is made visible with a drop of food coloring placed at the bottom of the beaker. The thickness of this convecting layer may be measured with a ruler at any time, and compared to that predicted from a model based on density differences. Temperatures in the bottom convecting layer, and in the fluid above, are measured with two mercury thermometers. Data for this experiment can easily be taken in a 2hour laboratory session. Preparing the linear salt profile takes about 15 minutes, while checking the linearity requires another 15 minutes. The individual convection experiments take from 15 minutes to one hour, depending upon the rate of heating. Part I. The Linear Salt Profile A mixing procedure described by Oster and Yamamoto (3) may be used to produce a continuously stratified salt solution in a beaker at room temperature. Two containers are connected by tubing as shown schematically in Figure 2. One contains a salt solution; the other (with the stirrer) initially contains an equal volume of fresh water. The rates of flow from each container are controlled by stopcocks, so water levels in each drop at the same rate. The combined effluent changes from fresh to salty and produces a linear vertical salt profile when introduced slowly at the bottom of the beaker. A glass tube used for filling is then carefully withdrawn from the beaker.
Figure 2. Diagram of the mixing apparatus.
Figure 3. Making a stratified fluid.
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In the Laboratory The actual apparatus (Fig. 3) used in our experiment consists of two plastic juice pitchers (cost $3 each) with nylon nipples (3 mm i.d.) epoxied in their bottoms. The rod on an inexpensive kitchen mixer (cost $15) was initially too short to reach the bottom of a pitcher. This rod was cut in half, and extended by epoxying the cut ends in a straight section of metal tube. Control of the flow is improved by maintaining the level of the fresher water about 20% higher during filling. This compensates for the greater density of the saltier water in the other pitcher, and for the distortion of the fluid surface by mixing. Part II. Heating the Fluid from Below
Conditions for Layer Growth The dependence of fluid density ρ on temperature T and salt concentration S is given by ρ = ρ0 (1 – α ∆ T + β ∆ S)
(1)
where ρ0 is an initial density, α is the coefficient of thermal expansion, β is the coefficient of saline contraction, and ∆ T and ∆ S are changes from the initial state. In the present situation, the fluid is initially at rest with uniform temperature and has a constant salt gradient dS/dz, where z is the vertical coordinate, taken positive upwards. When the fluid is heated at a constant rate from below, a well-mixed convecting layer of thickness h develops at the bottom of the beaker. Convection is made visible by circulating impurities in the water. This bottom layer is assumed to have constant temperature and salt concentration throughout. The overlying fluid remains at its original temperature, and retains the original linear salt profile. As the heating continues, the temperature of the convecting layer increases. The warmed fluid below becomes more buoyant than the unheated, fresher fluid just above the interface at the top of the convecting layer. Fluid mixes across the interface when the local vertical density gradient becomes marginally unstable (dρ/dz slightly > 0). Equation 1 then implies α∆T = β∆S
(2)
which is the condition for the density steps due to temperature and salt being equal and opposite (Fig. 4). Turner (4) explored other possible relations between ∆T and ∆S, but rejected them in favor of eq 2.
Rate of Layer Growth Heat Q added to the convecting layer is given by Q = mc∆T
since heating began. The rate of heating is given by H = Q/At
(4)
where A is the horizontal area of the fluid, and t is the elapsed time. The volume of fluid in a convecting layer of thickness h is V = Ah, and the fluid density ρ= m/V. Equations 3 and 4 may be combined to give H = ρhc (∆T/ t)
(5)
The difference in salt content between fluid in the convecting layer and that just above is ∆S = (dS/dz)h/2
(6)
Substituting eqs 5 and 6 into eq 2 gives h2 = K 1t
(7)
where K1 = 2αH/(ρcβ dS/dz). Equation 7 predicts that the height of the convecting layer is proportional to t1/2. Substituting eqs 6 and 7 into eq 2 gives (∆T) 2 = K2t
(8)
where K2 = (βH dS/dz) / (2αρc). Equation 8 predicts the difference in temperature between the convecting layer and the fluid above is also proportional to t1/2. Measurement Results and Discussion
(3)
where m is the mass of fluid in the convecting layer, c is its specific heat capacity, and ∆T is its temperature change
Figure 4. Profile of density at height Z above the beaker bottom due to (a) temperature increase from heating at a constant rate, (b) mixing of salt in the convecting layer, and (c) adding curves in (a) and (b) when the density steps ∆ρ are equal and opposite.
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Figure 5. Profile of salt concentration S at height Z above the beaker bottom.
