Capítulo 1. Fundamentación teórica
1.5 Metodologías, Lenguajes y Herramientas de Desarrollo
1.5.9 Frameworks utilizados en la solución
In the design and testing of large equipment involvingfluid flow, it is customary to build small models geometrically similar to the larger prototypes. Experimental data achieved for the models are then scaled to predict the performance of full-sized prototypes according to the requirements of geometric, kinematic, and dynamic similarity. The following examples will illustrate the manner of utilizing model data to evaluate the conditions for a full-scale device.
Example 2
A cylindrical mixing tank is to be scaled up to a larger size such that the volume of the larger tank isfive times that of the smaller one. What will be the ratios of diameter and height between the two?
Geometric similarity between tanks A and B in Figure 11.2 requires that Da
The volumes of the two tanks are
Vap
4D2aha and Vbp 4D2bhb
The scaling ratio between the two is stipulated asVb
Va
Figure 11.2 Cylindrical mixing tanks for Example 2.
and we get
Db
Da
2
hb
ha 5
We now substitute the geometric similarity requirement that gives Db
Da
3
Lb
La
3
5 and the two ratios of interest become
Db
DaLb
La 51=3 1:71
Example 3
Dynamic similarity may be obtained by using a cryogenic wind tunnel in which nitrogen at low temperature and high pressure is employed as the workingfluid. If nitrogen at 5 atm and 183 K is used to test the low-speed aerodynamics of a prototype that has a 24.38 m wing span and is tofly at standard sea-level conditions at a speed of 60 m/s, determine
1. The scale of the model to be tested
2. The ratio of forces between the model and the full-scale aircraft
Conditions of dynamic similarity should prevail. The speed of sound in nitrogen at 183 K is 275 m/s.
For dynamic similarity to exist, we know that both model and prototype must be geometrically similar and that the Reynolds number and the Mach number must be the same. A table such as the following is helpful.
Model Prototype
Characteristic length L 24.38 m
Velocity v 60 m/s
Viscosity m 1.789 10 5Pa?s
Density r 1.225 kg/m3
Speed of sound 275 m/s 340 m/s
The conditions listed for the prototype have been obtained from Appendix I. Equating Mach numbers, we obtain Mm Mp
v 275
34060 48:5 m=s Equating the Reynolds numbers of the model and the prototype, we obtain
Rem Rep
r48:5L
m 1:225?60?24:38
1:789?10 5 1:002 108
11.5 Model Theory ◀ 149
Using equation (7-10), we may evaluatem for nitrogen. From Appendix K, e/k 91.5 K and s 3:681 A for nitrogen so that
The density may be approximated from the perfect gas law r P
Solving for the wing span of the model, we obtain
L 3:26 m 10:7 ft
The ratio of the forces on the model to the forces experienced by the prototype may be determined equating values of Eu between the model and the prototype. Hence
F
where ARis a suitable reference area. For an aircraft, this reference area is the projected wing area. The ratio of model force to prototype force is then given by
Fm
where the ratio of reference areas can be expressed in terms of the scale ratio. Substituting numbers, Fm
The forces on the model are seen to be 8.9% the prototype forces.
▶ 11.6
CLOSURE
The dimensional analysis of a momentum-transfer problem is simply an application of the requirement of dimensional homogeneity to a given situation. By dimensional analysis, the work and time required to reduce and correlate experimental data are decreased substantially by the combination of individual variables into dimensionlessp groups, which are fewer in number than the original variables. The indicated relations between dimensionless parameters are then useful in expressing the performance of the systems to which they apply.
It should be kept in mind that dimensional analysis cannot predict which variables are important in a given situation, nor does it give any insight into the physical transfer
mechanism involved. Even with these limitations, dimensional analysis techniques are a valuable aid to the engineer.
If the equation describing a given process is known, the number of dimensionless groups is automatically determined by taking ratios of the various terms in the expression to one another. This method also gives physical meaning to the groups thus obtained.
If, on the contrary, no equation applies, an empirical method, the Buckingham method, may be used. This is a very general approach, but gives no physical meaning to the dimensionless parameters obtained from such an analysis.
The requirements of geometric, kinematic, and dynamic similarity enable one to use model date to predict the behavior of a prototype or full-size piece of equipment. Model theory is thus an important application of the parameters obtained in a dimensional analysis.
