For a venturi meter given below, the volumetric flow rate is defined in terms of the geometrical parameters, the density of working fluid (p), and density of the manometer liquid (pm) as 4. Q = f(D,, D2, A2, g, h, Pm. Pr) %3D Write down the balance equations and show your work to end up with an expression for the volumetric flow rate in terms of the variables defined above.
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- When a liquid in a beaker is stired, whirlpool will form and there will be an elevation difference h, between the center of the liquid surface and the rim of the liquid surface. Apply the method of repeating variables to generate a dimensional relationship for elevation difference (h), angular velocity (@) of the whirlpool, fluid density (p). gravitational acceleration (2), and radius (R) of the container. Take o. pand R as the repeating variables.Water X 50-mm inside diameter The volume flow rate of the system The pressure at the point "A" The pressure at the point "B" 25-mm diameter As a consultant at one of the country's leading soft drink companies, you are tasked with evaluating the system depicted to determine the following parameters using Bernoulli's Equation if the values assigned to "X" and "Y" are 3m and 0.5m respectively: i. ii. iii.The power P required for operating a pump depends on the diameter of the pump D, the pressure rise Ap across the pump, the fluid density p, the fluid viscosity and the Theorem, and D, Q and p as the repeating variables, the dimensionless group associated with μ is given by volume flow rate Q through the pump. Using Buckingham's μDªQbpº. Determine the values of a, b and c. 1. [0.99, 1.01 2. [-1.01, 0.99 3. [-1.01, -0.99 a= b = C=
- 1- The thrust (P l ) of a propeller depends upon diameter (D); speed (u) through a fluid density (p); revolution per minute (N); and dynamic viscosity (u) Show that: P = (p D² u²) f P Du [; where fis any function.3- Use dimensional analysis to show that in a problem involving shallow water waves, both the Froude number and the Reynolds 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's to get the parameters into the following form: Fr= √=f(Re) where Re=pch μ h Too 8 P₂ μ1. The thrust of a marine propeller Fr depends on water density p, propeller diameter D, speed of advance through the water V, acceleration due to gravity g, the angular speed of the propeller w, the water pressure 2, and the water viscosity . You want to find a set of dimensionless variables on which the thrust coefficient depends. In other words CT = Fr pV2D² = fen (T₁, T₂, ...Tk) What is k? Explain. Find the 's on the right-hand-side of equation 1 if one of them HAS to be a Froude number gD/V.
- b) When a liquid in a beaker is stirred, whirlpool will form and there will be an elevation difference h, between the center of the liquid surface and the rim of the liquid surface. Apply the method of repeating variables to generate a dimensional relationship for elevation difference (h), angular velocity (o) of the whirlpool, fluid density (p), gravitational acceleration (g), and radius (R) of the container. Take o, p and R as the repeating variables.a)Weber number (We) represents the ratio of disruptive hydrodynamics forces to the stabilizing surface tension force. It is an important dimensionless parameter applied during the analysis of thin film flows, which examines the ratio between inertia force and surface tension force acting on a fluid element. Using The Buckingham Pi Theorem, generate the formula for We. We is a function of fluid density (ρ), fluid velocity (v), characteristic length (l) and surface tension (σs), which can be mathematically written as: ?? = ?(?,?,?,??) b)Froude number (Fr) is an important dimensionless parameter used in open channel flow. Give the physical and mathematical definition of Fr. Prove that Fr is dimensionless.Ql: The viscosity in industrial measurement continue to use the CGS system of Lunits, since centimeters and grams vield convenient numbers for many fluids. The absolute viscosity () unit is the poise, I poise = 1 gtem. s). The kinematic viscosity (v) unit is the stohes, I stokes = 1 em /s. Water at 20C has u = 001 poise and also V= 0.01 stokes. Express these resalts in (a) SI and (h) BG tanits.
- The conduction heat transfer in an extended surface, known as a fin, yields the following equation for the temperature T, if the temperature distribution is assumed to be one-dimensional in x, where x is the distance from the base of the fin, as shown in figure: To Fin >X T(x) h,T Heat Loss d²T_hp (T-T) = 0 dx² ΚΑ dT dx = 0 Here, p is the perimeter of the fin, being 2R for a cylindrical fin of radius R; A is the cross-sectional area, being R2 for a cylindrical fin; k is the At x = 0: T = T₁ dT At x=L: :0 dx thermal conductivity of the material; T is the ambient fluid temperature; and h in the convective heat transfer coefficient. The boundary conditions are as follows: where L is the length of the fin. Solve this equation to obtain 7(x) by using Euler's method for R=1cm, h= 20 W/m².K, k = 15 W/m-K, L = 25 cm, T₁ = 80°C, and T = 20°C.Under a laminar flow, the liquid flows through small holes. It has a triangular cross-section, width b and length L, where the volumetric flow rate Q of the flow is a function of viscosity. ,pressure reduction per unit length p/L and width b 1) Use the PI theory to write the relationship as a dimensionless variable. 2) if the width b is doubled by viscosity And the pressure drop per unit length p/L is the same. I want to know how the flow rate Q will change.Thermal conductivity k is a measure of the ability of a material to conduct heat. For conduction heat transfer in the x-direction through a surface normal to the x-direction, Fourier’s law of heat conduction is expressed as: Q=-kA.dT/dx where ?̇ is the rate of heat transfer and A is the area normal to the direction of heat transfer. Determine the primary dimensions of thermal conductivity (k). Look up a value of k and verify that its SI units are consistent with your result. Write a set of primary SI units for k.