The example used thus far illustrates the logic inFigure 2.2,but it is deceptive in its simplicity. It is, in fact, unusual that the conservation equations alone lead so directly to a model that can be employed for analysis and design. We illustrate this important point here with a slight variant of the tank problem, as shown inFigure 2.8.There is no inflow (qf= 0); the tank empties by gravity-driven flow through a small hole, or orifice, in the base. The area of the orifice is A0. As before, there is no heating or cooling, and the temperature of the liquid remains constant. Our objectives might be the answer to some or all of the following questions:
How long will it take the tank to drain?
How does the height of liquid vary with time?
How does the flow rate through the orifice vary with the depth of liquid?
This tank draining is a physical situation with which most people have had direct experience; if not, it simply requires punching a hole in a large can, filling it with water, and then observing the behavior as the liquid flows out. Observation shows
us that the level in the tank decreases with time, the flow rate of liquid through the orifice varies with the height of the liquid and with the size of the exit orifice, and the tank empties completely in a finite time. Little more can be said with verbal statements, and we turn again to the conservation equation. The situation is identical to that leading to Equation2.5, so with qf= 0 we obtain
dh dt = −qe
A. (2.11)
In contrast to the preceding example, Equation2.11is a single equation involving two quantities that we do not know: the liquid height, h, and the effluent rate, qe. (Henceforth, since there is only one flow rate and there is no possibility of confusion, we will denote the effluent flow rate simply by q.) Since we have two unknowns and only one equation we must seek a second relation. This relation can, in fact, be established with some approximations from the principle of conservation of momentum or from the principle of conservation of energy, either of which can be used to derive a fundamental relation in fluid mechanics known as the Bernoulli equation; indeed, you may have already been introduced to the Bernoulli equation in a physics course. It is often the case, however, that we are unwilling or unable to apply further conservation equations at a convenient level of complexity, and we shall presume somewhat artificially that such is the situation here. Our additional relationship between q and h, the constitutive relationship, must then be obtained by intuition and/or experiment. We anticipate that a relationship obtained in this way will be rather less general than one based on fundamental conservation principles, and we must use great care in applying the results to situations that differ very much from the conditions of any experiments that we have performed.
Now, we know that the flow occurs through the orifice because the pressure in the liquid at the base of the tank is greater than the pressure of the atmosphere, thus forcing the liquid out, and that the greater the pressure difference the greater the flow. We can express the general relationship as q= q(p), by which we mean that q, the flow rate, is a function ofp, the pressure change across the orifice. If the top of the tank is open, then the pressure there, too, is atmospheric. The pressure in the liquid at the bottom of the tank exceeds the pressure of the atmosphere by the weight per unit area of the liquid column, which is proportional to the height of liquid. The pressure difference, or driving force for flow, is therefore proportional to h. It is the functional relationship of q to h, denoted as q(h), that we seek as our second relation to supplement Equation 2.11.The approach that we shall take is to postulate the form of the dependence of q on h (our constitutive relation), solve the model Equation2.11for h, and then check the prediction of the model with the experimental data. If the model and data do not agree, we will use the way in which they disagree as an aid in postulating a new dependence. (This is the Revision step inFigure 2.1.)
Table 2.1shows some data of liquid (water) height versus time for three experi-mental runs in a draining tank. (Height was taken as the independent variable in the experiment, since it is easier to read the time for a given height than vice versa. The
Table 2.1. Liquid heightversus time for the tank emptying experiment. Tank diameter= 27.3 cm (10.75 in.), tank height
= 30.5 cm (12 in.), orifice diameter = 1.55 cm (0.61 in.) Height of liquid Time Height of Liquid Time (centimeters) (seconds) (centimeters) (seconds)
data were originally recorded in even inches, and the centimeter values are rounded to one decimal place.) The data are plotted inFigure 2.9.In most cases the three data points cannot be distinguished on this scale, and only a single point is shown. There are also two lines inFigure 2.9;the lower line is a straight line drawn through the first three data points and extrapolated, while we will discuss the upper line a bit later.
The data clearly indicate that the slope (i.e., the rate of change of height), which is proportional to the flow rate, decreases in magnitude with decreasing height. This is consistent with our understanding of the physical process: We know the liquid will flow out more slowly for a small height than a large one, and there can be no flow if there is no liquid height at all. This last observation seems trivial but in fact has a
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Figure 2.9. Liquid height as a function of time for a draining tank.
profound implication, for it tells us that whatever the relation between q and h may be, q must vary as a positive power of h for sufficiently small h.