▶
PROBLEMS
11.1 The power output of a hydraulic turbine depends on the diameter D of the turbine, the densityr of water, the height H of water surface above the turbine, the gravitational acceleration g, the angular velocityw of the turbine wheel, the discharge Q of water through the turbine, and the efficiency h of the turbine. By dimensional analysis, generate a set of appropriate dimension-less groups.
11.2 Through a series of tests on pipeflow, H. Darcy derived an equation for the friction loss in pipeflow as
hL fL D
v2 2g;
in which f is a dimensionless coefficient that depends on (a) the average velocity u of the pipe flow; (b) the pipe diameter D;
(c) the fluid density r; (d) the fluid viscosity m; and (e) the average pipe wall uneveness e (length). Using the Buckinghamp theorem,find a dimensionless function for the coefficient f.
11.3 The pressure rise across a pump P (this term is propor-tional to the head developed by the pump) may be considered to be affected by thefluid density r, the angular velocity w, the impeller diameter D, the volumetric rate offlow Q, and the fluid viscositym. Find the pertinent dimensionless groups, choosing them so that P, Q, andm each appear in one group only. Find similar expressions, replacing the pressure risefirst by the power input to the pump, then by the efficiency of the pump.
11.4 The maximum pitching moment that is developed by the water on aflying boat as it lands is noted as cmaxThe following are the variables involved in this action:
a angle made by flight path of plane with horizontal b angle defining attitude of plane
M mass of plane L length of hull r density of water g acceleration of gravity
R radius of gyration of plane about axis of pitching
a. According to the Buckinghamp theorem, how many inde-pendent dimensionless groups should there be characteriz-ing this problem?
b. What is the dimensional matrix of this problem? What is its rank?
c. Evaluate the appropriate dimensionless parameters for this problem.
11.5 The rate at which metallic ions are electroplated from a dilute electrolytic solution onto a rotating disk electrode is usually governed by the mass diffusion rate of ions to the disk. This process is believed to be controlled by the following variables:
Dimensions k mass-transfer coefficient L/t
D diffusion coefficient L2/t
d disk diameter L
a angular velocity 1/t
r density M/L3
m viscosity M/Lt
Obtain the set of dimensionless groups for these variables where k, m, and D are kept in separate groups. How would you accumulate and present the experimental data for this system?
11.6 The performance of a journal bearing around a rotating shaft is a function of the following variables: Q, the rate offlow lubricating oil to the bearing in volume per unit time; D, the bearing diameter; N, the shaft speed in revolutions per minute;m, the lubricant viscosity;r, the lubricant density; and s, the surface tension of the lubricating oil. Suggest appropriate parameters to be used in correlating experimental data for such a system.
11.7 The mass M of drops formed by liquid discharging by gravity from a vertical tube is a function of the tube diameter D, liquid density, surface tension, and the acceleration of gravity.
Problems ◀ 151
Determine the independent dimensionless groups that would allow the surface-tension effect to be analyzed. Neglect any effects of viscosity.
11.8 The functional frequency n of a stretched string is a function of the string length L, its diameter D, the mass density r, and the applied tensile force T. Suggest a set of dimensionless parameters relating these variables.
11.9 The power P required to run a compressor varies with compressor diameter D, angular velocityw, volume flow rate Q, fluid density r, and fluid viscosity m. Develop a relation between these variables by dimensional analysis, wherefluid viscosity and angular velocity appear in only one dimensionless parameter.
11.10 A large amount of energy E is suddenly released in the air as in a point of explosion. Experimental evidence suggests that the radius r of the high-pressure blast wave depends on time t as well as the energy E andr the density of the ambient air.
a. Using the Buckingham method,find the equation for r as a function of t,r, and E.
b. Show that the speed of the wave front decreases as r increases.
11.11 The size d of droplets produced by a liquid spray nozzle is thought to depend upon the nozzle diameter D, jet velocity V, and the properties of the liquidr, m, and s. Rewrite this relation in dimensionless form. Take D,r, and V as repeating variables.
11.12 Identify the variables associated with Problem 8.13 and find the dimensionless parameters.
11.13 A car is traveling along a road at 22.2 m/s. Calculate the Reynolds number
a. based on the length of the car
b. based on the diameter of the radio antenna
The car length is 5.8 m and the antenna diameter is 6.4 mm.
