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Homework 4 Fluid Mechanics 2022: K For Hydrogen at 20 C and 1 Atm If 1.2E 7 M. Use

This document contains 6 questions regarding fluid mechanics. Question 1 asks to group quantities into the dimensionless Brinkman number. Question 2 asks to rewrite the thrust relationship of a propeller as a dimensionless function. Question 3 asks to determine the thermal conductivity of hydrogen using given values for air. Question 4 provides the differential energy equation for incompressible flow through a porous medium and asks about the dimensions of a term and how to nondimensionalize the equation. Question 5 asks to determine the minimum scale ratio of a model to ensure a minimum Weber number. Question 6 provides a relationship for windmill power and asks to write it dimensionlessly and determine the power and rotation rate of a prototype.

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57 views1 page

Homework 4 Fluid Mechanics 2022: K For Hydrogen at 20 C and 1 Atm If 1.2E 7 M. Use

This document contains 6 questions regarding fluid mechanics. Question 1 asks to group quantities into the dimensionless Brinkman number. Question 2 asks to rewrite the thrust relationship of a propeller as a dimensionless function. Question 3 asks to determine the thermal conductivity of hydrogen using given values for air. Question 4 provides the differential energy equation for incompressible flow through a porous medium and asks about the dimensions of a term and how to nondimensionalize the equation. Question 5 asks to determine the minimum scale ratio of a model to ensure a minimum Weber number. Question 6 provides a relationship for windmill power and asks to write it dimensionlessly and determine the power and rotation rate of a prototype.

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엄기웅
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Homework 4

Fluid mechanics 2022

Q1. The transfer of energy by viscous dissipation is dependent upon viscosity , thermal
conductivity k, stream velocity U, and stream temperature To. Group these quantities, if
possible, into the dimensionless Brinkman number, which is proportional to .

Q2. The thrust F of a propeller is generally thought to be a function of its diameter D and
angular velocity , the forward speed V, and the density  and viscosity  of the fluid.
Rewrite this relationship as a dimensionless function.

Q3. To good approximation, the thermal conductivity k of a gas (see Ref. 21 of Chap. 1)
depends only on the density , mean free path , gas constant R, and absolute temperature T. For
air at 20C and 1 atm, k  0.026 W/mK and  6.5E−8 m. Use this information to determine
k for hydrogen at 20C and 1 atm if  1.2E−7 m. Use  = 1.205 kg/m3 and R = 287 m2/s2K
for air and  = 0.0839 kg/m3 with R = 4124 m2/s2K for hydrogen.

Q4. The differential energy equation for incompressible two-dimensional flow through a
“Darcy-type” porous medium is approximately

  p T   p T  2T
cp + cp +k 2 =0
 x x  y y y

where  is the permeability of the porous medium. All other symbols have their usual meanings.
(a) What are the appropriate dimensions for  ? (b) Nondimensionalize this equation, using (L,
U, , To) as scaling constants.

Q5. A prototype spillway has a characteristic velocity of 3 m/s and a characteristic length of
10 m. A small model is constructed by using Froude scaling. What is the minimum scale ratio
of the model which will ensure that its minimum Weber number is 100? Both flows use water
at 20C. Use  = 998 kg/m3 and Y = 0.073 N/m.

Q6. The power P generated by a certain windmill design depends upon its diameter D, the air
density , the wind velocity V, the rotation rate , and the number of blades n.
(a) Write this relationship in dimensionless form. A model windmill, of diameter 50 cm,
develops 2.7 kW at sea level when V = 40 m/s and when rotating at 4800 rev/min. (b) What
power will be developed by a geometrically and dynamically similar prototype, of diameter 5
m, in winds of 12 m/s at 2000 m standard altitude? (c) What is the appropriate rotation rate of
the prototype? Note that at 2000 m altitude,  = 1.0067 kg/m3. At sea level,  = 1.2255 kg/m3.

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