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?

Dr. M. Khosravy 18

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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