11.14 In natural-convection problems, the variation of density due to the temperature difference DT creates an important buoyancy term in the momentum equation. If a warm gas at TH moves through a gas at temperature T0and if the density change is only due to temperature changes, the equation of motion becomes
Show that the ratio of gravity (buoyancy) to inertial forces acting on afluid element is
Lg
where L and V0are reference lengths and velocity, respectively.
11.15 A 1/6-scale model of a torpedo is tested in a water tunnel to determine drag characteristics. What model velocity corresponds
to a torpedo velocity of 20 knots? If the model resistance is 10 lb, what is the prototype resistance?
11.16 During the development of a 300-ft ship, it is desired to test a 10% scale model in a towing tank to determine the drag characteristics of the hull. Determine how the model is to be tested if the Froude number is to be duplicated.
11.17 A 25% scale model of an undersea vehicle that has a maximum speed of 16 m/s is to be tested in a wind tunnel with a pressure of 6 atm to determine the drag characteristics of the full-scale vehicle. The model is 3 m long. Find the air speed required to test the model andfind the ratio of the model drag to the full-scale drag.
11.18 An estimate is needed on the lift provided by a hydrofoil wing section when it moves through water at 60 mph. Test data are available for this purpose from experiments in a pressurized wind tunnel with an airfoil section model geometrically similar to but twice the size of the hydrofoil. If the lift F1is a function of the densityr of the fluid, the velocity u of the flow, the angle of attackq, the chord length D, and the viscosity m, what velocity offlow in the wind tunnel would correspond to the hydrofoil velocity for which the estimate is desired? Assume the same angle of attack in both cases, that the density of the air in the pressurized tunnel is 5.0 10 3slugs/ft3, that its kinematic viscosity is 8.0 10 5ft2/s, and that the kinematic viscosity of the water is approximately 1.0 10 5ft2/s. Take the density of water to be 1.94 slugs/ft3.
11.19 A model of a harbor is made on the length ratio of 360:1.
Storm waves of 2 m amplitude and 8 m/s velocity occur on the breakwater of the prototype harbor. Significant variables are the length scale, velocity, and g, the acceleration of gravity. The scaling of time can be made with the aid of the length scale and velocity scaling factors.
a. Neglecting friction, what should be the size and speed of the waves in the model?
b. If the time between tides in the prototype is 12 h, what should be the tidal period in the model?
11.20 A 40% scale model of an airplane is to be tested in aflow regime where unsteady flow effects are important. If the full-scale vehicle experiences the unsteady effects at a Mach number of 1 at an altitude of 40,000 ft, what pressure must the model be tested at to produce an equal Reynolds number? The model is to be tested in air at 70°F. What will the timescale of the flow about the model be relative to the full-scale vehicle?
11.21 A model ship propeller is to be tested in water at the same temperature that would be encountered by a full-scale propeller. Over the speed range considered, it is assumed that there is no dependence on the Reynolds or Euler numbers, but only on the Froude number (based on forward velocity V and propeller diameter d). In addition, it is thought that the ratio of forward to rotational speed of the propeller must be constant (the ratio V/Nd, where N is propeller rpm).
a. With a model 041 m in diameter, a forward speed of 2.58 m/s and a rotational speed of 450 rpm is recorded.
What are the forward and rotational speeds corresponding to a 2.45-m diameter prototype?
b. A torque of 20 N?m is required to turn the model, and the model thrust is measured to be 245 N. What are the torque and thrust for the prototype?
11.22 A coating operation is creating materials for the elec-tronics industry. The coating requires a specific volumetric flow rate Q, solution densityr, solution viscosity m, substrate coating velocity v, solution surface tension s, and the length of the
coating channel L. Determine the dimensionless groups formed from the variables involved using the Buckingham method.
Choose the groups so that Q,s, and m appear in one group only.
11.23 A pump in a manufacturing plant is transferring viscous fluids to a series of delivery tanks. This critical transfer requires careful monitoring of the solution mass flow rate, the power (work) that the pump adds to thefluid, the internal energy of the system, and the viscosity and density of the solution. Determine the dimensionless groups formed from the variables involved using the Buckingham method. Carefully choose your core group based on the description of the system.
Problems ◀ 153