Liquid Jet Device A device shown in the figure is used to pr

Liquid Jet Device: A device shown in the figure is used to produce a liquid jet. The liquid in the tank is pressurized, causing the liquid to be ejected from a tube (diameter D and length L) as a free jet with average velocity V. We shall assume that V is a function of P, D, and L, as well as liquid properties including density rho, dynamic viscosity mu, and the liquid-gas surface tension sigma [units N/m]. How many dimensionless groups are needed to describe the operation of this device? Find a suitable set of dimensionless groups for this system. Prove that your dimensionless groups are really dimensionless by showing that the units cancel A prototype device is designed to produce a jet of acetonitrile with a velocity V_P =15 m/s. The prototype\'s pipe diameter (D_P) is 0.2 mm and pipe length (L_P) is 5 cm. To complete the design, we need to know the pressure P_P in the prototype to produce the acetonitrile jet. You propose building and testing a dynamically similar model device that uses water. What should be the model\'s pipe diameter (D_m) and pipe length (L_m)? What will be the velocity of the water jet produced by the model device? (physical properties of acetonitrile: density: 779 kg/m^3; viscosity: 0.37 times 10^-3 N middot s/m^2; surface tension: 2.87 times 10^-2 N/m) After building and testing the model device, we find that a pressure P_m = 150 kPa produces the required water jet velocity found in part (c). What is your estimate of the pressure P_P required in the prototype device (D_P= 0.2 mm and (L_P =5 cm) to produce a acetonitrile jet with V_P=15 m/s.

Solution

Given the basic units (M,L,T) (ie 3)

and the 6 derived variables, ( Q,d, rho,mu,sigma,p) there are 3 groupings (6-3) by Buckingham Pi

The groups are taken as Q d rho mu

Q d rho p

Q d rho sigma

equating powers to rhs 0,0,0 for M,.L, T

gives ( mu/(Qdrho), ( pD⁴/( rho Q² ), sigma D³ /( Q ²rho)  

c) Q = V.A, and take D = L length of pipe

the group pD⁴/rho Q² is the pressure divided by velcity squared

Using the groups with viscosity and surface tension, can get diameter, and subsequently get velocity

knowing dia and velocity can get pressure , and length

d) do your own

Best