1. 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
is the average heat transfer coefficient
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2. Heat Transfer Coefficient
For flow over a flat plate:
How can we estimate heat transfer coefficient?
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3. The Velocity Boundary Layer
Consider flow of a fluid over a flat plate:
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
For Newtonian fluids:
where Cf is the local friction coefficient
and
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4. The Thermal Boundary Layer
Consider flow of a fluid over an isothermal flat plate:
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:
At the plate surface (y=0) there is no fluid motion – Conduction heat transfer:
and
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5. 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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6. Laminar and Turbulent Flow
Transition criterion:
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7. Boundary Layer Approximations
Need to determine the heat transfer coefficient, h
In general, h=f (k, cp, r, m, V, L)
We can apply the Buckingham pi theorem, or obtain exact solutions by applying the continuity, momentum and energy equations for the boundary layer.
In terms of dimensionless groups:
(x*=x/L)
Local and average Nusselt numbers (based on local and average heat transfer coefficients)
where:
Prandtl number
Reynolds number
(defined at distance x)
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8. Example (Problem 6.27 textbook)
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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9. Summary
In addition to heat transfer due to conduction, we considered for the first time heat transfer due to bulk motion of the fluid
By applying the overall energy balance, we derived the thermal energy equation, which can be used for solving problems involving convection.
Useful in problems were we need to know the temperature distribution within a 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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