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- Measurements at a certain point of a pipe have been done where the following parameters were recorded: Fluid of density = 887 kg/m3, Fluid velocity = 4 m/s, Pressure= 11.3 KN/m2 If the total energy per unit weight at this point = 32 m, then the potential energy is:Volumetric flow rate, Q, of a pump is a function of impeller diameter d, fluid velocity V,pressure difference P, fluid density and viscosity .a) (15P) Determine dimensionless groups.b) (20P) A test is to be performed for a large pump that delivers 1.5 m3/s of water from 40 cmdiameter impeller with pressure rise of 400 kPa. For the test, a model with an 8 cm diameter impelleris to be used. What flow rate should be used and what would be the pressure rise in model pumpImportant variables in a particular water machine are: Fluid density (ρ), impeller angular velocity (ω), impeller diameter (D), volume flow rate (), fluid viscosity (µ), head (gH). In fact, (gH) shows the energy that the fluid unit mass receives from or gives to the machine, independent of gravitational acceleration (g). Therefore, gH value is used instead of H. Important sizes in terms of the operation of the water machine are power (P), (gH) and efficiency ( h). Derive dimensionless numbers by typing these two parameters based on the above variables and applying Buckingham's p method (repeating variables method).
- % ParametersD = 0.1; % Diameter of the tube (m)L = 1.0; % Length of the tube bundle (m)N = 8; % Number of tubes in the bundleU = 1.0; % Inlet velocity (m/s)rho = 1.2; % Density of the fluid (kg/m^3)mu = 0.01; % Dynamic viscosity of the fluid (Pa.s) % Define the grid size and time stepdx = D/10; % Spatial step size (m)dy = L/10; % Spatial step size (m)dt = 0.01; % Time step size (s) % Calculate the number of grid points in each directionnx = ceil(D/dx) + 1;ny = ceil(L/dy) + 1; % Create the velocity matrixU_matrix = U * ones(nx, ny); % Perform the iterationsfor iter = 1:100 % Calculate the velocity gradients dUdx = (U_matrix(:, 2:end) - U_matrix(:, 1:end-1)) / dx; dUdy = (U_matrix(2:end, :) - U_matrix(1:end-1, :)) / dy; % Calculate the pressure gradients dpdx = -mu * dUdx; dpdy = -mu * dUdy; % Calculate the change in velocity dU = dt * (dpdx / rho); % Update the velocity matrix U_matrix(:, 2:end-1) = U_matrix(:, 2:end-1) + dU; % Apply…Volumetric flow rate, Q, of a pump is a function of impeller diameter d, fluid velocity V, pressure difference DP, fluid density r and viscosity m. a)Determine dimensionless P groups. b)A test is to be performed for a large pump that delivers 1.5 m3 /s of water from 40 cm diameter impeller with pressure rise of 400 kPa. For the test, a model with an 8 cm diameter impeller is to be used. What flow rate should be used and what would be the pressure rise in model pump. Note: The model fluid is water at same temperature as the water in the prototype.Here, a 1:10 scale prototype of a propeller on a ship is to be tested in a water channel. What would the rotating speed of the model be if the rotational speed of the p propeller is 2000 rpm, and if: (a) the Froude number governs the model-prototype similarity(b) Reynolds number governs the similarity
- Use dimentional analysis to evaluate that in a problem involving shallow water waves (figure 6), both the Froude number and the Reynold's number are relevant dimensionless parameters. The wave speed, c of waves on the surface of a liquid is a function of depth h, gravitational acceleration g, fluid density, p and fluid viscosity μ. Manipulate your II's to get the parameters into the following form: Fr = c/(gh)1/2= f(Re) , where Re = pch/μAt a sudden contraction in a pipe the diameter changes from D, to D2. The pressure drop, Ap, which develops across the con- traction is a function of D, and Dz, as well as the velocity, V, in the larger pipe, and the fluid density, p, and viscosity, u. Use D, V, and u as repeating variables to determine a suitable set of dimensionless parameters. Why would ti be incorrect to include the velocity in the smaller pipe a sa nadditional variable?A small wind turbine is tested in a wind tunnel using the following model parameters: ρ = 1.2 kg/m3 , µ = 1.81x10-5 Ns/m2 , v = 12 m/s, A = 0.03 m2 and Cp is measured as 0.42. Assuming dynamic similarity exists, calculate the power output of a full-size turbine of diameter 60 m operating in air of density 1.3 kg/m3 and viscosity µ = 1.73x10-5 Ns/m2 with wind speeds of 15 m/s
- Independent variables in a turbomachine are the impeller diameter, D, angular speed, ω, and fluid viscosity and density. Dependent properties are volumetric flow rate, Q, head, H (energy per unit mass), and power input, P. Use ρ, ω, and D as repeating variables in a dimensional analysis. a) Determine the dimensionless ratios that characterize this problem. b) Under what conditions will flows in two different machines be similar? c) Determine the speed of operation for machine 2 for the same flow as machine 1 if D2/D1 = 2 and viscous effects are unimportant. What will be the head ratio under these conditions? Note:hand written solution should be avoided.The pressure difference ∆p produced by a water pump, and the power P required to operate it, each depend on the size of the pump, measured by the diameter D of the impeller, the volume flow rate ˙q, the rate of rotation ω, the water density ρ and dynamic viscosity µ. (a) Express the non-dimensional pressure difference and power as separate functions of the other non-dimensional groups. (b) Tests on a model pump are performed at 0.5 × full scale, at a rotation rate that is 2 × the full-scale value. To achieve dynamic similarity in the model test: (i) what would the volume flow rate of the water need to be in the model test compared to the full-scale? (ii) What would the pressure difference be compared to the full scale? (iii) What would the power consumption be relative to the full scale?11.8 The functional frequency η of a stretched string is a function of the string length L, its diameter D, the mass density ⍴, and the applied tensile force T. Suggest a set of dimensionless parameters relating these variables.