Concept explainers
The Rosin-Rammler-Bennet (RRB) equation is used to describe size distribution in fine dust.
(a) Numerically calculate the mass density distribution
(b) Using your results from part (a ), calculate the mode size of the mass density distribution-that is, the size at which the derivative of
(c) Find the surface area per mass of the dust
The equation is valid only for spherical particles. Assume a density
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Chapter 24 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
- Cardiologists use the short-range scaling exponent α1, which measures the randomness of heart rate patterns, as a tool to assess risk of heart attack. The article “Applying Fractal Analysis to Short Sets of Heart Rate Variability Data” compared values of α1 computed from long series of measurements (approximately 40,000 heartbeats) with those estimated from the first 300 beats to determine how well the long-term measurement (y) could be predicted the short-term one (x). Following are the data (obtained by digitizing a graph). Short Long 0.54 0.55 1.02 0.79 1.4 0.81 0.88 0.9 1.68 1.05 1.16 1.05 0.82 1.05 0.93 1.07 1.26 1.1 1.18 1.19 0.81 1.19 0.81 1.2 1.28 1.23 1.18 1.23 0.71 1.24 Note: This problem has a reduced data set for ease of performing the calculations required. This differs from the data set given for this problem in the text. Compute the least-squares line for predicting the long-term measurement from the short-term measurement.…arrow_forwardKinetic energy of a fluid flow can be computed by 1 pv · vdV, where p(x, y, z) and v(x, y, z) are the pointwise fluid density and velocity, respectively. Fluid with uniform density flows in the domain bounded by x2 + z² = 6 and 0 < y<. = The velocity of parabolic flow in the given domain is v(x, y, z) 6 (6 – x² – 2²)j. Find the kinetic energy of the fluid flow.arrow_forwardThe friction force F on a smooth sphere falling in water depends on the sphere speed V, the sphere density ρ_s and diameter D_s, the density ρ and dynamic viscosity μ of the water, gravitational acceleration g. Choose the correct repeating variables. sphere diameter, dynamic viscosity, gravitational acceleration density, gravitational acceleration, dynamic viscosity friction force, gravitational acceleration, density density, sphere speed, sphere diameter friction force, sphere speed, sphere diameterarrow_forward
- A physics lab consists of a large ball attached to a wire. Students hold on to one end of the wire, then whirl the ball around in circles and count the number of rotations per second. One group finds these numbers: ball mass= 320 gram, wire length= 1.3m, number of rotations/sec=2.5. The wire is made of steel with a diameter of 1mm and a Young's modulus of 20x10^10 N/m^2.How much does the wire stretch due to the tension on it? Should the students correct their data for the wire stretching?arrow_forward2. Casson model was discussed in class in the context of blood rheology. This phenomenological model is often used to describe the shear stress vs. shear rate relationship in colloidal suspensions where particle aggregation might cause the measured viscosity to increase at low shear rates. In an experiment, data for the shear stress and the applied shear rate S were fitted to the Casson model written below (in a slightly different form compared to that given in the lecture notes): √t = √²₁+√as. (1) The best least square fit parameters to the experimental data were found to be 40 mPa for the yield stress to and 2.5 mPa s for the parameter a, which is referred to as the plastic viscosity. a. Using Eq. (1), derive an expression for the fluid viscosity u as a function of S. b. Plot the viscosity of the fluid as a function of S for 0.1s¹ ≤S≤ 10 s¹. c. Based on class discussion on fluid classification, how would you characterize this fluid?arrow_forwardThe modulus k of a coil spring (force required to stretch the spring a unit distance) can be expressed in equation form as k=(Gr^4)/(4R^3n) in which r and R are lengths and n is a dimensionless number. Determine the dimensions of G (a property of the spring material) .arrow_forward
- A closed thermodynamic system consists of a fixed amount of substance (i.e. mass) in which no substance can flow across the boundary, but energy can. For a closed themodynamic system we cannot add energy to the system, via substance (E ) (1.e. matter which contains energy is not allowed across the boundary) Across the Boundaries E° = No Q = = Yes W mass NO CLOSED = Yes SY STEM m = constant | energy YES Figure 1.1. If the substance inside the thermodynamic system shown in figure 1.1. (i.e. piston cylinder device) is air, is the system a Fixed closed system Moveable closed system A. В.arrow_forward100 10 Circular disk Sphere 0.1 0.01 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 Reynolds number Re = U d/v A spherical weather balloon of 2 m diameter is filled with hydrogen. The total mass of the balloon skin and the instruments it carries is 3.7 kg. At a certain altitude the density of air is 1.0 kg/m3 and is 10 times the density of hydrogen in the balloon; the viscosity of air is 1.8x10-5 Ns/m3. Determine the steady upward velocity of the balloon. m/s Drag coefficlent Caarrow_forwardIn medical literatures, local blood perfusion rate is typically presented as xx ml/(min 100g tissue), in another word, it represents xx ml of blood supplied to a tissue mass of 100 g per minute to satisfy its nutritional needs. As we learned from the course lectures, the local blood perfusion rate appearing in the Pennes bioheat equation is in a unit of 1/s, or can be interpreted as xx ml of blood supplied to a tissue volume of 1 ml per second. The following lists the blood perfusion rates in various organs or structures in a human body from medical textbooks: brain (50 ml/(min 100g tissue)), kidney (35 ml/(min 100g tissue)), and muscle at rest (3 ml/(min 100g tissue)). Please convert the above local blood perfusion rates into values with the unit of 1/s, therefore, they can be used in the Pennes bioheat equation. The tissue density in a human body is 1050 kg/m³.arrow_forward
- You are developing a porous membrane for use in a dialysis system. The membrane must be able to retain both protein and glucose on the inlet side and allow other, smaller molecules to flow through. You have found that the membrane is 0.25 mm thick and contains long, rectangular pores with a width of 0.1 microns. 57% of the 50 cm^2 membrane surface area is covered with pores. A test fluid (viscosity = 1.5 cP, density = 1015 kg/m^3) is passed through the membrane. You can assume that the test fluid has a composition similar to that of blood plasma. An initial test is run at physiological conditions, and you observe that the flow rate of fluid through the membrane is 500 cm^3/min. Given this data, what must the hydrodynamic pressure drop across the membrane in your test system be in pascals?arrow_forwardWhen a fluid flows over a surface, the shear stresS I (N/m) at the surface is given by the expression du dy surface where u is the viscosity (N-s/m), u is the velocity parallel to the surface (m/s), and y is the distance normal to the surface (cm). Measurements of the velocity of an air stream flowing above a surface are made with an LDV (laser-Doppler-Velocimeter). The values given below were obtained. y u y и 0.0 0.00 2.0 88.89 1.0 55.56 3.0 100.00 At the local temperature, u = 0.00024 N-s/m. In four decimal places, calculate du/dy at the surface using all the points.arrow_forwardIn the development of a new piece of equipment, it is necessary to measure the shear stress (7) between two parallel plates where the lower plate is stationary and the upper plate moves at a velocity (V) of 2m/sec. The plates are 0.025m apart (h). The shear stress 7 is estimated by 2m/sec = 0.025m V 80μ. The viscosity (4) in N – sec/m² is a function of temperature. Laboratory measurements have determined µ at the temperatures shown in the table below: T(°C) | (µ)(N – sec/m² 5 0.08 20 0.015 30 0.009 50 0.006 55 0.0055arrow_forward
- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning
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