Heat Transfer Coefficient

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 14

2. Heat Transfer Coefficient • For flow over a flat plate: • How can we estimate heat transfer coefficient? 15

3. 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: where Cf is the local friction coefficient and 16

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 17

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 18

6. Laminar and Turbulent Flow Transition criterion: 19

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) 20

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? 21

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 22