Fundamentals Of Thermal-fluid Sciences In Si Units
Fundamentals Of Thermal-fluid Sciences In Si Units
5th Edition
ISBN: 9789814720953
Author: Yunus Cengel, Robert Turner, John Cimbala
Publisher: McGraw-Hill Education
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Chapter 15, Problem 99RQ
To determine

The error involved in each case and the accuracy of Stokes law.

Expert Solution & Answer
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Explanation of Solution

Given:

The density of aluminum balls is, ρs=2600 kg/m3.

Temperature of glycerin is, T=22°C.

The density of glycerin is, ρf=1274 kg/m3.

The viscosity of glycerin is, μ=1 kg/ms.

The diameter of the aluminum balls are, D=2,4,10 mm.

The terminal velocities are, Vexp=3.2,12.8,60.4mm/s.

Calculation:

From the force balance,

  FD=WFB3πμVD=ρsgVρfgV3πμVD=(ρsρf)gπD36V=(ρsρf)gD218μ

Case 1: D=2 mm and Vexp=3.2mm/s.

The terminal velocity is,

  V=(2600 kg/m31274 kg/m3)(9.81m/s2)(0.002 m)218(1 kg/ms)V=0.00311m/sV=3.11mm/s

The error in terminal velocity is,

  Error=VexpVVexp×100%=3.23.113.2×100%=2.8%

Thus, the error in the terminal velocity is 2.8%.

The Reynolds number is,

  Re=ρfVDμ=(1274 kg/m3)(0.0032m/s)(0.002m)1 kg/ms=0.0081

Hence, Stokes law is accurate.

Case 2: D=4 mm and Vexp=12.8mm/s.

The terminal velocity is,

  V=(2600 kg/m31274 kg/m3)(9.81m/s2)(0.004 m)218(1 kg/ms)V=0.0124m/sV=12.4mm/s

The error in terminal velocity is,

  Error=VexpVVexp×100%=12.812.412.8×100%=3.1%

Thus, the error in the terminal velocity is 3.1%.

The Reynolds number is,

  Re=ρfVDμ=(1274 kg/m3)(0.0128m/s)(0.004m)1 kg/ms=0.0651

Hence, Stokes law is accurate.

Case 3: D=10 mm and Vexp=60.4mm/s.

The terminal velocity is,

  V=(2600 kg/m31274 kg/m3)(9.81m/s2)(0.010 m)218(1 kg/ms)V=0.0777m/sV=77.7mm/s

The error in terminal velocity is,

  Error=VexpVVexp×100%=60.477.760.4×100%=28.6%

Thus, the error in the terminal velocity is 28.6%.

The Reynolds number is,

  Re=ρfVDμ=(1274 kg/m3)(0.0604m/s)(0.010m)1 kg/ms=0.770<1

Hence, Stokes law is inaccurate.

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Chapter 15 Solutions

Fundamentals Of Thermal-fluid Sciences In Si Units

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