Post on 07-Feb-2018
Islamic Azad University Karaj Branch
Dr. M. Khosravy
Chapter 6 Introduction to Convection
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Problems involving conduction: Chapters 2-3
Chapter 3: • Obtained temperature profiles for 1-D, SS conduction, with and without generation • We wrote the 1-D, SS problems in terms of resistances in series • We defined an overall heat transfer coefficient, as the inverse of the total resistance
Transient problems: Chapter 5
Obtained temperature as a function of time for cases where resistance to conduction was negligible
Energy Conservation
Dr. M. Khosravy 2
Dr. M. Khosravy
• In Chapters 1-5 we used Newton’s law of convection:
! h was provided ! we did not consider any temperature variations within the fluid
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Dr. M. Khosravy
Chapter 6: • We will apply dimensional analysis to the boundary layer to find a functional dependence of h • In subsequent chapters we will use this information to obtain temperature distributions within the fluid.
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Introduction to Convection
• Convection denotes energy transfer between a surface and a fluid moving over the surface.
• The dominant contribution is due to the bulk (or gross) motion of fluid particles.
• In this chapter we will – Discuss the physical mechanisms underlying convection – Discuss physical origins and introduce relevant
dimensionless parameters that can help us to perform convection transfer calculations in subsequent chapters.
• Note similarities between heat, mass and momentum transfer.
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Dr. M. Khosravy
Heat Transfer Coefficient Recall Newton’s law of cooling for heat transfer between a surface of arbitrary shape, area As and temperature Ts and a fluid:
! Generally flow conditions will vary along the surface, so q” is a local heat flux and h a local convection coefficient.
! The total heat transfer rate is
where average heat transfer coefficient
(6.1)
(6.2)
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Dr. M. Khosravy
Heat Transfer Coefficient • For flow over a
flat plate:
! How can we estimate the heat transfer coefficient?
(6.3)
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Dr. M. Khosravy
The Velocity Boundary Layer
The flow is characterized by two regions: – A thin fluid layer (boundary layer) in which velocity gradients and
shear stresses are large. Its thickness d is defined as the value of y for which u = 0.99
– An outer region in which velocity gradients and shear stresses are negligible
Consider flow of a fluid over a flat plate:
For Newtonian fluids: and where Cf is the local
friction coefficient 8
Dr. M. Khosravy
The Thermal Boundary Layer
• The thermal boundary layer is the region of the fluid in which temperature gradients exist
• Its thickness is defined as the value of y for which the ratio:
Consider flow of a fluid over an isothermal flat plate:
At the plate surface (y=0) there is no fluid motion – The local heat flux is:
and (6.5) (6.4)
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Laminar and Turbulent Flow
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Transition criterion at Recritical:
Transition criterion at Recritical:
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Dr. M. Khosravy
Example Consider airflow over a flat plate of length L=1m under conditions for which transition occurs at xc=0.5 m.
(a) Determine the air velocity (T=350K). (b) What are the average convection coefficients in the laminar region and
turbulent region as a function of the distance from the leading edge?
Clam=8.845 W/m3/2.K Cturb=49.75 W/m1.8.K
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Dr. M. Khosravy
Boundary Layers - Summary
• Velocity boundary layer (thickness d(x)) characterized by the presence of velocity gradients and shear stresses - Surface friction, Cf
• Thermal boundary layer (thickness dt(x)) characterized by temperature gradients – Convection heat transfer coefficient, h
• Concentration boundary layer (thickness dc(x)) is characterized by concentration gradients and species transfer – Convection mass transfer coefficient, hm
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! Need to determine the heat transfer coefficient, h
! Must know T(x,y), which depends on velocity field
(6.5)
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Functional form of the solutions
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• From dimensional analysis, or solution of boundary layer equations:
(6.6)
where Nu is the local Nusselt number (6.7)
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Functional form of the solutions
where:
Prandtl number
Reynolds number
The average Nusselt number, based on the average heat transfer coefficient is:
(6.8)
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Physical meaning of dimensionless groups
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See Table 6.2 textbook for a comprehensive list of dimensionless groups
Dr. M. Khosravy
True or False • A velocity boundary layer always forms when a stream with free
velocity V! comes into contact with a solid surface.
• Similarly a thermal boundary layer will always form when a stream with free stream temperature T ! comes into contact with a solid surface.
• The critical Reynolds number for laminar to turbulent transition is the same for flow inside a pipe and for flow over a plate
• The Nusselt number is the same as the Biot number.
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Example
• An object of irregular shape has a characteristic length of L=1 m and is maintained at a uniform surface temperature of Ts=400 K. When placed in atmospheric air, at a temperature of 300 K and moving with a velocity of V=100 m/s, the average heat flux from the surface of the air is 20,000 W/m2. If a second object of the same shape, but with a characteristic length of L=5 m, is maintained at a surface temperature of Ts=400K and is placed in atmospheric air at 300 K, what will the value of the average convection coefficient be, if the air velocity is V=20 m/s?
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Dr. M. Khosravy
Summary
• In addition to heat transfer due to conduction, we considered for the first time heat transfer due to bulk motion of the fluid
• We discussed the concept of the boundary layer • We defined the local and average heat transfer
coefficients and obtained a general expression, in the form of dimensionless groups to describe them.
• In the following chapters we will obtain expressions to determine the heat transfer coefficient for specific geometries
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