Solutions Manual for Fluid Mechanics: Fundamentals and Applications Second Edition Yunus A. Çengel & John M. Cimbala McGraw-Hill, 2010

Chapter 8 Internal Flow




Solutions Manual for

Fluid Mechanics: Fundamentals and Applications
Second Edition

Yunus A. Çengel & John M. Cimbala

McGraw-Hill, 2010




Chapter 8
Internal Flow



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8-1
PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and
educators for course preparation. If you are a student using this Manual, you are using it without permission.

, Chapter 8 Internal Flow

Laminar and Turbulent Flow



8-1C
Solution We are to define and discuss hydraulic diameter.

Analysis For flow through non-circular tubes, the Reynolds number and the friction factor are based on the hydraulic
4 Ac
diameter Dh defined as Dh = where Ac is the cross-sectional area of the tube and p is its perimeter. The hydraulic
p
4 Ac 4πD
diameter is defined such that it reduces to ordinary diameter D for circular tubes since D h = = =D.
p πD

Discussion Hydraulic diameter is a useful tool for dealing with non-circular pipes (e.g., air conditioning and heating
ducts in buildings).


8-2C
Solution We are to define and discuss hydrodynamic entry length.

Analysis The region from the tube inlet to the point at which the boundary layer merges at the centerline is
called the hydrodynamic entrance region, and the length of this region is called hydrodynamic entry length. The entry
length is much longer in laminar flow than it is in turbulent flow. But at very low Reynolds numbers, Lh is very small
(e.g., Lh = 1.2D at Re = 20).

Discussion The entry length increases with increasing Reynolds number, but there is a significant change in entry
length when the flow changes from laminar to turbulent.



8-3C
Solution We are to discuss why pipes are usually circular in cross section.

Analysis Liquids are usually transported in circular pipes because pipes with a circular cross section can withstand
large pressure differences between the inside and the outside without undergoing any significant distortion.

Discussion Piping for gases at low pressure are often non-circular (e.g., air conditioning and heating ducts in buildings).




8-2
PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and
educators for course preparation. If you are a student using this Manual, you are using it without permission.

, Chapter 8 Internal Flow
8-4C
Solution We are to define and discuss Reynolds number for pipe and duct flow.

Analysis Reynolds number is the ratio of the inertial forces to viscous forces, and it serves as a criterion for
determining the flow regime. At large Reynolds numbers, for example, the flow is turbulent since the inertia forces are
large relative to the viscous forces, and thus the viscous forces cannot prevent the random and rapid fluctuations of the
fluid. It is defined as follows: a
VD
(a) For flow in a circular tube of inner diameter D: Re =
ν b
VD h
(b) For flow in a rectangular duct of cross-section a × b: Re =
ν
4 Ac 4ab 2ab
where Dh = = = is the hydraulic diameter.
p 2( a + b) ( a + b) D

Discussion Since pipe flows become fully developed far enough downstream, diameter is the
appropriate length scale for the Reynolds number. In boundary layer flows, however, the boundary layer
grows continually downstream, and therefore downstream distance is a more appropriate length scale.



8-5C
Solution We are to compare the Reynolds number in air and water.

Analysis Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water than for
air (at 25°C, νair = 1.562×10-5 m2/s and νwater = μ ⁄ρ = 0.891×10-3/997 = 8.9×10-7 m2/s). Therefore, noting that Re = VD/ν,
the Reynolds number is higher for motion in water for the same diameter and speed.

Discussion Of course, it is not possible to walk as fast in water as in air – try it!



8-6C
Solution We are to express the Reynolds number for a circular pipe in terms of mass flow rate.

Analysis Reynolds number for flow in a circular tube of diameter D is expressed as V
VD m m 4m μ
Re = where V = Vavg = = = and ν = m
ν ρ Ac ρ (π D ) ρπ D 2 ρ
Substituting, D
VD 4mD 4m 4m
Re = = = . Thus, Re =
ν ρπ D ( μ / ρ ) π D μ
2
π Dμ

Discussion This result holds only for circular pipes.



8-7C
Solution We are to compare the pumping requirement for water and oil.

Analysis Engine oil requires a larger pump because of its much larger viscosity.

Discussion The density of oil is actually 10 to 15% smaller than that of water, and this makes the pumping requirement
smaller for oil than water. However, the viscosity of oil is orders of magnitude larger than that of water, and is therefore the
dominant factor in this comparison.




8-3
PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and
educators for course preparation. If you are a student using this Manual, you are using it without permission.

, Chapter 8 Internal Flow
8-8C
Solution We are to discuss the Reynolds number for transition from laminar to turbulent flow.

Analysis The generally accepted value of the Reynolds number above which the flow in a smooth pipe is turbulent is
4000. In the range 2300 < Re < 4000, the flow is typically transitional between laminar and turbulent.

Discussion In actual practice, pipe flow may become turbulent at Re lower or higher than this value.


8-9C
Solution We are to compare pipe flow in air and water.

Analysis Reynolds number is inversely proportional to kinematic viscosity, which is much smaller for water than for
air (at 25°C, νair = 1.562×10-5 m2/s and νwater = μ ⁄ρ = 0.891×10-3/997 = 8.9×10-7 m2/s). Therefore, for the same diameter and
speed, the Reynolds number will be higher for water flow, and thus the flow is more likely to be turbulent for water.

Discussion The actual viscosity (dynamic viscosity) μ is larger for water than for air, but the density of water is so
much greater than that of air that the kinematic viscosity of water ends up being smaller than that of air.


8-10C
Solution We are to compare the wall shear stress at the inlet and outlet of a pipe.

Analysis The wall shear stress τw is highest at the tube inlet where the thickness of the boundary layer is nearly
zero, and decreases gradually to the fully developed value. The same is true for turbulent flow.

Discussion We are assuming that the entrance is well-rounded so that the inlet flow is nearly uniform.


8-11C
Solution We are to discuss the effect of surface roughness on pressure drop in pipe flow.

Analysis In turbulent flow, tubes with rough surfaces have much higher friction factors than the tubes with smooth
surfaces, and thus surface roughness leads to a much larger pressure drop in turbulent pipe flow. In the case of
laminar flow, the effect of surface roughness on the friction factor and pressure drop is negligible.

Discussion The effect of roughness on pressure drop is significant for turbulent flow, as seen in the Moody chart.




8-4
PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and
educators for course preparation. If you are a student using this Manual, you are using it without permission.