A liquid delivery system is being designed such that ethylene glycol flows out of a hole in the bottom of a large tank, as shown in the figure. The designers need to predict how long it will take for the ethylene glycol to completely drain. Since it would be very expensive to run tests with a full-scale prototype using ethylene glycol, they decide to build a one quarter scale model for experimental testing, and they plan to use water as their test liquid. The model is geometrically similar to the full-scale unit. a. Use dimensional analysis to develop a dimensionless expression for the time to empty a container tempty as a function of the liquid density ρ, viscosity µ, gravity constant g, hole diameter d, and tank diameter D, and initial height h above the bottom of the tank. Use ρ, g, and h as repeating variables. b. The temperature of the ethylene glycol in the prototype tank is 60°C, at which ν = 4.75*10–6 m2 /s. At what temperature should the water in the model experiment be set in order to ensure geometric similarity and dynamic similarity with the dimensionless number in part (a)? Note that most of the Π groups developed in part (a) are essentially requirements for geometric similarity. c. The experiment is run with water at the proper temperature as calculated in part (b). It takes 4.53 min to drain the model tank. Predict how long it will take to drain the ethylene glycol from the full-scale tank.
A liquid delivery system is being designed such
that ethylene glycol flows out of a hole in the bottom of a large
tank, as shown in the figure. The designers need to predict how
long it will take for the ethylene glycol to completely drain.
Since it would be very expensive to run tests with a full-scale
prototype using ethylene glycol, they decide to build a one
quarter scale model for experimental testing, and they plan to
use water as their test liquid. The model is geometrically
similar to the full-scale unit.
a. Use dimensional analysis to develop a dimensionless expression for the time to empty a
container tempty as a function of the liquid density ρ, viscosity µ, gravity constant g, hole
diameter d, and tank diameter D, and initial height h above the bottom of the tank. Use ρ,
g, and h as repeating variables.
b. The temperature of the ethylene glycol in the prototype tank is 60°C, at which ν =
4.75*10–6 m2
/s. At what temperature should the water in the model experiment be set in
order to ensure geometric similarity and dynamic similarity with the dimensionless
number in part (a)? Note that most of the Π groups developed in part (a) are essentially
requirements for geometric similarity.
c. The experiment is run with water at the proper temperature as calculated in part (b). It
takes 4.53 min to drain the model tank. Predict how long it will take to drain the ethylene
glycol from the full-scale tank.
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