Three-Tank Experiment Overview The three-tank experiment focuses on application of the mechanical balance equation to a transient flow. Three tanks are interconnected by Schedule 40 pipes of nominal diameter 3/4" leading from the bottom of each tank to the other two tanks (see Figure 1). Two of the tanks are identical and have the diameter of 11". The diameter of the third tank is 13". The tanks will be filled with water to different heights, which will drive flow of water between the tanks. The purpose of this experiment is to investigate time dependence of water levels in the tanks as the system reaches equilibrium. The experimental observations should be compared with theoretical predictions which will be obtained by solving a system of nonlinear differential equations. Therefore, students should review numerical methods (e.g., Euler, Runge-Kutta) for integration of differential equations. Figure 1. Three-tank system. 1
Theory The governing equations in this experiment are obtained from applying force and mass balances on the fluid flow between tanks. Mass balance for the flow of water leaving one of the i-th tanks is: dh i dt = Q i A i, (1) where h i is the level of water in the i-th tank, Q i is the flow rate of water leaving or entering the i- th tank, and A i is the cross-sectional area of the i-th tank. A more useful relationship can be obtained by rewriting the flow rate in terms of the average fluid velocity v i of water in the pipe leading to the i-th tank, dh i dt = A p v A i, (2) i where A p is the pipe area. To obtain the average velocity, we need to apply the mechanical balance to the pipe system. For simplicity, let us first perform this analysis for a 2-tank system shown in Figure 2. After that, we will extend the analysis to the 3-tank system. Figure 2. Schematics of the two-tank system. Two-Tank System The mechanical balance between points 1 and 2 of the 2-tank system is P 1 P 2 ρ = 1 2 K 12v 2 12, (3) where K 12 is the friction coefficient of the pipe connecting the two tanks, v 12 is the fluid velocity in this pipe, P i is the hydrostatic pressure at the bottom of the i-th tank (i = 1, 2), and ρ is the water density. The hydrostatic pressure difference is Substituting this into Eq. (3) and solving for v 12 we get: P 1 P 2 = ρg(h 1 h 2 ) (4) 2
v 12 = 2g(h 1 h 2 ) K 12. (5) This equation assumes that h 1 h 2. Note that if the h 1 < h 2, Eq. (5) will yield an imaginary number for v 12, whereas physically it should be a negative value. To account for this, Eq. (5) should be rewritten as follows: where v ij = Sign(h i h j ) 2g h i h j K ij, (6) 1, x > 0 Sign(x) = { 0, x = 0 1, x < 0 is the sign function. The indices ij in Eq. (6) denote the location of the pipe network (for example, K 12, is the friction coefficient of the pipes between tanks 1 and 2). Eqs. (5) and (6) adhere to the following sign convention: v ij > 0 if water flows from tank i to tank to tank j. Now that we have an expression for the velocity between two tanks, we can plug it into the mass balance Eq. (2) to get a system of first-order differential equations describing the water level in each of the tanks. For simplicity, let us consider flow between two tanks of the same diameter. In this case, the mass balance simplifies to Integrating this equation, we obtain where (7) dh 1 dt = dh 2 dt. (8) h 1 (t) = h 2 (t) + C (9) C = h 1 (0) + h 2 (0) (10) is the integration constant related to the initial levels h 1 (0) and h 2 (0). Substituting Eq. (9) into Eq. (6) and then substituting the resulting expression into the mass balance Eq. (2) for tank 2 yields: dh 2 dt = A p A 2 v 12 = A p A 2 Sign(h 1 h 2 ) 2g C 2h 2 K 12 (11) Assuming that K 12 is independent of the fluid velocity, the differential equation (11) can be solved analytically to obtain an expression for dependence of h 2 on time. Plotting a linearized model of the result is useful to obtain the friction coefficient from the slope of the graph. This 3
friction coefficient can be used as an approximation in the three-tank system to simplify calculations. Three-Tank System In the three-tank system, water leaving one tank flows into two other tanks, depending on the levels of water in the tanks. Performing a mass balance on the tee below tank 1, we get: v = v 12 + v 13 (12) Since the area of the pipe is the same throughout the system, the flow rate reduces to just the average velocities. v 12 is the velocity of water going from tank 1 to tank 2 and similarly, v 13 is the velocity of water going from tank 1 to tank 3. Plugging in Eq. (6) for both v 12 and v 13 and then substituting Eq. (12) into our mass balance Eq. (2) for each of the tanks will yield three first-order differential equations. The friction coefficients in the pipe network are given by: K ij = K ij,f + K ij,s (13) K f is a constant and it is the summation of the friction contributions from the fittings, K s is the skin friction of the pipe. The skin friction coefficient for flow through a pipe is: K s = 4 L f, (14) D where L is the length of the pipe, D is the diameter of the pipe, and f is the Fanning friction factor. The flow between the tanks can be assumed to be turbulent for the entire duration of the experiment (see prelab questions). Therefore, we can use Blasius equation for the Fanning friction factor: f = 0.079Re.25 (15) Substituting Eq. (15) into Eq. (14) and then plugging in the friction coefficients into the system of equations will yield three non-linear first-order differential equations. Solution of this system requires a numerical solution using, e.g., Euler or Runge-Kutta method. The friction coefficients depend on the velocity of the system, which means a solver is needed to first obtain the velocity to obtain the friction coefficients. This must be done at all water levels and seeded in during the numerical solver process which is difficult to manage. Instead, friction coefficients obtained from the two-tank analysis can be used as an approximation. Since the pipe networks between tanks 1 and 2 and tanks 1 and 3 are different, two different coefficients must be obtained. 4
Objectives 1. Perform 2-tank flow experiments and obtain friction coefficients in pipes connecting each pair of tanks. 2. Perform 3-tank flow experiments at various initial water levels in the tanks. 3. Make a theoretical prediction for transient flows in the 3-tank system. 4. Compare the experimental results with the theoretical prediction. 5