Determine the dimension {MLTO} of the following quantities: ди (b) (a) pu- ах а?т (p – Po) dA (c) pcp* әх ду ди (d) SSSP dx dy dz дt All quantities have their standard meanings; for example, p is density.
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- Q3: Analyze the primary dimensions of the universal ideal gas constant Rµ using the ideal gas law, Pv = nR, T where P is pressure, v is volume, T is absolute temperature, and n is the number of moles of the gas.Books on porous media and atomization claim that the viscosityμ and surface tension Y of a fl uid can be combinedwith a characteristic velocity U to form an important dimensionlessparameter. ( a ) Verify that this is so. ( b ) Evaluatethis parameter for water at 20°C and a velocity of3.5 cm/s. Note: You get extra credit if you know the nameof this parameter.When a capillary tube of small diameter D is inserted into a container of liquid, the liquid rises to height h inside the tube (Fig.). h is a function of liquid density ? , tube diameter D, gravitational constant g, contact angle ?, and the surface tension ?s of the liquid. (a) Generate a dimensionless relationship for h as a function of the given parameters. (b) Compare your result to the exact analytical equation for h. Are your dimensional analysis results consistent with the exact equation? Discuss.
- The output power W. of a spinning shaft is a function of torque T and angular velocity ? . Use dimensional analysis to express the relationship between W., T, and ? in dimensionless form. Compare your result to what you know from physics and discuss brieflyWrite the primary dimensions of the universal ideal gas constant Ru. (Hint: Use the ideal gas law, PV = nRuT where P is pressure, V is volume, T is absolute temperature, and n is the number of moles of the gas.)Using primary dimensions, verify that the Rayleigh number is indeed dimensionless. What other established nondimensional parameter is formed by the ratio of Ra and Gr?
- The height H that fluid rises in a liquid barometer tubedepends upon the liquid density ρ , the barometric pressurep , and the acceleration of gravity g . (a) Arrange these fourvariables into a single dimensionless group. (b) Can youdeduce (or guess) the numerical value of your group?Use dimensional analysis to prove this equation and show how units are equivalent: v = _/ N / (kg/m)Write the primary dimensions of each of the following variables from the field of thermodynamics, showing all your work: (a) energy E; (b) specific energy e = E/m; (c) power W . .
- Directions: Solve the following problems completely, and make a detailed solution. Any incompletesolution arriving at such answers will never be considered. A barrel contains a 15 cm layer of oil of density 5.88 kN/m3 floating on water that is 30 cm deep. What is the gauge pressure in mmHg at the oil-water interface?We define the specific ideal gas constant Rgas for a particular gas as the ratio of the universal gas constant and the molar mass (also called molecular weight) of the gas, Rgas = Ru/M. For a particular gas, then, the ideal gas law is written as follows: PV = mRgasT or P = ρRgasT where P is pressure, V is volume, m is mass, T is absolute temperature, and ? isthedensityoftheparticulargas.Whatare the primary dimensions of Rgas? For air, Rair = 287.0 J/kg·K in standard SI units. Verify that these units agree with your result.The Reynold's number of a sphere falling in air is 1E6. If the sphere's radius is 1ft, what is its velocity? Density of air is 0.00234 slug/ft³ and viscosity of air is 3.8E-7 lbf-sec/ft²A. 2.5 ft/sB. 5.1 ft/sC. 40.6 ft/sD. 81.2 ft/s