Linear salt profiles were produced in a 2000-mL beaker by mixing 1000 mL of saturated salt solution (S = 26.4 g/100 g and ρ = 1.19 g/cm 3) with 1000 mL of tap water (S = 0 and ρ = 1.00 g/cm3) at room temperature. A refractometer (ATAGO S-28) was used to measure the salt content of samples taken by pipet from various depths in the fluid. The salt profile shown in Figure 5 was checked about 10 min after it was produced. The good linearity indicates the stratified fluid was ready for immediate use, with no additional time required for local diffusion to straighten the profile. With 2000 mL of fluid in the beaker, the fluid depth was 16.0 cm, and the salt gradient dS/dz = 1.65 g/100 g/cm. By using a saturated salt solution, a known salinity gradient may be produced without using a refractometer (cost $200). The heating rate for the hot-plate/stirrer (Corning PC351) was measured at four settings (Nos. 1, 3, 5, and 7) with 2000 mL of fresh water stirred by a magnet rotating at low speed. Equation 5 was used to determine heating rates from
Journal of Chemical Education • Vol. 74 No. 8 August 1997
In the Laboratory
Figure 6. h2 vs. t for four heater settings.
Figure 7. (∆T)2 vs. t for four heater settings.
slopes of temperature-vs.-time curves. A sample calculation using the slope 0.0392 °C/s for hot-plate setting #7 is
long heating period. This conclusion is supported by a 12 °C increase in the temperature of the upper fluid noted at the end of the experiment. At the highest heating rate (No. 7), the slope h2 /t is 40% higher than expected. This may be due to additional convective entrainment of fluid from the upper layer into the bottom layer, in violation of eq 2. In this case, the temperature of the upper fluid increased by less than 1 °C during the entire experiment, indicating that radiation effects were small. It should also be noted that the thick glass bottom of the beaker introduces an initial time delay in heating the fluid. This delay is especially evident at the lowest heating rate (No. 1 in Fig. 7), but appears to have little effect on the time-averaged rates of convecting layer growth and heating. In conclusion, this experiment offers a simple, inexpensive, and visually instructive method for modeling the convection process. Students gain experience with a technique for constructing a density-stratified fluid and are introduced to the notion of marginal stability by considering the competing effects of heat and salt on water density. In contrast, Count Rumford’s historic experiments were limited to temperature variations within a fluid with a fixed fibrous structure, and are not so easily modeled in terms of density differences.
(0.0392 °C) 1g H = ρhc ∆T = 3 16.0 cm 4.19 J = 2.63 J2 s g °C t cm cm s
Heating rates obtained for settings Nos. 1, 3, and 5 are 0.240, 0.846, and 1.55 J/ cm 2/s, respectively. The thickness h of the convecting bottom layer was measured at various times for each of four hot-plate settings. As predicted by eq 7, plots of h2 versus t (Fig. 6) exhibit long-term linearity with correlation coefficients R2 > .92. Departure from linearity at somewhat irregular time intervals is related to the formation and subsequent disappearance of additional convecting layers within the bottom layer. Turner (4) examined the stability of these additional layers. Using ∆T as the temperature change in the bottom layer since heating began, plots of (∆T)2 versus t (Fig. 7) are linear (R2 > .97) in agreement with eq 8. Making these plots may have less pedagogical value than those of Figure 6. However, it is very important for students to observe that the temperature of the fluid just above the bottom layer (or layers) remains close to its initial value. This illustrates that heat from the hot-plate is confined to the convecting bottom layer by the initial density (salt) stratification. It is possible to perform these experiments qualitatively, without reference to systematic errors inherent in the experimental design. The study of these errors is beyond the scope of this paper. However, the numerical results may be checked with either eq 7 or eq 8. The constant K1 was calculated from α = 0.00034/°C and β = 0.0065g/100 g, as determined in the laboratory to within about 10%. Using h 2/t from the slopes of the lines in Figure 6, the ratio (h 2/t) ÷ K1 was found to be 0.61, 0.71, 1.1, and 1.4 for heater settings Nos. 1, 3, 5, and 7, respectively. The value of this ratio should ideally be unity. If K1 is considered constant, then h 2/t is 40% lower than expected at the lowest heating rate (No. 1). This decrease in layer growth rate is probably due to radiative heat loss from the convecting layer, during the
Acknowledgments I am very grateful to Raymond D’Ambra for assistance with these experiments. Supplies were provided by the Roger Williams University Research Foundation. Literature Cited 1. Brown, S. C. Benjamin Thompson—Count Rumford (Count Rumford on the Nature of Heat); Pergamon: Elmsford, NY, 1967; pp 30–32. 2. Prout, W. Bridgewater Treatise; W. Pickering: London, 1834; Vol. 8, p 65. 3. Oster, G.; Yamamoto, M. Chem. Rev. 1963, 63, 257–268. 4. Turner, J. S. J. Fluid Mech. 1968, 33, 183–200 (see pp 185– 186).
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